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CFD-Based Surrogate Modeling of Liquid Rocket Engine Components via Design Space Refinement and Sensitivity Assessment

Permanent Link: http://ufdc.ufl.edu/UFE0021303/00001

Material Information

Title: CFD-Based Surrogate Modeling of Liquid Rocket Engine Components via Design Space Refinement and Sensitivity Assessment
Physical Description: 1 online resource (238 p.)
Language: english
Creator: Mack, Yolanda T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: cfd, design, dsr, injector, optimization, refinement, rocket
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Computational fluid dynamics (CFD) can be used to improve the design and optimization of rocket engine components that traditionally rely on empirical calculations and limited experimentation. CFD based-design optimization can be made computationally affordable through the use of surrogate modeling which can then facilitate additional parameter sensitivity assessments. The present study investigates surrogate-based adaptive design space refinement (DSR) using estimates of surrogate uncertainty to probe the CFD analyses and to perform sensitivity assessments for complex fluid physics associated with liquid rocket engine components. Three studies were conducted. First, a surrogate-based preliminary design optimization was conducted to improve the efficiency of a compact radial turbine for an expander cycle rocket engine while maintaining low weight. Design space refinement was used to identify function constraints and to obtain a high accuracy surrogate model in the region of interest. A merit function formulation for multi-objective design point selection reduced the number of design points by an order of magnitude while maintaining good surrogate accuracy among the best trade-off points. Second, bluff body-induced flow was investigated to identify the physics and surrogate modeling issues related to the flow?s mixing dynamics. Multiple surrogates and DSR were instrumental in identifying designs for which the CFD model was deficient and to help to pinpoint the nature of the deficiency. Next, a three-dimensional computational model was developed to explore the wall heat transfer of a GO2/GH2 shear coaxial single element injector. The interactions between turbulent recirculating flow structures, chemical kinetics, and heat transfer are highlighted. Finally, a simplified computational model of multi-element injector flows was constructed to explore the sensitivity of wall heating and improve combustion efficiency to injector element spacing. Design space refinement using surrogate models and a multi-objective merit function formulation facilitated an efficient framework to investigate the multiple and competing objectives. The analysis suggests that by adjusting the multi-element injector spacing, the flow structures can be modified, resulting in a better balance between wall heat flux and combustion length.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yolanda T Mack.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Shyy, Wei.
Local: Co-adviser: Haftka, Raphael T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021303:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021303/00001

Material Information

Title: CFD-Based Surrogate Modeling of Liquid Rocket Engine Components via Design Space Refinement and Sensitivity Assessment
Physical Description: 1 online resource (238 p.)
Language: english
Creator: Mack, Yolanda T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: cfd, design, dsr, injector, optimization, refinement, rocket
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Computational fluid dynamics (CFD) can be used to improve the design and optimization of rocket engine components that traditionally rely on empirical calculations and limited experimentation. CFD based-design optimization can be made computationally affordable through the use of surrogate modeling which can then facilitate additional parameter sensitivity assessments. The present study investigates surrogate-based adaptive design space refinement (DSR) using estimates of surrogate uncertainty to probe the CFD analyses and to perform sensitivity assessments for complex fluid physics associated with liquid rocket engine components. Three studies were conducted. First, a surrogate-based preliminary design optimization was conducted to improve the efficiency of a compact radial turbine for an expander cycle rocket engine while maintaining low weight. Design space refinement was used to identify function constraints and to obtain a high accuracy surrogate model in the region of interest. A merit function formulation for multi-objective design point selection reduced the number of design points by an order of magnitude while maintaining good surrogate accuracy among the best trade-off points. Second, bluff body-induced flow was investigated to identify the physics and surrogate modeling issues related to the flow?s mixing dynamics. Multiple surrogates and DSR were instrumental in identifying designs for which the CFD model was deficient and to help to pinpoint the nature of the deficiency. Next, a three-dimensional computational model was developed to explore the wall heat transfer of a GO2/GH2 shear coaxial single element injector. The interactions between turbulent recirculating flow structures, chemical kinetics, and heat transfer are highlighted. Finally, a simplified computational model of multi-element injector flows was constructed to explore the sensitivity of wall heating and improve combustion efficiency to injector element spacing. Design space refinement using surrogate models and a multi-objective merit function formulation facilitated an efficient framework to investigate the multiple and competing objectives. The analysis suggests that by adjusting the multi-element injector spacing, the flow structures can be modified, resulting in a better balance between wall heat flux and combustion length.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yolanda T Mack.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Shyy, Wei.
Local: Co-adviser: Haftka, Raphael T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021303:00001


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CFD-BASED SURROGATE MODELING OF LIQUID ROCKET ENGINE COMPONENTS
VIA DESIGN SPACE REFINEMENT AND SENSITIVITY ASSESSMENT




















By

YOLANDA MACK


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007




































O 2007 Yolanda Mack





































To my son, Trevor.









ACKNOWLEDGMENTS

I would like to thank the people in my life who have supported me during my graduate

studies. In particular, I would like to thank my advisor, Dr. Wei Shyy, for his support and for

pushing me to strive for excellence. I also thank Dr. Raphael Haftka for his guidance through

the years on my research. I thank the members of my supervisory committee members, Dr.

Corin Segal, Dr. William Lear, Dr. Andreas Haselbacher, and Dr. Don Slinn, for reviewing my

work and offering suggestions during my studies. I thank Dr. Nestor Queipo for inviting me to

explore new methods and new ideas that have been invaluable in my work. I thank Dr.

Siddharth Thakur and Dr. Jeffrey Wright for their troubleshooting assistance over the years. I

thank Mr. Kevin Tucker and the others that I have collaborated with at NASA Marshall Space

Flight Center along with the Institute for Future Space Transport under the Constellation

University Institute Proj ect for providing the motivations for my work as well as financial

assistance. I would also like to thank Zonta International and the South East Alliance for

Graduate Education and the Professoriate for their financial support and recognition.

I thank Antoin Baker for his love and support through any difficulty as we pursued our

graduate degrees. I would like to thank my father, Calvin Mack, for his gentle guidance, and my

mother, Jacqueline Mack, to whom I express my deepest gratitude for her unconditional love and

assistance over the years. I thank my sisters, Brandee and Cailah Mack, and other members of

my family for their undying support and for forever believing in me.











TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ................ ...............8............ ....

LIST OF FIGURES .............. ...............10....


AB S TRAC T ........._. ............ ..............._ 15...

CHAPTER


1 INTRODUCTION ................. ...............17.......... ......

1.1 Thrust Chamber Characteristics .............. ...............18....
1.1.1 Rocket Engine Cycles............... ...............19.
1.1.2 Engine Reliability Issues .............. ...............20....
1.2 Current and Past Design Practices ................. ...............21........... ..
1.2. 1 Rocket Engine Inj ector Design ................. ...............22.............
1.2.2 Rocket Engine Turbine Design .............. ...............24....
1.3 CFD-Based Optimization .............. .. .......... ......... ... ........2
1.3.1 Optimization Techniques for Computationally Expensive Simulations ................25
1.3.2 Design Space Refinement in CFD-Based Optimization Using Surrogate
Model s............... ...............29.
1.4 Summary ................. ...............3.. 0......... ....
1.5 Goal and S cope ................. ...............3.. 1............

2 OPTIMIZ ATION FRAMEWORK ................. ...............3.. 5......... ....


2.1 Optimization Using Surrogate Models .............. ...............36....
2.1.1 Design of Experiments .............. .. ....... ...............3
2. 1.2 Surrogate Model Identifieation and Fitting .............. ...............38....
2. 1.2. 1 Polynomial response surface approximation............... .............3
2. 1.2.2 Kriging .............. ..... ... ..............4
2. 1.2.3 Radial basis neural networks ................. ...............44........... ..
2.1.3 Surrogate Model Accuracy Measures .............. ...............47....
2.1.3.1 Root mean square error ................. ...............47...
2. 1.3.2 Coefficient of multiple determination ................. ................ ......... .48
2. 1.3.3 Prediction error sum of squares ................. ........... ... ............ ...4
2.2 Dimensionality Reduction Using Global Sensitivity Analysis............... ................4
2.3 Multi-Obj ective Optimization Using the Pareto Optimal Front ................. ................. .52
2.4 Design Space Refinement Techniques .............. ........ .............5
2.4. 1 Design Space Reduction for Surrogate Improvement ................. ............. ......56
2.4.2 Smart Point Selection for Second Phase in Design Space Refinement..................59
2.4.3 Merit Functions for Data Selection and Reduction .............. .....................6
2.4.4 Method of Alternative Loss Functions ................ ...............65...............












3 RADIAL TURBINE OPTIMIZATION .............. ...............75....


3 .1 Introducti on ............ ..... ._ ...............76..
3.2 Problem Description ............ ..... .__ ...............79..
3.2.1 Verification Study .............. ...............79....
3.2.2 Optimization Procedure ............ ..... ._ ............... 1...
3.3 Results and Discussion ............_...... .._ .. ... .._ .. ..... .......8
3.3.1 Phase 1: Initial Design of Experiments and Construction of Constraint
Surrogates ................ .......... .... .........
3.3.2 Phase 2: Design Space Refinement ................ ... ..........__ ........__ .........8
3.3.3 Phase 3: Construction of the Pareto Front and Validation of Response
Surfaces............... .. .. .. .. .... .............................8
3.3.4 Phase 4: Global Sensitivity Analysis and Dimensionality Reduction Check...._....87
3.4 M erit Function Analysis .............. ...............88...
3.4.1 Data Point Selection and Analysis .............. ...............88....
3.4.2 Merit Function Comparison Results ................. ...............89...............
3.5 Conclusion ................ ...............91................


4 MODELING OF INJECTOR FLOWS .............. ...............105....


4.1 Literature Review ............... ...............106...
4.1.1 Single-Element Injectors .............. ...............106....
4. 1.2 Multi-Element Inj ectors .........._..... .... ...._._ ........._ ..........10
4. 1.3 Combustion Chamber Effects and Considerations ........._...... ......._._..........108
4. 1.4 Review of Select CFD Modeling and Validation Studies ................ ................1 12
4.2 Turbulent Combustion Model ................. ...............117...............
4.2. 1 Reacting Flow Equations ................. ...............117........... ...
4.2.2 Turbulent Flow Modeling ................. ...............120...............
4.2.3 Chemical Kinetics .............. ... ...............124..
4.2.4 Generation and Decay of Swirl ............... ...............125..
4.3 Simplified Analysis of GO2/GH2 COmbusting Flow ....._____ .........__ ................127

5 SURROGATE MODELING OF MIXING DYNAMIC S ................ .........................13 7


5 .1 Introducti on ................. .......... ...............137 ....
5.2 Bluff Body Flow Analysis .................. ......... ... ... ...............138.
5.2. 1 Geometric Description and Computational Domain ................. .....................139
5.2.2 Obj ective Functions and Design of Experiments ................ .......................140
5.3 Results and Discussion ..........._.._ ....._. ...............142.
5.3.1 CFD Solution Analysis............... ...............14
5.3.2 Surrogate Model Results .............. ...............143....
5.3.3 Analysis of Extreme Designs .............. ...............145....
5.3.4 Design Space Exploration .............. ...............147....
5.4 Conclusions............... ..............14


6 INJECTOR FLOW MODELING ............ .......__ ...............157.












6. 1 Introducti on ................. ...............157........... ...
6.2 Experimental Setup............... ...............158
6.3 Upstream Injector Flow Analysis ................ ...............159........... ...
6.3.1 Problem Description ................. ...............159........... ...
6.3.2 Results and Discussion ................. ...............160........... ...
6.3.3 Conclusion ................. ........... ...............162 .....
6.4 Experimental Results and Analysis .............. ...............163....
6.5 Injector Flow Modeling Investigation ................. ...............166..............
6.5.1 CFD Model Setup ........._..... .... ....___ ......_._... ... .........6
6.5.2 CFD Results and Experimental Comparison of Heat Flux ........._._... ............... 168
6.5.3 Heat Transfer Characterization............... ...........16
6.5.4 Species Concentrations............... ............17
6.6 Grid Sensitivity Study ........._.___..... ._ __ ...............171..
6.7 Conclusion ........._.___..... .__ ...............173....


7 MULTI-ELEMENT INJECTOR FLOW MODELING AND ELEMENT SPACING
EFFECT S............... ...............19


7. 1 Introducti on ................. ...............191........... ...
7.2 Problem Set-Up .................... ...............193.
7.3 Feasible Design Space Study ................. ...............195........... ...
7.4 Design Space Refinement ................. ...............199...............
7.5 Conclusion ................ ...............203................


8 CONCLUSIONS .............. ...............219....


8. 1 Radial Turbine Efficiency and Weight Optimization ........................... ...............220
8.2 Bluff Body Mixing Dynamics ............ ............ ...............221.
8.3 Single-Element Inj ector Flow Modeling .....__ ................ ............... 222 ...
8.4 Multi-Element Inj ector Flow Modeling. ......... ........____ ......... ..........22
8.5 Future Work............... ...............224.


REFERENCE LIST .............. ...............225....


BIOGRAPHICAL SKETCH .............. ...............238....










LIST OF TABLES


Table page

1-1 Summary of CFD-based design optimization applications with highlighted surrogate-
based optimization techniques. ............. ...............32.....

2-1 Summary of DOE and surrogate modeling references. ........._..__......_. ..............67

2-2 Design space refinement (DSR) techniques with their applications and key results.........68

3-1 Variable names and descriptions. ............. ...............92.....

3-2 Response surface fit statistics before (feasible DS) and after (reasonable DS) design
space reduction............... ...............9

3-3 Original and final design variable ranges after constraint application and design
space reduction............... ...............9

3-4 Baseline and optimum design comparison. ............. ...............93.....

4-1 Selected inj ector experimental studies ................. ...............128..............

4-2 Selected CFD and numerical studies for shear coaxial inj ectors. ........._..... .........._....129

4-3 Reduced reaction mechanisms for hydrogen-oxygen combustion. ............. .................129

5-1 Number of grid points used in various grid resolutions. ........._._ ....... .... ............149

5-2 Data statistics in the grid comparison of the CFD data. ................ ................. .... 149

5-3 Comparison of cubic response surface coefficients and response surface statistics. .......150

5-4 Comparison of radial-basis neural network parameters and statistics. .................. .......... 150

5-5 RMS error comparison for response surface and radial basis neural network. ...............150

5-6 Total pressure loss coefficient and mixing index for extreme and two regular designs
for multiple grids............... ...............151.

5-7 Total pressure loss coefficient and mixing index for designs in the immediate
vicinity of Case 1 using Grid 3......... ........ ......... ................ ...............151

6-1 Flow regime description. ............. ...............174....

6-2 Effect of gri d re soluti on on wall heat flux and combustion l ength ................. ...............17 5

6-3 Flow conditions............... ..............17










7-1 Flow conditions and baseline combustor geometry for parametric evaluation. .............204

7-2 Kriging PRESSrms error statistics for each design space iteration. ................ ...............205











LIST OF FIGURES


Figure page

1-1 Rocket engine cycles............... ...............33.

1-2 SSME thrust chamber component diagram .............. ...............33....

1-3 SSME component reliability data ................. ...............34...............

1-4 Surrogate-based optimization using multi-fidelity data ................. ................. ...._.34

2-1 Optimization framework flowchart. ........_................. ...............69. ....

2-2 DOEs for noise-reducing surrogate models. .............. ...............70....

2-3 Latin Hypercube Sampling. ............. ...............70.....

2-4 Design space windowing .............. ...............71....

2-5 Smart point selection............... ...............7

2-6 Depiction of the merit function rank assignment for a given cluster. .............. .... ..........._72

2-7 The effect of varying values of p on the loss function shape.............__ ........._ ......72

2-8 Variation in SSE with p for two different responses ........... _...... __ ........_......73

2-9 Pareto fronts for RSAs constructed with varying values of p............... ..................7

2-10 Ab solute percent difference in the area under the Pareto front curves ................... ...........74

3-1 Mid-height static pressure (psi) contours at 122,000 rpm. ............. ......................9

3-2 Predicted Meanline and CFD total-to-static efficiencies. ................. .................9

3-3 Predicted Meanline and CFD turbine work. ............. ...............95.....

3-4 Feasible region and location of three constraints ..........._._ ......_.._ .. ......_.._......9

3-5 Constraint surface for Cx2 Utip = 0.2............... ...............96..

3-6 Constraint surfaces for B1 = 0 and B1 = 40. ................ ............... ......... ....... ..96

3-7 Region of interest in function space ................. ...................... ..................97

3-8 Error between RSA and actual data point............... ...............97.

3-9 Pareto fronts for p = 1 through 5 for second data set............... ...............98..











3-10 Pareto fronts for p = 1 through 5 for third data set .............. ...............99....

3-11 Pareto Front with validation data ................. ...............99...............

3-12 Variation in design variables along Pareto Front ................. ............... ......... ...10

3-13 Global sensitivity analysis .............. ...............100....

3-14 Data points predicted by validated Pareto front compared with the predicted values
using six Kriging models based on 20 selected data points ................. ............. .......101

3-15 Absolute error distribution for points along Pareto front............... ...............101.

3-16 Average mean error distribution over 100 clusters ................. ................ ......... .102

3-17 Median mean error over 100 clusters............... ...............10

3-18 Average maximum error distribution over 100 clusters .............. ....................10

3-19 Median maximum error over 100 clusters ................. ...............104.............

4-1 Coaxial inj ector and combustion chamber flow zones. ................ ........................130O

4-2 Flame from gaseous hydrogen gaseous oxygen single element shear coaxial
inj ector. ............. ...............13 0....

4-3 Multi-element inj ectors ................. ...............13. 1..............

4-4 Wall burnout in an uncooled combustion chamber. ............. ...............132....

4-5 Test case RCM-1 inj ector. ................ ...............132........ ...

4-6 Temperature contours for a single element injector. ............. ...............133....

4-7 CFD heat flux results as compared to RCM-1 experimental test case. ................... ........134

4-8 Multi-element inj ector simulations ....__. ................. ........._.._ ....... 13

4-9 Fuel rich hydrogen and oxygen reaction with heat release..........._.._.. ............. .......136

5-1 M odified FCCD. ............. ...............15 1....

5-2 Bluff body geometry ................. ...............151......... .....

5-3 Computational domain for trapezoidal bluff body. ....._____ ..... ...___ ................. 152

5-4 Computational grid for trapezoidal bluff body. ................ .....___............... ....15

5-5 Bluff body streamlines and vorticity contours ................. ...............152........... ..










5-7 Comparison of response surface (top row) and radial basis neural network (bottom
row) prediction contours for total pressure loss coefficient............... ...............5

5-8 Comparison of response surface (top row) and radial basis neural network (bottom
row) prediction contours for mixing index (including extreme cases) ................... .........154

5-9 Comparison of response surface (top row) and radial basis neural network (bottom
row) prediction contours for mixing index (excluding extreme cases) ...........................155

5-10 Variation in obj ective variables with grid refinement ........._._. ......___ ............... 155

5-11 Difference in predicted mixing index values from response surface (top row) and
radial basis neural network (bottom row) prediction contours constructed with and
without extreme cases ................. ...............156...... ......

5-12 Comparison of response surface and radial basis neural network prediction contours
for mixing index at B* = 0 and h* = 0 ..........._ ....._.._ ...............15

6-1 Blanching and cracking of combustion chamber wall due to local heating near
inj ector el ements. ........... ..... ._ ............... 175...

6-2 Hydrogen flow geometry. ............. ...............176....

6-3 Hydrogen inlet mesh. .............. ...............176....

6-4 Z-vorticity contours............... ...............17

6-5 Swirl number at each x location ................. ...............177........... ..

6-6 Average axial velocity u and average tangential velocity vs with increasing x...............178

6-7 Reynol ds numb er profile es ........._.._ ...... .___ ............... 178..

6-8 Non-dimensional pressure as a function of x ................. ...............178.............

6-9 Hydrogen inlet flow profiles ................. ...............179........... ...

6-10 Combustion chamber cross-sectional geometry and thermocouple locations. ................179

6-11 Estimated wall heat flux using linear steady-state and unsteady approximations. ..........180

6-12 1-D axisymmetric assumption for heat conduction through combustion chamber wall .180

6-13 Estimated wall temperatures using linear and axisymmetric approximations .................18 1

6-14 Temperature (K) contours for 2-D unsteady heat conduction calculations ................... ..181

6-15 Experimental heat flux values using unsteady assumptions ................. .....................182











6-16 Computational model for single-element inj ector flow simulation .............. ..............182

6-17 Velocity contours vx(m/s) and streamlines............... ..............18

6-18 Temperature (K) contours............... ...............18

6-19 CFD heat flux values as compared to experimental heat flux approximations. ..............183

6-20 Wall heat transfer and eddy conductivity contour plots. ............. .....................8

6-21 Streamlines and temperature contours at plane := 0............... ...............184...

6-22 Heat flux and y+ profiles along combustion chamber wall. ........._.._.._ ....._.._.. ......185

6-23 Temperature and eddy conductivity profiles at various y locations on plane := 0.........185

6-24 Mass fraction contours for select species............... ...............186

6-25 Mole fractions for all species along combustion chamber centerline (y = 0, z = 0) ........187

6-26 Select species mole fraction profiles............... ...............18

6-27 Sample grid and boundary conditions............... ..............18

6-28 Computational grid along symmetric boundary. ............. ...............189....

6-29 Wall heat flux and y+ values for select grids ........... ......__ ... ..___.......18

6-30 Comparison of temperature (K) contours for grids with 23,907, 3 1,184, 72,239, and
103,628 points, top to bottom, respectively ................. ...............190........... ..

7-1 Injector element subsection............... ..............20

7-2 Design points selected for design space sensitivity study .............. ......................0

7-3 Effect of hydrogen mass flow rate on obj ectives ................. ...............206............

7-4 Oxygen iso-surfaces and hydrogen contours. ............. ...............207....

7-5 Hydrogen contours and streamlines ................. ...............207........... ...

7-6 Maximum heat flux for a changing radial distance r* ................... ............... 20

7-7 Heat flux distribution. .............. ...............208....

7-8 Maximum heat flux as a function of aspect ratio ................. ...............209............

7-9 Design points in function and design space. .............. ...............209....

7-10 Merit function (M~Fz) contours for ........._ ....... ...............210.










7-11 Kriging surrogates based on initial 16 design points. ................... ................1

7-12 Kriging surrogates and merit function contours for 12 design points in Pattern 1 ..........211

7-13 Kriging surrogates and merit function contours for 12 design points in Pattern 2..........212

7-14 Kriging fits for all 15 design points from Pattern 1 .............. ...............213....

7-15 Kriging fits for all 12 design points from Pattern 2. ................................... 21

7-16 Pareto front based on original 16 data points (dotted line) and with newly added
points (solid line). ............. ...............214....

7-17 Approximate division in the design space between the two patterns based on A) peak
heat flux .............. ...............215....

7-18 Variation in flow streamlines and hydrogen contours in design space. ................... ........215

7-19 Location of best trade-off points in each pattern group in design space. ................... .....216

7-20 Injector spacing for selected best trade-off design point .............. ....................21

7-21 Predicted heat flux profies for baseline case (case 1), best trade-off from Pattern 1
(case 19), and best trade-off from Pattern 2 (case 18). ................ ............... ...._...217

7-22 Heat flux profies for different grid resolutions. The coarser grid was used to
construct the surrogate model.. ............ ...............218.....

7-23 Peak heat flux and combustion length as a function of hydrogen mass flow rate...........218









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CFD-BASED SURROGATE MODELING OF LIQUID ROCKET ENGINE COMPONENTS
VIA DESIGN SPACE REFINEMENT AND SENSITIVITY ASSESSMENT

By

Yolanda Mack

August 2007

Chair: Wei Shyy
Cochair: Raphael Haftka
Major: Aerospace Engineering

Computational fluid dynamics (CFD) can be used to improve the design and optimization

of rocket engine components that traditionally rely on empirical calculations and limited

experimentation. CFD based-design optimization can be made computationally affordable

through the use of surrogate modeling which can then facilitate additional parameter sensitivity

assessments. The present study investigates surrogate-based adaptive design space refinement

(DSR) using estimates of surrogate uncertainty to probe the CFD analyses and to perform

sensitivity assessments for complex fluid physics associated with liquid rocket engine

components.

Three studies were conducted. First, a surrogate-based preliminary design optimization

was conducted to improve the efficiency of a compact radial turbine for an expander cycle rocket

engine while maintaining low weight. Design space refinement was used to identify function

constraints and to obtain a high accuracy surrogate model in the region of interest. A merit

function formulation for multi-obj ective design point selection reduced the number of design

points by an order of magnitude while maintaining good surrogate accuracy among the best

trade-off points. Second, bluff body-induced flow was investigated to identify the physics and









surrogate modeling issues related to the flow' s mixing dynamics. Multiple surrogates and DSR

were instrumental in identifying designs for which the CFD model was deficient and to help to

pinpoint the nature of the deficiency. Next, a three-dimensional computational model was

developed to explore the wall heat transfer of a GO2/GH2 Shear coaxial single element inj ector.

The interactions between turbulent recirculating flow structures, chemical kinetics, and heat

transfer are highlighted.

Finally, a simplified computational model of multi-element injector flows was constructed

to explore the sensitivity of wall heating and improve combustion efficiency to inj ector element

spacing. Design space refinement using surrogate models and a multi-objective merit function

formulation facilitated an efficient framework to investigate the multiple and competing

objectives. The analysis suggests that by adjusting the multi-element injector spacing, the flow

structures can be modified, resulting in a better balance between wall heat flux and combustion

length.










CHAPTER 1
INTRODUCTION

The Space Shuttle Main Engine (SSME) was developed as a reusable launch engine.

However, even with noticeable success, the complexity of the SSME drove up the maintenance

costs of the engine due to the harsh operating environment among the engine components. These

additional maintenance costs were unforeseen in the original design and analysis efforts.l For

this reason, NASA is striving to design a simpler, more reliable engine.2,3 Today, there are more

computational, diagnostic and experimental tools available to assist in streamlining the design

process than when the SSME was designed. For example, flow and combustion processes can

now be modeled using advanced computational fluid dynamics (CFD) tools.4,5 Using CFD tools

can help narrow down the design of an engine component before a prototype is built and tested.

Often, designs are based on past experience, intuition, and empirical calculations. Without a

comprehensive and methodical process, the designs may not be satisfactorily refined without

tedious and inefficient trial-and-error exercises. Optimization tools can be used to guide the

selection of design parameters and highlight the predicted best designs and trade-off designs. In

this study, CFD and optimization tools are used to develop further capabilities in the design and

analysis of a radial turbine and an inj ector: critical liquid rocket engine components.

Current design practices for inj ector and turbine designs include past experience, intuition,

and empirical calculations. It is difficult to fully analyze, design, and optimize the combustion

chamber using experimental information alone. Because of measurement difficulties in

combustion chambers due to high temperatures and pressures, much is still too poorly

understood of the flow dynamics to accurately predict the response of future inj ector designs.

Using CFD can be used as a tool to help analyze and understand the flow dynamics within the

combustion chamber to fill in knowledge gaps left by the inj ector experiments. This research









effort will use CFD as an analysis and design tool for rocket engine components to improve

component design and provide information on the system processes that may be difficult to

obtain experimentally. CFD validation efforts are ongoing to Eine tune CFD codes and improve

their predictive ability. To use CFD effectively as a design tool for rocket engine component

design, an efficient design methodology must be used. A CFD-based optimization methodology

can be used to analyze, design, and optimize combustion chambers by using the increased

knowledge provided by CFD simulation to predict better designs. Through successive

refinement of the design process, or design space refinement (DSR), the design optimization

methodology is streamlined and improved to make it suitable for complex CFD-based design. In

summary, this research effort seeks to improve the design efficiency of complex rocket engine

components using improved CFD-based optimization and design space refinement techniques.

This chapter presents a general introduction to the issues that motivate the work, including

background information of rocket engine cycles and individual component reliability. Past and

current design practices for select rocket engine components are described and their limitations

are noted. Next, CFD-based optimization and DSR are introduced as methods for improving

upon traditional experimentally-based analysis and design. Finally, a roadmap of the various

chapters is presented.

1.1 Thrust Chamber Characteristics

A rocket engine thrust chamber is comprised of a main inj ector with hundreds of injector

elements, a combustion chamber, and a nozzle. Supporting elements include turbopumps and

possibly a preburner or gas generator. In addition to the fuel tanks, piping, and valve system, a

rocket engine is complex, and all components must be designed for maximum efficiency.









1.1.1 Rocket Engine Cycles

Rocket engines carry both the fuel and the propellant with the vehicle. For chemical

rockets, the propellant is a large part of the total mass of the vehicle. The basic goal in rocket

engine design is to obtain the highest thrust possible with the lowest total vehicle weight.6 As a

general obj ective, it is desirable to have a specific impulse that is as large as possible. Hydrogen

and oxygen are often chosen as the fuel and propellant because they have very high values of

specific impulse. While the goal of maximizing the specific impulse is straightforward, many

rocket engines designed to help achieve the goal are inherently complex and expensive. The

need for better performance drives the complexity.

Pressure-fed engines are relatively simple;' two high pressure tanks are connected directly

to the combustion chamber, and valves are used to regulate the flow rate. Pressure-fed engines

have low costs and good reliability because they are simple with few components. However,

pressure-fed engines require heavy, massive tanks to provide the required pressure. For this

reason, turbopumps were used in other rocket engine cycles to deliver the needed pressure,

allowing for lighter fuel and propellant tanks. Additionally, gas generators or preburners were

required to supply hot gases to the turbines. This led to a need for additional components, where

each component could compromise engine reliability.

In a gas-generator cycle (Figure 1-1B) a portion of the fuel is burned before it reaches the

combustion chamber. The exhaust from the gas generator is used to drive a turbine, which is

used to drive the pumps. After the exhaust leaves the turbine, it is dumped.'0 Gas generator

engine cycles provide improved thrust and specific impulse as compared to a pressure-fed engine

cycle.ll

The expander cycle (Figure 1-1A) eliminates the need for a gas generator, thus resulting in

a simpler and more reliable system. The expander cycle is the simplest of the pump-fed engine









cycles,12 and it offers a high specific impulse.13 Current expander cycle rocket engines in use are

Pratt and Whitney's RL-60 and the Ariane 5 ESC-B. In the expander cycle, the unburned fuel is

preheated by being passed through tubes used to cool the combustion chamber. This heated gas

is passed through the turbine used to drive the fuel and oxidizer pumps. The fuel is then sent to

the combustion chamber where it is combusted. Using preheated fuel instead of hot combustion

exhaust leads to a low turbine failure rate.14 One disadvantage to the expander cycle is that the

engine is limited by the amount of power that the turbine can deliver. In this case, power output

can be increased by improving the turbine drive, or increasing the amount of heat transferred to

the unburned fuel used to drive the turbine."

In the Space Shuttle main engine (SSME) cycle shown in Figure 1-2, a portion of the fuel

and oxygen is used to power the turbopumps. The SSME has two separate gas generators; the

exhaust of one powers the turbopump that delivers fuel, while the other powers the turbopump

that delivers oxygen. Unlike the gas generator cycle, the fuel-rich exhaust is not dumped.

Instead, the remaining fuel and oxygen also enter the combustion chamber and are combusted.

For this reason, the gas generators in the SSME are referred to as prebumners. A simple

preburner cycle is shown in Figure 1-1C. Using preburners maximized the specific impulse of

the engine.16 Thus, the SSME has very high performance, but is also highly complex. For this

reason, it is very expensive to operate and maintain.

1.1.2 Engine Reliability Issues

One of NASA' s goals in reusable launch vehicles is to improve engine reliability. After

each shuttle flight, the engines were removed from the Space Shuttle and inspected. Parts of the

engine were replaced as necessary. Improving engine reliability and engine component life can

help drive down the costs of reusable launch vehicles. Jue and Kuckl examined the reliability of

the various Space Shuttle Main Engine (SSME) components by determining the probability for









catastrophic failure of various components as shown in Figure 1-3. The components with the

lowest reliabilities are the high-pressure fuel turbopump (HPFTP), the high-pressure oxidizer

turbopump (HPOTP), and the large throat main combustion chamber (LTMCC). The

turbopumps and the combustion chamber are components common to all rocket engine cycles

(with the exception of a pressure fed cycle in which turbopumps are absent). Improving these

components can therefore cause large improvements in overall engine reliability. Using an

expander cycle, for example, can improve the turbopump reliability by providing a milder

turbine environment, thus improving the life of the turbine blades. Improving the inj ector

configuration can increase the reliability of the main combustion chamber (MCC) by helping to

alleviate local hot spots within the combustion chamber and prevent wall burnout.

To help address the issue of engine reliability, NASA is gathering data for the combustion

chamber design of the next generation of reusable launch vehicles. Specifically, NASA is

overseeing tests on several gas-gas H12/02 rocket inj ectors by Marshall et al.17 Particularly, the

inj ector tests are being run to supply data for the purpose of CFD validation. In this way, direct

comparisons can be made between the experimental and computational data. Also, by steering

the experiments, NASA can ensure that any specifically needed data is extracted in sufficient

amounts. Efforts are also in place to improve the life and efficiency of additional thrust chamber

components, such as the turbine which drives the fuel pump.

1.2 Current and Past Design Practices

Inj ectors of the past and even of today are designed primarily through the use of empirical

models.'" Inj ectors are also frequently designed based on the results of simple small-scale tests,

which are then scaled to a full-scale model.19 Many inj ectors are also designed based on cold

(non-reacting) flow tests.20 These cold flow tests are used primarily to test the mixing

capabilities of an inj ector design. Then correlations were developed to apply them to hot









(reacting) flow tests.21 Various design parameters are tested experimentally by making the thrust

chamber apparatus of modular design. In this way, the inj ector can be analyzed by switching out

inj ectors, changing combustion chamber length, and changing nozzles in addition to varying the

chamber pressure, mass flow rates, etc.17,22 The prospects of CFD-based design have been

explored only recently. This is due to modeling difficulties as a result of the presence of

turbulence in combination with reacting flow and possibly including multiphase and supersonic

flow. CFD validation efforts are still ongoing.

The design of turbines for rocket engine applications is more advanced than inj ector

design. This is because turbines are designed straightforwardly for its aerodynamic and stress

properties; the turbines for rocket engines have no added complications due to reacting and/or

multiphase flow. Turbines are now routinely designed with the aid of CFD, allowing for

sophisticated flow prediction and designs.

1.2.1 Rocket Engine Injector Design

Osborn et al.23 TepOrted on early water cooled combustion chambers that experienced

significant wall erosion near the injector. This erosion was common, and trial and error

experimentation often resulted in the quick destruction of much of the thrust chamber.

Peterson et al.24 COnstructed an inj ector with 137 elements based on tests from a smaller

inj ector with only 37 elements. It was noted that due to improved cooling in the larger scale

inj ector, the number of inj ector elements, and hence, inj ector spacing, could be increased. This

in turn meant that the mass flow rate per element was increased, requiring an increase in the size

of the inj ector orifices to regulate pressure. Separate thrust chambers were built to distinguish

differences in performance for cooled and un-cooled thrust chambers.

The early optimization efforts of an inj ector element may have consisted of a dimensional

analysis of geometric design variables and flow properties, as in the Apollo Service Propulsion









System Injector Orifice Study Program.25 In this study, a dimensional analysis along with

Buckingham's pi theory reduced the number of design variables from 14 to 8, which still made

testing all possible combinations an intractable problem using experimental techniques alone.

Thus, an experimental design analyses was applied to reduce the number of tests. Limited by the

use of experiments only, six critical design variables were chosen and were tested at two levels

per variable. After 32 cases had been built and tested, the sensitivity of the response to the

design variables required a reevaluation of the optimization process. However, the results of the

analysis eventually yielded information about the relationship between the flow behavior and the

chosen geometry parameters. Curve fits were then developed based on the optimization results

for later use.

Gill26 described the movement from injector designs based on historical data and the

injector designer' s experience and intuition to computer-aided designs. These early analyses

included one-dimensional approximations such as equilibrium reactions. These equilibrium

approximations were based on the assumption of "fast-chemistry," and were used to characterize

the combustion field to aid the design process. Gill's research looked at the effects of the

inj ector geometry and inj section pattern on the combustion process. Performance of the inj ector

had to be maintained while increasing the life of the inj ector by preventing burning through or

erosion of the combustor components.

Most experimental techniques were insufficient to predict for many conditions that could

reduce inj ector or combustion chamber life. For this reason, researchers now look to employ

CFD techniques to aid in injector design. For example, Liang et al.27 developed semiempirical

calculations to aid in the computer modeling of liquid rocket combustion chambers. Sindir and

Lynch28 Simulated a single element GO2/GH2 inj ector using an 8-species, 18 reaction chemical









kinetics model with the k-E turbulence model and compared the velocity profile to experimental

measurements with good results. Tucker et al.5 stressed the importance of supplying

experimental results to aid in the task of CFD validation.

Shyy et al.29 and Vaidyanathan et al.30 identified several issues inherent in CFD modeling

of inj ectors, including the need for rigorous validation of CFD models and the difficulty of

simulating multi-element injector flows due to lengthy computational times. They chose to

simulate single inj ector flow, as this allowed for the analysis of key combustion chamber life and

performance indicators. The life indicators were the maximum temperatures on the oxidizer post

tip, inj ector face, and combustion chamber wall, and the performance indicator was the length of

the combustion zone. These conditions were explored by varying the impinging angle of the fuel

into the oxidizer. The studies were successful in using CFD to specifically access the effects of

small changes in inj ector geometry on combustion performance.

1.2.2 Rocket Engine Turbine Design

Turbines are developed based on basic knowledge of the fuel characteristics, estimated

stresses, and turbine pressure. The turbines are driven using hot exhaust gas from the

combustion process. Turbines were and still are designed by using empirical relations to

calculate a preliminary starting point for the design. The designs rely both on past information

from similar turbine designs and on experimental correlations. Unlike for injector design,

computers have been used in the design process for rocket engine turbines for over 40 years.

Beer 1965 used the aid of computers to estimate the velocity distribution across a turbine blade

to help improve turbine design. Simple 1-D codes are routinely used to provide preliminary

information on design temperatures, stresses, overall flow geometry, velocities, and pressures.31

The prediction of turbine performance was later improved by the consideration of aerodynamic

effects that lead to pressure losses.32









Current design methodologies now combine past design practices with CFD analyses.

Experimental correlations and simple computer programs are used to determine the preliminary

design. CFD can then be used to fine-tune or analyze the design.33 In particular, the CFD

analyses are used to shape the blade such that problems such as large pressure losses or

unsteadiness are minimized.34

1.3 CFD-Based Optimization

The designs of many aerospace components are moving from the experimental realm to the

computational realm. Some products can now be completely designed on computers before they

are built and tested, including aircraft such as the Boeing 777-300,35 Bombardier Learj et 45,36

and Falcon 7X.37 Using CFD in coordination with optimization techniques provides a means of

effectively handling complex design problems such as rocket engine components.

1.3.1 Optimization Techniques for Computationally Expensive Simulations

Scientific optimization techniques have evolved substantially and can now be effectively

used in a practical environment. When appropriately combined with computer simulations, the

entire design process can be streamlined, reducing design cost and effort. Popular optimization

methods include gradient-based methods,38-4 adjoint methods,44,45 and surrogate model-based

optimization methods such as response surface approximations (RSA),46 Kriging models,47-4 Of

neural network models.'o Gradient-based methods rely on a step by step search for an optimum

design using the method of steepest descent on the objective function according to a convergence

criterion. Adj oint methods require formulations that must be integrated into the computational

simulation of the physical laws. In both gradient-based and adj oint methods, the optimization is

serially linked with the computer simulation.

At times, gradient-based optimization is not feasible. For example, Burgee et al.5 found

that an extremely noisy obj ective function did not allow the use of gradient-based optimization,









as the optimizer would often become stuck at noise-induced local minima. Also, for simulating

designs with long run times, it may be more time efficient to run simulations in parallel, as this

will allow many simulations to run simultaneously. For these problems, optimization based on

an inexpensive surrogate is a good choice. Surrogate-based optimization (SBO) uses a

simplified, low-order characterization of the design space instead of relying exclusively on

expensive, individually conducted computational simulations or experimental testing. Surrogate-

based optimization allows for the determination of an optimum design while exploring multiple

design possibilities, thus providing insight into the workings of the design. A surrogate model

can be used to help revise the problem definition of a design task by providing information on

existing data trends. Furthermore, it can conveniently handle the existence of multiple desirable

design points and offer quantitative assessment of trade-offs, as well as facilitate global

sensitivity evaluations of the design variables to assess the effect of a change in a design variable

on the system response.52,53 The SBO approach has been shown to be an effective approach for

the design of computationally expensive models such as those found in aerodynamics,54

structures," and propulsion.56 The choice of surrogates may depend on the problem. For

example, Shyy et al.5 compared quadratic and cubic polynomial approximations to a radial basis

neural network approximation in the multi-obj ective optimization of a rocket engine inj ector.

Simpson et al.5 suggested using Kriging as an alternative to traditional quadratic response

surface methodology in the optimization of an aerospike nozzle. Dornberger et al.59 COnducted a

multidisciplinary optimization on turbomachinery designs using response surfaces and neural

networks.

To construct the surrogate model, a sufficient number of different designs must be tested to

capture how a system response varies with different design parameters. The process of selecting









different designs for testing is termed Design of Experiments (DOE). Designs may be tested

mainly at the design parameter extremes as in central-composite designs (CCD) and face-

centered cubic designs (FCCD), or they may be tested across the full range of design parameter

values as in multi-level factorial designs46 Or Latin hypercube sampling.60 The design space is

defined by the physical range of parameters, or variables, to be explored. Variable ranges are

often determined based on experience with an existing design, but in many cases, the choice of

variable ranges is an educated guess. The simplest choice is to set simple range limits on the

design variables before proceeding with the optimization, but this is not always possible.

Simpson et al.61 USed a hypercube design space with design variables range selection based on

geometry to prevent infeasible designs in the SBO of an aerospike nozzle. Rodriguez 62

successfully optimized a complex jet configuration using CFD with simple geometry-based

bounds on the design variables. Vaidyanathan et al.63 chose variable ranges based on variations

of an existing design in the CFD-based optimization of a single element rocket inj ector.

Commonly, the design space cannot be represented by a simple box-like domain. Constraints or

regions of infeasibility may dictate an irregular design. Designs that work best for irregular

design spaces include Latin Hypercube Sampling60 and orthogonal arrays.46 Once the design

space is bounded, the optimization commences. In order to reduce computational expense, it is

very important for a surrogate to be an accurate replacement for an expensive CFD simulation.

The application of the SBO approach in the context of CFD-based optimization with

complex flow models can present a significant challenge, in that the data required to construct

the surrogates is severely limited due to time and computational constraints of the high fidelity

CFD runs. In some cases, the problem of high computational cost in SBO can be addressed by

performing an optimization on a low fidelity model and translating the result to a high fidelity









model. Doing so requires additional analyses to ensure that the low-fidelity model is a

qualitatively similar to the high fidelity model. In other instances, low-fidelity data may be

combined with high-fidelity data to reduce the overall number of expensive high-fidelity runs.

The low-fidelity data may employ a correction surrogate for improved accuracy, as shown in

Figure 1-4.

Han et al.64 perfOrmed a surrogate-based optimization on a low-fidelity 2-D CFD model

and tested the results on a more expensive 3-D model in the optimization of a multi-blade fan

and scroll system and verified the 2-D model using physical experiments as well as comparisons

to previous studies and the 3-D model. Keane65 built an accurate surrogate model of a transonic

wing using high-fidelity CFD data supplemented with data from a low fidelity empirical model.

Alexandrov et al.66 USed variation in mesh refinements as variable fidelity models for a wing

design. Knill et al.67 Saved 255 CPU hours by supplementing high-fidelity data with a low-

fidelity aerodynamic model in the optimization of a high speed civil transport wing. Vitali et

al.68 USed surrogates to correct a low-fidelity model to better approximate a high fidelity model in

the prediction of crack propagation. Thus, the accuracy of the low-fidelity models can be

improved by combining the low-fidelity model with the surrogate correction, and a new

surrogate could be constructed based on the composite model. Venkataraman et al.69 also used a

surrogate correction on a low-fidelity model to aid in the optimization of shell structures for

buckling. Balabanov et al.70 applied a surrogate correction to a low-fidelity model of a high

speed civil transport wing. Haftkan1 and Chang et al.72 Scaled surrogate models with local

sensitivity information to improve model accuracy for analytical and structural optimization

problems.









With continuing progress in computational simulations, computational-based optimization

has proven to be a useful tool in reducing the design process duration and expense. However,

because of the complexity of many design problems such as is involved with liquid rocket engine

components, CFD-based optimization can be a daunting task. In some cases, a surrogate model

is fit to the points and the predictive capability for the unknown cases is found to be

unacceptably low. One solution is to try an alternative surrogate model--perhaps one with more

flexibility--and approximate the response again. Another remedy is to refine the design space

based on the imperfect but useful insight gained in the process. By reducing the domain size,

design space refinement often naturally leads to the improvement of a surrogate model.

1.3.2 Design Space Refinement in CFD-Based Optimization Using Surrogate Models

The design space is the region in which obj ective functions are sampled in search of

optimal designs, where the region is bounded by defined constraints. Design space refinement

(DSR) is the process of narrowing down the search by excluding regions as potential sites for

optimal designs because (a) they obviously violate the constraints, or (b) the obj ective function

values in the region are poor. In the optimization of a high speed civil transport wing, Balabanov

et al.70 discovered that 83% of the points in their original design space violated geometric

constraints, while many of the remaining points were simply unreasonable. By reducing the

design space, they eliminated their unreasonable designs and improved the accuracy of the

surrogate model. Roux et al.73 found that the accuracy of a polynomial RSA is sensitive to the

size of the design space for structural optimization problems. They recommended the use of

various measures to find a small "reasonable design space."

It is often advantageous to refine the design space to meet the needs of the design problem.

For example, design constraints can prohibit the use of a simple hypercube design space.74,75 In

other instances the fitted surrogate is simply inaccurate. For a high-dimensional design space,









adequately filling a hypercube design space region with points becomes cost prohibitive.

Balabanov et al.70 noted that to sample only at the vertices of a box-shaped design requires 2n

points, where n is the number of design variables. In contrast, a simplex sampled at the vertices

would require only n + 1 points. In particular, for n = 25, a box domain has a volume 1010 times

larger than a corresponding ellipsoid. Several techniques are used to improve the optimization

process and surrogate accuracy by reducing the domain over which the function is approximated.

Madsen et al.74 refined the design space to an ellipsoid shape in the shape optimization of a

diffuser to prevent the unnecessary CFD analysis of physically infeasible designs. In the

optimization of a super-detonative ram accelerator, Jeon et al.76 found that a surrogate model that

was a poor fit due to sharp curvature in the response and was able to achieve a better result by

performing a transformation on the design space, hence, smoothing out the curvature. Papila et

al.7 explored the possibility of reducing errors in regions the design space by constructing a

reduced design space on a small region of interest. Balabanov et al.70 eliminated suspected

infeasible design regions from the design space using a low-fidelity analysis prior to optimizing

using a high-fidelity CFD analysis in the design of a high speed civil transport wing. Papila et

al.7 refined the design space by expanding the variable bounds to account for optimal designs

that lay on the edge of the design space in the shape optimization of a supersonic turbine.

1.4 Summary

Rocket engine component design is a complex process that can be made more

straightforward using CFD-based optimization techniques. The performance and efficiency of

key components can be improved using the SBO methods. The application of the SBO approach

to CFD problems can be challenging due to time and computational constraints. Tools including

low-fidelity analyses and design space refinement can help reduce computational expense. In










particular, design space refinement techniques can be employed to facilitate a more efficient

optimization process by reducing the number of necessary computationally expensive CFD runs.

1.5 Goal and Scope

The research goal is to investigate adaptive DSR using measures based on model

uncertainty from varying surrogate models and focusing on regions of interest to improve the

efficiency of the CFD-based design of rocket engine components. Chapter 2 reviews the

proposed DSR techniques as well as the optimization framework that is used to complete the

optimizations. This includes the use of new optimization tools to accelerate the optimization

process. In Chapter 3, DSR techniques are applied in the low-fidelity optimization of a radial

turbine for liquid rocket engines. The preliminary design of a radial turbine is a logical next

step, because 1) it is a new design where the optimal design space size and location is unknown,

and 2) a 1-D solver is used to provide function evaluations allowing for the collection of a large

number of data points. The DSR techniques are used to determine both an applicable design

space size by excluding infeasible regions and the region of interest by excluding poorly

performing design points. The optimization framework techniques are used to complete the

optimization process. The optimization techniques then combined with CFD with a Einal

obj ective of improving the rocket engine injector design process. The governing equations for

combusting flow and a background on rocket engine inj ectors are given in Chapter 4. The

optimization methods are demonstrated using an analysis of a trapezoidal bluff body to improve

combustor mixing characteristics in Chapter 5. The bluff body analysis is used to show that

DSR can identify regions of poor simulation accuracy. Chapter 6 presents the CFD modeling

effort for a single element inj ector. The CFD results are compared to an experimental work for

accuracy. Then, the flow dynamics leading to wall heat transfer are analyzed, providing

information that would be difficult to obtain using an experimental analysis alone. Finally, in
























Author Application Method highlighted Key results


9


Chapter 7, CFD-based optimization techniques are applied to improve the design process of a

multi-element rocket inj ector. In this case, the optimization techniques must be applied in the

context of a CFD-based design process where a limited number of function evaluations are

available. Chapter 8 concludes the effort.


Table 1-1. Summary of CFD-based design optimization applications with highlighted surrogate-
based optimization techniques.


Shyy et al.


Simpson et al."0'



Dornberger et
al.59

Rodriguez62


Vaidyanathan et
al.63
Han et al.64



Keane65


Alexandrov et
al.66
Knill et al.6


Balabanov et
al. '


Madsen et al. 4


Jeon et al. 6


Papila et al. 7

Balabanov et
al. 0

Papila et al. 8


Rocket engine
injector
Aerospike nozzle



Turbomachinery
designs
Jet configuration


Rocket engine
injector
Multi-blade
fan/scroll
system
Transonic wing


Wing design


High speed civil
transport wing

High speed civil
transport wing


Diffuser


Ram accelerator


Supersonic turbine


Civil transport
wing
Supersonic turbine


Surrogate comparison


Surrogate comparison,
design space
selection

Surrogate comparison


Design space selection


Design space selection


Low-fidelity
optimization


Multi-fidelity
optimization
Multi-fidelity
Optimization
Multi-fidelity
optimization
Multi-fidelity analysis
with surrogate
correction

Design space
refinement

Design space
refinement

Design space
refinement

design space
refinement


Polynomial approximation compared to neural
network surrogate model.

Kriging model compared to quadratic response
surface. Simple design variable ranges
selected using geometry.
Multidisciplinary optimization using response
surfaces and neural networks.

Optimization using simple geometry-based
bounds on design variables

Design variables chosen from variations of
existing design

Optimized low-fidelity 2-D model to predict for
high-fidelity 3-D model


Supplemented high-fidelity CFD data with low-
fidelity empirical model
Varied mesh density for variable fidelity models


Saved 255 CPU hours using low-fidelity
supplemental data
Used coarse and fine finite element models as
low- and high-fidelity models


Removed infeasible design space regions to
prevent unnecessary CFD analyses
Performed transformation on design space for
better response surface fit
Refine design space to improve regions with
high error
Used low-fidelity model to refine design space
before optimizing using high-fidelity model

Design space expansion to explore optimal
designs at edge of design space


Design space
refinement






































MOVy


al '


OXlduz


oxidizer pump


oxiize


fuel pump


Figure 1-1. Rocket engine cycles.
cycle.


A) Expander cycle, B) gas generator cycle, C) pre-burner


I POTP


Figure 1-2. SSME thrust chamber component diagram showing low-pressure fuel turbopump
(LPFTP), low-pressure oxidizer turbopump (LPOTP), high-pressure fuel turbopump
(HPFTP), high-pressure oxidizer turbopump (HPOTP), main fuel valve (MFV), main
oxidizer valve (MOV), combustion chamber valve (CCV), and the main combustion
chamber (MCC). Picture reproduced from [81].


PFTP


HPOTP


HPFTP
GCV













0.00015

















mode


Hig-fdeit i ;Eb mmBt~ P
model
Surrogate K~







Figure 1-3.S Asses surrogante refia ine design space rpodcdfrm[



Lwfidelitymoe

Figure 1-4. Surrogate-based op imizatio sn ut-ieiydta ut-ieiyaayi

may requireasroaemdlfrcretngtelwfdlt oe obte

appoxiiehymatlw/cre thhighfdltymdl










CHAPTER 2
OPTIMIZATION FRAMEWORK

The optimization framework steps include 1) modeling of the obj ectives using surrogate

models, 2) refining the design space, 3) reducing the problem dimensionality, and 4) handling

multiple obj ectives with the aids of Pareto front and a global sensitivity evaluation method.

Several procedures may require moving back and forth between steps. Figure 2-1 illustrates the

process used to develop optimal designs for the problems involving multiple and possibly

conflicting objectives. The steps of the framework are detailed in the following sections.

The first step in any optimization problem is to identify the performance criteria, the

design variables and their allowable ranges, and the design constraints. This critical step requires

expertise about the physical process. A multi-objective optimization problem is formulated as

Minimize F(x), where F = f,: j= 1,Ml; x = x,:Vi =1, N

Subj ect to

C(x)< 0, where C=c :Vp=1,P
H(x) = 0, where H= hk : k =1, K

Once the problem is defined, the designs are evaluated through experiments or numerical

simulations. For numerical simulations, the type of numerical model and design evaluation used

varies with the goals of the study. For a simple preliminary design optimization, the use of an

inexpensive 1-D solver may be sufficient. However, for the final detailed design, more complex

solvers may be needed.82,78 The choice of a model has an important bearing on the

computational expense of evaluating designs. When obtaining many design points is time-

prohibitive, it is often more prudent to use an inexpensive surrogate model in place of the

expensive numerical model.









2.1 Optimization Using Surrogate Models

This step in the framework involves developing alternate models based on a limited

amount of data to analyze and optimize designs. The surrogates provide fast approximations of

the system response making optimization and sensitivity studies possible. Response surface

approximations, neural network techniques, spline, and Kriging are examples of methods used to

generate surrogates for simulations in the optimization of complex flows52 inVOlving applications

such as engine diffusers,74 TOcket inj ectors,63 and supersonic turbines.78,82

The maj or benefit of surrogate models is the ability to quickly obtain any number of

additional function evaluations without resorting to more expensive numerical models. In this

aspect, surrogate models can be used for multiple purposes. Obviously, they are used to model

the design obj ectives, but they can also be used to model the constraints and help identify the

feasible region in design space. Key stages in the construction of surrogate models are shown in

Figure 2-1.

2.1.1 Design of Experiments

The search space, or design space, is the set of all possible combinations of the design

variables. If all design variables are real, the design space is given as x E R" where N is the

number of design variables. The feasible domain S is the region in design space where all

constraints are satisfied.

For adequate accuracy, the data points used in the surrogate model must be carefully

selected. The proper data selection is facilitated by using Design of Experiments (DOE). One

challenge in design optimization despite the type of surrogate used is called the "curse of

dimensionality." As the number of design variables increase, the number of simulations or

experiments necessary to build a surrogate increases exponentially. This must be taken into

consideration in choosing a DOE. The key issues in the selection of an appropriate DOE include









(i) the dimensionality of the problem, (ii) whether noise is important source of error, (iii) the

number of simulations or experiments that can be afforded, (iv) the type of surrogate used to

model the problem, and (v) the shape of the design space. If noise is the dominant source of

error, DOEs that reduce sensitivity to noise are commonly used. These include central composite

designs (CCD) or face-centered cubic designs (FCCD) for box-shaped domains. Design

optimality designs such as D- or A-optimal83 (Myers and Montgomery 2002, pp. 393 395)

designs are useful for irregular shaped domains and high dimensional domains when minimizing

noise is important. Specifically, these designs can be used to reduce the number of points in an

experimental design for a given accuracy. When noise is not an issue, space-fi11ing designs such

as Latin-Hypercube Sampling (LHS)60 Or Orthogonal Arrays (OAs)46 are preferred to efficiently

cover the entire design space.

Central composite designs and face-centered cubic designs are intended to minimize the

presence of noise in the response. They are comprised of a single point at the center of the

design space with the remaining points situated along the periphery of the design space, as

shown in Figure 2-2. The center point detects curvature in the system while all other points are

pushed as far as possible from the center provide noise-smoothing characteristics. The FCCD is

actually a modified version of the CCD with the axial points moved onto the edges of a square.

An FCCD is required when the limits of the design space are strictly set, requiring a square

design space.

The CCD and FCCD are most popular for response surface approximations. They are

commonly used to reduce noise in the response by placing most of the points on the boundary of

the design space. For responses that do not contain noise, such as data obtained using

deterministic computational simulations, a space-fi11ing design may be required for accuracy.









While CCD and FCCD only contain points at the center and edges of the design space, space-

Eilling designs distribute points throughout the design space. The most common type of space-

Eilling design is Latin-Hypercube Sampling (LHS). The LHS technique consists of dividing the

design space Ntimes along each variable axis creating Nksub sections where N is the number of

data points and k is the number of design variables. A total of Ndata points are randomly placed

such that only one point exists in each subsection interval. This procedure can possibly leave

holes in the design space as shown in Figure 2-3A. A criterion to maximize the minimum

distance between points can help spread points more evenly and prevent bunching. Orthogonal

array LHS84 is another method of improving the distribution of data points within the design

space. It insures a quasi-uniform distribution of points across the design space. The design

space is uniformly divided into larger subsections and LHS is applied while ensuring one point

lies in each large subsection (Figure 2-3B).

2.1.2 Surrogate Model Identification and Fitting

There are many types of surrogate models to choose from. There are parametric models

that include polynomial response surfaces approximations (RSA) and Kriging models, and there

are non-parametric models such as radial basis neural networks (RBNN). The parametric

approaches assume the global functional form of the relationship between the response variable

and the design variables is known, while the non-parametric ones use different types of simple,

local models in different regions of the data to build up an overall model. Response surface

approximations assume that the data is noisy and thus it fits the data with a simple, smooth

global function to minimize the root-mean-square (rms) error. Kriging and RBNN use functions

localized to the data points. Ordinary Kriging, the type of Kriging used in this study, uses all of

the data points for that purpose and interpolates over all of them, while RBNN uses a subset of

the points and minimizes the rms error. The subset is determined based on the desired minimum









size of the error. RSA requires a solution of linear equations, as does Kriging for a given

correlation function. In Kriging, the process of finding the correlation function is an

optimization problem that is done either with cross-validation or minimum likelihood estimates.

On the other hand, RBNNs let the user experiment with one parameter that defines the

correlation (SPREAD). However, the process of minimizing the rms is an optimization problem.

2.1.2.1 Polynomial response surface approximation

Response surface approximations (RSA) were developed for experimental design

processes. They consist of a polynomial curve fit to a set of data points. Usually the polynomial

is a 2nd-order model, but higher order models may also be used. A 2nd-order model requires a

minimum of(k +1I)(k + 2)/2 design points, where k is the number of design variables. Thus, a

2nd-order model with three variables would require at least 10 points. A 3rd-order model with

three variables would require twice that amount. A 2nd-order model provides good trade-off in

terms of accuracy and computational expense. For RSAs to be effective in design optimization,

the sampled points must be near the optimum design. In this case, it is often necessary to

perform a screening exercise on the data to ensure that the design space is selected in the

appropriate region. Once the proper design region has been narrowed down, the region can be

populated with data points using a DOE such as CCD or FCCD.

The popular polynomial RSAs have been used extensively in design optimization for

numerous applications using varying DOEs with varying results. Rais-Rohani and Singhss used

polynomial RSAs to increase the structural efficiency of composite structures while maximizing

reliability. They found that the accuracy and efficiency ofRSAs can vary depending on the

choice of DOE. Polynomial RSAs have been successful in smoothing responses that contain

noise. In the aerodynamic design of a subsonic wing, Sevant et al.86 CHCOuntered numerical









noise resulting in local maxima. Polynomial RSAs were used to smooth the noise. Sturdza et

al.8 used a response surface to optimize the shape of a fuselage to retain laminar flow across a

wing. Clues to the accuracy of RSAs can be found through the simultaneous use of multiple

RSAs. Kurtaran et al.8 used linear, elliptic, and quadratic RSAs to optimize for

crashworthiness. Kurtaran et al. found that the differences between the predictive capabilities of

the various RSAs decreased as the design space was reduced. Hosder et al.89 USed RSAs in

conjunction with techniques to reduce the design space and perform multi-fidelity analyses in the

multidisciplinary optimization of aircraft. Additional references and their key results are given

in Table 2-1.

The polynomial RSA assumes that the function of interest f can be represented as a linear

combination of 4 basis functions z, and an error term e. For a typical observation i, a response

can be given in the form of a linear equation as


f(,()= S,,z B e (e,)=O 0 V(e)= cr (2-1)


where the errors 8, are considered independent with an expected value equal to zero and a

variance equal to o The coefficients a, represent the quantitative relation among basis functions

z,. Monomials are the preferred basis functions.

The relationship between the coefficients B, and the basis functions z, is obtained using N,

sample values of the response f for a set of basis functions z,(') such that the error in the

prediction is minimized in a least squares sense. For N, sample points, the set of equations

specified in Equation 2-1 can be expressed in matrix form as

f = XP+ e E (E)=0 V (E) = 0- (2-2)










where X is a Ns x No matrix of basis functions, also known as a Gramian design matrix, with the

design variable values as the sampled points. A Gramian design matrix for a quadratic

polynomial in two variables (Ns = 2; Nc = 6) is given by

1 x,, x21 121 11x, 21
1x12 X22 212 X12 22 X2

X ~ ~ =~~ X21 (2-3)

1 x,, x2; 12 2 2



The vector b of the estimated coefficients, which is an unbiased estimate of the coefficient vector

(3 and has minimum variance, can then be found by


b =(XTX) XTf (2-4)

At a new set of basis function vector z for design point P, the predicted response and the variance

of the estimation are given by


.,(z)=~n b z ') and V .,(z,) = c2 T XT)z (2-5)
J=1

2.1.2.2 Kriging

In cases where a given surrogate performs poorly, it is necessary to use a different type of

surrogate model, because the highly nonlinear nature of some processes may not be captured by

surrogates such as RSAs. Kriging is a popular geostatistics technique named after the pioneering

work of D.G. Krige47 and was formally developed by Matheron.90 The Kriging method in its

basic formulation estimates the value of a function or response at some location not sampled as

the sum of two components: the linear model (e.g., polynomial trend) and a systematic departure

representing low (large scale) and high frequency (small scale) variation components,

respectively. Kriging has the added flexibility of being able to either provide an exact data fit or









to act as a smoothing function. It can be used to approximate a highly nonlinear response with

fewer points than a comparable high-order response surface. Koch et al.91 asserts that using

Kriging may be preferable over RSAs for design problems with a large number of design

variables. Mahadevan et al.92 found that Kriging was more accurate than an RSA and used fewer

function evaluations for engineering reliability estimates.

Kriging has the capability of overcoming the limit of relatively small design space that is

inherent in RSAs, and can be used effectively for larger design spaces. It is gaining popularity in

CFD-based optimization as an alternative to RSAs. Forsberg and Nilsson93 found that Kriging

provided a better approximation as compared to an RSA in the structural optimization of

crashworthiness. However, Jin et al.94 found that Kriging was very accurate, but also very

sensitive to noise as compared to other surrogates in the data fitting of several analytical test

problems. Rijpkema et al.95 found that Kriging was better able to capture local details in an

analytical test function than an RSA, but also cautioned against the unintentional fitting of noise.

Chung and Alonso96 COmpared Kriging to RSAs and found that while both can be accurate,

Kriging is better at fitting functions with several local optima. Simpson et al.97 found that

Kriging and response surface performed comparably in the optimization of an aerospike nozzle.

Kanazaki et al.98 USed Kriging to reduce the computational cost involved in optimizing a three-

element airfoil using genetic algorithms by using the Kriging surrogate for function evaluations.

Jouhaud, J. -C. et al.99 USed Kriging to adaptively refine the design space in the shape

optimization of an airfoil.

Kriging is represented by a trend perturbed by a "systematic departure." In this study,

ordinary Kriging is used, where the constant trend is represented by the sample mean. Other

types of Kriging include simple Kriging that uses a constant trend of zero, and universal Kriging









uses a linear trend. The systematic departure component represents the fluctuations around the

trend with the basic assumption being that these are correlated, and the correlation is a function

of distance between the locations under consideration. More precisely, it is represented by a zero

mean, second-order, stationary process (mean and variance constant with a correlation depending

on a distance) as described by a correlation model. Hence, these models suggest estimating

deterministic functions as

y(x) = p~r+ (x) E() =0 cov(E(x(")), e(x('))-A V i, j (2-6)

where pu is the mean of the response at sampled design points, and e is the error with zero

expected value and with a correlation structure that is a function of a generalized distance

between the sample data points.

cov e(Ex I), e(x I)) = c R x', x') (2-7)

In this study, a Gaussian correlation structure'oo is used


R =ep- k X/k- X (2-8)


where R is the correlation matrix among the sample points, N, denotes the number of dimensions

in the set of design variables x, o- identifies the standard deviation of the response at sampled

design points, and Ok is a parameter which is a measure of the degree of correlation among the

data along the kth direction. Specifically, the parameters pu, e, and a are estimated using a set ofN

samples (x, y) such that a likelihood function is maximized.100 Given a probability distribution

and the corresponding parameters, the likelihood function is a measure of the probability of the

sample data being drawn from it. The model estimates at non-sampled points are

j)= + rTR' -1 ) (2-9)









where is the estimated mean, r identifies the correlation vector between the set of prediction

points and the points used to construct the model, y a vector of the response at the sampled

points, and 1 denotes an Ns-vector of ones. The mean is estimated by

p= 1 R'1) 1 I'R y (2-10)

and the standard deviation of the response is estimated as


<^r =(lLT 'yp (2-11)


On the other hand, the estimation variance at non-sampled design points is given by

1-1'R-'r) (-2
V(y(x))=o cr -rTR 'r+ (-2
1 R 11

In this study, the model estimates, estimation variance, and standard deviation of the response

are calculated using the Matlab toolbox DACE.101

2.1.2.3 Radial basis neural networks

Neural networks are another alternative to traditional response surface methods.

Specifically, neural networks are able to fit data with a highly complex and nonlinear response.

Artificial neural networks are made of interconnected local models called neurons. A high

number of neurons can improve the accuracy of the fit, while using fewer neurons improves the

smoothing qualities of the model. Mahajan et al.102 USed neural networks (NN) to optimize a

mechanically aspirated radiation shield for a meteorological temperature sensor to gain

information on the workings of the temperature sensor system and the performance of the

radiation shield. Papila et al.78 used radial basis neural networks to supplement the data used to

construct an RSA in the optimization of a supersonic turbine for rocket propulsion.

Charalambous et al.103 USed neural networks in the prediction of bankruptcy and stressed that the









prediction ability of a neural network is problem dependent. Hiisken et al.104 USed neural

networks in turbine blade optimizations. Shi and Hagiwaralos used neural networks to maximize

the energy dissipation of the crashworthiness of a vehicle component. Pidaparti and Palakall06

compared the prediction capability of neural networks with experimental results in the prediction

of crack propagation in aging aircraft. Chan and Zhu107 demonstrated the ability of neural

networks in modeling highly nonlinear aerodynamic characteristics with many variables.

Brigham and Aquinolos used neural networks to accelerate their optimization search algorithm

by providing inexpensive function evaluations.

Neural networks are made up of a distribution of artificial neurons, whose function is

modeled after biological neurons. A shape function is applied at each neuron with the

combination being a complete function curve. Each neuron is given a weight in the network that

defines that individual neuron's strength. Neural networks are often difficult to train due to the

required specification of several parameters that can affect the fit of the model. Care must also

be taken, as there is sometimes danger in over-fitting the data, resulting in poor prediction

capability away from the known data points. Nonetheless, the models are extremely flexible and

can provide an exact data fit, or meet a user-defined error criterion to filter noise to prevent data

over-fitting. Neural networks use only local models, and the degree of effective influence of

each local model, or neuron, is a selection made by the user.

Radial basis neural networks are two-layer networks consisting of a radial-basis function

(RBF) and a linear output layer. The radial basis function is given by

f= radbas c xb) (2-13)

where b is the bias, c is the center vector associated with each neuron, x is the input design

vector, and the function radba~s is the Matlab function name for an RBF, or a local Gaussian









function with centers at individual locations from the matrix x. The radial basis function is given




rradbas (n)- = ep -72) (2-14)

where n is given by the term in parenthesis in Equation 2-13. Based on these equations, the

function prediction is given by


j) ,~=1 w, ep -b (2-15)

where w, is the weight vector for each neuron i, and N is the number of neurons. The Gaussian

RBF decreases monotonically with the distance from its center. The bias b is set to

0.8326/SPREAD where SPREAD is the spread constant, a user defined value that specifies to the

radius of influence for each neuron where the radius is equal to SPREAD/2. Specifically, the

radius of influence is the distance at which the influence reaches a certain small value. If

SPREAD is too small, then prediction is poor in regions that are not near the position of a

neuron. If SPREAD is too large, then the sensitivity of the neurons will be small. Because

RBNNs are based on the combination of multiple local approximations, a space-filling DOE is

necessary. If large spaces exist in the DOE, it may be difficult to overlap information from

nearby points. The capability of the RBNN to fill the information gaps in the design space is

based on the size SPREAD. If SPREAD is too small, the RBNN surrogate may not provide any

predictive ability in some regions within the large space. For this region, there must be

coordination between the spacing of data points within the design space and the size of

SPREAD.

Increasing the number of neurons will improve the accuracy of the surrogate. Neurons are

added to the network one by one until the sum of the squares of the errors (SSE) is reduced

enough to reach a specified error goal. If the error goal is set to zero, neurons will be added until









the network exactly predicts the input data. However, this can lead to over-fitting of the data

which may result in poor prediction between data points. On the other hand, if error goal is large,

the network will be under-trained and predictions even on data points will be poor. For this

reason, an error goal is judiciously selected to provide early stopping of the network training to

prevent over-fitting and increase the overall prediction accuracy.

The purpose of the user is obviously to determine good values of SPREAD and the error

goal, GOAL. This can be done by reserving a small number of data points as test data. The

remaining data is designated as the training data. First, the network is trained using the training

data and the Matlab function newrb. Then, the network is simulated at the test points and the

error is calculated between the actual and predicted response. The process is repeated for a small

number of values with selected range of SPREAD and a small set of values for GOAL. The

values of SPREAD and GOAL that produce the smallest errors in the test data are identified,

then the ranges of SPREAD and GOAL are reduced and new values are selected. The process

iteration continues until the SPREAD and GOAL combination is identified that minimizes the

error in the test data. The ideal combination can usually be identified within three iterations. In

this way, the training data is used to train the neural network, and the test data is used to tune it.

2.1.3 Surrogate Model Accuracy Measures

Once the surrogate models are available, it is imperative to establish the predictive

capabilities of the surrogate model away from the available data. Several measures of predictive

capability are given below.

2.1.3.1 Root mean square error

The error 8, at any design point i is given by

E, = J; I (2-16)










whereJ; is the actual value and j; is the predicted value. For Ns design points, the root mean

square (rms) error a is given by



r 1= (2-17)
SN,

A small rms error indicates a good fit. The rms error is a standard measure of fit that may be

used to compare the performances of surrogates of different types.

The adjusted rms error oa is used to adjust the rms error based on the number of parameters

in a polynomial RSA and is given by



cr (2-18)
a (N q)

where Ns is the number of data points, and q is the number of terms in the polynomial

approximation. A polynomial with a large number of terms is penalized and deemed less

desirable for a polynomial approximation with fewer terms for the same number of data points.

For a good fit, oa should be small compared to the data value ranges.

For Kriging, the standard error measurement is given by the estimated standard deviation

of the response given by Equation 2-11i. The square root of the estimated standard deviation of

the response can be compared to the rms error of polynomial RSAs.

2.1.3.2 Coefficient of multiple determination

The, adj uLuse coeficientII of mIultle determIIinaionLV R)4d defines the prediction capability of

the polynomial RSA as



R2 = 1- %cr=1a ( 1-)\2 Where f= ,= (2-19)









For a good fit, R~ should be close to 1.

2.1.3.3 Prediction error sum of squares

When there are an insufficient number of data points available to test the RSA, the

prediction error sum of squares (PRESS) statistic is used to estimate the performance of the

surrogate. A residual is obtained by fitting a surrogate over the design space after dropping one

design point from the training set. The value predicted by the surrogate at that point is then

compared with the expected value. PRESS is given by


PRESS=~Jf f-/ (2-20)


and PRESSms is given by



PRESS -'=1 (2-21)


where j* is the value predicted by the RSA for the ith point which is excluded while generating

the RSA. If the PRESSms value is close to cra, this indicates that the RSA performs well.

2.2 Dimensionality Reduction Using Global Sensitivity Analysis

At the beginning of a design optimization, several variables are chosen as design variables

with the assumption that they are important to the optimization. However, having large numbers

of design variables can greatly increase the cost of the optimization. Section 2. 1.2 introduced the

concept of the "curse of dimensionality," meaning that as the number of design variables

increase, the number of data points required to obtain a good approximation of the response can

increase exponentially. It is of great benefit, therefore, to simplify the design problem by

identifying variables that are unimportant and removing them from the analysis. The most

efficient way of doing this is to perform a sensitivity analysis. A global sensitivity analysis can









provide essential information on the sensitivity of a design obj ective to individual variables and

variable interactions. By removing the variables that have negligible influence on the design

obj ective, the dimensionality of the problem can be reduced.109,110,111

Global sensitivity analyses enable the study of the behavior of different design variables.

This information can be used to identify the variables which are the least important and thereby

can reduce the number of variables. A surrogate model f(x) of a square integrable obj ective as a

function of a vector of independent input variables x ( xl a [0,1] Vi = 1, N) is assumed and modeled

as uniformly distributed random variables. The surrogate model can be decomposed as the sum

of functions of increasing dimensionality as

.f(x) =. f+ Cr;(x,)+/ Cf;, (x, x3)+ ---+.t;, (xl,Zx,.,x,) (2-22)


where / = = of dx If the following condition


dxk = 0(2-23)


is imposed for k = ii, ..., i,, then the decomposition described in Equation 2-22 is unique.

In the context of a global sensitivity analysis, the total variance denoted as YV) can be

shown to be equal to


V(f)= IV + V +---+V (2-24)


where V ( f) = EjT i f -f, and each of the terms in Equation 2-24 represent the partial variance

of the independent variables (y,) or set of variables to the total variance. This provides an

indication of their relative importance. The partial variances can be calculated using the

following expressions:









V = V (E [f | x, ])
V, = V(E[ f | x,, x, ]) V V, (2-25)
Vyk = V(E[ f |x, ,x ,x ] ) -V, -Vzk Vk V Vk

and so on, where V and E denote variance and the expected value respectively. Note that


E[S f ,] = fadx, and V(E[ f | ,]) = /2 x Now the sensitivity indices can be computed

corresponding to the independent variables and set of variables. For example, the first and

second order sensitivity indices can be computed as


S, SI (2-26)
y (f)' y (f)

Under the independent model inputs assumption, the sum of all the sensitivity indices is equal to

one.

The first order sensitivity index for a given variable represents the main effect of the

variable but it does not take into account the effect of interaction of the variables. The total

contribution of a variable on the total variance is given as the sum of all the interactions and the

main effect of the variable. The total sensitivity index of a variable is then defined as


Stotl ,pr ] r ~e (2-27)
V( f)

To calculate the total sensitivity of any design variable x,, the design variable set is divided

into two complementary subsets of x, and Z (Z = x,,Vj = 1,N; j +i) The purpose of using these

subsets is to isolate the influence of x, from the influence of the remaining design variables

included in Z. The total sensitivity index for x, is then defined as


S'' total~' (2-28)
SV(f)










V1totax= V+Vz,z (2-29)

where V, is the partial variance of the obj ective with respect to x, and Vz,z is the measure of the

obj ective variance that is dependent on interactions between x, and Z. Similarly, the partial

variance for Z can be defined as Vz. Therefore, the total obj ective variability can be written as

V = V, + Vz + y6,z (2-30)

Sobolll2 prOposed a variance-based non-parametric approach to estimate the global sensitivity

for any combination of design variables using Monte Carlo methods that is also amenable.

While Sobol used Monte Carlo simulations to conduct the global sensitivity analysis, the

expressions given above can be easily computed analytically once the RSA is available. In the

present study, the above referenced expressions are evaluated analytically using polynomial

RSAs of the objective functions. No accommodations are made for irregular-shaped domains, so

the analytical treatment works best in a well-refined design space where the assumption of a box-

like domain will not introduce significant errors. Using a polynomial RSA as the function f(x),

the approximation can be decomposed as in Equation 2-22 and the sensitivity indices can be

obtained.

2.3 Multi-Objective Optimization Using the Pareto Optimal Front

After developing a computationally inexpensive way of evaluating different designs, the

final step is to perform the actual optimization. In the case of a single objective, this requires a

simple search of design space for the minimum value of the obj ective. For two or more

objectives, additional treatment is needed. Highly correlated objectives can be combined into a

single objective function. When the objectives are conflicting in nature, there may be an infinite

number of possible solutions that will provide possible good combinations of obj ectives. These

solutions are known as Pareto optimal solutions. While there are numerous methods of solving









multi-obj ective optimization problems, the use of evolutionary algorithms (EAs) is a natural

choice to get many Pareto optimal solutions in a single simulation due to its population based

approach and ability to converge to global optimal solutions.

A feasible design x l) dominates another feasible design X(2) (denoted by x l) < X(2)), if both

of the following conditions are true:

1. The design x(1) is no worse than x(2) in all objectives, i.e., ~f, (x )< fJ x(2') for all j -
1,2,..., M obj ectives.

x /I~X(2) -> Vj eMf(L~) x ; fX(2~) OrT Vj et h x )I <:f (X(2))(-1

2. The design x l) is strictly better than X(2) in at least one obj ective, or ~f, x ) < ~f, x(2) foT
at least one je {1t, 2,...,M} ).

x(1 < X(2)-> Aj E IfJ((x) )< fJ(X(2)) (2-32)

If two designs are compared, then the designs are said to be non-dominated with respect to

each other if neither design dominates the other. A design x e S where S is the set of all

feasible designs, is said to be non-dominated with respect to a set A c- S if Bae A: a < x

Such designs in function space are called non-dominated solutions. All the designs x (x e S)

which are non-dominated with respect to any other design in set S, comprise the Pareto optimal

set. The function space representation of the Pareto optimal set is the Pareto optimal front.

When there are two obj ectives, the Pareto optimal front is a curve, when there are three

obj ectives, the Pareto optimal front is represented by a surface and if there are more than three

obj ectives, it is represented by a hyper-surface.

In this shtdy, an elitist non-dominated sorting genetic algorithm NSGA-IIll3 and a parallel

archiving strategy to overcome the Pareto drift problem114 are used as the multi-obj ective










optimizer to generate Pareto optimal solutions. The description of the algorithm is given as

follows:

1. Randomly initialize a population (designs in the design space) of size npop.

2. Compute objectives and constraints for each design.

3. Rank the population using non-domination criteria. Many individuals can have the same
rank with the best individuals given the designation of rank-1. Initialize an archive with all
the non-dominated solutions.

4. Compute the crowding distance. This distance finds the relative closeness of a solution to
other solutions in the function space and is used to differentiate between the solutions on
same rank.

5. Employ genetic operators-selection, crossover, and mutation--to create intermediate
population of size npop.

6. Evaluate objectives and constraints for this intermediate population.

7. Combine the two (parent and intermediate) populations, rank them, and compute the
crowding distance.

8. Update the archive:

9. Compare archive solutions with rank-1 solutions in the combined population.

10. Remove all dominated solutions from the archive.

11. Add all rank-1 solutions in the current population which are non-dominated with respect to
the archive.

12. Select a new population npop from the best individuals based on the ranks and the
crowding distances.

13. Go to step 3 and repeat until the termination criteria is reached, which in the current study
is chosen to be the number of generations

2.4 Design Space Refinement Techniques

Given knowledge about a problem, it is desirable to perform CFD simulations with (a)

design parameters set in a region that is known to provide good results in order to refine the

design (exploitation) or (b) to set the design parameters in an unexplored region and use









additional simulations to decide whether these designs have potential (exploration). The

approaches of exploitation versus exploration are the basis of most global search algorithms.

In the context of design space refinement, a more modest goal is set by using exploitation

to identify the bad regions of a design space:

* A "reasonable design space" approach uses simple analysis models or inexpensive
constraints to crop areas of the design space that give blatantly poor results.

* A surrogate based on a small number of points can be used in the identification of "bad"
regions. In this case, the surrogate may not be accurate enough to identify the good
regions, but is sufficient to identify regions that are obviously bad.

Exploration is needed to identify poorly represented areas of the design space by using error

indicators to identify regions where the surrogate model fits poorly. Exploration is necessary

even in regions with apparent moderate performance, as the improved surrogate may reveal that

the performance in the region is better than expected. The design space refinement process using

the ideas of exploitation and exploration should provide more accurate surrogate models and

reduce the simulation of poorly performing designs.

Design space refinement often leads to irregular domains. Most DOE techniques are

intended for box-like domains. For noise-dominated problems and RSAs, alphabet designs (A-

optimal, D-optimal, G-optimal, etc.) are available.46 For problems dominated by bias errors,

space filling designs such as LHS are normally used, generating them first in an enclosing box

and throwing out points outside the domain.

Surrogate models have been shown to be an effective means of reducing the number of

computationally and time intensive function evaluations required in CFD-based optimization.

However, to use surrogates effectively, it must first be ensured that the surrogate models are high

accuracy representations of the system response. An inaccurate surrogate model can prevent

further analysis and can cast doubt on the validity of the data. Accuracy can suffer if the choice









of DOE was inappropriate for the type of surrogate chosen or if the design space was too large.

The advantages of design space refinement offer promising ways to deal with an initial

inaccurate surrogate model. Current techniques often include forms of design space reduction or

windowing to obtain a reasonable design space.

2.4.1 Design Space Reduction for Surrogate Improvement

Reducing the design space by reducing the ranges of the design variables often leads to

better accuracy in the surrogate model. This is often due to the fact that the response in the

reduced design space has less curvature. This feature makes the response easier to approximate.

Papila et al.78 reduced the ranges of the design variables to perform a preliminary optimization

on a supersonic turbine for the rocket engines of a reusable launch vehicle using RSAs. The goal

of the study was to maximize the turbine efficiency and vehicle payload while minimizing the

turbine weight. The preliminary design used a simplified model to provide the system response.

A FCCD was applied for each turbine stage. A composite response surface was constructed

using desirability functions, and an optimum design was located. The error between the actual

response and the predicted response at the located optimum design was high, indicating poor

RSA accuracies. As a result, Papila et al. reduced the size of the design space by about 80%.

The smaller design space resulted in substantial improvement in the RSA prediction accuracies.

Reducing the size of the design space in this manner is depicted in Figure 2-4, and is an example

of design space exploitation, and can lead to a good accuracy surrogate in a small region of

interest.

Shaping the design space based on feedback from the obj ective function values allows the

designer to concentrate solely on the region of interest. Balabanov et al.70 pointed out that effort

could be saved if the design space was shaped such that it only considered reasonable designs

rather than using the standard box-shaped design space. The study by Balabanov et al. used this









method in the RSA of a high-speed civil transport model. The approach removes large areas of

the standard box-shaped design leaving more of a simplex or ellipsoid design. Inexpensive, low-

fidelity models are used to predict the system response, and the result is used to shape the design

space into a reasonable design space. Points that would lie on the edge of the standard box in a

traditional DOE procedure are moved inward to the boundary of the identified reasonable design

space. By concentrating the data points on the region of interest, the accuracy of the response

surface within that region is improved. The technique was used to separately estimate the

structural bending material weight and the lift and drag coefficients. In these cases, the design

space was reduced at least by half, resulting in improvements in RSA accuracy.

Roux et al.73 employed techniques such as intermediate response surfaces and

identification of the region of interest to improve surrogate accuracy in structural optimization

problems. Linear and quadratic response surfaces were used to fit the data. The design space

was iteratively reduced around the suspected optimum by fitting response surfaces on smaller

and smaller sub-regions. It was found that surrogate accuracy was improved more for carefully

selected points within a small sub-region than for a large number of design points within a large

design space. Both Balabanov et al.70 and Papila et al.82 COmbine multiple obj ectives into a

composite objective function before optimizing. However, this procedure can limit design

selection to one or a few designs, when multiple other desirable designs might exist.

Papila et al.7 pointed out that the inaccuracy in the initial response surface may result in

windowing to a region that does not actually contain the optimum design. Papila et al. argued

that particular attention must be paid to the regions of higher error in an RSA. Papila et al.

explains that because a quadratic RSA is a low-order polynomial, the model can often be

inadequate. This is termed as "bias error," and can obviously be reduced by using higher-order









approximating polynomials. The study by Papila et al. seeks to improve the accuracy of RSAs

by using a design space windowing technique to focus only on the region of interest. At the

same time, it seeks to exclude regions of the design space with high bias errors. Papila studied

the effects of fitting three different RSAs: a standard quadratic RSA, a RSA within the region of

interest using the design space windowing technique, and an RSA using the design space

windowing technique while excluding regions with high error. Papila found that the design

space windowing technique improved the accuracy of the RSA over the global RSA. However,

using the design space windowing technique excluding regions of high error did not improve

accuracy over a standard design space windowing technique.

Sequential response surfaces115,116 inVOlVe SeVeral design space windowing steps. This

method is used in the search of a single optimum by coupling surrogate modeling with the

method of steepest descent. After a surrogate is fitted, a new design space box is constructed at a

new location by stepping in the direction of the steepest descent. A new DOE is applied, and a

new surrogate is constructed for the new design space region. The obvious disadvantage to this

method is that it could potentially require a large number of data points for several different local

optimizations. Rais-Rohani and Singhss used sequential response surfaces to increase the

structural efficiency in various problems. In the study, Rais-Rohani and Singh employ global

and local response surface models. They noted that the global response surface model required a

large number of data points for sufficient accuracy. For the local model, a linear response

surface model was fit to the local regions. A total of ten design subspaces were needed before

the optimum was reached. The two methods obtained similar optimum points, indicating

comparable accuracy. However, the global response surface required 1000 to 3000 design

points, while the local response surface method required from 69 to 406 data points. The study









shows that using a large overall design space region over a small, local design space is not

always the best choice. Goel et al.11 reached a similar conclusion in the optimization of a

diffuser vane shape. Goel et al. found that surrogates constructed within a smaller, refined

design space were significantly more accurate than surrogates constructed over the unrefined

design space.

It has been shown that design space refinement techniques can increase the accuracy of

RSAs. One obvious omission in current design space refinement techniques is a method of

refining the design space in the presence of multiple obj ectives when a satisfactory weighting

criterion is not known. In the use of RSAs for multi-obj ective optimization, it is difficult to

locate a design space that can simultaneously represent the multiple obj ectives while retaining

accuracy for each obj ective. One solution is to ensure that the design space is large enough to

encompass all of the effects the design. This course of action, as Papila et al.78 noted, can

introduce substantial error into RSA. Using an alternative surrogate such as Kriging or neural

networks can alleviate this problem. However, doing so may require beginning the analysis from

scratch to obtain a data set using a DOE that is amenable to the desired surrogate model.

2.4.2 Smart Point Selection for Second Phase in Design Space Refinement

A popular technique in optimization is the use of the Efficient Global Optimization (EGO)

algorithm proposed by Jones et al.ll This technique uses the variance prediction provided by

Kriging to predict the location of the optimum point. The procedure used to predict the optimum

point is shown briefly in Figure 2-5 and is described in detail in Section 2.4.3, and involves

maximizes the expected improvement of a sampled function. The optimization process usually

proceeds by sampling the function at a limited number of points. Kriging used as the surrogate

model fit to the data points, and the variance and expected improvement across the design space

is calculated. The optimization then proceeds in a linear fashion, using the value of the expected









improvement to select the optimum. The predicted optimum is sampled, and then added to the

data set. The iteration continues until convergence is reached.

Various attempts have been made to improve upon the process to make it suitable for

multi-objective optimization. Jeong and Obayashill9 USed EGO in the multi-objective

optimization the pressure distribution at two operating conditions of a two-dimensional airfoil

shape. Genetic algorithms were used to develop the predicted Pareto front using the value of the

expected improvement as the fitness criteria. The Pareto front identifies points that are non-

dominated in the expected improvement of two obj ectives. Jeong and Obayashi choose three new

data points based on the original Kriging model: 1) the predicted point with the maximum

expected improvement in the first obj ective which lay at one end of the Pareto Front, 2) the

predicted point with the maximum expected improvement in the second obj ective at the other

end of the Pareto front, and 3) a point with an expected improvement that lies in the middle of

the Pareto front. The process was iterated multiple times until a converged optimum was

achieved in both objectives. This straightforward optimization was concerned with and was

successful in finding a single optimum, but did not address competing obj ectives.

KnowleS120 Sought modify EGO for the use of computationally expensive multi-obj ective

problems. The new algorithm called ParEGO by simultaneously using Pareto front-based

optimization with the EGO algorithm, similar to the procedure used by Jeong and Obayashi.119

In this case, the goal was to develop an accurate Pareto front using a smaller number of function

evaluations than methods typically used to generate Pareto fronts such as the NSGA-IIll3

algorithm. Again, EGO was used to select the points. However, this procedure relies on a few

hundred function evaluations. This type of evaluation is only possible with relatively

inexpensive function evaluations or a surrogate model that is already sufficiently accurate.









Farhang-Mehr and Azarml21 USed a different selection technique for constructing surrogate

models for crash analyses. The technique also required a standard DOE and Kriging fit to

initialize the response surface. The method then added points based on the predicted

irregularities in the response surface. This method allowed for the response surface to be

improved while requiring potentially fewer points overall. Thus, the design space was

adaptively refined using points that were selected based on the predicted obj ective function,

itself. Farhang-Mehr and Azarm cited the benefits of their approach in reducing the total number

of points needed for optimizations based on computationally expensive simulations. The

procedure is good for improving surrogate accuracy, but does not take into account the value of

the function, meaning that points may be added in regions of non-interest.

2.4.3 Merit Functions for Data Selection and Reduction

Using the concepts such as design space reduction (Section 2.4.1) along with the benefits

of innovative point selection as in the previous section, merit functions, facilitated by Kriging,

can be smoothly integrated into the multi-objective DSR process. Merit functions are statistical

measures of merit that use information of the function values and model uncertainty in the

surrogate model to indicate the locations where the function values can be improved and

uncertainty reduced. In this research effort, merit functions are used to select data points such

that the accuracy of the surrogate model can be improved using a minimal number of data points.

The merit functions rely on predictions of the function values and function uncertainty and

attempt to balance the effects of the two. Kriging is selected as the surrogate model for the

optimization due to its inherent ability to provide estimates of both the function value and the

prediction variance. Although several merit functions are available, the strengths of a given

merit function over another for the purpose of multi-obj ective optimization with an extremely

limited number of function evaluations has not been determined.









Four different merit functions are considered. Because ordinary Kriging assumes that the

response data is normally distributed, a statistical lower bound can be calculated. The first merit

functions seeks to minimizing the statistical lower bound (M~F;) given by

MEF;(x)=9(x)-k-V(x) (2-33)

where k is an adjustable scaling factor that is set to 1 in this study.. The variance Vis predicted as

given in Equation 2-12. The statistical lower bound is simply the predicted function value minus

the estimated standard error in the surrogate prediction at a given location, where the estimated

error is a function of the distance between data points. Locations far away from any data points

will have higher estimated error, or variance. The second merit function is maximizing the

probability of improvementl22 (M~F2). The function is used to calculate the probability of

improving the function value beyond a value T at any location. A point is selected where the

probability of improvement is the highest.


M~F2 X) = O~ (2-34)

where Tis a target value given by

T= f ,-P ~f, (2-35)

where foun is the current predicted optimum, O is the normal cumulative distribution function,

and P is an adjustable scaling factor set to 0.1 in this study, which corresponds to an

improvement of 10% over the current best function value. Depending on the value of P, using

the criterion of maximizing the probability of improvement (M~F2) Can TOSults in a highly

localized search. JoneS123 Suggests that selecting several targets can help the search proceed in a

more global manner. The third merit function is maximizing the expected improvements

(M~F3). Maximizing the expected improvement (M~F3) is the criterion used in the Efficient Global









Optimization (EGO) algorithm developed by Jones et al.ll and has been shown to be effective in

searching globally for the optimum. It calculates the amount of improvement that can be

expected at a given location.


ME x)=(L )(x) "-f)x (x) x) (x9)x (-6

where


the expected value of the minimum function valuel22 (MF4). The expected value is simply the

predicted value at a given location minus the expected improvement. The criterion M~F4 is

similar to M~F3, but the search tends to be more local. Sasena et al.122 Suggests that M~F4 Should

be used only when there is confidence that the optimum region has been found.


MF4 X = V(x) V (x) (2-37)



The procedure is developed such that it accounts for the presence of multiple obj ectives.

In this way, a single point can be selected to improve the response surface across multiple

obj ectives. The procedure involves applying a DOE with a large number of points across the

design space. The points are then divided into clusters such that the number of clusters equals

the number of desired new points. Clustering the data prior to selection ensures that the data are

spread sufficiently across the design space, reducing the possibility of choosing two points that

are very near each other. For the purpose of multi-obj ective optimization, the characteristics for

each obj ective need to be combined before the best points can be selected. The points are

selected based on a simple weighting function. In this study, the data is clustered based on

proximity using the Matlab function kmeans. For a given cluster m, a rank R is given to each

point n











Fk~max Fk~~min (2-38)


where F is the value of the selection criterion (merit function, function value, etc.), F"'min is the


point with the smallest value ofF within the cluster nz, n is a point from the cluster nz, and Fk,max

and Fk,min are the maximum and minimum values F, respectively, within the entire data set for

obj ective k. For M~F; and M~F4, the point in each cluster with the minimum value R is selected.

For MFz and M~F3, the point in each cluster with the maximum value R is selected. The full

procedure for using merit functions to select points is as follows:

1. Construct a response surface using Kriging for each obj ective using an initial sampling.

2. Determine if the design space size should be reduced. Construct a DOE using a large
number of points M~within the new design space.

3. Choose a merit function and calculate the merit function value for each obj ective k at each
point M~in the new DOE using the predicted function value and variance.

4. Calculate R at each point M~ as determined by Equation 2-3 8.

5. Choose the desired number of points na to be sampled from the new DOE. The number of
points na is selected by the user according to the number of sample points that can be
afforded by the simulation or experiment. The quantity M~ should be much larger than nt.

6. Group the points M~into na number of clusters.

7. Select the point that has the best value of R from each cluster.

The final set of na design points will satisfy three basic criteria:

1. The new points will be evenly spread across the new design space

2. The selected points will be selected such that they are predicted to improve the function
values in each objective

3. The selected points are those in each cluster predicted to best reduce the errors in the
surrogate models for each objective.









2.4.4 Method of Alternative Loss Functions

By modifying the least squares loss function, it is possible to fit several similar, but slightly

different polynomial response surfaces to the same set of data. For adequately refined design

spaces, it can be shown that the difference among these polynomials is negligible. For design

spaces that require some type of refinement, large differences can exist. This property of using

the differences in different surrogates highlight potential problems in the surrogate model or in

the data is not new." However, the proposed method has an advantage in that surrogates of the

same type can be compared requiring no change in the data set. This may enable a quantitative

measurement in the degree of refinement needed in a problem.

For a polynomial response surface the prediction at any point i is given as


9,= [b,xx, (2-39)


where k is the number of terms in the polynomial response surface and x,, are values from the

matrix X from Equation 2-3. For example, at any point i, a quadratic response surface with two

variables can be given as

f = bo + b, x, + b2x 12 b3 2 + b4 1 2 + b, x2 (2-40)

or simplified to

f = b + b,x, +b2x 2+b3x 3+b4x 4+b,x, (2-41)

The notation used for these and the following equations are adapted from Myers and

Montgomery.46 To determine the coefficients b,, the loss function to be minimized can be

defined as


L =~ fE, 72 -_ 9] 72 bo C bx, (2-42)
=1- =1- =1- ]=1









where e is the error given by the difference between the actual response y and the predicted

response B, and k is the number of terms in the response surface equation. Figure 2-7 illustrates

the effect of changing the value of p in the loss function formulation. For a traditional least

squares loss function, p = 2. The coefficient vector b, must be determined such that the loss

function is minimized. To do so, the partial derivatives must be set equal to zero

S-=p ~sgn(e,) e, =

(2-43)
L=-p ~sg~n(e,)e 1, x,=0,j12..k


resulting in a set of simultaneous equations that must be solved where sgn(e,) = 8, /| 8, | The

coefficient vector b can be determined using any method for solving simultaneous nonlinear

equations. In this study, the Matlab function fsolve can be used to obtain the coefficients.

Preliminary tests were conducted on data from the preliminary optimization of a radial

turbine. Details of this optimization are given in Chapter 3. This optimization problem involved

the refinement of the design space for a problem with two objectives and involved screening data

(Data Set 1) to determine the feasible design space (Data Set 2). Figure 2-8 shows the results of

response surfaces fit to Data Set 2 and illustrates how the sums of the squares of the errors

(~SSE~= Ce, ) of a quadratic response surface vary with p for two objectives. In general, SSE

increases as p increases. The amount of change in SSE depends on the size of the design space.

This can be better illustrated through use of Pareto fronts.

Figure 2-9 shows the Pareto fronts constructed using response surfaces from the two

objectives for coordinating values of p. In this case, data is used from the preliminary

optimization of a radial turbine, and the objectives are the turbine weight, Wotor, and the total-to-

static efficiency, rls. Figure 2-9A shows a large difference in the Pareto fronts for an identical











data set and identical forms of the RSAs. The only difference was the value of p in the loss

function. After the design space was refined (Data Set 3) to improve the accuracies of the RSAs,

the Pareto fronts based on varying values of p are quite similar. To study the quantitative

differences, the curves were integrated to obtain the area under the Pareto front curves, and the

results were compared to the Pareto front using RSAs obtained from a standard least squares

regression. Figure 2-10 shows that the differences in the Pareto fronts from the unrefined design

space were very large. In contrast, the Pareto fronts from the refined design space differed by

less that Hyve percent. This may be a useful tool in determining whether a design space is

adequately refined.

Table 2-1. Summary of DOE and surrogate modeling references.
Author Application DOE Surrogate(s) Key results
Han et al.64 Multiblade CCD Quadratic RSA using CCD required no special
fan/scroll RSA treatment, in this case
Sevant et al.86 Subsonic wing CCD RSA CCD used to ensure equal variance from
center of design space. RSA used to
smooth numerical noise

Rais-Rohani Composite LHS, OA, RSA Accuracy and efficiency of RSA varies
and Singh"' structures random depending on choice of DOE
Kurtaran et al."" Crash-worthiness Factorial Linear, Difference between RSAs reduces as
design elliptic, design space size reduces


with D-
optimal
D-optimal


IMSEa



D-optimal

LHS



Full factorial


quadratic
RSA
RSA



1st-order RSA,
Kriging


RSA, Kriging

RSA, MARSb,
RBF ,
Kriging
RSA, Kriging


Hosder et al.89



Mahadevan et
al.92


Forsberg and
Nilsson93
Jin et al.94



Rijpkema et
al.95


Aircraft
configuration


Engineering
reliability
analysis
Crash-worthiness


Analytic test
functions


Analytic test
functions


A 30 variable optimization required
design space refinement and multi-
fidelity analysis
For very small number of points,
Kriging performed better than 1st
order RSA

Kriging performs better than RSA in this
case

Kriging more susceptible to noise than
other surrogates


Kriging better captures local
perturbations, but causes problems
when strong noise exists





DOE
CCD, MPDe


Table 2-1. Cc
Author
Chung and
Alonso96

Simpson et al.9'

Mahajan et
al.'"

Papila et al.'"


continued.
Application
Supersonic
business jet


Surrogate(s)
RSA, Kriging

RSA, Kriging


Key results
Kriging better at fitting when there are
several local optima
Kriging and RSA produce comparable
results
Gain information on system trends and
shield performance
NN data used to supplement CFD data
for RSA
Maximized vehicle energy dissipation


Aerospike nozzle OA

Radiation shield Full


Supersonic FCC
turbine I


factorial NN


:D, OA,
D-optimal


NN, RSA


Shi and
Hagiwara 0


Crash-worthiness


CCD with D- RSA, NN
optimal


alMSE integrated mean square error92
bMARS multivariate adaptive regression splines'2
"MPD minimum point design96


Table 2-2. Design space refinement (DSR) techniques with their applications and key results.


Author
Balabanov et
al. "
Roux et al.7


Papila et al.8

Rais-Rohani and
Singh8
Bosque-Sendra
et al.'2


Papila et al.n



Jeong and
Obayashill9

Knowles "



Farhang-Mehr
and Azarm 2


Application
Transport wing


Structural
optimization
Supersonic
turbine
Structural
optimization
Chemical
applications


Preliminary
turbine
design
2D airfoil shape



Arbitrary
functions


Crash analysis


DSR Technique
Reasonable design
space
Design space
windowing

Design space reduction


Sequential response
surface

Sequential response
surface


Design space
windowing


Smart point selection
USing EGO

Pareto front refinement
using EGO
(ParEGO)
Smart point selection
based on function
irregularities


Key Results
Unreasonable designs eliminated and
surrogate accuracy improved

Polynomial RSA accuracy sensitive to
design space size
Design space reduction substantially
improved RSA accuracy
Successive small design spaces more
efficient than large design space
Box-Behnken design used to move and
resize design space while including
previously used points
Windowing improved RSA accuracy
over global RSA


Used expected improvement uncertainty
parameter for Pareto front fitness to
select new design points
ParEGO more efficient than random
search for Pareto front construction


Increasing point density in irregular
function regions improves surrogate
accuracy using minimal points

















































Figure 2-1. Optimization framework flowchart.















69













MI I


b


.


*


,-e--


III


Figure 2-2. DOEs for noise-reducing surrogate models. A) Central composite design and B)
face centered cubic design for two design variables. The extreme points are selected
around a circle for CCD and a square for FCCD.


Figure 2-3. Latin Hypercube Sampling. A) LHS with holes in the design space. B) Orthogonal
array LHS can help address this issue by filling the design space more evenly.

















































Figure 2-5. Smart point selection. The Einal Kriging approximation of the actual function is
shown. The new point added in each cycle is shown along with the three points at the
function ends and center that comprised the original Kriging approximation. In this
case, points are selected based on the location of the minimum merit function value.


Figure 2-4. Design space windowing showing optimum based on original design space and the
Einal optimum based on the refined design space.


- actual function
--~kriging model
---- meritfunction
original data points
O new datapoints


O


I~ Or-01~












F
F J
Jmax /pl $


Cluster mI


FIm
),max


I?


~F~m_ Fm am Fm
1 11 ,n If 1,1B


F .....~....
J~mln


Fi, mm F, max

Figure 2-6. Depiction of the merit function rank assignment for a given cluster given by
Equation 2-38 for two objectives where an" = Fn" F"'


Figure 2-7. The effect of varying values of p on the loss function shape. A traditional least
squares loss function is shown by p = 2.












SSE for objective 1


SSE for objective 2


0.16

0.14

0.12


0.080


0.06
1 2 3 4 5


1 2 3 4 5


Figure 2-8. Variation in SSE with p for two different responses in the preliminary optimization
of a radial turbine for the unrefined design space (Data Set 2).


- p=3
p=4


p=4


018~


:::~


012~


008


S2 25


4 OB 08 1 12 14
Rotor Wt


18 2 22 24


Rotor Wt


Figure 2-9. Pareto fronts for RSAs constructed with varying values of p for A) the original
design space and B) the refined design space. The blue dots represent actual
validation data along the Pareto front constructed on the refined design space using
standard least squares regression.











Absolute percent area difference (compared to p = 2)


10






1 15 2 25 335 4 4.5 5



Figure 2-10. Absolute percent difference in the area under the Pareto front curves for the
original feasible design space (RS 2) and the refined design space (RS 3) for various
values of p as compared to a traditional least squares loss function (p = 2).










CHAPTER 3
RADIAL TURBINE OPTIMIZATION

Surrogates can be used to obtain necessary information about the design space. This

information can be used to help refine the design space to improve the accuracy of the surrogate

model. The following case study demonstrates how response surfaces can be used to provide

constraint information necessary in refining the design space to prevent infeasible CFD runs. It

also shows ways in which the design space can be refined to a small region of interest by using

Pareto fronts to simultaneously satisfy two obj ectives.

A response surface-based dual-obj ective design optimization was conducted in the

preliminary design of a compact radial turbine for an expander cycle rocket engine. The

optimization obj ective was to increase the efficiency of the turbine while maintaining low

turbine weight. Polynomial response surface approximations were used as surrogates and the

accuracy of such approximations improves by limiting the size of the domain and the number of

variables of each response of interest. This was done in three stages using an approximate, one-

dimensional model. In the first stage, a relatively small number of points were used to identify

approximate constraint boundaries of the feasible domain and reduce the number of variables

used to approximate each one of the constraints. In the second stage, a small number of points in

this approximate feasible domain were used to identify the domain where both obj ectives had

reasonable values. The last stage focused on obtaining high accuracy approximation in the

region of interest with a large number of points. The approximations were used to identify the

Pareto front and perform a global sensitivity analysis. Substantial improvement was achieved

compared to a baseline design.

A second study was conducted on the radial turbine data after the first optimization study

was completed. The research effort was also used to test strategies of improving the efficiency










of the DSR process by reducing the necessary number of points for use in the multi-obj ective

optimization of computationally expensive problems. These improved DSR strategies were

applied to the optimization of a compact radial turbine, which was originally optimized using

traditional response surface methodology and Pareto fronts. Merit functions were used to reduce

the number of points in the analysis, while ensuring that the accuracy of the surrogate was

maintained. In particular, the use of merit functions in a multi-obj ective optimization process

was explored. This research seeks to explore the benefits of using merit functions to select

points for computationally expensive problems. The purpose of the proposed analysis procedure

is twofold. First, it must be ensured that using a given selection criterion, such as merit functions

to select the data points, results in better surrogate accuracy than using a seemingly more obvious

method, such as simply choosing points with the best function values. Second, it must be

determined whether the use of one merit function for the purpose of computationally expensive

multi-obj ective optimization is superior over the other merit functions.

3.1 Introduction

In rocket engines, a turbine is used to drive the pumps that deliver fuel and oxygen to the

combustion chamber. Cryogenic fuel is heated, resulting in a phase change to a gaseous state.

The increased pressure drives the turbine. However, the survivability of turbine blades limits the

degree to which the fuel can be heated. There is a limited amount of heat available from the

combustion process with which to preheat the fuel, resulting in low chamber pressure and

temperature. This can be an advantage because lower fuel temperatures can improve turbine

reliability. However, the low chamber pressure means that the turbine work output is also

limited.

Turbine work can be increased in two ways: increasing the available energy in the drive

gas or improving the efficiency by which the turbine can extract the available energy. Increasing









the available energy for a given fluid is accomplished by increasing the turbine inlet temperature.

To increase the turbine inlet temperature in an expander cycle, a higher heat flux from the thrust

chamber to the cooling fuel is needed. Obtaining this higher heat flux is problematic in several

ways. First, materials and manufacturing development is necessary to produce a thrust chamber

with high heat flux capability. This work is an ongoing area of technology development.

Second, to enable a higher heat exchange, increased surface area and contact time between the

thrust chamber and cooling fuel is needed. These requirements lead to a larger, and heavier,

thrust chamber. In addition, significantly raising the turbine inlet temperature defeats the

expander cycle' s advantage of maintaining a benign turbine environment. Immich et al.126

reviewed methods of enhancing heat transfer to the combustion chamber wall in an expander

cycle. The authors tried three methods of enhancing heat flux to the unburned fuel: 1) Increase

the length of the combustion chamber, 2) increase the combustion chamber wall surface area by

adding ribs, and 3) increase the combustion chamber wall roughness. The authors also

mentioned that in the future they would experimentally investigate the influence of the

distribution of the inj ector element distribution and the effect of the distance of the inj ector

element to the wall on the heat transfer to the wall. However, results of the future analysis are

not available.

The second approach to increasing turbine work is to improve turbine efficiency. If the

turbine inlet temperature is held constant, an increase in turbine work is directly proportional to

efficiency increase. If the required work can be achieved with moderate efficiency, an

improvement in that efficiency can be traded for reduced inlet temperatures, providing better

design environment margins. One way to improve turbine efficiency is to use a radial turbine.

Radial inflow turbines perform better than axial turbines at high velocity ratios, exhibit better









tolerance to blade incidence changes, and have lower stresses than axial designs. Radial turbines

have been used successfully in automotive applications, but are not often used in rocket engines

due to their relatively large size and weight. The size of the compact radial turbines makes them

applicable to rocket engine cycles when a high velocity ratio is involved.

This research will focus on improving turbine efficiency. The radial turbine design must

provide maximum efficiency while keeping the overall weight of the turbine low. This

necessitates a multi-objective optimization. A response surface analysis46 prOVides an efficient

means of tackling the optimization problem.

The research presented here represents the preliminary optimization of a radial turbine

using a simplified 1-D radial turbine model adapted from the 1-D Meanlinel27 COde utilized by

Papila et al.82 The Meanline code provides performance and geometry predictions based on

selected input conditions. It is an approximate and inexpensive model of the actual processes.

Because it is an approximate model, there is some degree of uncertainty involved, but it can

provide a good starting point in the design process.

Using response surface analysis, an accurate surrogate model was constructed to predict

the radial turbine weight and the efficiency across the selected design space. The surrogate

model was combined with a genetic algorithm-based Pareto front construction and facilitates

global sensitivity evaluations. Because the radial turbine represented a new design, the feasible

design space was initially unknown. Techniques including design constraint boundary

identification and design space reduction were necessary to obtain an accurate response surface

approximation (RSA). The analysis used the optimization framework outlined in Chapter 2. The

framework steps included in the analysis are 1) modeling of the obj ectives using surrogate

models, 2) refining the design space, 3) reducing the problem dimensionality, and 4) handling









multiple obj ectives with the aids of Pareto front and a global sensitivity evaluation method.

Finally, the ability of Merit Functions to improve the efficiency of the optimization process was

demonstrated.

3.2 Problem Description

The radial turbine performance was simulated using the 1-D Meanline code. Using a 1-D

code allowed for the availability of relatively inexpensive computations. To determine whether

using the 1-D code was feasible for optimization purposes, a 3-D verification study was

conducted. Once the Meanline code was verified, the radial turbine optimization could proceed.

3.2.1 Verification Study

Three-dimensional unsteady Navier-Stokes simulations were performed for the baseline

radial turbine design, and the predicted performance parameters were compared with the results

of the Meanline analysis. Simulations were performed at three rotational speeds: the baseline

rotational speed of 122,000 RPM, a low speed of 103,700 RPM and a high-speed of 140,300

RPM. A 2-vane/1-rotor model was used. The simulations were run with and without tip

clearance, and the computational grids contained approximately 1.1 million grid points.

The PHANTOM code was used to perform the numerical simulations.128 The governing

equations in the PHANTOM code are the three-dimensional, unsteady, Navier-Stokes equations.

The equations have been written in the Generalized Equation Set (GES) format,129 enabling it to

be used for both liquids and gases at operating conditions ranging from incompressible to

supersonic flow. A modified Baldwin-Lomax turbulence model is used for turbulence closure.130

In addition to the perfect gas approximation, the code contains two options for the fluid

properties. The first option is based on the equations of state, thermodynamic departure

functions, and corresponding state principles constructed by J. C. Oefelein at Sandia Corporation

in Livermore, California. The second option is based on splines generated from the NIST









Tables.131 A detailed description of the code/algorithm development, as well as its application to

several turbine and pump test cases, is presented in Venkateswaran and Merklel29 and Dorney et

al.132

Figure 3-1 shows static pressure contours (psi) at the mid-height of the turbine for the

baseline rotational speed of 122,000 RPM. This figure illustrates the geometry of the turbine, and

the contours indicate that the pressure decrease is nearly evenly divided between the vane and

the rotor. In fact, the reaction was approximately 0.60 for each of the three 3-D simulations as

compared to 0.55 for the 1-D simulation. Figure 3-2 contains the predicted total-to-static

efficiencies from the Meanline and CFD analyses. The CFD results include values with and

without tip clearance. In general, fair agreement is observed between the Meanline and CFD

results. The trends are qualitatively similar, but the Meanline analysis predicts higher

efficiencies. There is approximately a four-point difference in the quantitative values. The

quantitative differences in the results are not surprising considering the lack of experimental data

available to anchor the Meanline code. The differences in the predictions with and without tip

clearance decrease with increasing rotational speed. Figure 3-3 shows the predicted work from

the Meanline and CFD analyses. The trends are again similar between the Meanline and CFD

analyses, but the Meanline values are consistently 5 6% higher than the CFD values.

The similar trends between the 1-D Meanline code and 3-D CFD analyses indicate that the

optimization can be confidently performed on the 1-D Meanline code. It can be expected that for

a given turbine speed the Meanline code will over-predict the total-to-static efficiency by an

expected degree. The predicted optimum point based on the 1-D Meanline code will likely yield

overly optimistic results, but the predicted degree of improvement should translate to the 3-D

CFD analysis.










3.2.2 Optimization Procedure

The purpose of the design optimization is to maximize the turbine efficiency while

minimizing the turbine weight. A total of six design variables were identified. The ranges of the

design variables were set based on current design practices. Additionally, five constraints were

identified. Two of the five constraints are structural constraints, two are geometric constraints,

and one is an aerodynamic constraint. The aerodynamic constraints are based on general

guidelines. The descriptions of all objectives, variables, and constraints are given in Table 3-1.

It is unknown in what way the constraints depend on the design variables. It is possible

that certain combinations of design variables will cause a constraint violation. It is also unknown

whether the selected ranges result in feasible designs. The response surface analysis can help

clarify these unknown factors.

The design of experiments (DOE) procedure was used to select the location of the data

points that minimize the effect of noise on the fitted polynomial in a response surface analysis.

A face-centered cubic design was used to generate a total of 77 data points within the selected

ranges.

3.3 Results and Discussion

3.3.1 Phase 1: Initial Design of Experiments and Construction of Constraint Surrogates

The values of the obj ective functions were obtained using the Meanline code. Of the 77

solutions from the initial DOE, seven cases failed and 60 cases violated one or more of the five

constraints, resulting in only 10 successful cases. Before the optimization could be conducted, a

feasible design space needed to be identified. Because there was limited information on the

dependencies of the output constraints on the design variables, response surface analyses were

used to determine these dependencies. Response surfaces were used to properly scale the design

variable ranges and identify irregular constraint boundaries.









Quadratic response surfaces were fit to the output constraints. The change in each response

surface with respect to changes in some of design variables was small, indicating that the

response surfaces were largely insensitive to certain design variables. The accuracy of the

response surfaces was not compromised when the identified design variables were neglected.

Therefore, for each response surface, the effects of variables that contributed little to the

response surface were removed. In this way, each response surface was simplified. The

simplified dependencies are shown in Equation 3-1.

AN 2 = AN 2 (AnsqrFrac)
Tip, Spd = Tip, Spd (U/C isen)
(3-1)
Cx2/Utip, = Cx2/Utip, (RPM~,U/C isen, AnsqrFrac)
A = A (React, U/C isen, Tip, inll
Rsex/Rsin = Rsex/Rsin (AnsqrFrac, U/C isen, Dhex%)

The information obtained from the response surfaces about the output constraint

dependence were further used to develop constraints on the design variables. A quadratic

response surface was constructed for each design variable as a function of the output constraint

and the remaining design variables, for example,

React = React ( fly, U/Cisen, Tip Flw) ( 2

The most accurate response surfaces (R2adj > 0.99) were used to determine the design variable

constraints. The output constraints were in turn set to the constraint limits. For example, a

constraint on React was applied to coincide with the constraint fly > 0:

p, = P1 (React, U/Cisen,Tip, Flw)2 0
-> Ract <: eact (/3 = 0, C Cisen,Tip Flw) (3 -3)

Constraint boundary approximations were developed in this manner for each constraint.

As can be seen from Equation 3-1, two of the Hyve constraints (AN2 and Tip Spd) were simple

limits on a single variable. For the single variable constraints, the variable ranges were simply










reduced to match the constraint boundaries. The remaining constraints were more complex.

However, one of the complex constraints (Rsex Rsin) was automatically satisfied by the

reduction of the variable ranges for the single variable constraints. The feasible region is shown

in Figure 3-4.

The remaining constraints involved three variables each. It was discovered that many low

values of RP2~violated the Cx2 Utip constraint. The region of violation was a function of RPM,

U/C isen, and AnsqrFrac as shown in Figure 3-5. The B1 constraint was found to be the most

demanding and resulted in a feasible design space as shown in Figure 3-6. Much of the original

design space violated this constraint. It was also discovered that the constraint surface

representing the bounds for B1 > 40 lay outside of the original design variable range for React. In

this case, the lower bound for React was sufficient to satisfy this constraint.

The predictive capability of the constraints was tested using the available data set. The

design variable values were input into the RSAs for the constraints. Using the RSAs, all points

that violated the output constraints were correctly identified. Now that the feasible design space

was accurately identified, data points could be placed in the feasible data region. The results and

summary of the prediction of constraint violations are as follows:

1. Quadratic response surfaces were constructed to determine the relationship between the
output constraints and the design variables.

2. The variable ranges were adjusted based on information from the constraint surfaces.

3. A 3-level factorial design (729 points) was applied within new variable ranges.

4. Points that violated any constraints (498 / 729 points) were eliminated based on RSAs of
constraints.

Using the response surface constraint approximations, 97% of the 23 1 new data points

predicted to be feasible lay within the actual feasible design space region after simulation using

the Meanline code. The points that were predicted to be feasible, but were actually found to be










infeasible, often violated the B1 > 0 requirement by a slight amount. The resulting data set

contained 224 data points.

3.3.2 Phase 2: Design Space Refinement

Plotting the data points in function space revealed additional information about the

location of the design space as shown in Figure 3-7. A large area of function space contained

data points with a lower efficiency than was desired. There also existed areas of high weight

without improvement in efficiency. Additionally, the fidelity of the response surface for rls was

apparently compromised by the existence of a design space that was too large. These undesirable

areas could be eliminated, and the response surface fidelity could be improved by refining the

design space. The density of points could then be increased within the region of interest,

eliminating the possibility of unnecessary investigation of undesirable points. The region of

interest is shown in Figure 3-7.

To further test the necessity of a reduced design space, Hyve RSAs each were used to fit the

data for Wroto and rl, in the original feasible design space. These response surfaces were

constructed using the general loss function given in Equation 2-42 for p = 1...5. The RSA

constructed using the least square loss function (p = 2) was used as a reference point. As seen in

Figure 3-8, regardless of the RSA used, the error in the RSA at the data points is high at high

Wrotor and low rls. To improve RSA performance, additional data points could be added in these

regions, or these data regions could be eliminated.

Pareto fronts were constructed for each RSA set. The results are shown in Figure 3-9. In

this case, the Pareto fronts differ by as much as 20%. Because the results differ significantly

depending on which RSA is used, this also indicates that further design space refinement is

necessary.









The design variable bounds were further reduced to match the new design space. The new

design variable ranges are given in Table 3-3. Quadratic response surfaces were constructed for

the turbine weight, Wrotor, and the turbine total-to-static efficiency, rls, using the original feasible

design space to screen data points. Points predicted to lie outside of the refined design space

would be omitted. For the refined design space, a third set of data was required. As seen in

Figure 3-7, using a factorial design tended to leave holes in function space. It was possible that

this could hamper construction of an accurate Pareto front. To prevent this, Latin Hypercube

Sampling was used to help close possible holes for the third data set. The points were efficiently

distributed by maximizing the minimum distance between any points. Of the best points from the

second data set, it only RPM, Tip Flw, and U/C isen varied, while React and Dhex% remained

constant at their lowest values, and AnsqrFrac remained constant at its maximum value among

these points. To ensure that this effect was captured in the third data set, additional points were

added using a 5-level factorial design over these three variables. The remaining variables were

held constant according the values observed in the best trade-off points. Quadratic response

surfaces previously constructed for the turbine weight, Wrotor, and the turbine total to static

efficiency, rls, were used to screen the potential data points. Points predicted to lie outside of the

newly refined design space would be omitted from the analysis. In summary,

1. Only the portion of design space with best performance was reserved to allow for a
concentrated effort on the region of interest and to increase response surface fidelity.

2. Latin Hypercube Sampling (181 / 300 feasible points) was used over all six variables and
was supplemented by a 5-level factorial design used over RPM, Tip Flw, and U/C isen
(1 19 / 125 feasible points) to improve resolution among the best trade-off designs.

3. Points that were predicted to violate constraints or lie outside of region of interest were
omitted.

The combination of the DOEs resulted in a total of 323 feasible design points.









3.3.3 Phase 3: Construction of the Pareto Front and Validation of Response Surfaces

As in the second data set, five RSAs each with p = 1...5 were used to fit the data for oro;o

and rl, for the third data set. Pareto fronts were constructed for each RSA set and are shown in

Figure 3-10. In this case, the Pareto fronts differed by a maximum of only 5%. Because the

difference in the response surfaces for varying values of p is small for the third data set, the

design space was determined to be adequately refined.

Function evaluations from the quadratic response surfaces (p = 2) were used to construct

the Pareto Front shown in Figure 3-11. Within the Pareto front, a region was identified that

would provide the best value in terms of maximizing efficiency and minimizing weight. This

trade-off region was selected for the validation of the Pareto front. The results of the subsequent

validation simulations indicated that the response surfaces and corresponding Pareto front were

very accurate. A notable improvement was attained compared to the baseline radial turbine

design. The design selected optimum design had the same weight (Woro;- as the baseline case

with approximately 5% improvement in efficiency. The specifications for the optimum design

are given in Table 3-4.

Within the best trade-off region, only RPM~ and Tip Flw vary along the Pareto front as seen

in Figure 3-12. The other variables are constant within the trade-off region and are set to their

maximum or minimum value. This indicates that increasing the range of one of these variables

might result in an increase in performance. The minimum value of the variable React was chosen

as the only variable range that could reasonably be adjusted. The validation points were

simulated again using a reduced React value. Reducing the minimum value of React from 0.45 to

0.40 increased the maximum efficiency only for oro;- > 1. The maximum increase in efficiency

improved from 4.7% to 6.5%, but this increase occurred outside of the preferred trade-off region.









It was further determined that using a React value of 0.40 resulted in a turbine design that was

too extreme.

Based on the results of the current study, several observations can be made concerning the

design of high-performance radial turbines.

1. The more efficient designs have a higher velocity ratio based on the rotational speed as
shown by U/C isen in Figure 3-12. This is a result of the inlet blade angle being fixed at
0.0 degrees due to structural considerations. Classic radial turbine designs have velocity
ratios of approximately 0.70.

2. The more efficient designs have a smaller tip radius and larger blades (AnsqrFrac) as
compared to designs with a larger radius and smaller blades. Increasing the annulus area
leads to higher efficiencies. In this case, the annulus area (AnsqrFrac) is set to the highest
value allowable while respecting stress limitations.

3. The higher performing designs have higher rotational speeds (as compared to the original
minimum RPM value of 80,000 given in Table 3-1) as a result of the smaller radius and
large blades. The higher rotational speeds also lead to more efficient pump operation.

3.3.4 Phase 4: Global Sensitivity Analysis and Dimensionality Reduction Check

A global sensitivity analysis was conducted using the response surface approximations for

the final design variable ranges given in Table 3-4. The results are shown in Figure 3-13. It was

discovered that the turbine rotational speed RP2~had the largest impact on the variability of the

resulting turbine weight Wrotor. The effects of the rotational speed RPM~ along with the isentropic

velocity ratio U/C isen make up 97% of the variability in Wrotor. All other variables and variable

interactions have minimal effect on Wrotor. For the total to static efficiency, rls, the effects of the

design variables are more evenly distributed. The reaction variable, React, has the highest overall

impact on rls at 28%. This information is useful for future designers. For future designs, it may

be possible to eliminate all variables except RPM~ and U/C isen when evaluating Wrotor, whereas

for rlts, it may be necessary to keep all variables.









It must be noted that the local variability at any given point can vary significantly from the

global sensitivity values due to the nonlinear nature of the RSAs. However, the global sensitivity

analysis can be scaled to the region of interest to explore more localized effects.

3.4 Merit Function Analysis

Merit functions were used in an attempt to reduce the number of points in the DSR

analysis as compared to the original radial turbine optimization study. They were used to select

data points such that the accuracy of the surrogate model can be improved using a minimal

number of data points. The merit functions rely on predictions of the function values and

variance and attempt to balance the effects of the two. Kriging was selected as the surrogate

model for the optimization. The merit function analysis demonstrates how the merit functions,

facilitated by Kriging, can be integrated into the DSR process.

3.4.1 Data Point Selection and Analysis

First, a Kriging model is fit to the 231 points that comprised the feasible design space

constructed as the result of the first cycle in the optimization process. This feasible design space

is designated DS1. Next, the 323 points from the refined design space constructed during the

second cycle of the optimization process are used as a databank. This reasonable design space is

designated DS2, but no function values are initially made at these points. Out of the databank of

data point locations from DS2, points are clustered based on proximity, and the point from each

cluster that is predicted to have the best characteristics based on merit function values is selected

for function evaluation. It is assumed that the points in DS2 lay very near the suspected optima.

The procedure used in the analysis is given as follows:

1. Fit Kriging model to points in the feasible design space (DS1) for each obj ective.

2. Using Kriging model from DS1, predict function value 9 and prediction variance V(x) at
each point in reasonable design space (DS2) for each obj ective.









3. Calculate merit function values at each point in DS2 based on Eqs. (2-33) through (2-37).

4. Normalize variables by the design variable range and cluster the data points from DS2 into
m groups, based on proximity.

5. Select 6 sets of m points each:

a. Choose the data points with the best rank R from each cluster and predict obj ective
value at those points for each of the four merit functions.

b. Choose the points with the best rank R based on the minimum function value fnun
from each cluster using Equation 2-3 8.

c. Choose the data points that lie closest to the cluster centers.

6. Evaluate obj ectives at each point in each of the six data sets.

7. Fit Kriging model to the points in each of the six data sets.

8. Use Kriging model to predict obj ective values along Pareto front for each set.

9. Calculate the errors in the obj ective values for each scenario.

10. Repeat steps 1 through 9 for 100 times such that the random variation that occurs when
clusters are selected sets is minimal.

11. Compare accuracy of different selection criteria among the 6 selection criteria.

12. Repeat steps 1 through 11 using m = 20, 30, 40, 50 clusters (with one data point selected
per cluster) to check sensitivity to number of data points.

13. Compare accuracy among the 6 selection criteria for different numbers of data points.

3.4.2 Merit Function Comparison Results

The points along the validated Pareto front given by the blue curve in Figure 3-11 were

predicted by each of the six new Kriging approximations based on M~F;, MlFz, MlF3, MlF4, fnun, Of

the cluster centers. The differences were small for data sets containing 50 data points, so the

results shown in Figure 3-14 and Figure 3-15 are illustrated using the results of the smallest data

set. The results for a selected cluster set and 20 data points are shown in Figure 3-14. Although

only 20 points were selected by maximizing the probability of improvement (M~Fz) and used to

generate the Kriging model, the predicted points only vary slightly from the actual values along










most of the Pareto front. This indicates that the 20 data points selected by M~Fz actually lie on or

very near the Pareto front. Because Kriging interpolates the data points, the prediction of points

near the 20 data points is very good which leads to the construction of an accurate Pareto front.

It is likely that the prediction of points away from the Pareto front would be very poor. Other

selection criteria were not as successful in predicting points along the Pareto front.

The absolute scaled error at each point is calculated as


|elscaled =(3-4)
ymax ymm

and the absolute combined error is given as


le comlbined Cl scaled (3 -5)


The error distribution for data points along the Pareto front for each selection criterion is given in

Figure 3-15. The use ofM~Fz results in the lowest combined error. Thus, it is demonstrated

qualitatively that selecting points using M~Fz results in the highest accuracy for data sets with a

low number of points.

Due to the random nature of the Matlab function kmeans that is used to generate the

clusters, the cluster sets can be slightly different depending on the starting point of the search,

especially if the data set being clustered is well distributed. For this reason, 100 different cluster

sets were used to reduce effects due to cluster selection. The mean and maximum error

distributions were compared for various numbers of data points and the different selection

criteria and are shown in Figure 3-16 and Figure 3-18, respectively. Figure 3-17 and Figure 3-19

provide direct comparisons of all selection criteria using the median values from Figure 3-16 and

Figure 3-18, respectively. By using the probability of improvement MFz as a selection criterion,

the overall accuracy was maintained to a surprising degree. Except for MF;, using merit









functions as selection criteria appeared to result in better accuracy than simply selecting the point

with the minimum function value or centers from each cluster. Using M~F; gave results that were

very similar to using foun. When the maximum error was considered, however, all merit

functions except M~F2 TOSulted in higher maximum errors as the number of points decreased.

Selecting points based on maximizing the probability of improvement resulted in the best

performance given a small number of data points as compared to simply choosing the minimum

function value or cluster centers. The results in the use of the remaining merit functions were

mixed. Maximizing the probability of improvement, maximizing the expected improvement, or

minimizing the expected value of the minimum function value resulted in lower average errors

than selection based on minimizing the statistical lower bound, minimum function value, or

cluster centers. However, simply selecting the cluster centers seemed to limit the maximum

error better than any merit function except maximizing the probability of improvement. In this

analysis, it was possible to reduce the number of points in the final optimization cycle by 94%

while keeping the accuracy of points along the Pareto optimal front within 10% with an average

error of only 3% or less.

3.5 Conclusion

An optimization framework can be used to facilitate the optimization of a wide variety

design problems. The liquid-rocket compact radial turbine analysis demonstrated the

applicability of the framework:

* Surrogate Modeling. The radial turbine optimization process began without a clear idea
of the location of the feasible design region. RSAs of output constraints were successfully
used to identify the feasible design space.

* Design Space Refinement. The feasible design space was still too large to accommodate
the construction of an accurate RSA for the prediction of turbine efficiency. A reasonable
design space was defined by eliminating poorly performing areas thus improving RSA
fidelity.










* Dimensionality Reduction. A global sensitivity analysis provided a summary of the
effects of design variables on obj ective variables, and it was determined that no variable
could be eliminated from the analysis.

* Multi-objective Optimization using Pareto Front. Using the Pareto front information
constructed using genetic algorithms, a best trade-off region was identified within which
the Pareto front and the response surfaces used to create the Pareto front were validated.
At the same weight, the RS optimization resulted in a 5% improvement in efficiency over
the baseline case.

* Merit Functions. A methodology was presented to use merit functions as a selection
criteria to reduce the number of points required for multi-objective optimization. Using
merit functions gives the ability of dramatically reducing the number of points required in
the final cycle of multi-obj ective optimization.

Through this study, a number of aspects from the framework were demonstrated, and the

benefits of the various steps were made apparent. The framework provides an organized

methodology for attacking several issues that arise in design optimization.

Table 3-1. Variable names and descriptions.
Objective Description Baseline design
variables
Wrotor Relative measure of "goodness" for overall weight 1.147
Y7ts Total-to-static efficiency 85%
Design variables MIN Baseline MAX
RPM~ Rotational Speed 80,000 122,000 150,000
React Percentage of stage pressure drop across rotor 0.45 0.55 0.70
U/C isen Isentropic velocity ratio 0.50 0.61 0.65
Tip Fhr Ratio of flow parameter to a choked flow 0.30 0.25 0.48
parameter
Dhex % Exit hub diameter as a percent of inlet diameter 0.10 0.58 0.40
4nsqrFrac Used to calculate annulus area (stress indicator) 0.50 0.83 1.0
Constraints Desired range
Tip Spd' Tip speed (ft/sec) (stress indicator) 5 2500
4AN Annulus area x speed (stress indicator) 5 850
P; Blade inlet flow angle 0 I P; 5 40
2C Uti Recirculation flow coefficient (indication of > 0.20
pumping upstream)
E le anRatio of the shroud radius at the exit to the shroud < 0.85
radius at the inlet












































Table 3-4. Baseline and optimum design comparison.
Objectives Description Baseline Optimum
Wrotor Relative measure of "goodness" for overall 1.147 1.147
weight
Grls Total-to-static efficiency 85.0% 89.7%

Design Variables Baseline Optimum


Table 3-2. Response surface fit statistics before (feasible DS) and after (reasonable DS) design
space reduction.


Feasible DS


Reasonable DS
Wrotor grs
0.996 0.995
0.996 0.994
0.0235 0.00170
1.04 0.844
310 310


Wrotor
0.987
0.985
0.094
1.04
224


0.917
0.905
0.020
0.771
224


R2ad,
Root Mean Square Error
Mean of Response
Observations


Table 3-3. Original and final design variable ranges after constraint application and design space
reduction.


Design
variable
RPM ~


Original ranges
MIN MAX
80,000 150,000


Final ranges
MIN MAX
100000 150,000


Description
Rotational Speed
Percentage of stage pressure drop across
rotor


React


0.45


0.68

0.63

0.65


0.4


0.85


0.45


0.57


U/C isen Isentropic velocity ratio

Tip Flw Ratio of flow parameter to a choked flow
parameter
Dhex% Exit hub diameter as a % of inlet
diameter


0.56 0.63


0.53


0.1 0.4


AnsqrFrac


Used to calculate annulus area (stress
indicator)


0.68


0.85


RPM


122,000
0.55


124,500
0.45
0.63
0.30


0.10


Rotational Speed
Percentage of stage pressure drop across rotor
Isentropic velocity ratio
Ratio of flow parameter to a choked flow
parameter
Exit hub diameter as a % of inlet diameter

Used to calculate annulus area (stress indicator)


React


U/C isen

Tip Flw

Dhex %

AnsqrFrac


0.61
0.25


0.58
0.83


0.85













3200.0





;-/ .- 1300.














Figure 3-1. Mid-height static pressure (psi) contours at 122,000 rpm.


0.922


0.99


0.880


0.86"


0.84r


0.822


0.8e


0.78e


0.76s


0.74r,
100


- II!!!! c


110 120 130


103 P


Figure 3-2. Predicted Meanline and CFD total-to-static efficiencies.














tllc. I!!!!!!c


E


100 110 120
103 RPM


130 140 150


Figure 3-3. Predicted Meanline and CFD turbine work.


Rse/Rs,, > 0.85
0.95

0.9 lAN2 < 850
0.85

0.8

0.7 5

07

0.65

0 6

0,55

.5~ 0 55
U/C isen


Infeasible


Tip Spd
> 2500







06 063 ().65


Figure 3-4. Feasible region and location of three constraints. Three of five constraints are
automatically satisfied by the range reduction of two design variables.


















Cx2 Utip > 0.2




/ 0.5

0.8-,lr Cx2/Uti <0.2 as~
0 5 06 3U/C isen
AnsqrFrac


Figure 3-5. Constraint surface for Cx2 Utip = 0.2. At higher values of AnsqrFrac and U/C isen,
lower values of RPM~ are infeasible.



0.7

0.65


Inlfea ible~
0.55 Fiile"'ible ~

a:0.5 eact > 0.45

0.45

0.4-


0.5


Figure 3-6. Constraint surfaces for ft
variable ranges.


0 and B1 = 40. Values of ft > 40 lay outside of design












Wrotor vs. rts


Approximate region of
r! m iterest
Note: Maximum Etats 90%


03C


B 0.25

3 0.2


0 05 1 15 7 PE 3
Wrotor


35 4 45


Figure 3-7. Region of interest in function space. (The quantity 1
readability.)


yrls is used for improved plot


1.2






0.8


~0.6


0.4


0.2


0


0 1 2 3 4 5


Figure 3-8. Error between RSA and actual data point. A) Wrotor and B) rts at p = 1, 2, and 5.




















0.16

0.14 -

0.12 -

0.1 -

"0.08 -

0.06 -

0.04 -

0.02 -

0-
0.55


0.6 0.65 0.7 0.75 0.8 0.85
17ts


0.9 0.95


Figure 3-8. Continued.


0. 18



0. 16



0. 14 -



0. 12


0.08
0


0.5 1 1 .5 2 2.5
W


Figure 3-9. Pareto fronts for p = 1 through 5 for second data set. (The quantity 1 as is used for
improved plot readability.) Pareto fronts differ by as much as 20%.































rotor


Figure 3-10. Pareto fronts for p = 1 through 5 for third data set. (The quantity 1 rls is used for
improved plot readability.) Pareto fronts differ by no more than 5%.


1 1.5
W


Figure 3-11. Pareto Front with validation data. Deviations from the predictions are due to
rounded values of the design variables (prediction uses more significant digits). The
quantity 1 as is used for improved plot readability.


















~1






0.3 -


0.5 1 1.5 2


t; Region



05S 1 15S 2


0.4
05S 1 15 2


0.4


S0.3


0.
0.2 rade-off

01 region
0.5 1 1.5 2


0.5 1 1.5 2


Figure 3-12. Variation in design variables along Pareto Front.


Dhex %
3%
AnsqrFrac
5%

React*RPM
2%
Other
53%


Figure 3-13. Global sensitivity analysis. Effect of design variables on A) Wrotor and B) rls.













-- Actual
MF 1
MF 2
MF 3
MF 4
fmin
-centers
MF3


S0.14


0.12 E


I I I I I I


04 06 08 1 12
Wroo


14 16 18


Figure 3-14. Data points predicted by validated Pareto front compared with the predicted values
using six Kriging models based on 20 selected data points. The validated Pareto front
is labeled "Actual."


Wrtr





- (

- I


combined

























Absolute Error distribution


i



-j -




Absolute Error Distribut on


S015

01


111 015



01





O5


Figure 3-15. Absolute error distribution for points along Pareto front using 20 selected data
points each where the points were selected using (1) MF;, (2) M~F2, (3) M~F3, (4) M~F4,
(5) fou,, and (6) cluster centers.















Average Error over 100 Clusters




















Merit function 2


Average Error over 100 Clusters
015





01





O 05


O



Merit function 3



Average Error over 100 Clusters
015





01





O 05






Cluster centers


0 16





0 1


Average Error over 100 Clusters
0 16r


01




LUs


Mmn obl value


Average Error over 100 Clusters
015











01 3






Merit funct on 1



Average Error over 100 Clusters
015










01 t

"~ I





Merit funct on 4


Figure 3-16. Average mean error distribution over 100 clusters for (1) 50 points, (2) 40 points,

(3) 30 points, and (4) 20 points. As number of points decreases, merit functions 2, 3,

and 4 perform better than M~F;, f,,,,,, or cluster centers with M~F2 Showing the best

performance .









































102

















+MF 1
MF 2
+MF3
MF 4
-afmin
-*-centers



















50 40 30 20 10

Number of Points


0.045


0.04






0.035 E




0.025






0.015


Figure 3-17. Median mean error over 100 clusters. Merit function 2 shows the best performance

as the number of points decreases.


viax Error over 100 Clusters
04 .

035~

03 -

0 25





015






Merit function 1



blax Error over 100 Clusters
04 .

035

03~

0 25~

t o

0 ~ ~ 15-

01 -




01234
Merit function 4


Max Error over 100 Clusters
0d .


Max Error over 100 Clusters
nd.


035E


035E


03


025


O 15



0005


+ -7





1234
Merit funcilan 3


02


]05 -

0 234
Merit funct on 2



Max Error over 100 Clusters
04


Max Error over 100 Clusters
04


035~


1 '-


1~ I





Mmn obj value


+







1234
Cluster centers


lu

0 15

01


00


lu

0 15

01



0


Figure 3-18. Average maximum error distribution over 100 clusters for (1) 50 points, (2) 40

points, (3) 30 points, and (4) 20 points.




103














-*-MF 1
MF 2 //10.18
-AMF3


0.12


~cete' 0.14

0.08



0.06

0.04

0.02


50 40 30 20 10
Number of Points



Figure 3-19. Median maximum error over 100 clusters. Merit function 2 shows the best
performance as the number of points decreases.










CHAPTER 4
MODELINTG OF INJECTOR FLOWS

The combustion chamber of a liquid rocket engine is cooled by transferring heat to the

unburned fuel that circulates around the outside of the combustion chamber via a series of tubes

or coolant channels. Rocket inj ectors deliver the fuel and oxidizer to the combustion chamber.

Commonly, the inj ector face is made up of a series of inj ector elements arranged in concentric

rows. The row of inj ectors near the chamber wall can cause considerable local heating that can

reduce the life of the combustion chamber. The local heating near each inj ector element takes the

form of a sinusoidal wall heat flux profile caused by the interactions of the outer row elements.

Often, this effect is not considered during the design process. Reducing the intensity of this local

heating is of prime importance. For example, in the design of the Space Shuttle Main Engine

(SSME), the effect of the local heating in the combustion chamber was not considered during the

design process. After construction, it was discovered that hotspots along the chamber wall

severely reduced the expected chamber life. This resulted in an unforeseen increase in the

reusability operating costs of the engine. Local hotspots can even cause actual burnout of the

chamber wall. By accurately predicting potentially detrimental phenomena in advance, it may be

possible that issues such as wall burnout can be avoided. CFD modeling and validation efforts, in

conjunction with the experimental data, can assist in the understanding of combustor flow

dynamics, eventually leading the way to efficient CFD-based design.

This chapter outlines current and proposed methodologies in the CFD-based optimization

of liquid rocket engine components. First, a summary is presented of past and present inj ector

analysis techniques. Then, the basic governing equations are presented that are applicable in the

simulation of turbulent reacting flow. Finally, a simplified one-dimensional analysis is









demonstrated to help define the relationship between flow parameters and heat transfer due to

reacting flow.

4.1 Literature Review

Inj ector research has been ongoing for over 40 years. Inj ector inlet flow has a significant

influence on combustor performance. The understanding of the inj ector characteristics is critical

in determining the nature of the flow within the combustion chamber. In particular, the inlet

flow geometry and inj ector outlet diameter have large influences on flow in the combustion

chamber.26 However, at times, the research has raised more questions than answers. Work must

still be done to explain certain elements of the flow within the combustion chamber. A general

diagram of inj ector flow is given in Figure 4-1.

Historically, inj ectors have been designed using experimental techniquesl33 and empirical

calculations. A design was built and tested, and then improvements were made based on the

results. For example, Calhoon et al.134 extensively reviewed standard techniques for inj ector

design, including the analysis of cold-fire and hot-fire testing to study general injector

characteristics. The results of these tests were used along with a number of additional multi-

element cold-fire tests to design the full inj ector. The full injector was then fabricated and tested

for performance and heat flux characteristics and combustion stability. However, the

experimental design techniques were insufficient to predict for many conditions that could

reduce inj ector or combustion chamber life.

4.1.1 Single-Element Injectors

A significant portion of inj ector research has been conducted using experiments consisting

of a single inj ector element. The single element analysis is often used as a starting point in

modeling full combustor flow. A test firing of a single element inj ector is shown in Figure 4-2.

Hutt and Cramerl35 found that if it is assumed that all of the inj ectors in the core are identical,









then the measure of energy release from a single element' s flow Hield is a good approximation of

the efficiency of the entire core. In particular, the mixing characteristics of a single inj ector can

be used to determine the element pattern and spacing needed for good mixing efficiency.

Single-element inj ectors are often used in inj ector experimentation to approximate the

mixing effects of the full inj ector. Calhoon et al.134 tested a multiple element inj ector chamber

such that the radial spacing of the elements could be varied. In cold flow testing, it was found

that mixing increased dramatically when multiple elements were used. However, it was

determined that the element spacing only had a slight influence on the mixing efficiency. Yet, as

the distance between the elements increased, the mixing efficiency increased. This was due to

large recirculation regions that brought flow from the well-mixed far region to the near Hield.

When recirculation effects are compensated for, Calhoon et al. found that there is no significant

effect of multiple element interactions compared to single element mixing. This, in theory,

indicates that a satisfactory combustion analysis can be done using only a single injector.

4.1.2 Multi-Element Injectors

A multi-element inj ector face is made up of an array of inj ector elements, and usually

contains from seven to hundreds of injector elements as shown in Figure 4-3. Small changes in

the design of the inj ector and the pattern of elements on the inj ector face can significantly alter

the performance of the combustor. Elements must be arranged to maximize mixing and ensure

even fuel and oxidizer distribution. Gill26 found that the element diameters and diameter ratios

largely influence mixing in the combustion chamber, and that small diameters lead to overall

better performance. The type of elements need not be consistent across the entire inj ector face.

The outer elements must be chosen to help provide some wall cooling in the combustion

chamber. Gill suggests that using a coaxial type inj ector for the outer row of inj ectors provides

an ideal near-wall environment.









Heating effects on the combustion chamber wall due to the arrangement of inj ector

elements is of prime importance. Inj ector placement can result in high local heating on the

combustion chamber wall. Figure 4-4 shows the effect of local heating that resulted in the

burnout of an uncooled combustion chamber. Rupe and Jaivinl36 found a positive correlation

between the temperature profile along the wall and the placement of inj ector elements. Farhangi

et al.137 inVCStigated a gas-gas inj ector and measured heat flux to the combustion chamber wall

and injector face. It was found that the mixing of the propellants controlled the rate of reaction

and heat release. Farhangi et al. suggested that the injector element pattern could be arranged in

a way that moved heating away from the injector face by delaying the mixing of the propellants.

4.1.3 Combustion Chamber Effects and Considerations

Several studies have suggested that pressure has little influence on the combustion

dynamics. Branam and Mayerl38 Studied turbulent length scales by injecting cryogenic nitrogen

through a single inj ector to help in understanding cryogenic rocket propellant mixing efficiency.

It was found that changes in pressure and injection velocity had very little effect on flow

dynamics. Similarly, Calhoon et al. 134 found that the pressure effects on chamber heat flux were

very small. Quentmeyer and Roncacel39 found no change in the heat flux data with a change in

pressure when calculating the heat flux to the wall of a plug flow calorimeter chamber. In

addition, Wanhainen et al.140 found that changing the chamber pressure had no effect on

combustion stability. While, Mayer et al.141 found that pressure is relatively constant throughout

the combustion chamber, while Moonl42 asserts that even the very small pressure gradients that

exist near the inj ector exist can significantly alter velocity profiles and mixing in the combustion

chamber. The relatively constant pressure indicates that density gradients in the combustion

chamber must be due to temperature, rather than pressure. Thus, the flow can be considered

incompressible. In addition to pressure independence, Preclik et al.143 experimentally measured









heat fluxes alon the combustion chamber wall. It was found that the wall heat fluxes were

largely independent of the mixture ratio in the analysis of inj section patterns and velocities on

wall heat flux when the oxygen mass flow rate was held constant. Conley et al.144 also

discovered a negligible influence on wall heat flux due to the mixture ratio.

Several analyses consistently found that there is a sharp jump in the heat flux a certain

ways down the combustion chamber wall. The explanations for the jump have been varied.

Quentmeyer and Roncacel39 found a sharp jump in heat flux 3 cm from the inj ector face. They

took this to mean that at this point, the reactants had mixed sufficiently for rapid combustion to

begin. Calhoon et al.134 found a sharp increase in heat flux at 4.44 cm from the injector face. It

was believed that this was due to the deterioration of the cooling effect of the fuel near the wall.

Reedl45 Sought to understand mixing and heat transfer characteristics in small film cooled

rockets using a gas-gas hydrogen/oxygen combustor. Reed also found a local maximum heat

flux in the flow. Similar to Calhoon et al., Reed proposed that the local maximum heat flux was

due to a breakdown of cooling due to increased mixing. Reed also suggested that the peak heat

flux has to do with the mixture ratio and the amount of available oxygen present to react with the

cooling flow.

One characteristic of reacting flow in a rocket engine combustion chamber is that the

reaction happens very quickly, and is limited by the amount of turbulent mixing in the

combustion chamber. The combusting shear area is highly turbulent, so mixing, and thus

reaction, can occur very quickly. Foust et al.146 inVCStigated the use of gaseous propellants for

the main combustion chamber by exploring the mixing characteristics, combustion length, flame

holding, injector face heating, and the potential for combustion chamber wall cooling. Foust

found that reaction occurs almost immediately after inj section, and the high velocity gaseous









hydrogen j et decelerates rapidly after leaving the inj ector, then reaccelerates due to heat addition.

Morren et al.147 found that the temperature profies along the combustion chamber wall were

caused exclusively by mixing and shear layer effects, and not by pressure gradients. de Grootl48

found that for gaseous inj section, a high degree of mixing is critical. Additionally, combustion

must occur away from inj ector face to reduce injector face wall heating.

Several efforts have been made to study inj ector dynamics in detail, such that the results

could be used to improve combustion chamber performance. Experiments have only been able to

provide a limited picture due to measurement difficulties posed by the high temperature, high

pressure, and multiple species flow. Rupe and Jaivinl36 Studied the effect of inj ector proximity

to the wall on combustion chamber heat flux. However, the available experimental techniques

were not enough to fully understanding the boundary layer phenomena and were insufficient to

allow accurate future predictions on heat flux based on the results. Strakey et al.149

experimentally examined the effects of moving the inner oxygen post off-axis and away from the

combustion chamber wall to study the effects on wall heat transfer. While an attempt was made

to balance between wall heating and combustion performance, the experimental results were not

enough to accurately predict whether the wall heat transfer would be sufficiently low for the

optimized designs. Smith et al.15o conducted hot fire tests of a 7-element gas-gas coaxial inj ector

for different pressures to assess the effect on inj ector face heating, but were unable to accurately

evaluate chamber performance due to the presence on interior boundary layer cooling. CFD

promises to fi11 in the gaps of knowledge left by experimental evaluations.

The experimental procedures are currently directed at providing validation data for CFD

simulations. Experimental measurements are invaluable in the ongoing task of CFD validation.

A benchmark inj ector experiment was run by Marshall et al.1 for the purpose of CFD validation.









In this study, wall heat flux measurements were taken for a gas-gas single element shear coaxial

inj ector element. A single-element inj ector was used rather than a multi-element inj ector

because the experimental setup is simpler and cheaper. The study was conducted using hot and

ambient temperature gaseous oxygen and gaseous hydrogen at four chamber pressures.

Preburners were used to supply the hot gases. The preburners were used to simulate flow in a

staged-combustion cycle, such as the SSME. They found that for a given geometry and injection

velocity, the heat flux along the wall is pressure independent.

Conley et al.22 inVCStigated a GO2/GH2 COaxial shear inj ector to examine the heat flux to

the combustion chamber wall for the use of CFD validation. The experimental setup included

optical access and the ability to measure heat flux, temperature, and pressure in the combustion

chamber. In the experiment, Conley et al. investigated the effects of combustion chamber length,

pressure, and mass flow rate on the heat flux to the wall. A square chamber was used to allow

for a window to be placed on the combustion chamber. The optical access was in place for the

purpose of providing information about combustion dynamics by viewing the flame at various

pressures. This was accomplished using OH-Planar Laser Induced Fluorescence (OH-PLIF) to

show the primary reaction region, or flame. The edges of the square combustion chamber were

rounded to reduce stress on the chamber due to pressure. Conley et al. found no dependence of

the heat flux on pressure when the heat flux was scaled by the mass flow rate. Specifically,

Conley found that when the heat flux could be scaled by the inlet hydrogen mass flow as

qscle = q/ i, the heat flux profile along the wall collapsed to a single curve for a given

combustion geometry. Conley concluded from the experiments that the amount of heat flux

along the combustion chamber wall was primarily a function of the combustion chamber

geometry.









4.1.4 Review of Select CFD Modeling and Validation Studies

The maj ority of CFD validation efforts to date have concentrated on single element coaxial

injectors. Work has focused on LOX/GH2 Or GO2/GH2 COmbustion. Most efforts focused on

matching species or velocity profies. Only recently have efforts been in place to work towards

using CFD as a combustor design tool. These recent efforts focus on accurately modeling the

heat characteristics of combustors.

Liang et al.27 Sought to improve the effects of multi-phase modeling that included liquid,

gas, and liquid droplets. A 2-D axisymmetric model was used along with a chemical model

consisting of a 9-equation kinetic model along with 4 equilibrium equations. The equilibrium

equations were included to help anchor the flame during the computation by allowing for

instantaneous reactions. In this case, equilibrium reactions were necessary due to the coarseness

of the grid used. The turbulence model used is the eddy viscosity model. In this case, no effort

is made to couple combustion with the turbulence equations. For the simulation of a gas-gas

inj ector, a mixture fraction of unity was used for the propellants, and the combustion chamber

was originally filled with oxygen. An artificial ignition region was placed near the oxygen post

tip. The computation was run for a physical equivalent of 10ms. An attempt was made to use

the 2-D model to simulate LOX/GH2 multi-element inj ector flow by changing the walls from a

no-slip to a slip boundary condition. Local peak temperature is around 2000K, and the average

temperature is around 1500K. Liang et al. mentioned that, based on the CFD results, atomization

might be a rate-controlling factor, but there was insufficient experimental data to confirm this

fact. Liang et al. notes that grid resolution has a significant effect on flame ignition and flame

steadiness, but no grid sensitivity study is conducted to explore the effects. Temperature,

velocity, and mass fraction contours were obtained, but no comparison was made to

experimental results, so the accuracy of the simulation is unknown.









Foust et al.146 mOdeled GO2/GH2 COmbustion using an 18 reaction finite-rate chemical

model. The chemistry parameters were determined based on the temperature field. The k-E

turbulence model was used. Preliminary computations were done to obtain the boundary

conditions upstream of the injector. The final computational model extends slightly upstream of

the injector exit. The model is 2-D Cartesian, and the grid was coarse at 101x51. The

computation found good agreement in the species concentration profiles and the velocity

profiles. Using CFD, Foust confirmed the flame holding ability of the oxidizer post tip over

wide ranges of equivalence ratios for a given inj ector.

Cheng et al.15 looked to develop a CFD spray combustion model to help understand the

effects on wall erosion. The Finite Difference Navier-Stokes (FDNS) solver used a k-E model

with wall functions, and the real fluids models were used for the multi-phase flow. Heat and

mass transfer between phases was neglected. The velocity and species concentrations were

solved based on a constant pressure assumption, and then the density and temperature ere

determined based on the real fluids model. The pressure was then corrected based on the newly

determined density. Finite-rate and equilibrium chemistry models were used. The GO2/GH2

computation used the 4-equation equilibrium chemistry model for hydrogen-oxygen combustion.

The shear layer growth was well predicted by the model. Good comparisons were made between

the CFD and experiment for velocity and species profiles. Disagreement in the H20 species

profile was attributed to experimental measurement error. It is assumed that the grid was two-

dimensional, however, no grid information is provided.

Schley et al.152 COmpared the Penn State University (PSU) code used by Foust et al.146

and the NASA FDNS code used by Cheng et al.15 with a third computational code by called

Aeroshape 3D153 (AS3D). The purpose in the CFD development was to reduce combustor









development time and costs. The CFD computations were all based on the experiment

performed by Foust et al.146 Schley noted that the experimental combustion chamber had a

square cross-section with rounded corners, while all three research groups chose to treat it as an

axisymmetric computational domain. None expected the non-asymmetry to impact the

comparison, as other errors were thought to be more significant. The FDNS and AS3D codes

included the nozzle in their computations, while the PSU code did not. The FDNS code did not

include any upstream inj ector analysis, while the others did at least some simulation of the flow

within the injector. All three codes used the k-E turbulence model. Coarse grids were used, as

the researchers found that the solution increased in unsteadiness with increasing grid resolution.

Despite the different codes used and the different treatments in boundary conditions, all

computations agreed reasonably with each other and the experiment.

Ivancic et al.154 USed the AS3D code to do simulate LOX/GH2 combustion using a

coaxial shear injector. Ivancic et al. assumed chemical equilibrium for the combustion modeling

with no coupling between turbulence and chemistry. A real gas model was used for the oxygen.

An adaptive grid was used with 150,000 grid elements and was grid independent based on finer

meshes. The simulated shear layer thickness and the flame location were not consistent with the

experimental results. It is suggested that the discrepancy is due to the 2-D simulation of 3-D

effects. Another reason for the discrepancy was the overly simplified combustion model.

Lin et al.15 specifically investigates the ability of CFD to predict wall heat flux in a

combustion chamber for a shear coaxial single element inj ector based on the experimental results

from the test case RCM-1 of the 3rd Rocket Combustion Modeling (RCM) Workshop held in

Paris, France in March, 2006. The experimental setup of the RCM-1 test case is shown in Figure

4-5. Lin et al. uses the FDNS code introduced by Cheng et al.15 The model was simulated










using two codes: FDNS, a Einite-volume pressure-based solver for structured grids, and loci-

CHEM, a density-based solver for generalized grids. The FDNS code has long been used by

NASA for reacting flow simulation, but is cumbersome to use for complex geometries. Loci-

CHEM, on the other hand, is a new code that is still in development. Because it is applicable to

structured or unstructured grids, loci-CHEM is convenient for complex geometries. The code

used a 7-species, 9-reaction Einite rate hydrogen-oxygen combustion model. The computational

grid encompassed the pre-inj ector flow, combustion chamber, and nozzle. Computations are 2-D

axisymmetric, and consisted of coarse grids with 61,243 points using wall functions, or Eine grids

with 117,648 points where the turbulence equations were integrated to the wall using Menter' s

baseline model. The mass flow rates are fixed at the inlet to the computational zone with a

uniform velocity profile. The experimental temperature profile is applied along the wall of the

computational domain. The computational domains were initially filled with steam, and for the

Loci-CHEM code, an ignition point was specified. The temperature contours for each code by

integrating the turbulence equations to the wall are given in Figure 4-6. The best matching of the

heat flux values came by using the loci-CHEM code integrated to the wall using Menter' s SST

model. Prediction of the heat flux was good in the recirculation zone and poor beyond the

reattachment point, as shown in Figure 4-7A. Lin et al. cited the need for additional validation

and improved turbulence models to improve results.

West et al.156 expanded on the work by Lin et al.15 using an updated version of loci-

CHEM. West et al. performed a detailed grid refinement study on the CFD model of the RCM-1

test case. In particular, West et al. found that a structured grid was prohibitive, in that a large

number of grid points were required for adequate resolution near the combustion chamber wall.

For this reason, West et al. used hybrid grids consisting of a structured grid near the wall and









triangle elements away from the wall. The structured grid had 51,000 points, while the hybrid

grids had between 500,000 and 1,000,000 points. West et al. separately explored the effects of

grid resolution in the boundary layer region, the non-boundary layer region, and the region in the

vicinity of the flame. The effects of grid resolution were small, and the coarsest hybrid grid was

found to be sufficient to accurately predict the peak heat flux. The study also compared three

turbulence models. The use of Menter's SST model provided slightly better results than

Menter' s BSL model and Wilcox k-mo model. The basic k-mo was found to provide the worst

approximation, as the prediction of the peak heat flux was in the wrong location, indicating that

the model would not be appropriate for a design problem. All turbulence models were found to

overpredict the heat flux downstream of the peak heat flux.

Thakur and Wrightl5 tested the code loci-STREAM against the same RCM-1 test case

used by Lin et al.15 A 2-D axisymmetric model was used for the computational domain. A 7-

species, 9-reaction finite rate hydrogen-oxygen combustion model was used. The CFD model

was initiated with steam filling the combustion chamber. Similarly to Lin et al., different grids

were investigated: a fine grid with 104,000 grid points and a coarse grid with 26,000 points for

integrating the turbulence equations to the wall, and a grid with 104,000 points that remains

coarse near the combustion chamber wall to use with wall function formulations. The maximum

chamber temperature was 3565 K, and the temperature contours are shown in Figure 4-6. The

pressure was found to be approximately constant from the inj ector inlet to the nozzle throat.

Thakur and Wright found that the use of wall functions lowered the predicted heat flux at the

wall, regardless of whether the grid was fine or coarse. It is suspected that this may be due to

errors in the computational code, as this same phenomenon was corrected in Lin et al. 1 The

heat flux profiles obtained when integrating to the wall are similar to the results by Lin et al., as









shown in Figure 4-7. It was also determined that the wall heat flux was not sensitive to the grid

distribution far from the combustion chamber wall, as long as the flame region is well resolved.

Improving the fidelity of CFD computations is ongoing. Currently, work is advancing

with the use of three-dimensional CFD models and simulation of multi-element inj ector flow

simulations. Tucker et al.15 reports on several works-in-progress at Marshall Space Flight

center. This includes a simplified CFD model that uses a 7-element inj ector element flow along

with bulk equilibrium assumption in the remainder of the 17 degree subsection to help

approximate full multi-element combustor dynamics of the Integrated Powerhead

Demonstratorl59 IPD inj ector. Sample results are given in Figure 4-8A and Figure 4-8B.

Additional simulations are being conducted on the Modular Combustor Test Article (MCTA) to

supplement the IPD inj ector studies as shown in Figure 4-8C.

4.2 Turbulent Combustion Model

Combusting flows are highly complex, simultaneously involving chemical kinetics, fluid

dynamics, and thermodynamics.160 For example, in a rocket combustion chamber, heat transfer

to the walls depends on the boundary layer characteristics.161 Turbulence enhanced mixing along

with the chemical kinetics can affect the rate of combustion.162 This section reviews the basic

governing equations, chemical kinetics, and equilibrium equations needed for analyzing

combusting flow.

4.2.1 Reacting Flow Equations

Because several species exist in combusting flow, the properties of each fluid must be

accounted for. Mass fractions are defined by


Yk k'1 (4-1)


where mk is the mass of species k in a given volume, and m is the total mass of gas.









The total pressure and density are

"R R
P =E k Whrep p T and pk k, T (4-2)
k=1k=1' nW Wk

where Wk is the molecular weight of species k, R is the ideal gas constant, Tis the temperature,

and Wis the mean molecular weight given by


W = ( rk (4-3)


The mass enthalpy of formation of species k at temperature To (the enthalpies needed to

form 1 kg of species k at the reference temperature) is a property of the substance and is

designated by Ah~,k (USually To = 298. 15K). The enthalpy of species k is then given by


hk pk~dT + Ah (4-4)
sensible

where Cpk is the specific heat of species k. The heat capacities at constant pressure of species k

are usually not constant with temperature in combusting flows. In fact, large changes are

possible. They are usually tabulated as polynomial functions of temperature.

The diffusion coefficient of species k in the rest of the mixture is Dk. Different gases

diffuse at different rates. In CFD codes, this is usually resolved by using simplified diffusion

laws such as Fick' s law:

dY
Yiz" = Ykd" 1- pDk k (4-5)
dxL~

In any chemical reaction, species must be conserved. This conservation is expressed in

terms of the stoichiometric coefficients where v'k, and vIk, designate reactants and products,

respectively:










k=1 k=1

The mass rate of reaction ch is given by:


rhk Wk ky ] (4-7)


where N is the number of species, M~is the number of reactions, and Q, is the rate of progress of

reaction j which can be written as


Q,= K~ k- ,k k' (4-8)
k=1Wk :'k=1 W~k

where Kf and Kr are the experimentally determined forward and reverse reaction rate coefficients.

To solve for reacting flow, the conservation of mass, species, momentum, and energy is

required. In this case, the governing equations in Cartesian coordinates are the conservation of

mass, species, momentum, and energy, respectively.

dp 8 pu,)
+ = 0 (4-9)
at 8x,




S( pk ) C3P122 ap dI

+ + (4-11)
8 t "x Dx D


(pH -p)+-d(puH)= d4 '+ 2ZJ (4-12)
at 8x D D

where p is given by Equation 4-2, Vk is the diffusion velocity of species k, and

H = h +-_uru, (4-13)











k=1 (4-14)

k=1 k=1 k=1

where h, is the sensible enthalpy. The heat flux q, can be written as

pu Sh pu aH p 8(-~
q((L 24"1 (4-15)
SPrL a ] PrL 8 ] PrL a ]

4.2.2 Turbulent Flow Modeling

The conservation equations given in equations (4-9) (4-12) are applicable for laminar

flow. For turbulent flow, the Reynolds Averaged Navier Stokes methods (RANS) use ensemble

averaging to obtain the time-averaged form of the conservation equations. The flow properties

are decomposed into their mean and fluctuating components where Reynolds averaging is used

for the pressure and density components, while Favre averaging is used for the velocity, shear,

temperature, enthalpy, and heat flux components. The mean flow governing equations are given

as the conservation of mass, species, momentum, and energy:


+ = 0 (4-16)
at ax,



S(y~ t drx (i> k k k) dk,,)~-~ k =1,N (4-17)


+~~,~ i l=, +- r- p1"u (4-18)
at Dx, Dx


St~ P H+pu"u,"-P) + pl 2H +& -- pu,"u,
(4-19)
C + pu,"h") + dr u(? ,, pu,"u") + ,r, u"(1uu









where a tilde and double prime denote a Favre-averaged mean and its fluctuation, respectively,

and a bar and a prime denote a Reynolds-averaged mean and its fluctuation, respectively.

The values of several terms in the mean averaged conservation equations are unknown,

and must be modeled instead. Specifically, the terms are


pk = pu,"u, (4-20)


where k is the kinetic energy per unit volume of the turbulent velocity fluctuations, and the

Reynolds-stress tensor pu,"ul" that is denoted by


po, = puu,"u (4-21)

These terms can be modeled using the Boussinesq eddy viscosity models (EVM). The EVM

models the Reynolds-stresses by

du SH 2 85", 2
p5", = pu,"ul" = pu, i + p, 2 pk 3, (4-22)
8x, x 3 8xt 3

where Prt is the turbulent, or eddy, viscosity.

The turbulent heat transport term pu,"h" is modeled as


-pu,"h"= h (4-23)
P r, Dx,

where Prt is the turbulent Prandtl number, and the eddy conductivity is given by

p, cp
kt (4-24)
Pr,


In this study, the Prandtl number is kept constant at Prt = 0.9. The term iiEz,- pu j~ul"u," from

equation (4-19) is modeled as










z,"r,, pu" (J u,"," = p + (-25
ok (425

In the species equation, the term pu"Yk is modeled as


-pu,"Yk ~
Sc, Dx
kt 1 (4-26)

The final forms of the conservation equations for mass, species, momentum and energy are given

respectively as follows:


+ = 0 (4-27)
at ax



+- pH, +fk k, k=l 1N (4-28)



+t +- (pcp, '+: 3 (4-29)







Dx S, x,3 dx, B xL~, crdk 1

where

& h"+ 0,0 +k (4-31)

and the turbulent eddy viscosity Prt is modeled as a product of the velocity scale 5-and a length

scale -emultiplied by a proportionality constant C

pu, = cy( (4-32)









The velocity and length scales can be determined using the k-e modell63 for turbulence

modeling. This model is based on idealized assumption of homogenous isotropic turbulence and

consists of a transport equation of kinetic energy of turbulence k and second for rate of

dissipation of turbulent energy e. Note that in all subsequent equations, bars and tildes have been

dropped for convenience.


(pk)+(purk) = pXrG +k1 +Pk- pe (4-33)

8t 8r 8 366

(pe+-pue)= p++C~k C~p (4-34)
dt Dx x, G 6 2 k k

where the production Pk of k from the mean flow shear stresses is defined as

du 6udX ( -3u
Pk ~t : 2 4-35


and the velocity and length scales are given respectively by

r=J (4-36)


= "' (4-3 7)


and the constants are given by

C, = 0.09; C, = 1.44; C2 = 1.92; ok = 1.0; o, = 1.3 (4-38)

For turbulent flows, the sharp variations in velocity and temperature profiles in the near

wall region require special treatments for the wall boundary conditions. Two common methods

are the use of a low Reynolds number k-e model and the use of wall functions. Examples of low

Reynolds number k-e models can be found in Patel et al.,164 Avva et al.,165 and Chien.166 Wall

functions assume that the turbulence calculations stop at some distance away from the wall. It









applies log-law relations near the wall.163 Further description can be found in Popel67 and

Tennekes and Lumley.168

4.2.3 Chemical Kinetics

In rocket combustion, the fuel and oxidizer enter the combustion chamber in a non-

premixed fashion. The resulting diffusion flames must form in the midst of turbulence

fluctuations, leading to a highly complex flow. One way of handling this is by making the

assumption of "fast chemistry." This assumption assumes that the reaction rates of the species

are very fast compared to the mixing time. This means that the instantaneous species

concentrations and temperatures are functions of conserved scalar equations, such as the mixture

fraction. The various average thermodynamic variables can be obtained from scalar statistics

using probability-density functions. Further details on using the conserved-scalar approach to

account for turbulence-combustion interactions can be found in Libby and Williams.169

The method used in this research is the finite rate chemistry assumption. In particular, a

kinetic-controlled model is used that depends primarily on chemical kinetics as described

previously. This method involves solving the conservation equations describing the convection,

diffusion, and reaction sources. Examples can be found in Thakur et al.170 Turbulence-

combustion models struggle in forms of accuracy and level of description, so new models are

continually being developed.

Chemical reaction rates control the rate of combustion and are related to flame ignition and

extinction. Chemical kinetics map the pathways and rates of reaction from reactants to products.

The global reaction of interest in this effort is the hydrogen-oxygen combustion reaction. Within

the global reaction

2H2, +0 2H2,0 (4-3 9)









there are several elementary reactions. A global reaction can possibly be made up of dozens of

elementary reactions. However, the effects of some mechanisms are small at certain

temperatures and pressures. In this case, the number of reaction mechanisms used to represent a

global reaction can be reduced. If necessary, the number of reactions can be reduced as to

contain only the key reactions. Often, the reaction mechanisms are "optimized" for this purpose.

In this work, nine elementary reactions are considered for the reaction of hydrogen and oxygen

to water. The reaction mechanisms were supplied by NASA Marshall. Each of the nine

reactions has a rate coefficient K given by

K(T)= ATb OXp(-E, /R T) (4-40)

where Rz, is the universal gas constant and A, b, and EA are experimentally determined empirical

parameters given in Table 4-3.

4.2.4 Generation and Decay of Swirl

Swirl number is the ratio of axial flux of angular momentum to the axial flux of axial

momentum


S = ry-d (4-41)
R uv -dA


where R is the hydraulic radius, and a ~is the angular velocity. The swirl number can be

approximated as


S =4 (4-42)


where Wis the mean angular flow velocity and Uis the mean axial velocity. A swirl number of

S < 0.5 indicates weak to moderate swirl.









For the case of annular flow, it can be expected that the rate of swirl decay will be much

greater than that of pipe flow due to the increased fluid drag at the center core. If the difference

between the diameters becomes sufficiently small, the flow is approximately equal duct flow. In

the case of the hydrogen inlet flow, the outer to inlet diameter ratio is 1.22. Any swirl should

quickly be eliminated for diameter ratios close to unity.

For confined swirling flows, Weske and Sturovel7 examined the decay of swirl and

turbulence in a pipe flow. They found that the turbulence field decayed quickly, and that the rate

of decay was a strong function of swirl number. Yajnik and Subbaiahl72 found that swirl decay,

in general, was not significant for pipes less than 5 pipe diameters long.

In general, it has been found that swirl decays exponentially. For laminar swirling pipe

flow (Re < 4000), Talbotl73 determined a theoretical formulation for decay of swirl in a pipe.

The experimental values were found to fall within the theoretical values for Re from 100 to

4000. The solution to the swirl equation for pipe flow is

v(r, z) = v(r,0)e" (4-43)

where : is the non-dimensional length given by : = L/Dh and a is not a function of the initial

flow distribution v(r, 0) and is given by twjo theoretical limits: P Re = 22.2 and P, Re = 74.3 .

The decay of swirl Ds can therefore be given by

D 1v(r,z) =1-ep (4-44)


If it is simply assumed that the pipe flow example can be extended to annular flow, it can

be estimated that the swirl decay in the hydrogen pipe will be between 47% and 88% without

considering the effects of a fuel baffle. In actuality, this amount would be greater for annular

flow due to the growth of the boundary layer on the inner and outer surfaces of the annular pipe.









If the amount of swirl generated in the pipe is small, it can be assumed that there will be

negligible swirl at the exit of the hydrogen inlet.

4.3 Simplified Analysis of GO2/GH2 COmbusting Flow

As shown earlier, several factors are involved in inj ector and injection pattern design that

are of interest in this research effort:

* A single inj ector can be analyzed as an approximation to full injector analysis

* The outer injector elements must be chose such that cooling is enhanced at the combustion
chamber wall

* Inj ector placement can result in high local heating

* Pressure effects on combustion chamber wall heat flux are very small

* Wall heat fluxes are largely independent of the mixture ratio

* A sharp increase in heat flux occurs consistently at a certain distance away from the
inj ector face

* Analysis of the flow upstream of the inj section may be necessary to determine the effect on
combusting flow

A preliminary combustion analysis can give some insight into the nature of H2/02 reacting flow.

In particular, the relationship between temperature, mass flow rate, pressure, heat flow rate, and

equivalence ratio can be explored using a simplified version of the combustion chamber.

A system can be considered wherein hydrogen and oxygen are inj ected such that the molar

flow rate of hydrogen exceeds that of oxygen (Figure 4-9). The reaction is given as

20H2 + 02 4 2H2 + 2(#-1)H2 (4-45)

and the following assumptions are made:

* Constant pressure
* Complete combustion with no dissociation
* Fuel-rich combustion ($> 1)
* Steady-state
* All combustion products exit at the same temperature











For this system the heat output in watts is given by


+1 a ,


(4-46)


where the species enthalpies Ah are equal to the molar enthalpies in J/mol divided by the

molecular weight of each species. Using this equation, several observations can be made. First,

the heat flow rate is dependent only on the species enthalpies at the inlet and outlet temperatures,

the mass flow rates, and the equivalence ratio. Second, it must be noted that enthalpies are

largely independent of pressure. This would indicate that the heat flow rate is also independent

of pressure when the inlet temperatures, outlet temperatures, and mass flow rates are held

constant.

Table 4-1. Selected injector experimental studies.
Author Inj ector type Propellants Key results
Hutt and Single element LOX/GH2 IHROVative measurements of LOX swirl
Cramer oxidizer-rich spray
(1996)135 COaxial swirl
inj ector
Preclik Multiple-element LOX/LH2 Wall heat flux measurements
(1998)143 COaxial injector
Farhangi et GO2/GH2 Design, develop, and demonstrate a
al. (1999)137 Subscale hydrogen/oxygen gaseous


injector
Evaluate performance of gas-gas
injector.
Characterization of liquid j et flow

Wall heat flux measurements


Wall heat flux independent of pressure,
inj section velocity


Smith et al.
(2002)1so
Mayer
(2002)141
Marshall et
al. (2005)
Conley et al.
(2007)22


Multiple-element
coaxial for FFSC
Non-reacting j et


Single-element
shear coaxial

Single-element
Shear coaxial


GO2/GH2


LN2


GO2/GH2


GO2/GH2


q = lilH, hH,


1 -1
MHoO + 2 AhH,






















































Table 4-3. Reduced reaction mechanisms for hydrogen-oxygen combustion.
Reaction A ((m3/gmOl)n-1/s) B EA (kJ/gmol)
H2 + 02 ++ OH + OH 1.70 x10'0 0.00 2.41 x104
H2 +OH ++H20 +H 2.19x1010 0.00 2.59x103
OH +OH ++OH +H 6.02x109 0.00 5.50x102
H + O ++ OH +H 1.80 x107 1 00 4.48x103


Table 4-2. Selected CFD and numerical studies for shear coaxial iniectors.


Author Propel-
lants


P T(K) Summary/
(MPa) parameters
compared
37.9 94.4/298, Developed
118/ 154 three-phase
COmbustion
model

1.3 297 Velocity and
species
COmpanison


Grid spec. Turbulence
model


Accuracy


Liang et
al.
(1986)27

Foust et
al.
(1996)146


Cheng et
al.
(1997) 5

Ivancic
et al.
(1999)154
Lin et al.
(2005)'"'


West et
al.
(2006)156


LOX/GH2


Eddy- Unknown. Cited inadequate
viscosity experimental knowledge.



k-E Good species matching with
in-house experiment.
Velocity matching degrades
downstream.

k- E Good species matclung to
experiment by Foust et al.
Poor H20 species
matching.
k- E Fair matching of radial OH
distribution to in-house
experiment.
Menter Good peak heat flux
BSL prediction. Overpredicts
downstream heat flux. Low
grid dependence.


GO2/
GH 2


2-D
Cartesian
101x51

Axisym.
151x81



Axisym.
1.5 x10'
elements

Axisym.
61,243-
117,648
points
Axisym
structured/
hybrid
50,000 -
1,000,000
points


LOX/GH2,
GO2/GH2


LOX/GH2



GO2/GH2




GO2/GH2


3.1, 117/309,
1.3 290/298


Velocity and
species
comparison


6.0 127/125 Time/length
scale
inVCStigation
5.2 767/798 Heat flux
comparison
(RCM-1)

5.2 767/798 Heat flux
comparison
(RCM-1)


Wilcox k-
co, Menter
BSL and
SST


Good peak heat flux
prediction when adequate
grid resolution in high
gradient areas. Menter's
SST model provides
superior peak heat flux
prediction.


Thakur GO2/GH2
and
Wright
(2006)'5


5.2 767/798 Heat flux
comparison
(RCM-1)


Axisym.
26,000-
104,000
points


Menter Good peak heat flux
SST prediction. Overpredicts
downstream heat flux.


8.37x103
0.0
5.94x104
0.0
0.0


H+0 + + t OH + O
H + O+M ++OH +M
O+O +M +O2 +M
H+H +M ++ H2 + M
OH + H + M ++ H20 + M


1.22x1014 -0.91
1.00x1010 0.00
2.55x1012 -1.00
5.00x109 0.00


8.40 x10 5


-2.00











heat tansfe


Ol OXygen-rich core



recirculation


heat transfer


recirculation


I
n/ shear layer


reattachmentpm





homogenous products


shear layer


Figure 4-2. Flame from gaseous hydrogen gaseous oxygen single element shear coaxial
inj ector.


Figure 4-1. Coaxial injector and combustion chamber flow zones.



























`1.
:v NAS
j ~


Figure 4-3. Multi-element injectors. A) Seven-element shear coaxial injector and B) multi-
element IPD159 inj ector. Both inj ectors contain smaller holes at the outer edge that

inj ect unburned fuel for wall cooling.





































Figure 4-4. Wall burnout in an uncooled combustion chamber.


147.3mm(5W.90" -I 1194mm(4 7U")
S268 70mm (10 5017')
~ 285.75mm (111.2517)
~~ 3111.Bam (12271")
~ 336.55mm (13 250")


Figure 4-5. Test case RCM-1 injector. Test rig A) schematic and B) photo.





Figure 4-5. Continued.


C i;


Figure 4-6. Temperature contours for a single element injector. A) FDNS and loci-CHEM,15 B)
loci-STREAM, and C) loci-CHEM156 COdes. Plots are not drawn to same scale.













































O Test Dtar
Chem-3 BSL
-- Chem-3SST T) :r
-I--:- ---- Chem-r3 KW t~~~~~~~~~~l~~~~~



012345878910
Axial Distance from Injector Face (inch)


r of riodes ( 104K)
onl chamber wall


o iI:







X D mance frm Injector Face., in. B


Figure 4-7. CFD heat flux results as compared to RCM-1 experimental test case. A) FDNS and
loci-CHEM codes," B) loci-STREAM code," and C) loci-CHEM code using
hybrid grids and various turbulent models.156



























134




















































Figure 4-8. Multi-element injector simulations. Reproduced from [158]. A) Integrated
Powerhead Demonstrator (IPD) injector showing computational domain and B) a
close-up of temperature contours. C) Normalized temperature contours are shown for
Modular Combustor Test Article (MCTA) inj ector.


1
~ ;jdiltj
o :oal~


I ,,,,,,





ZH2 ,n, TH2-



02,2n, O2,2 "


H2,u vi ou

H20,out, out


Figure 4-9. Fuel rich hydrogen and oxygen reaction with heat release.











































136










CHAPTER 5
SURROGATE MODELING OF MIXING DYNAMIC S

A bluff body-induced flow is used as a model problem to help probe the physics and

surrogate modeling issues related to the mixing dynamics. Understanding of the mixing

characteristics in the wake of the bluff body has important implications to reacting flow, such as

in the shear reacting layer of inj ector flows. The sensitivity of the mixing dynamics to the

trailing edge geometry of the bluff body can be explored using CFD-based surrogate modeling.

Plausible alternative surrogate models can lead to different results in surrogate-based

optimization. The current study demonstrates the ability of using multiple surrogate models to

discover the inadequacy of the CFD model. Since the cost of constructing surrogates is small

compared to the cost of the simulations, using multiple surrogates may offer advantages

compared to the use of a single surrogate. This idea is explored for a complex design space

encountered in a trapezoidal bluff body. Via exploration of local regions within the design

space, it is shown that the design space has small islands where mixing is very effective

compared to the rest of the design space. Both polynomial response surfaces and radial basis

neural networks are used as surrogates, as it is difficult to use a single surrogate model to capture

such local but critical features. The former are more accurate away from the high-mixing

regions, while the latter are more accurate near these regions. Thus, surrogate models can

provide benefits in addition to simple model approximation.

5.1 Introduction

The case study focuses on the mixing and total pressure loss characteristics of time

dependent flows over a 2-D bluff body. Bluff body devices are often used as flameholding

devices such as in afterburner and ramjet systems. Bluff body devices should have good mixing

capability and low pressure loss across them. Challenges exist in that the bluff body flow is









unsteady and difficult to predict. There is recirculating flow in the near-wake region that decays

to form a well-developed vortex street in the wake region. The instantaneous loss and degree of

mixing changes over time and must be well resolved for accurate solutions.

An earlier investigation of the optimization of a trapezoidal bluff body was conducted by

Burman et al.138 The study uses a relatively coarse mesh (9272 computational cells) that was

selected by performing a grid sensitivity study on a single case based on the drag coefficient. The

effects of grid resolution on the measure of mixing were not investigated, so it is possible that

not all of the flow effects were captured. For higher Reynolds number flow, as observed by

Morton et al.,174 grid resolution can have a marked effect on the prediction of unsteady flows and

can substantially affect the fidelity of the surrogate model.

Surrogate models are used to approximate the effect of bluff body geometry changes on

total pressure loss and mixing effectiveness. The time-averaged flow field solutions are

compared by looking for common trends and correlations in the flow structures.

5.2 Bluff Body Flow Analysis

In practical considerations, such as the aforementioned afterburner and ramj et combustor,

the flow around many bluff body devices is turbulent. To simplify the analysis, an effective

viscosity is often estimated based on engineering turbulence closures. The effective Reynolds

number is therefore considerably lower than the nominal Reynolds number. For the purpose of

this study, the flow is modeled using a Reynolds number in the range of the effective Reynolds

number without resorting to turbulence models. Furthermore, without either enforcing the wall

function or suppressing high gradient regions, both typically observed in engineering turbulent

flow computations, the fluid flow tends to exhibit more unsteady behavior, rendering the flow

computations more interesting and challenging. The bluff body geometry along with the flow









area is described in detail. The computational domain and bluff body geometry specifications

are based on the study performed by Burman et al.17

5.2.1 Geometric Description and Computational Domain

The dimensions of the bluff body are given in Figure 5-2. The variables of interest are B,

b, and h. Altering the variables changes the slant angles of the upper and lower surfaces. The

area, A, of the bluff body is held constant and equal to unity. The frontal height is kept constant

at D= 1. The value of H can be calculated from the aforementioned variables and constants

using the equation

H= 2A+lah b-) l(D+B) (5-1)

The computational domain consists of a trapezoidal bluff body within a rectangular channel.

The flow area is illustrated in Figure 5-3. The fluid is incompressible, and the flow is laminar

with a Reynolds number of 250 given by


Re = "D(5 -2)


where D is the frontal height of the bluff body, v is the kinematic viscosity, and L,. is the

freestream velocity. The upper and lower boundaries have "slip", i.e., zero gradient, boundary

conditions. The left boundary is the inlet, and the right boundary is the outlet.

The 2-D, unsteady Navier-Stokes equations are solved using a CFD code called

STREAM.176 The time dependent calculations were solved using the PISO (Pressure Implicit

with Splitting of Operators) algorithm, and convective terms were calculated using the second-

order upwind scheme. Other spatial derivatives are treated with the second-order central

difference schemes. The grid was constructed using ICEM-CFDin7 software. The grid is non-

uniform, with a higher density of grid points near the body and in the wake area as shown in

Figure 5-4. Grids of varying resolution were used to determine the grid that provided the best









balance in terms of solution run time and solution convergence in terms of grid resolution as

detailed in Goel et al.m7 Multiple grid refinements were investigated to identify the adequate

resolution for this problem. The number of grid points in each grid is given in Table 5-1. While

the difference in the total number of grid points between Grid 1 and Grid 2 is not large, the grid

density in the near field and wake region for Grid 2 is higher than Grid 1. In Grid 3 and Grid 4,

both the near field grid density and far field grid density are high, making them more refined.

Overall, Grid 1 has the poorest resolution and Grid 4 has the best resolution. Grid 2 was selected

for the analysis as it appeared to offer the best trade-off in terms of grid resolution and

computational run time. The run time for a single case was approximately eight hours on a 16

CPU cluster with Intel Itanium processors (1.3 GHz) and 16GB of RAM.

5.2.2 Objective Functions and Design of Experiments

The two obj ectives are the total pressure loss coefficient, CD, and the mixing index, MI1..

The total pressure loss coefficient at any time instant is equal to the sum of the pressure and

shear forces on the body divided by the drag force. The total pressure loss coefficient is

averaged between time t and time t* + T. It is assumed that Tis large compared to the

oscillatory time period. The total pressure loss coefficient is given by


CDft x pU ~ xtzc) dS dt (5-3)


where p is the pressure, zx is the viscous stress tensor, p is the fluid density, and D is the frontal

height of the trapezoidal bluff body.

The measure of mixing efficacy is given by the laminar shear stress plus the unsteady

stress. The mixing index, M.I., is chosen as the integral of the mixing efficacy over the entire

computational domain averaged over a time range T. It is given by










M~.I. =t pt pV d (5-4)


where pu is the fluid viscosity, Uis the time-averaged velocity, and u' and v' are the fluctuation

velocities (u' = u U, v' = v v). Equation 5-4 is related to the full Reynolds momentum

conservation equation. The mixing is thus described as the combination of the momentum

transfer due to viscosity and the momentum transfer by the fluctuation velocity field. The

magnitude of the product of the fluctuation velocities is used to keep the sign of this term

consistent throughout the data points. Only the magnitude of the mixing index is important, so

the absolute value of the time-averaged quantity represents the mixing index value. The

constraints are given as

B+b<1
0.5 (5-5)
0.0 < b < 0.5
0.0
The first constraint in Equation 5-5 is incorporated to keep the frontal height equal to unity so

that the Reynolds number remains constant. The remaining constraints maintain convexity of the

geometry. The constraints are identical to those used in Burman et al.m7

A total of 52 data points were used in the analysis. The design of experiments (DOE)

procedure was used to select the location of the data points that minimize the effect of noise on

the fitted polynomial in a response surface analysis. A modified face-centered composite design

(FCCD) is used to select 27 of the 52 data points. The face-centered composite design usually

uses the corners and center faces of a cube for point selection, but in this case, half of the cube

would violate the first constraint in Equation 5-5. This means that four points from the FCCD

are infeasible and must be removed from the design. Additional points were added on the









interior of remaining triangular section to compensate for the missing points. The remaining 25

data points were selected using Latin-hypercube sampling to fi11 the design space.

5.3 Results and Discussion

The CFD solutions were obtained for the 52 data points. These solutions were compared

and categorized based on the mean flow Hield. Response surface and radial basis neural network

prediction models were constructed from the data. Comparisons of fit were made between the

surrogates.

5.3.1 CFD Solution Analysis

Figure 5-5A shows typical instantaneous vorticity characteristics for a given trapezoidal

bluff body geometry. The flow exhibits characteristic vortex pair shedding. This oscillatory

behavior is reflected in the instantaneous total pressure loss coefficient and mixing index values.

The vortex pair shedding is asymmetric due to the asymmetry of the bluff body. However, in

order to evaluate the overall device performance, the time-averaged flow is a more concise

quantity. Figure 5-5C shows the time-averaged solution for the same geometry. In the time

averaged solution it can be seen that the oscillatory behavior in the wake has been completely

averaged out.

Table 5-2 provides the range of the total pressure loss coefficient and mixing index for the

52 design points. From Table 5-2, it is evident the maximum mixing index value lies well

outside of the range indicated by the standard deviation. This indicates evidence of possible

outliers. Along closer inspection, it was discovered that three of the 52 cases had mixing index

values that lay far beyond the range bounded by the standard deviation. These were termed

"extreme cases." A complete analysis of the extreme cases is given in Goel et al.m7

The flow Hields from the CFD runs for geometries representing the full expanse of the

design space can be divided into several maj or groups as illustrated in Figure 5-6. The flow









fields are categorized based on the location and size of mean recirculation regions near the bluff

body. These groups will provide a means of reference for describing general trends in the

sample data. Geometries in Group F had the lowest total pressure loss coefficient with an

average value of 1.85. Group B had the highest mixing index with an average value of 655. It

can be predicted that the best designs in terms of high mixing and low pressure drop, and thus

the focus of a possible future design refinement, may lie in the region including and between

Group B and Group F.

5.3.2 Surrogate Model Results

When constructing the response surface models of the two obj ectives, a cubic response

surface model was found to perform better than a simpler quadratic model. The full cubic model

is given by

j) = p, + PB* + PAb* + PAh* + pB*2 + P5B'b* + PAb*2 + 7B h*
+ PAh'b* + PAh*2 10pB*3 1 ~1B*2b* + P42B'b*2 1 P3b*3 1 P4B*2h* (5 -6)
+ P,4B'b'h* + P, b*2h* + P,6B'h*2 17Pb'h*2 18Pb'h*2 19Ph*3

where a starred value indicates the variable is normalized between 0 and 1. Insignificant terms

were then removed from some cubic response surfaces, resulting in reduced cubic models. The

models used in this analysis are given in Table 5-3.

Problems were discovered in the mixing index response surface model fit:

* The response surface has a low R2ad, Value and a high RMS error, so the fit is very poor.
* Roughly half of the coefficients in the response surface are very large (> 1000).
* Three of the large coefficients are positive and four are negative.

The last two symptoms indicate a polynomial with sharp gradients and oscillatory behavior. It

was suspected that these symptoms were due to behavior exhibited by the extreme cases. The

extreme cases were removed in an attempt to improve the response surface fit for the mixing

index. As seen from Table 5-3, after removal of the extreme cases the response surface fit










improved to 0.921, and the magnitude of the highest term dropped from 4780 to 1550. The

number of terms needed for the response surface fit also reduced from 16 to 14 and the RMS

error reduced from 39 to 12.

Radial basis neural networks were also constructed to model the data to determine whether

it was a better surrogate model for the bluff body. For the analysis, 10 of the 52 points were

reserved as test cases for cross-validation. The values of SPREAD and GOAL that minimize the

error in the test cases are given in Table 5-4. The neural network also had problems in fitting the

data when all 52 data points were used. Indications of those problems are given as follows.

* The optimum value of SPREAD for the mixing index is quite low.
* A large number of neurons are needed to fit the data.

This behavior is indicative of large local variations in the data. When the same three points that

were removed in the response surface analysis were removed in the neural network analysis, the

number of neurons needed to fit the data reduced from 35 to 21. The RMS error was also

reduced and the optimum spread constant increased from 0.14 to 0.35.

The response surface and neural network prediction surfaces for the total pressure loss

coefficient are very similar. The most noticeable difference in the total pressure loss coefficient

predictions occurs at high h values. This behavior can be observed in Figure 5-7. For example,

at h* = 1, B* = 0, and b* = 0.5, the response surface predicts a relatively high value for the total

pressure loss coefficient, while the neural network predicts a relatively low total pressure loss

coefficient value.

The differences between the prediction models for the mixing index are striking, as seen in

Figure 5-8. The neural network clearly shows three points with mixing index values that are

considerably higher than that of the surrounding area. The response surface attempts to fit the

same three points, but the effects are wider spread. The result is that prediction is poor in areas










near these points. The large differences between the response surface and neural network

surrogates are useful in that they can alert us to the fact that both models may have problems in

some parts of the design space, so that at a minimum they serve as a warning signal.

The three points with disproportionately high values were the ones identified as extreme

cases in the previous analysis. When these extreme cases are removed, the response surface and

neural network prediction surface are more consistent with each other as seen in Figure 5-9. Yet,

large differences still exist in the overall shape of the prediction model surfaces.

The RMS error values shown in Table 5-4 indicate that, overall, the neural network is able

to fit the data better than the response surface. The points reserved as test points in the

construction of the neural network should have an RMS error that is near the RMS error for all

52 points. For the mixing index, the RMS errors for the test points in the neural network

prediction surface are lower than the RMS errors of the response surfaces as seen in Table 5-5.

The fit of the neural network to the data that includes extreme cases is only fair, but is

considerably better than that of the response surface. Removal of the extreme cases further

improves the fit. The error in the test data is considerably reduced and is, in fact, lower than the

total RMS error. The RMS error for the total pressure loss coefficient approximation in the test

data is also less than the total RMS error and is considerably smaller than the response surface

RMS error. This indicates that better overall prediction ability is gained for using the radial basis

neural network.

5.3.3 Analysis of Extreme Designs

A grid resolution study was conducted by Goel et al.m7 Because the quality of the response

surface for the mixing index from Grid 2 was poor, an outlier analysis was conducted to identify

possible outliers. Three cases were detected as potential outliers. The suspect outlier designs

were analyzed to identify possible issues. The results for the grids used had adequate time









convergence, and these three potential outlier cases did not indicate any abnormal behavior. In

addition to the three potential outliers, two other cases were also analyzed to observe the trends

of grid refinement on other solutions. The total pressure loss coefficient and mixing index for

these cases are given in Table 5-6. The total pressure loss coefficient did not change

significantly but the mixing index further increased. Also the contrast of the outlier designs with

the other designs became even sharper. This indicated that the cases detected as potential

outliers might represent a true phenomenon. These designs are therefore designated as extreme

designs rather than outliers.

Six additional designs between extreme Case 1 and Case 3 (see Table 5-6) were analyzed

using Grid 2. Designs for Case 1 and Case 3 were mirror symmetric. Since B and h were at the

lower extremes, b = 0.25 was the axis of mirror symmetry. This means that the designs with b =

0. 05 and b = 0. 45 were mirror images and so on. The total pressure loss coefficient and mixing

index for mirror symmetric designs were expected to be the same and observed to be almost the

same. The mixing indices for these intermediate designs did not exhibit the trend of high mixing

indices. The time-averaged flow fields also did not reveal any abnormality. The sudden change

in behavior near the extreme designs was not captured, and higher resolution was required.

To investigate this sharp change in the mixing index, additional simulations were

conducted in the vicinity of the extreme design Case 1. Grid 3 was used to conduct the

simulations to get a more accurate estimate of the total pressure loss coefficient and mixing

index. In these designs, h was set at the lower limit and B was also fixed at the lower limit for

most cases. Only b was varied to differentiate between designs. It was observed that the mixing

index was very high for the designs near the extreme case up to b = 0.02 and there was a sudden

drop in the mixing index for design with b = 0.03. The results are shown in Table 5-7.









Extreme design cases 1, 3, 52 and regular cases 7 and 17 were analyzed using Grid 4 to

assess the convergence trends with respect to grid size. The results for these representative cases

for different grids are shown in Figure 5-10. The mixing indices for different cases showed

trends of achieving asymptotic behavior with improvements in grid resolutions. However, it is

apparent that the Grid 2 solution has not reached grid convergence, and this may be contributing

to the problem with the response surface fit.

5.3.4 Design Space Exploration

The extreme cases cannot be dismissed as outliers, as Goel et al.m7 determined, as they

were found to be properties of the simulated flow. Thus the behavior of the prediction surfaces

constructed using the extreme cases needed to be further investigated. The RMS errors indicate

that when the extreme cases are included in the surrogate model construction, the radial basis

neural network fits the data better than the response surface. In reality, by inspection of Figure

5-8, the neural network appears quite deficient in areas away from the outlier. The fits of the

response surface and radial basis neural networks constructed without the extreme cases were

determined to be quite good. Thus, these surfaces could be compared to the prediction surfaces

constructed with the extreme cases to determine areas where the prediction was poor by using

the equation

MI.I. = ~.I.with extremes M.I.without extremes (5-7)

which leads to the contour plots in Figure 5-11. In areas away from the extreme cases, the

change in the mixing index MII. for the response surface is small. In contrast, even away from

the extreme cases, there are areas of relatively high MIL.. This suggests that the response

surface is better than the neural network at predicting the response away from the extreme cases.










To test if the response surface fit was better away from the extreme cases, an exploration

of the design space was conducted. Figure 5-12 plots the response surface and radial basis

neural network predicted values as compared to the actual CFD data from

Table 5-7 along the line M~.L = f(B* = 0, b*", h* = 0). Of the cases used to construct the

surrogate models, only the extreme cases (B,b,h) = (0,0,0) and (B,b,h) = (0,0,1) fall on this line.

The radial basis neural network predicts both of these points very well, while the response

surface under-predicts the mixing index at b = 1. The variation in the data is very high, with a

sharp drop in the mixing index value over a short distance near the extreme cases and very

smooth behavior away from the extreme cases. Neither surface captures the complete behavior.

In this case, the response surface predicted the interior points very well, as expected, but

performed poorly near the extreme cases. The neural network, on the other hand, better predicts

the trends in the data near the extreme cases.

Areas of large differences within the design space between the two surrogates correspond

to regions where the accuracy in the CFD model is compromised. Particularly, these regions are

concentrated around the extreme designs. These designs were found by Goel et al.m7 to have not

reached grid convergence. The results indicate that the extreme designs may require a greater

degree of grid refinement than other design points.

5.4 Conclusions

Polynomial response surface and radial basis neural networks were used as surrogates for

flow over a trapezoidal bluff body. Both the response surface and the radial basis neural network

approximations adequately predicted the total pressure loss coefficient. However, both

surrogates poorly fit the mixing index and were substantially different from each other. The

large differences in the surrogate approximations served as a warning signal and prompted

further investigation. Inspection of the surrogate models' statistics revealed that the fitting










problems were due to high variations in the data in localized regions. Removing three extreme

cases greatly improved the fits. Further investigation revealed that the CFD simulations were not

converged, and may have been contributing to the inability of the surrogates to properly

approximate the model.

Simulations of complex flows can sometimes result give unexpected results. Here, three

cases had considerably higher mixing indices than the other cases. This behavior could not be

accurately captured by a single surrogate model alone. The radial basis neural network was

found to better approximate the response near the extreme cases due to its local behavior, while

the response surface provided better prediction of the response away from the extreme cases.

Refinement of the design space near the extreme designs revealed that the differences in the

surrogate models corresponded to the locations of the low accuracy CFD designs. Thus,

surrogate models and design space refinement techniques can be used to identify the presence

and location of inaccurate CFD models.

Table 5-1. Number of grid points used in various grid resolutions.
Grid # Number of grid points
1 37,320
2 44,193
3 74,808
4 147,528


Table 5-2. Data statistics in the grid comparison of the CFD data.
CD .
Max 2.21 857
Min 1.79 515

Range (max-min) 0.42 342
Mean 2.03 617
Standard deviation 0.11 66











Table 5-3. Comparison of cubic response surface coefficients and response surface statistics.
Terms CD
Intercept 1.95 801 576
B* -0.368 -1490 -60.6
B* 0.672 -562 795
H* 0.337 -186 830
B*2 1.24 2900 394
B'b* 3.29 4780 0
B*2 0 515 -1550
B'h* 0.507 842 0
H'b* -0.633 494 -556
H*2 -0.700 0 -39.9
B*3 -0.844 1660 -360
B*2b* -4.01 -4620 -902
B'b*2 -4.08 -3660 0
B*3 -0.686 0 765
B*2h* -0.597 -715 -80.7
B'b'h* -0.740 -1000 319
B*2h* 0 -769 484
B'h*2 0.517 0 0
B'h*2 0 268 0
H*3 0.340 0 0
Response Surface Statistics
R2ad, 0.963 0.654 0.921
RM~S error 0.021 39 12
# of terms 17 16 14
Mean response 2.03 617 604
a extreme cases omitted
normalized variable

Table 5-4. Comparison of radial-basis neural network parameters and statistics.
Parameters CD
SPREAD 0.65 0.14 0.35
GOAL 0.0055 3000 5500
# of neurons 29 35 21
rms error 0.0109 15.6 11.6
a extreme cases omitted

Table 5-5. RMS error comparison for response surface and radial basis neural network.
Obj ective Quadratic RS rms Cubic RS rms RBNN rms error at RBNN rms error at
function error error test points all points
CD 0.035 0.022 0.0096 0.014
ML 54 41 29 16
M~1a 16 13 10 12
a extreme cases omitted











Table 5-6. Total pressure loss coefficient and mixing index for extreme and two regular designs
for multiple grids.


Case # B b H CD
Grid 1
la 0.500 0.000 0.000 2.00
3a 0.500 0.500 0.000 2.00
7 0.500 0.250 0.250 2.09
17 0.875 0.125 0.375 1.93
52a 0.690 0.200 0.460 1.98
a suspected outliers


Grid 1
477
477
538
434
502


Grid 2
1.95
1.96
2.17
1.92
2.15


Grid 3
1.95
1.96
2.14
1.91
2.14


Grid 2
801
805
635
541
857


Grid 3
850
853
643
579
977


Table 5-7. Total pressure loss coefficient and mixing index for designs in the immediate vicinity
of Case 1 using Grid 3.
Case B b h CD -
1 0.50 0.00 0.00 1.95 850
I 0.50 0.01 0.00 1.95 858
II 0.50 0.02 0.00 1.94 886
III 0.50 0.03 0.00 1.96 639
IV 0.50 0.04 0.00 1.98 650


Figure 5-1. Modified FCCD.


Figure 5-2. Bluff body geometry. The three design variables are B, b, and h. The parameter D is
the frontal area which is kept constant at 1.












6.5

D= 1


Figure 5-3. Computational domain for trapezoidal bluff body.







>0






0 10 20 30


Figure 5-4. Computational grid for trapezoidal bluff body.












Figure 5-5. Bluff body streamlines and vorticity contours. A) Typical instantaneous vorticity
contours, B) typical instantaneous streamlines, and C) time-averaged streamlines for
flow past a trapezoidal bluff body (B = 0.5, b = 0.25, h = 0.25) at Re = 250.













---- ~

//

///// -=


C


Figure 5-5. Continued.


F'''d' '
*~1:~

rS I 'r
~: '
i~J r


~i''''


~-;

/I ,,

Dc.-zpl
ul-~i
i I


1 ;.
1 ui.s~e
r
.s
I""`



,BT~j
L~-~1L
nr I. esf

...~,
~~~'''''~r.. .'-
~---
1:11
,

j


,' C
c,im
: H1-UB
~---- ----IO)..

''
Ofi O~


Figure 5-6. Flow characterization in design space on the B-b axis. At b = 0, the angle of the
lower bluff-body surface is zero. At B = 0, small angles exist on lower and upper
bluff body surfaces and the sum of the magnitude of these angles is at a maximum.

Along the line B b = 1, the angle of the upper surface is equal to zero.








153














c, RS (h = 0) c, RS (h = 0 5) c, Rs (h = 1)
2 2 22






O1 0511 1 0' 05 01 0



19 ':2 11 22







O 05 1 0 05 0 OS 1
B B 8






MI RS (h = 0) 750 MI (h = 0 5) -re 1 M R (h = 1) 5

70 0L 8 2 1 -_O









D 0 5 1 S 0 0 5 1 6 0 0 5 1 0







Mt / N (h = 0) 0Mt / N (h = 0 5) g 1 Mt / N (h = 1)70











D 0 5 1 SO- OO O5 1 60- 0 0 5 1 0
B B B



row prdcincnorsfrmxn ne (inldn exrm caes at h'(A= 0 (left),(A= 1
0. (cntr) an 1 rgt. omlzd aibe aeson












Mt PS (h = 01


Mt PS (h = 0 5)


Mt PS ( = 1)


M / NN (h = O 5)




1 '



0 OS 1


M / NN ( = 1)
1s


Figure 5-9. Comparison of response surface (top row) and radial basis neural network (bottom
row) prediction contours for mixing index (excluding extreme cases) at h* = 0 (left),
0.5 (center), and 1 (right). Normalized variables are shown.


...eas 17
Cas 17


Grnd#

A) CD and B) M1.I


A # B
Figure 5-10. Variation in obj ective variables with grid refinement.


O 05


0 05













AM / RS (h =O)


aM / RS (h = 0 6)


aM / RS (h= 1)


08

06

04

02

0



06




04

02


B
AM / NN (h= O)


Ub0
B
AM / NN (h = 0 5)


B
aM / NN (h= 1)
1
OB


sl
II


05


Figure 5-11. Difference in predicted mixing index values from response surface (top row) and
radial basis neural network (bottom row) prediction contours constructed with and

without extreme cases at h* = 0 (left), 0.5 (center), and 1 (right). Normalized variables
are shown.


B=0, h=0
050
-RS
NN
800c a CFD data


S750


~~700 0


'E 650


600


550
0 0.2 0.4 0.6 0.8




Figure 5-12. Comparison of response surface and radial basis neural network prediction
contours for mixing index at B* = 0 and h* = 0. The CFD data is additional data and

is not used to construct the surrogate models.










CHAPTER 6
INJECTOR FLOW MODELING

6.1 Introduction

Heating effects due to the arrangement of inj ector elements is of prime importance.

Inj ector placement can result in high local heating on the combustion chamber wall. A multi-

element inj ector face is made up of an array of inj ector elements. The type of elements need not

be consistent across the entire inj ector face. The outer elements must be chosen to help provide

some wall cooling in the combustion chamber. Small changes in the design of the inj ector and

the pattern of elements on the inj ector face can significantly alter the performance of the

combustor. The elements must be arranged to maximize mixing and ensure even fuel and

oxidizer distribution. For example, Gill26 found that the element diameters and diameter ratios

largely influence mixing in the combustion chamber, and that small diameters lead to overall

better performance. Gill suggests that using a coaxial type inj ector for the outer row of inj ectors

provides an ideal environment due to the outer flow being fuel. Rupe and Jaivinl36 found a

positive correlation between the temperature profile along the wall and the placement of inj ector

elements. Farhangi et al.137 inVCStigated a gas-gas inj ector and measured heat flux to the

combustion chamber wall and inj ector face. It was found that the mixing of the propellants

controlled the rate of reaction and heat release. Farhangi et al. suggested that the inj ector element

pattern could be arranged in a way that moved heating away from the inj ector face by delaying

the mixing of the propellants.

A CFD model is constructed based on experimental results provided by Conley et al.22 The

CFD model is used to help understand the flow dynamics within the GO2/GH2 liquid rocket

combustion chamber. In particular, the CFD model attempts to match the experimental heat flux

data at the wall. The CFD simulation is also used to help provide answers to questions that could









not be easily answered using an experiment alone. These issues include (1) the ability to show

flow streamlines, (2) two-dimensional wall heat flux effects, and (3) the relationships between

flow dynamics and heat flux to the combustion chamber wall. Finally, a grid sensitivity analysis

is conducted to determine the effects of grid resolution and density on wall heat flux and

combustion length.

6.2 Experimental Setup

The validation exercise is based on an experimental setup by Conley et al.22 The injector

itself was constructed of an outer hydrogen tube with an oxygen tube within. The oxygen tube is

concentric to the hydrogen tube and is stabilized by a baffle. This baffle keeps the oxygen tube

centered within the hydrogen tube and helps to evenly distribute the hydrogen flow. The

temperature and pressure of the propellants when they are inj ected is not known from the

experiments. In modeling flow in the combustion chamber, Schley et al.152 COmpared the results

from three different codes with experimental results with good agreement for gas-gas

combustion. One primary concern for all simulations was the lack of inlet turbulent conditions

for the combusting flow. All contributing parties simply made an approximation of suitable inlet

conditions. By analyzing the pre-injector flow, some of the guesswork can be removed. A pre-

inj ector analysis is used to determine the flow conditions within the inj ector itself, as well as to

determine how the flow conditions affect the flow entering the combustion chamber.

The combustion chamber is allowed to cool between tests. Conley et al.22 determined in a

preliminary analysis that the temperature along the combustion chamber wall would reach a

maximum of 450 K after 10 seconds of combustion. It was estimated that the actual wall

temperature reached up to 600 K. For a similar CFD simulation, this would mean that the

combustion chamber wall would be considerably cooler than the bulk flow, where the flame










temperature can reach up to 3000 K. Thus, the thermal boundary layer plays a significant part in

the estimation of the heat flux via CFD.

6.3 Upstream Injector Flow Analysis

Because it was determined that the geometry of the inj ector could have a large effect on

combustion dynamics, the flow upstream of the injector must be characterized for flow

properties and effects. A CFD study was conducted based on experimental results of a coaxial

inj ector built and tested by Conley et al.22 In particular, the CFD simulation will seek to glean

information from the inj ector that is difficult or impossible to measure using conventional

experimental methods. The simulation will investigate the possibility of swirl development

upstream of the injector, as well as determine the pressure drop across the injector. Finally, the

shape of the velocity profile at the inj ector exit is investigated.

6.3.1 Problem Description

The geometry consists primarily of an annular flow region with the hydrogen entering at a

direction perpendicular to the primary flow. A flow baffle, that exists in the experimental setup

and is meant to center the oxygen tube within the hydrogen tube, has been omitted in this

analysis. The flow undergoes a sharp reduction in area near the exit of the geometry. The flow

parameters of interest are those of the exiting flow.

For the hydrogen inlet flow, two separate cases were examined. The first case assumes

that the inlet flow is parallel to the walls, or a = 00. This assumes that the hydrogen is being

drawn from quiescent flow such as a tank, for example. The second case assumes that the flow

enters at a slight angle, or a = 5.70. This case represents a perturbation that may have existed in

the original flow. The two cases are compared and analyzed for similarities and differences as

well as for the potential for swirl development and subsequent decay.









The flow regime was assumed to be turbulent, compressible, and unsteady. The

computational domain and boundary conditions are given in Figure 6-2. The computational mesh

consisted of 2.36 x 10' nodes and the mesh density was increased near the no slip walls and areas

of flow curvature as shown in Figure 6-3. The computation was run using the loci-STREAMm5

code using six 1.3GHz processors. Each computation was run long enough to correspond to a

physical time of 1.6 seconds.

6.3.2 Results and Discussion

Figure 6-4 shows the vorticity contours for the hydrogen flow given both inlet conditions.

At x = 0, the flow makes a 900 turn while simultaneously flowing around the inner pipe. This

results in the development of two vorticies in the x-plane on either side of the y axis. These

vorticies are convected down stream while they gradually decay. This decay is due to a

relatively low Reynolds number combined with a long non-dimensional chamber length of 10.5.

However, in the converging section of the annular pipe, the remaining weak vorticies strengthen

due to a sharp increase in the local Reynolds number. Due to the small tangential velocity and

small hydraulic diameter, the viscous effects result in a dramatic decrease in x-plane vorticity,

thus reducing the overall swirl value to negligible levels. The non-dimensional length of the

portion of the pipe with a small hydraulic diameter is comparable to that of the larger portion of

the pipe at 10.7. This allows the flow to reach almost fully-developed flow conditions at the exit

of the pipe.

Figure 6-5 shows the overall swirl numbers that arise as a result of the individual vorticies.

The overall swirl is very weak (< 0.1) outside the entrance region at the x = 0 plane. The

magnitude increases momentarily at the converging section of the pipe, then peaks as viscous

effects overcome the accelerating effect cause by the convergence and retard the flow in the

tangential direction. The swirl number then reduces to a very low, and negligible, amount.










Figure 6-6 shows the average velocity for increasing x for each inlet condition. The average

mean velocity in the x direction, u, is nearly identical for each of the two inlet conditions.

However, the inlet conditions result in very different values for the velocity in the a direction, vs.

While the a = 00 inlet condition results in negligible flow rotation, the a = 5.70 causes a

noticeable, but still small maximum vs velocity of 0.6m/s. The tangential velocity vs continues to

decrease until the converging portion of the pipe where there is a sharp increase. As the pipe

area again becomes constant, the tangential velocity decreases again to a value near 0.3. An

examination of the Reynolds number reveals that practically no difference occurs in the

Reynolds numbers for the two inlet conditions, as shown in Figure 6-7A. The Reynolds number

in the tangential direction for the a = 00 inlet condition is nearly zero along the length of the

pipe. For the the a = 5.70 inlet condition, the Reynolds number in the tangential direction

continually decreases until its value nearly reaches that of the the a = 00 inlet condition. Based

on these results, it can be concluded that perturbing the inlet flow ultimately has little effect on

the resulting exiting flow.

Because little difference exists in the flow exiting the inj ector, the efforts can now focus on

the single case based on uniform inlet flow. There is a pressure drop of 5.5 x 10-3 MPa across the

inj ector, nearly all of which occurs in the converging region and final pipe section, as shown in

Figure 6-8. The hydrogen exit velocity profile, k, and co, are shown in Figure 6-9. The hydrogen

exit velocity profile is nearly symmetric with a maximum value of 60.6 m/s and a Reynolds

number of 2.8 x103. A more distinct asymmetry can be seen in the turbulent kinetic energy k

profile. Peaks in k occur near the pipe walls. Near the walls, the velocity gradients are high,

leading to a large production of k. At the walls, the velocity is zero, so the turbulent kinetic

energy is also set to zero. There is a decay of k away from the walls as the turbulent production










of k again becomes zero at the centerline where the u velocity gradient is equal to zero. The

production of turbulent dissipation a ~is also dependent on the velocity gradients. However, at

the wall, the value of turbulent dissipation is unknown, and must be guessed. The value of

turbulent dissipation at the wall is therefore set based on the cell distance to the wall. For an

infinitely small cell, the turbulent dissipation would be set to an infinite value. The free stream

turbulent dissipation boundary condition is based on the free stream velocity and characteristic

length. Therefore, there is a gradual reduction of ai from the walls to the free stream value.

Similar simulations were run for the oxygen flow. Full details are not given, as the

geometry was very simple. The results are those of standard turbulent pipe flow. The oxygen

flow Reynolds number is an order of magnitude higher at 3.7x 104 than the hydrogen flow due

largely to a larger hydraulic diameter. The maximum velocity for the oxygen flow was 25.4 m/s.

The shapes of the k and ai profies are similar to those of the hydrogen flow, but are of lower

magnitude than the hydrogen flow.

6.3.3 Conclusion

Regardless of the characteristics of the flow entering the hydrogen inlet, the swirl at the

outlet was found to be negligible. It was found that by biasing the inlet flow, a small amount of

swirl developed, but it was not enough to sustain through the final flow section. This flow

reduction had two effects: (1) the flow accelerated in the x-direction to such a degree such that

the swirl number became very small, and (2) the flow profie at the outlet was slightly

asymmetric. This asymmetry may or may not have a noticeable effect on the combustion

chamber dynamics.









6.4 Experimental Results and Analysis

No temperature measurements are taken directly at the combustion chamber wall. Instead,

equations were used to extrapolate temperature and heat flux values to the combustion chamber

wall. The heat flux is measured along one of the combustion chamber walls and is determined

using sensors that measure the temperature at two nearby locations, as shown in Figure 6-10. The

heat flux equation was calculated based on the time-dependent experimental results to account

for the unsteady effects of heat transfer.

The heat flux measured along one of the combustion chamber walls and is determined

using sensors that measure the temperature at two nearby locations. The steady-state (SS) heat

flux was then calculated using


qi\,,moss = y T (6-1)


where conduction coefficient k of Copper 1 10, the material used for the combustion chamber

wall, is k = 388 W/m-K Ay is the distance between the thermocouple holes as shown in Figure

6-10, and ATis the difference in temperature between the two thermocouple holes. The heat flux

profile using this definition was thought to be inaccurate. Thus, the equation was altered to

include a heat absorption correction which accounts for unsteady (US) effects of heat transfer.



Ay 2 At

where the subscript i denotes the thermocouple location closest to the combustion chamber wall

and o denotes the outer thermocouple location. The subscripts 1 and 2 represents an initial and

final time, respectively. The the density p and heat capacity c of Copper 110 are given by p =

8700 kg/m3 and c = 385 J/kg-K. The chamber wall temperatures were determined using the heat

flux data and the thermal conductivity of the wall.











Twall,hnear qlerS + 63

In the experimental documentation, the steady and unsteady heat fluxes are calculated using

Equations 6-1 and 6-2, respectively, and are shown in Figure 6-11.

Equations 6-1 through 6-3 assume that the heat flow is one-dimensional.179 For a square

duct, the one-dimensional linear assumption may not be correct. The actual equation would need

to be determined through a numerical solution of the two-dimensional heat conduction equation.

The correct heat flux equation would change the experimentally documented values of heat flux

at the wall. However, because at each cross section only two temperature measurements are

made, there is not enough data to do a full two-dimensional analysis of heat transfer through the

combustion chamber wall. Additional temperature measurements taken within the wall would be

required for an accurate estimation of wall heat flux. Therefore, a one-dimensional axisymmetric

approximation was included in addition to the one-dimensional linear approximation.

If it is assumed that the isotherms within the combustion chamber wall are axisymmetric, a

second one-dimensional assumption can be conducted. The geometry of the isotherms is given

in Figure 6-12. This is reasonable due to the approximate axisymmetry of the temperature

contours within the combustion chamber wall, and the high thermal conductivity of the copper.

For 1-D axisymmetric heat conduction, the steady state heat flux approximation is given by

kAT
q",,~s (6-4)


where rwan is the distance from the center of the combustion chamber to the combustion chamber

wall, and r, and ro are the distances from the center of the combustion chamber to the inner and

outer thermocouples, respectively, as shown in Figure 6-12. Using this equation, the heat flux at

the wall centerline can be determined based on the 1-D heat conduction equation, giving










4,,a~,,,, rwlannIn (ro /r,) (6-5



For the given combustion chamber diameter, this assumption would result in a wall heat flux

value that is everywhere 28% below the linear heat flux assumption. This is reasonable, as the

assumption of axisymmetric isotherms would result in the lowest possible heat flux given the

temperatures at the thermocouple locations. The temperature at the wall would be given by


T () Inr + To (6-6)
Taxlsymm ( o)


An unsteady approximation is determined by adding the second term from equation (6-2) to the

steady state heat flux approximations in Equation 6-5. Figure 6-13 shows the estimated wall

temperature profiles based on the experimental thermocouple readings. The estimated wall

temperatures are very similar between the linear and axisymmetric approximations, indicating

that the linear assumption used in the experimental documentation was reasonable in the case of

the wall temperature.

Finally, an approximation of the wall heat flux was conducted using a numerical two-

dimensional unsteady heat conduction analysis. Vaidyanathan et al.lso completed a similar

analysis on a different inj ector and provided the conduction analysis results for the inj ector by

Conley et al.22 The cross-sections at each thermocouple pair location were analyzed

independently. Because the experiment lasted only seven to eight seconds, it was estimated that

only negligible heat escaped through the outer combustion chamber wall. Therefore, the outer

wall was assumed to be adiabatic. The thermal propagation time scale for the combustion

chamber wall is equal to tp = e2/a = 3.13 s where -eis the thickness of the combustion chamber

wall equal to 0.019 m, and a is the thermal diffusivity and is equal to 1.158 x 10-4 m2/S. The










thermal propagation time scale is on the same order of magnitude as the experimental run time of

7.75 s, indicating that the adiabatic assumption is reasonable. The inner wall heat flux, taken to

be uniform along the inner combustion chamber wall, was adjusted until the computational and

experimentally measured temperatures obtained at 3.15 and 9.54 mm from the inner chamber

walls (see Figure 6-10), respectively, are matched. The resulting temperature contours are given

in Figure 6-14 for the prescribed boundary conditions for the combustion chamber cross-section

at x = 84 mm.

Significant differences are seen in the comparisons of the estimated wall heat fluxes shown

in Figure 6-15. The shapes of the profiles given by the linear and axisymmetric assumptions are

the same, albeit the axisymmetric assumption gives a proportionally lower heat flux value. The

peak heat flux value predicted using the axisymmetric assumption is similar to that predicted by

the 2-D unsteady conduction analysis. The location of the peak heat flux from the conduction

analysis is shifted from that of the 1-D approximations.

Conley et al.22 found that the point of maximum heat release using the linear

approximation is at 58 mm from the inj ector face, whereas the maximum temperature occurs

further downstream at 84 mm from the inj ector face. On the other hand, the point of maximum

heat release using the unsteady 2-D analysis gives a maximum heat flux at the same location as

that of the maximum temperature.

6.5 Injector Flow Modeling Investigation

A CFD modeling investigation was conducted based on the experimental results presented

by Conley et al.22 Of a single-element shear coaxial GO2/GH2 inj ector. The CFD model is used to

supplement the experimental results, and to gain insight into relevant flow dynamics. The

modeling effort is particularly useful in capturing the variation of heat flux along the combustion

chamber wall due to three-dimensional flow effects.









The CFD simulations were conducted using loci-STREAMm which is a pressure-based,

Einite-rate chemistry solver for combustion flows for arbitrary grids developed by Streamline

Numerics, Inc. at the University of Florida. The code numerically solves the 3-D unsteady,

compressible, Navier-Stokes equations. Menter' s SST model is used for turbulence closure. A

seven-species, nine-equation Einite rate chemistry models" was used for gaseous hydrogen and

gaseous oxygen.

6.5.1 CFD Model Setup

The computational domain, grid, and boundary conditions are shown in Figure 6-16. The

values for specific flow conditions are given in Table 6-1. The unstructured grid contains

1.7 x106 elements and 3.7 x105 nodes. The computational domain represents a three-dimensional,

one-eighth-section of the full combustion chamber. No-slip conditions are specified at the

combustion chamber wall, the inj ector face, and the oxygen post tip. Symmetry conditions are

specified along the planes of symmetry and along the centerline of the combustion chamber. The

nozzle is not included in the analysis; instead, a constant pressure outlet condition is specified, as

the pressure is nearly constant throughout the combustion chamber. The initial temperature was

set to 2000 K to place initial heat flux value within the range of the experimental measurements.

A short ignition region with T = 3000 K was used in the recirculation region of the shear

layer to start combustion. The ignition was turned off after the flame was self-sustaining, and the

computation was run until a steady-state condition was reached based on the boundary

conditions given. Applying an adiabatic condition to the combustion chamber wall was

considered, but this condition would erase the thermal boundary layer along the wall. The

thermal boundary layer is essential to determining the temperature gradient at the wall, and thus,

the heat flux. A constant temperature condition was considered a suitable alternative, and would

represent the case wherein the temperature along the wall was unknown. A constant temperature









of T = 500 K was specified along the combustion chamber wall. This temperature value was

arbitrarily selected as it is within the approximate range of the experimental value, but is not

equal to the average experimental temperature at the wall. Results when the experimental

temperature profile is used along the combustion chamber wall are not included, as there are no

obvious differences in the flow contours whether the experimental temperature profile or a

constant temperature is applied at the wall. The inj ector face and the oxidizer post tip region

between the hydrogen and oxygen inlets are set to adiabatic conditions.

6.5.2 CFD Results and Experimental Comparison of Heat Flux

In the computation, the heat flux at a location j is calculated using the following equation:

dT
q,"wit = -kflmld (6-7)


The calculated heat flux based on Equation 6-7 was calculated in the CFD simulation and then

compared to the experimental values.

Figure 6-17 shows the presence of a strong recirculation region with relatively slow fluid

flow. The hydrogen flow enters at a greater velocity than the oxygen flow, but quickly slows due

to mixing. Due to the low Mach number of 0.04 and the relatively constant pressure throughout

the combustion chamber, the density changes that occur within the combustion chamber are

essentially due to changes in temperature. Figure 6-18 shows the temperature contours along the

: = 0 plane. The combustion length, or the distance from the injector face where 99% of the

combustion is complete, is at 38 mm.

The steady state, one-dimensional heat transfer approximation results in a linear heat flux

and temperature relationship. If one accounts for the area variations via a steady state,

axisymmetric treatment, then there is a different heat flux and temperature profile relationship.

Finally, if one chooses to treat the outer wall boundary as adiabatic, to reflect the short










experimental run time, then one can develop an unsteady, two-dimensional representation

between the heat flux and the temperature distribution as described in Section 6.4. These three

cases, labeled as (i) linear one-dimensional, (ii) axisymmetric, and (iii) 2-D unsteady adiabatic

are shown in Figure 6-19 along with the computed result. Figure 6-20 shows the two-

dimensional nature of the heat flux along the wall. Overall, the agreement between the CFD

results and the experiment fair. The overall trends from the CFD data and the 2-D conduction

analysis are similar, but shifted. The shift in the CFD data as compared to the results of the 2-D

conduction analysis indicates a faster heat release in the CFD simulation than may actually occur

in the experiments. In all cases, the CFD data under-predicts the experimental data. Clearly, there

are substantial uncertainties in converting limited temperature measurements within the wall to

heat flux profiles on the wall. There is also uncertainty in the CFD results that may be due to the

resolution of the computational domain. The effects of the grid resolution on the experimental

inj ector element are explored in Section 6.6.

6.5.3 Heat Transfer Characterization

The three-dimensional simulation allows for a view of the heat flux characteristics across

the flat wall. Figure 6-20 shows the two-dimensional nature of the predicted heat flux along the

wall. Also shown in Figure 6-20 is the eddy conductivity. The location of peak heat flux can be

clearly seen in Figure 6-20A, and corresponds to the maximum eddy conductivity value along

the combustion chamber wall. The heat flux contours show where the heated gases are circulated

back towards the inj ector face located at the left of the plot. Additional measurements would be

required for more accurate comparisons of the two-dimensional heat flux and temperature.

Across x = 24 mm mark, there is a sudden jump in heat flux magnitude that corresponds to

the change in x-velocity from positive to negative as shown in Figure 6-21. The sudden jump in

heat flux is shown in Figure 6-22A. The locations of stagnation region resulting from the x-









velocity change and the streamline reattachment points are clearly apparent in the dips in the y+

values shown in Figure 6-22B. The effect of the velocity change is echoed in the temperature

values near the wall (y+ ~ 2.5) as seen in Figure 6-23A.

The mixing action of the shear layer transports heat from the flame to the wall. The point

of maximum heat flux occurs at 63 mm. This corresponds to the point of maximum eddy

conductivity that also occurs at 63 mm from the inj ector face, shown in Figure 6-23B. Similar to

the experimental results, the peak wall temperature does not occur in the same location as the

maximum heat flux. On the other hand, the maximum temperature along the wall occurs at 72

mm, but is essentially constant between 64 mm and 79 mm. This is immediately upstream of the

reattachment point which is 80 mm from the inj ector face. The disconnection between the peak

heat flux and the peak wall temperature can be related to the ratio of eddy viscosity to laminar

viscosity. The wall heat flux profile corresponds to the eddy viscosity, and similarly the thermal

conductivity. The temperature profile, on the other hand corresponds to the laminar, or molecular

viscosity.

6.5.4 Species Concentrations

From the temperature profile given in Figure 6-18 at y = 0, it can be seen that much of the

reaction occurs before 72 mm. This is also evident in the mass fraction contours in Figure 6-24.

The reaction proceeds beyond that at a much slower rate as the system moves toward

equilibrium. The species concentration through the flame center is shown in Figure 6-25.

Figure 6-25 shows the diffusion of hydrogen towards the centerline between 0 and 22mm. The

mole fraction of H2 inCreaSes at y = 0 due to the consumption of Oz. The y = 2 profile is located

just beyond the H2 inlet, so the increase in H2 reflects the spreading of the H2 Stream. Because

the reaction is fuel rich, the hydrogen is not completely consumed. There is almost the almost

complete disappearance of gaseous oxygen by approximately 53mm. This is also consistent with









the rise in temperature that occurs at this location. This consumption of oxygen and the high

temperatures gives rise to intermediate species.

Species concentrations and temperatures near the wall (y+ ~ 2.5) show a jump near 24mm

where vx = 0 similar to what was seen in the temperature and enthalpy profies. This effect is

still found somewhat away from the wall. Figure 6-26 shows the homogeneity of the combustion

products beyond approximately 72 mm. By this point, the products are comprised almost

completely of steam and unburned hydrogen.

6.6 Grid Sensitivity Study

For the grid sensitivity study, the inj ector element used is the same as that from the

modeling study along with boundary conditions that simulate a single inj ector element near a

combustion chamber wall. In preparation for the later parameter study in Chapter 8, the CFD

model attempts to approximate a near-wall inj ector element. The sub-domain is approximated

using a rectangular shaped computational domain with width and height of 4.03 mm, and a

length of 100 mm. Only one-half of the square cross section is simulated, with a symmetry plane

defined along the symmetric boundary.

The boundary conditions for all cases are given as shown in Figure 6-27. In the CFD

model, the distance to the wall is held equal to the distance to the inner slip. The purpose of the

inner slip boundary is to keep the computational domain approximately symmetric. A constant

temperature of T = 500 K was specified along the combustion chamber wall. The remaining

boundary conditions are the same as those used in Section 6.5.1. The inj ector face and the

oxidizer post tip region between the hydrogen and oxygen inlets were set to adiabatic conditions.

The nozzle is not included in the CFD model, and a constant pressure condition is specified at

the outlet. A constant mass flow rate for hydrogen and a constant mass flow rate for oxygen are










specified at the inlets, with gaseous oxygen entering through the center core of the inj ector and

hydrogen entering through the outer annular region.

The effects of modifying the grid resolution were investigated via a grid sensitivity study.

The computational grids investigated contained from 23,907 points for the coarse grid 103,628

points for the finest grid. The number of points and smallest grid density used for each grid is

given in Table 6-2, and each grid had a grid distribution as shown in Figure 6-28. A layer of

prism elements were used near the wall so that the turbulence equations could be integrated to

the wall without the use of wall functions which might reduce accuracy. The prism layers were

constructed with an exponential distribution with an increasing number of prism layers used as

the initial wall layer height was decreased. The specifications for the initial height of the prism

layers are given in Table 6-2.

The coarsest grid solution (grid 6) predicts a much sharper rise in the wall heat flux that of

the fine grid solution (grid 1), and underpredicts the peak heat flux as seen in Figure 6-29. This

indicates that performance could suffer when using a very coarse grid. The largest difference in

the results can be seen in the length of the combustion zone in Figure 6-30. The coarsest grid

solution (grid 6) predicts a much shorter combustion zone than the finest grid solution. Using the

coarse grid leads to much more smearing along the reaction zone than the two finest grid

solutions, grid 1 and grid 2. The species diffuse faster in grid 6 as compared to the grids 1 or 2.

However, the velocity field is less affected by the grid density than the species diffusion. Thus,

the mixing that is responsible for convecting heat to the combustion chamber wall leads to heat

flux profiles that are less sensitive to grid density. Significant differences are seen in the CFD

solutions as the grid resolution is increased to its finest level. Numerical unsteadiness was seen in









the finest grid solution (grid 1) shown in Figure 6-30. For this reason, the peak heat flux recorded

for grid 1 is not considered the "true" peak heat flux.

In this case, true grid independency cannot be confirmed. However, there is relative

consistency between the values of the peak heat flux, mean heat flux, and combustion length for

the intermediate grids. Overall, the intermediate grids with 31,184 nodes and 43,680 nodes

provide arguably the best tradeoffs between run time, numerical steadiness, and accuracy as

compared with other grid resolutions. A grid with a resolution between that of grids 1 and 2 was

used for the CFD modeling study in the Section 6.5. This indicates that significant improvement

in the heat flux profile would not necessarily be obtained by increasing the grid resolution.

For the same inj ector element and mass flow rates used for the grid sensitivity study and

for the modeling study, the reduction in the combustion chamber area appears to result in a

reduction of the combustion length. However, because of the numerical unsteadiness of the finest

grid, it is difficult to determine whether this is due to the combustion chamber geometry, or due

to inadequate grid resolution. It is possible that the combustion length may be, for the most part,

only a function of the inj ector geometry and mass flow rate, and not a function of the combustion

chamber geometry. The parameter study in Chapter 8 tries to determine some of the geometric

effects on inj ector flow.

6.7 Conclusion

The CFD model was able to qualitatively match the experimental results. Uncertainty

exists in the prediction of the peak heat flux and heat rise. However, the predicted location of the

peak heat flux was consistent with the experiment. The heat flux downstream of the reattachment

point was well predicted. In addition, the CFD results were able to provide additional insight into

the combusting flow. In particular, the relationship between the flow dynamics and heat transfer

could be characterized. It was found that the effect of the 3-D square geometry shaped the









recirculation in a way that noticeably affected the heat transfer profie. The heat flux along the

wall is further shaped due to the recirculation of fluid that has had time to cool along the

combustion chamber wall, resulting in a relatively cool region of fluid near the inj ector face.

Beyond the reattachment point, the homogenous fluid cools at a relatively constant rate. The heat

transfer within the combustion chamber can be characterized via the eddy conductivity which

further relates to the peak heat flux at the wall. Additional experimental temperature

measurements could improve the CFD modeling process by providing some measure of the 2-D

wall heat transfer effects. Additional measurements could also improve the accuracy of the

estimated wall heat flux and temperatures.

A grid sensitivity study on the inj ector element used for the CFD modeling remained

inconclusive. A grid independent solution could not be obtained by refining the grid. Numerical

unsteadiness resulted with the Einest grid. The coarsest grid predicted a sharper heat flux rise, a

lower peak heat flux, and a shorter combustion length than the Einer grids. With the exception of

the coarsest grid and the Einest grids, the predicted heat flux remained largely unaffected. This

indicates that a finer or coarser grid would likely have little effect of the heat flux predicted in

the modeling study. However, the predicted combustion length varied significantly based on the

grid resolution. Additional studies would be required to determine the reason for the numerical

unsteadiness.

Table 6-1. Flow regime description.
Inlet temperature, T 300 K
Exit pressure, pe 2.75 MPa
Mass flow rate, thz 0.187 g/s
Max Reynolds number, Rena 3000
Min Reynolds number, Renzn 960










Table 6-2. Effect of grid resolution on wall heat flux and combustion length.
Minimum element size in Run


Grid Grid
# points
1 103628
2 72239
3 43680
4 31184
5 25325
6 23907


shear layer/near wall
(mm)
0.15 /0.03
0.2 /0.03
0.5/ 0.01
0.5 /0.03
0.5/ 0.05
1 /0.03


time
(min)
10920
3150
744
588
390
392


Combustion
length (mm)
31.1
34.7
30.7
30.7
30.6
23.2


qnax
(MW/m2)
5.41
5.08
4.80
4.93
4.73
4.47


qnean
(MW/m2)
2.11
2.03
2.09
2.06
2.07
2.03


Table 6-3. Flow conditions.
Inlet hydrogen mass flow rate, viH,
Inlet oxygen mass flow rate, vio,

Equivalence ratio, ~
Chamber pressure, p
Initial combustion chamber temperature, T
Average Mach number, M ~


0.396 g/s
1.565 g/s
2.0
2.75 MPa
2000 K
0.04


Figure 6-1. Blanching and cracking of combustion chamber wall due to local heating near
inj ector elements.





NI 11 11


Figure 6-2. Hydrogen flow geometry.


Figure 6-3. Hydrogen inlet mesh.






















































0.10
0.08 I P= 00
0.06 -1 It 0- = 5.70



-0.04
0.06



-0.08 -
-0.10
-6 14 34 54 74 94


Figure 6-5. Swirl number at each x location.


NO


-5 0 Y L~~
5 10 15 100


10 15 100
B Y

Figure 6-4. Z-vorticity contours. A) B = 0 and B)B













CO ~ 0.70
-u, p = 0
501 -u, p =5.7 t0.60
v6, p = 0
0.50
40 ~v6, = 5.7.
0.40

= 30~ 0.30 >"

0.20
20
0.10
10 a '- ---. ,I\ 0.00

0 -0.10
-20 0 20 40 60 80 100



Figure 6-6. Average axial velocity u and average tangential velocity vs with increasing x. The
axial velocity is two orders of magnitude higher than the tangential velocity.

3000 700
Reu, p = 0' 600 Rev, P = 0
2500 Reu, P= 5.7" 0 Rev, P= 5.70
2000 500
20400
1500
300*
1000



0 0^
-20 0 20 40 60 80 100 120 -20 0 20 40 60 80 100 120
A x Bx

Figure 6-7. Reynolds number profiles. A) axial velocity and B) tangential velocity.

2.5E-03

2.0E-03

g 1.5E-03

8 1.0E-03
-turbulent, 1 = Oo
5.0E-04 -
-turbulent, = 5.7o
0.0E+00
-20 0 20 40 60 80 100 120


Figure 6-8. Non-dimensional pressure as a function of x.












1 60


~


1 00


S0 80


0 60 n .

040


0 20


0 00
0 00 20 00 40 00 60 00 80 00 0 5 10 15 20 25 1 E+04 1 E+05 1 E+06 1 E+07 1 E+08 1 E+09
A vx B k omega

Figure 6-9. Hydrogen inlet flow profiles. A) Axial velocity, B) turbulent production k, and C)
turbulent dissipation co.


Figure 6-10. Combustion chamber cross-sectional geometry and thermocouple locations. Units
are given in millimeters.




































Thermocouples





I I wall


-- Steady-state approximation
Unsteady approximation


C\I
E
~ 2.5
r


-


0 0.02 0.04 0.06 0.08
x(m)


0. 1 0. 12 0.14 0. 16


Figure 6-11. Estimated wall heat flux using linear steady-state and unsteady approximations.


Figure 6-12. 1-D axisymmetric assumption for heat conduction through combustion chamber
wall.


















180













linear
axisymmetric


500



450


0 0.02 0.04 0.06 0.08


0. 1 0. 12 0. 14 0. 16


Figure 6-13. Estimated wall temperatures using linear and axisymmetric approximations. Linear
and axisymmetric assumptions give nearly equal wall temperature estimations.
Temperature values at x = Om and x = 0.16m are estimated based on nearest
temperature values.


0 02E


002(

0 015

0 01

0 005




-0 005

-0 01

0 0 005 0 01 0 015 0 02 0 025 0 03


0 0 005 0 01 0 015 0 02 0 025 0 03

0 02


550


0 ... I i


C 0 0 005 0 01 0 015 0 02 0 025 0 03 D 0 0 005 0 01 0 015 0 02 0 025 0 03

Figure 6-14. Temperature (K) contours for 2-D unsteady heat conduction calculations at x =
0.084 m at A) 1 s, B) 2 s, C) 6.5 s, and D) 7.75 s. Units are given in meters. One-
quarter of the combustion chamber cross section is shown. The temperature scale is
consistent across plots.










Unsteady approximation

unsteady linear
unsteady axisymmetric
2-D unsteady adiabatic


C\I
E
Z 2
r


0 0.02 0.04 0.06


0.08 0.1
x(m)


0.12 0. 14


0.16


Figure 6-15. Experimental heat flux values using unsteady
mn meters.




p= constant


outlesymmetry







A


assumptions. The x values are given


H2 inlet
S02 inlet


"K77~""1


.r


~d~,


i'


Figure 6-16. Computational model for single-element injector flow simulation. A) Combustion
chamber boundary conditions, B) grid close-up, and C) representative combustion
chamber section.




182



































Figure 6-17. Velocity contours v,(m/s) and streamlines. The inj ector is located in the lower

right corner of the combustion chamber. The reattachment point is at x = 77 mm.









-10-

0 20 40 60 80 100 120 140 160


_I
~---
Ylh,_--
-- ~
: II
I~' I~.r~
F j~i ~i.~~


i 'I


Figure 6-18. Temperature (K) contours. Injector center is at x,y :
millimeters .


(0,0). Distances are given in


* -D linear
axisymmetric
*-unsteady adiabatic
-CFD


r
r \


*


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
x (m)

Figure 6-19. CFD heat flux values as compared to experimental heat flux approximations.


I i


















0 0.5 1 1.5 2 2.


10


>10

O 20 40 60 80 100 120 140 160
C x
Figure 6-20. Wall heat transfer and eddy conductivity contour plots. A) Heat flux along
combustion chamber wall (xz-plane at y = 12.7) B) eddy conductivity at y = 12.3, and
C) eddy conductivity at z = 0 with wall at y = 12.7 mm. Eddy conductivity contours
are oriented perpendicular to wall heat flux contours. Horizontal axis is x-axis.
(Solutions mirrored across x = 0). Length units are in mm. The inj ector is centered at
x,y,z = (0,0,0), and the flow proceeds from left to right.

vx= 0l I mrattachment






0 20 40 60 80

Figure 6-21. Streamlines and temperature contours at plane z = 0. All lengths are in millimeters.
















184


q (MW/m2)


0 20 40


60 80 100


120 140 160


Eddy Codctvl 2E


-10
O


20 40 60 80


140 160


100 120


























0002 004 006 008 01 012 014 016 0 002 004 006 008 01 012 014 016
A xem) B (m)

Figure 6-22. Heat flux and y+ profiles along combustion chamber wall. A) Heat flux profile
along the wall at := 0 showing sharp increase in heat flux corresponding to the
location of velocity vx = 0. B) The y+ values along the near wall cell boundary dip at
stagnation regions.










10'


I I
E50 1001 L. ED 10 150
AQ x (mm) B x (me

Figure~ 6-3 eprtr n dycnutvt rflsa aiu oain npae=0




Figur 6-A) Temperature. B) Eddy conductivity.poie tvrosyloain npaez=0



























185


Unsteady approximation


p vx = 0


If-reattachment point


vx = 0

















I-IIIIIIT
0 20 40 60 80 100 120 140 160



>10-


-10-

0 20 40 60 80 100 120 140 160



S10 -



-10-

0 20 40 60 80 100 120 140 160


10-



-10 ------~


0 20 40 60 80 100 120 140 160


Figure 6-24. Mass fraction contours for select species.






































021( H20




,o2

.O


0.8



c*
.9 0.6



O 0.4
E



0.2



0


1 00





1Q




O


E 1U"


1U'

1U"


x (mm)


B x (mm)
Figure 6-25. Mole fractions for all species along combustion chamber centerline (y = 0, z = 0).
A) uniform and B) log scaling.







187

































0 50 100 150
x I (m)


0.2






B
Figure 6-26.


x (mm)


Select species mole fraction profiles. A) H2 and B) H20 at plane z = 0.




































aI,

Fiue 62.Cmuainlgi ln ymtionay

230 30
6 18 18














5 35-d

cC'~~ 4



25

s 3 .,



2 1 5




0 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 0 01 0 02 0 03 0 04 0 05 0 06 0 07

Figure 6-29. Wall heat flux and yi values for select grids. The distance from the inj ector face x
is given in meters.
















189
















-


40








0 1 2 3 4





0 10 20 30 40








0 10 20 30 40


Figure 6-30. Comparison of temperature (K) contours for grids with 23,907, 3 1,184, 72,239, and
103,628 points, top to bottom, respectively. The finest grid shows time-dependent
oscillations.


O I-










CHAPTER 7
MULTI-ELEMENT INJECTOR FLOW MODELING AND ELEMENT SPACING EFFECTS

7.1 Introduction

Because of the arrangement of inj ector elements near the wall of the combustion chamber,

the heat flux peaks near each inj ector element. Chamber design and potential material failure is

based on peak heat flux, so the peak heat flux must be minimized to increase component life and

reliability. By controlling the inj section pattern, a layer of cool gas near the wall can be created.

One method of controlling the inj section pattern is to offset the oxidizer post tip away from the

wall. This results in a higher percentage of unburned fuel near the wall, and this technique was

used in the SSME. The cooler layer near the wall can possibly further minimize the peak heat

flux. Another method of controlling the heat flux is to change the spacing of inj ector elements

near the combustion chamber wall. This research effort looks to quantify selected geometry

effects by directly exploring the sensitivity of wall heating and inj ector performance on the

inj ector spacing.

Shyy et al.29 and Vaidyanathan et al.63 identified several issues inherent in CFD modeling

of inj ectors, including the need for rigorous validation of CFD models and the difficulty of

simulating multi-element injector flows due to lengthy computational times. They chose to

simulate single inj ector flow, as this allowed for the analysis of key combustion chamber life and

performance indicators. The life indicators were the maximum temperatures on the oxidizer post

tip, injector face, and combustion chamber wall, and the performance indicator was the length of

the combustion zone. These conditions were explored by varying the impinging angle of the fuel

into the oxidizer. In other words, the inj ector type was gradually varied from a shear coaxial

inj ector to an impinging inj ector element. Response surface approximations were used to model

the response of the inj ector system, and a composite response surface was built to determine the










inj ector geometries that provided the best trade-offs between performance and life. It was

determined that a shear coaxial inj ector provides the lowest maximum temperatures on the

inj ector face and combustion chamber wall, but had the lowest performance. An impinging

element provided the lowest maximum oxidizer post tip temperature and had high performance

due to the short combustion length, but because the maximum inj ector face and combustion

chamber wall temperatures were high, the life of the inj ector was compromised. The study

confirmed that small changes in the inj ector geometry can have a large impact on performance.

The optimization procedure was successful in specifically accessing the effects of a small change

in the inj ector geometry. This is important, as it allows for greater confidence in choosing the

next design. This information can be coupled with the experience of inj ector designers to

accelerate the design process.

The purpose of the proposed research is to improve the performance of upper stage rocket

engines by raising the temperature of the fuel entering the combustor. A multi-element inj ector

model is constructed. An effort is made to capture the 3-D effects of a full multi-element

injector face in a simplified 3-D model. The steps to improve the injector design are twofold: 1)

the mean heat flux to the combustion chamber wall must be increased to increase fuel

temperature and subsequently increase engine performance, and 2) the peak heat flux to the wall

must be decreased to diminish the effects of local heating on the chamber wall. The injector is

optimized to best satisfy the obj ectives.

The feasible design space is determined through exploration of the design space using

available design space refinement and selection techniques. After the feasible design space is

determined, the optimization framework described in Chapter 2 is used to guide the inj ector









element spacing study. The design space is refined as needed to obtain an accurate surrogate

model .

7.2 Problem Set-Up

This study explores the effects of geometry on the peak heat flux and combustor length.

Approximately 90% of the heat flux to the combustor wall is due to heat transfer from the outer

row of inj ector elements. The second row of inj ector elements is responsible for approximately

10% of heat flux to the wall. This information was used to simplify the computational model of a

GO2/GH2 Shear coaxial multi-element inj ector to make it suitable for a parametric study. The

inj ector face simplification of the full inj ector shown in Figure 7-1 is outlined below:

1. For the CFD model, only the outer row of elements is considered. Currently, the effects and
interactions of the outer element row with interior elements are simplified by imposing
symmetric conditions.

2. The multi-element injector effects are modeled by imposing symmetric conditions between
inj ectors with the computational domain sizes corresponding to the actual radial and
circumferential spacing of the full injector.

Thus, the CFD model is a scaled down version of what would occur in an actual combustion

chamber. The CFD model attempts to capture effects near the outer edge of the inj ector.

The variables are the circumferential spacing (number of inj ector elements in outer row),

wherein the total mass flow rate through the outer row inj ector elements is held constant, and the

radial spacing, such that the distance to the wall is held equal to the distance to the inner slip

boundary. By this assumption, the mass flow rate through a single injector is equal to the total

mass flow rate through the outer row of inj ectors divided by the number of inj ectors in the outer

row. The purpose of the inner slip boundary is to keep the computational domain approximately

symmetric, as the contributions and interactions from inner elements are neglected in the

analysis. It is desired that the CFD model be able agree qualitatively with related experiments as

well as establish the sensitivity of certain geometry changes on wall heating.









The CFD simulations were conducted using loci-STREAMm (Thakur and Wright 2006).

The CFD model attempts to approximate the single-element section near the chamber' s outer

wall given in Figure 7-1. The single-element section was approximated using a rectangular

shaped computational domain. Only one-half of the domain is simulated. The grid resolution

used is equivalent to the intermediate grid 3 from the grid sensitivity study of Section 6.6. This

grid was chosen for its balance of accuracy and relatively short run-time. The solutions are

expected to give good heat flux accuracy, but the combustion lengths are expected to be

somewhat under-predicted.

In summary, the obj ectives are to minimize the peak heat flux qmax(N*,r*) and minimize

the combustion length Le(N*,r*) where the independent variables are explored within the range

0.75 <; N* <1.25
0.25 <; r* <1.25 (7-1)
N' < Nmx /Nbaseline

where

N* = N/Nbaseline T lbaseline (7-2)

where N is the number of inj ector elements, Nna is the maximum possible number of inj ector

elements in the outer row, and r is given as shown in Figure 7-1. The combustion length is the

length at which combustion is 99% complete, and a shorter combustion length indicates

increased performance. A low peak heat flux would reduce the risk of wall burnout. Although

the number of inj ector elements in the outer row changes, the total combined mass flow rate to

the outer elements also remains constant, so the mass flow rates for the individual inj ectors are a

function of the number of inj ector elements N. The distance w of the computational domain in

Figure 7-1B is given by










w (N = Rtan(7-3)
r()= tN 2%)

where R is the distance from the center of the injector to the center of an outer element inj ector

element and D is the inj ector element diameter.

The baseline inj ector element geometry is based on the experimental test case RCM-1 from

Section 4. 1.4, while the overall inj ector and element spacing is based on the Integrated

Powerhead Demonstratorl59 (IPD) main inj ector. For this inj ector, the maximum number or

inj ector elements N;;; allowable in the outer row is 101. The flow conditions and inj ector

geometry for the baseline inj ector are given in Table 7-1. The inlet flow represents incompletely

burned flow from upstream preburners, so both the fuel and oxidizer contain some water as a

constituent.

7.3 Feasible Design Space Study

The design points based on r* and N* are shown in Figure 7-2. The first seven design

points were selected based on a preliminary design sensitivity study. Point 16 lies at the center of

the design space. Latin Hypercube Sampling was then used to select the seven remaining design

points within the variable ranges.

Figure 7-3 shows the values of the obj ectives at each design point. It can be seen that the

mean heat flux across the combustion chamber wall stays relatively constant across all of the

design points. The maximum heat flux, however, varies depending on the combustor geometry.

The design points have an average mean heat flux of 8.2 MW/m2 and an average maximum heat

flux of 37.3 MW/m2. In general, as the maximum heat flux becomes lower, the combustion

length is longer, indicating slower combustion. For design purposes, there would need to be

some balance between the maximum heat flux and combustion length. From Figure 7-3B, it can

be seen that case 6 shows good balance between minimizing the peak heat flux and minimizing










the combustion length, while case 12 gives arguably the worst characteristics due to its high

maximum heat flux and long combustion length. Figure 7-3 also shows the maximum heat flux

and combustion length as a function of the hydrogen mass flow rate. It is apparent that the

maximum heat flux is not a function of the mass flow rate alone, but that there must be other

influencing factors. There does seem to be an overall decrease in the maximum heat flux with the

mass flow rate, as well as an overall increase in combustion length with increasing mass flow

rate, which is a function of N*. No obvious relationship was seen between the maximum heat

flux or combustion length and the cross-sectional area. This indicates that, a smaller cross-

sectional area does not automatically translate into a longer combustion length, as one might

expect. This was also demonstrated by the grid sensitivity study.

The CFD results for four cases, representing the baseline case along with cases

demonstrating good or poor heat flux characteristics, are shown in Figure 7-4. It can be seen that

the flame is flattened somewhat based on the shape of the computational domain. Another maj or

observation is of the location of the recirculation region. Instead of hot gases being directed at

the wall, as in the baseline case and case 12, the recirculation region of case 2 and case 6 is

located between inj ector elements, and the streamlines are largely parallel to the combustion

chamber wall. This led to an observation that all of the CFD solutions resulted in one of two

scenarios. Eleven of the cases showed a sharp peak in the wall heat flux near the inj ector face

with the downstream heat flux considerably lower than the peak heat flux. This is termed

"Pattern 1," and corresponds to cases that have hot gases directed at the wall as in Figure 7-5A.

The remaining five cases had a relatively low heat flux near the combustion chamber wall that

grew gradually to a slight peak further downstream. This group is termed "Pattern 2," and

corresponds to the cases that have a recirculation zone that would exist between inj ector









elements. The two groups can be clearly seen in Figure 7-6 in the plot of maximum heat flux

versus r*. Figure 7-7 shows the heat flux profile and contours for case 12 of Pattern 1, and case 6

of Pattern 2. In Pattern 2, the heat flux is spread across the combustion wall, rather than being

localized as were the cases from Pattern 1. The best performing designs with regards to heat flux

appear to be those with small distances between the injector and the wall. However, of the cases

in Pattern 2, the worst cases are cases 7 and 9 that have smaller domain widths as compared to

the other cases. This would suggest that the peak heat flux might be a function of the domain' s

aspect ratio, rather than separately being a function of radial or circumferential spacing, alone.

The domain' s aspect ratio is given by AR = r "hy *, where w*"= whybaseline, and wbaseline IS

determined based on N = Nbaseline in Equation 7-3. It can be seen in Figure 7-8 that the value of

the peak heat flux appears to be strongly dependent on the aspect ratio. The designs with the

lowest peak heat flux also have the lowest aspect ratio. This indicates that it may be desirable to

have a low ratio of wall spacing (radial spacing) to inj ector-inj ector spacing (circumferential

spacing) to reduce the peak heat flux. Due to the presence of case 5, it is difficult to determine

whether the cases approximately follow a linear trend and case 5 is an outlier, or if the actual

trend is quadratic.

It appears that the peak heat flux is highly sensitive to the combustion chamber geometry

as evidenced by Figure 7-8. The relationship of the combustion length to geometry is less clear.

It appears that there is some sensitivity to the geometry due to the fact that equivalent inj ector

elements with equal mass flow rates can result in combustion lengths that can differ significantly

(see points 2, 4, and 6 in Figure 7-3D). In general, the cases in the region defined by Pattern 1

have shorter combustion lengths than those in Pattern 2. This indicates that increasing the size of

the recirculation zone, and hence the mixing, may increase the combustion efficiency. An









individual analysis on a single inj ector element of the effect of the combustion chamber

geometry on combustion length with constant mass flow rate may be necessary to determine the

relationship.

The best trade-off points are located in one or the other region of the design space, as

shown in Figure 7-9. Because the behavior of the design points in the different regions are very

different, this indicates the need for two separate design spaces. The dividing line for the new

design space is given approximately by the dashed line in Figure 7-9B. The design space will

also be expanded slightly to

0.60 < N* <1.25
(7-4)
0.20 <; r* <1.25

due to the best trade-off points of pattern 2 having low N* and r* values.

While it does appear that the combustion length is definitely affected by the geometry, the

results of the feasible design space study were not able to reveal the direct relationship between

combustion length and geometry. Based on a computational model for an inj ector element near

the combustion chamber wall of a multi-element inj ector, it was found that an inj ector element

that is located far from the combustion chamber wall does not always result in the best heat

transfer characteristics. The results of the feasible design space study suggest that by increasing

the spacing between inj ector elements of the outer row, while reducing the distance of the outer

row to the wall, that the heat transfer results could be better controlled. This configuration helps

to direct heat away from the wall, rather than towards it, and results in an even distribution of

heat across the combustion chamber wall. To a lesser degree, increasing the distance to the

combustion chamber wall can also result in reduced peak heat flux, but the required spacing may

possibly be prohibitively large.









7.4 Design Space Refinement

Using the merit function smart point selection procedure introduced in Section 2.4.3, along

with an assigned rank using the formulation for multi-obj ective problems introduced in Equation

2-38, additional points were added to the newly expanded design space. The merit function that

maximizes the probability of improvements M~F2 WAS chose for point selection, as it resulted in

the least error for a small number of sampling points as was shown the radial turbine analysis in

Section 3.4. The design space refinement proceeded over several steps. The steps are listed

briefly, then described in more detail:

1. Kriging RS was constructed for each objective over 16 initial design points. Based on the
results initial Kriging RS, the design space variable ranges were expanded due to favorable
function values on the edge of design space. Four points were selected via the multi-
obj ective merit function-based point selection criteria to add to the original 16 points for a
total of 20 designs.

2. The design space was separated based on flow patterns into two regions called Pattemn 1
and Pattemn 2. A Kriging RS was constructed for each obj ective for the 12 out of 20
designs in Pattern 1. Three new points were selected using M~F2 for a total of 15 designs in
Pattern 1.

3. A Kriging RS was constructed for each obj ective for the 8 out of 20 designs in Pattern 2.
Four new points were selected using M~F2 for a total of 12 designs in Pattemn 2.

4. A Kriging RS was constructed for each objective for the 15 designs in Pattern 1.

5. A Kriging RS was constructed for each objective for the 12 designs in Pattern 2.

6. For visualization of the full design space, a Kriging response surfaces were fit for each
obj ective to all 27 design points.

The error statistics for all six steps are given for each objective in Table 7-2.

A Kriging surrogate was fit to the entire design space before the design space separation.

The merit function value was calculated for points sampled across the design space as shown in

Figure 7-10. The densely sampled points were divided into eight clusters based on location. The

point with the best rank in each cluster was selected. Out of the eight resulting points, four were









chosen based on their favorable objective function values predicted by the surrogate model. The

Kriging surrogate results along with the four selected points are shown in Figure 7-11. While

Kriging can provide an adequate surrogate model for the peak heat flux, Figure 7-11B shows that

the design space is too large for an appropriate surrogate fit for the combustion length. When the

PRESSms is calculated as a percent of the obj ectives sampling range or the standard deviation of

the obj ectives, it can be seen that the PRES Sms error for the Le surrogate fit is about twice that of

the qmax surrogate. The design space was thus divited into two smaller regions to try to improve

the surrogate fits.

Fifteen of the 20 design points were located in the region for Pattern 1, while eight of the

20 design points were located in the region for Pattern 2. Kriging fits to the individual regions

increased the scaled prediction error in each region. In particular the Kriging scaled prediction

error for Pattern 2 increased significantly due to the small number and poor distribution of the

data points located in the region.

To improve the surrogate fits, additional data points were added to each region. Three

points were added to the Pattern 1 region, and four points were added to the Pattemn 2 region

based on merit function evaluations and favorable objective function values. The Kriging

surrogate fit and merit function contours for Pattemn 1 and Pattern 2 are given in Figure 7-12 and

Figure 7-13, respectively. Figure 7-12 indicates the locations of the three points added to the

design space in the region of Pattemn 1. Figure 7-13 indicates the locations of the four points

added to the design space in the region of Pattern 2. These points were chosen out of the points

selected by the merit function analysis based on high uncertainty and high promise of function

improvement. Regions where the predicted function values were very poor were avoided. This

would have the overall effect of refining the design space only in the region of the best trade-off


200










designs. The resulting Kriging fits including the new data points are shown for Pattern 1 and 2

in Figure 7-14 and Figure 7-15, respectively. The response surface for peak heat flux of Pattern

1 in Figure 7-14 shows smooth contours, indicating a good surrogate fit. The prediction errors of

the Kriging fits echo what can be casually observed from the contour plots. For Pattern 1, the

addition of points slightly decreases the scaled error in qmax, and significantly decreases the

scaled error in the prediction for Le. The prediction of the combustion length of Pattern 2 was

improved by separating the design space into two regions as evidenced by the scaled prediction

error. The fit of the peak heat flux became worse for Pattern 2 after splitting the design space

into two regions, but this may be due to the sparseness of data points in the region.

When the data points were plotted as combustion length versus peak heat flux in Figure

7-16, it was discovered that all of the newly added points dominated the points in the original

Pareto front. Overall, the heat flux was reduced by up to 20% and the combustion length was

reduced by up to 5% among the eleven new points as compared to the best trade-off points from

the original data set of 20 points. This indicates the success of using merit functions to select

points for design space refinement. This is a case where a surrogate based on a simple DOE can

be used to positively identify the "bad" regions so that they may be avoided. In this case, the

original surrogates were not accurate in regions away from the original design points,

particularly in the prediction of the combustion length, but they are sufficient to identify regions

that are obviously bad. By ignoring obviously bad regions of the design space, new data points

could be concentrated in the best trade-off region. The original surrogate model was also

successful in identifying the regions of the design space that may result in function improvement,

as six of the eleven new points lay in the expanded portion of the design space.









A Kriging fit to all points of the design space shown in Figure 7-17 show a smooth

response surface for the prediction of the peak heat flux. The highest peak heat fluxes are

predicted in Pattern 2. This is to be expected, as these cases have a recirculation zone that directs

hot gases to the combustion chamber wall. This feature allows for a new approximation of the

boundaries between the two regions of the design space. Cases that lie very near the division,

such as case 20 at N*= 0.6 and r* = 0.9, show characteristics of both regions with a recirculation

zone evenly situated about the inj ector, rather than in the vertical or horizontal directions, as

shown in Figure 7-18. The longest combustion lengths, or worst combustion efficiencies, occur

in the points in Pattern 2. This is likely due to the smaller cross-sectional area combined with

mass flow rates that are larger, in general, than the designs in Pattern 1. However, for very low

r* values, the combustion efficiency improves.

Based on Figure 7-16, cases 18 and 19 were chosen as the best trade-off designs. These

designs exist on opposite sides of the design space and in different pattern regions, as shown in

Figure 7-19, but have similar number of inj ectors in the outer row of the multi-element inj ector,

as shown in Figure 7-20. The cases also have similar heat flux values and combustion lengths.

Compared to the baseline case, case 19 predicts a 45% reduction in heat flux as compared to the

baseline case while maintaining the combustion length. Case 19 predicts a 38% reduction in the

heat flux and a 5% reduction in the combustion length as compared to the baseline case. Figure

7-21 shows the predicted heat flux profile for the selected trade-off cases as compared to the

baseline cases. The grid resolutions of the baseline case and one trade-off solution were

moderately increased and compared to the solutions used for the surrogate modeling. The

coarser grid solution required less than one day to obtain a solution, while increasing the number

of grid points by a factor of two increased the run time to approximately five days. A moderate


202









increase in the grid resolution has a minimal effect on the baseline peak heat flux, but increases

the downstream heat flux, as shown in Figure 7-22A. However, the combustion length is 15%

longer using a Einer grid as compared to the grid used to construct the surrogate model. For case

19, the trade-off case from Pattern 1, and increase in the grid resolution results in a 45%

increased prediction of peak heat flux over the coarser grid and a 10% increase in the combustion

length. When comparing the Eine grid solutions of the baseline case and case 19, the peak heat

flux is only reduced by 16% and the combustion length reduced by approximately 6%.

Significant differences in the prediction can occur as a result of the grid resolution. However,

the improvement in the Einer grid solutions indicates that the surrogate model can still provide

significant information on the general trends in the data with which to improve the design.

The peak heat flux has a strong dependence on the combustion geometry. The effects of

geometry can be stronger than the mass flow rate dependence. It was observed by Conley et al.22

(2005) that the heat flux of an inj ector was strongly dependent on the inj ector' s mass flow rate,

with the heat flux increasing proportionally with the square of the mass flow rate. The inj ector

spacing analysis, however found that the peak heat flux actually decreased with increasing mass

flow rate, as shown in Figure 7-23. This shows that there are definite effects of geometry on the

peak heat flux. In contrast, the combustion length predictably increases with the mass flow rate,

although the spread in the data indicates geometric sensitivity.

7.5 Conclusion

The optimization framework was used to determine the effects of combustion chamber

geometry on the wall heat flux and combustion length. Although, the accuracy of the surrogate

models may have been affected by the coarse grid resolutions used in the study, the surrogate

models provided an approximation of the design space that was sufficient enough to give

predictions of variable combinations that would result in improvement over the baseline case.


203










When additional points were selected based on large probabilities of improvement given as given

by the merit function M~Fz, improvements in the function values were obtained for every newly

selected variable combination as compared with the original data set. This indicates that the

search for best trade-off designs can proceed efficiently with careful selection of design points

using merit functions.

Based on a computational model for an inj ector element near the combustion chamber wall

of a multi-element inj ector, it was found that an inj ector element that is located far from the

combustion chamber wall does not always result in the best heat transfer characteristics. The

results of the multi-element inj ector spacing study suggest that by increasing the spacing

between inj ector elements of the outer row, while reducing the distance of the outer row to the

wall, that the heat transfer results could be better controlled. This configuration helps to direct

heat away from the wall, rather than towards it, and results in an even distribution of heat across

the combustion chamber wall. Sufficiently increasing the distance to the combustion chamber

wall can also result in reduced peak heat. In the future, further exploration into each phenomenon

should provide additional insight.

Table 7-1. Flow conditions and baseline combustor geometry for parametric evaluation.
Fuel annulus outer diameter (mm) 7.49
Fuel annulus inner diameter (mm) 6.30
Oxidizer post inner diameter (mm) 5.26
Oxidizer post tip recess (mm) 0.43
Combustion chamber diameter (mm) 262
rbasebne (mm) 6.59
Nbasehne 60
Total fuel mass flow rate in outer row (kg/s) 1.986
Total oxidizer mass flow rate in outer row (kg/s) 5.424
Fuel temperature (K) 798.15
Oxidizer temperature (K) 767.59
Chamber pressure (MPa) 5.42
Hz mass percentage of fuel 41.3
HzO mass percentage of fuel 58.7
Oz mass percentage of oxidizer 94.62
HzO mass percentage of oxidizer 5.38


204











Table 7-2. Kriging PRESSs error statistics for each design space iteration.


# of o, o,
# Data set pts qmea Lea
1 Original 16 6.93 5.33
2 Pattem 1 12 7.34 4.67
3 Pattern 1 15 4.28 5.90
4 Pattem 2 8 9.89 4.29
5 Pattern 2 12 4.26 6.06
6 Final 27 10.13 5.15
range(qmax) = max(qmax) min(qmax)
range(LC) = max(LC) min(Lc)
a a= do = square root of the process


PRESSms, PRESSms, PRESSms/
4nm Le range(qmx)
3.32 5.63 12.14
5.18 5.08 17.77
6.59 4.62 14.97
1.93 2.24 21.56
4.59 3.78 33.34
4.96 5.01 15.87


variance given in Equation 2-11


PRESSms/
range(LC)
25.39
34.67
12.26
31.54
20.72
22.59


PRESSms/
std(qmx)
0.41
0.67
0.40
0.65
1.03
0.51


PRESSms/
std(Lc)
1.02
1.04
0.35
1.05
0.66
0.97


~ "-


no-slip


H2 inlet


'02 pOst tip


slip


Figure 7-1. Inj ector element subsection. A) Outer row representation of baseline inj ector with
computational subsection shown at top and B) in close-up. C) Computational domain
and boundary conditions on inj ector element face for injector subsection. One
element in the outer element row is simulated. The circumferential spacing w is a
function of the number of inj ector elements in the outer row.


205















1.1C ---

z 1

0.9 -

0.8 --


02

0.4 0.6 0.8


Figure 7-2. Design points selected for design space sensitivity study. Point 1 indicates the
baseline case.


60

50

40 *

0- 30

20

10

0
0


100
,,
a, 95

~E 90

c 85


*** o


max

9mean


*13

*7
e6 *

*4


*15 *8
*10





*3 5


* *
*


5


...


30 40 ~~m2


50 60


10 15


point


100

S95

S90
* 15
4 O85


6 n 80
*2 0


50 12


e 16


11

101*8


S40
-,35

30

25

20
0.025


e 13* 15



*


S8

S. 114 *116 11
9
s1


5


* 13


75 L
0.025


0.03 0.035
mdot H2 (kg/s)


0.04 0.045


0.03 0.035
mdot H2 (kg/s)


0.04 0.045


Figure 7-3. Effect of hydrogen mass flow rate on obj ectives. A) Heat flux and combustion
length data for original sixteen points, B) Combustion length versus maximum heat
flux, and C) maximum heat flux and D) combustion length versus the hydrogen flow
rate.


206













-~-~




: ~C~i I





-


III
~ -1;--- Ii~i ...-i I










I I Ir


Figure 7-4. Oxygen iso-surfaces and hydrogen contours. A) Baseline case (case 1). B) Worst
overall case (case 12). C) Case with the lowest peak heat flux (case 2). D) Best
overall case (case 6). The combustion chamber wall is at the top of each figure.
Approximately half of the designs showed solutions similar to A) and B), while the
remainder was similar to C) and D). Solutions are mirrored across the :-plane.


25 30 35 40


ii


Figure 7-5. Hydrogen contours and streamlines. A) Case 12 and B) Case 6. The wall is located at
the top of each plot. Case 6 shows the persistence of a gaseous hydrogen layer near
the combustion chamber wall.


207












































































O 5 10 15 20 25 30 35 40


0 5 10 15 20 25 30 35 40 45 50


*12


*14
*16


35~ E/9


Pattern 1


*7
Pattern 2


25 1\*6

20
0.2


*13


0.4 0.6 0.8 1 1.2 1.4


Figure 7-6. Maximum heat flux for a changing radial distance r*. Point 1 is the baseline case.


-case 6




50 ae1



30


0 1 0 12 0 14 0 16


0 0 02 0 04 0 06 0 08
x (m)


q (MW/m2)


Figure 7-7. Heat flux distribution. A) Center of combustion chamber wall for case 6 and case 12.
B) Heat flux distribution along first one-third length of entire wall for case 12 (top)
and case 6 (bottom). Horizontal axis is x-axis.


208






















































A Imax\"'' B
Figure 7-9. Design points in function and design space. A) Combustion length versus maximum
heat flux, and B) the location of design points in the design space. The best trade-off
points within each pattern are circled by a dashed line.





















209


I0 *2 1 tsn
100 Pattern 2

951 *~ 13 i .15.8

90 1 *7 1$444
1 e6 I *9

\ 3
80 3 5
SPattern 1
75 *


.C C I-
74 0.6 0.8 1 12


45

E 40

M 35

30


*1
8 *3 *10

*15

*7


20
0 0.5 1 1.5
AR =r /w

Figure 7-8. Maximum heat flux as a function


2 2.5


of aspect ratio.


12

1.1



"a 1


I ers. I
07 L 03~ :05
--------r-- -----.--------.--------r-----
012
'' O 14
---0 --- ---'-- --- -[- Pattern 1

------~-- ---'< ----------iq- -- H --------
Pattern 2 `a3

0~13

06 | 02~ : 04


2


50 6


ag C

07
C 0


20


30 40
( MW/m2












MF(Lo)


MF(qmax)


1.2 E aT 1. -0 4975
-0 45
1. 1.1 CtI-0 498

1 -0 46
-0 4985
S0. -0 47 'z 0.
.-0 499
-0 48 0 5

5 -0 49 0 -e a 5 049

0 6 -0 5 0.6 g -0 5
0.2 0.4 6 0.11.2 0.2 0.4- (6 0.e 1 1.2

A B

Figure 7-10. Merit function (M~F2) COntours for A) qmax and B) Lc. Dark blue indicates regions
of lowest uncertainty. White points are data points. Black points are points chosen
using merit function analysis. Points circled by dashed lines indicate best trade-off
points. Circled black points are new points chosen for simulation.

Pp (MW/m2) ~Le (mm) C
50 100
*0 1 2

1 1 O45 1 1~


1 ~40 1 0 9

z 35 z 9


25 8


.~~~~0 _, ; I. J .. : 0 811 28


A 'B

Figure 7-11. Kriging surrogates based on initial 16 design points. A) qmax and B) Lc. White
points are data points. Black points are points chosen using merit function analysis.
Points circled by dashed lines indicate best trade-off points. Circled black points are
new points chosen for simulation.


210











qmax (MW/m2)


L (mm)


A4 B


1.1 1.1

1 1 ~s
09 "-0.46 ,,

O, -*Pji-O Q 8
-0.4



0 .6 -0. 0 00. 0 .81 1.
0.4 0. 0.8 1 1.2
C '

Figure~.4 7-12 Krgn urgtsadmrtfntincnor o 2dsg onsin Pattern1
Krgn ftfr )qaxadB)L n meit fucinvle o C mxadD
White pont ar aa ons Bakpinsaeponschsnuin eifnto
analsis Cice blc ons r ons hsnfriuain












qmax (MW/m2)


L (mm)


1~ -0' a494
12lr 1 -4
P II. rr 1 44-0 495

1 ** 0 4
-0 496
".~- -046
Z p -0 497
g, 1-0 47
0~ .1498


--0 49- 48 0. i -0 499

0 6 9 ~~-0 5 0.0.2 0. .4 0.5 0.6 0.8 -
0.2 0. a.4 0.5 0.6 PY0.8

C D

Figure 7-13. Kriging surrogates and merit function contours for 12 design points in Pattern 2.
Kriging fit for A) qmax and B) Lc. Merit function values for C) qmax and D) Lc. White
points are data points. Black points are points chosen using merit function analysis.
Circled black points are points chosen for simulation. Points in Pattern 1 region are
ignored.


212











qmax (MW/m2)


12


11




z
09


08


04 05 06 07 08 09 1 11 12 04 05 06 07 08 09 1 11 12

A B
Figure 7-14. Kriging fits for all 15 design points from Pattern 1. A) qmax and B) Lc.

qmax (MW/m2) Le (mm)


12


1 1



z 9a


32


30


28

26z


12 220


A B
Figure 7-15. Kriging fits for all 12 design points from Pattern 2. A) qmax and B) Lc.


213


L (mm)



O









105

*2
100~ 17


821
95 13*15 8
10 22
gSS *7~10
a, 5 12
E 90 *11 ~4
o 820 1
c 19.a 9
S 85-



80 --



15 20 25 30 35 40 45 50 55

qmax(MWim )

Figure 7-16. Pareto front based on original 16 data points (dotted line) and with newly added
points (solid line). Use of surrogate model plus merit functions as selection criteria
along with refinement of the design space resulted in all eleven new points (circled)
dominating all original sixteen points.


214











1.2 5 0

40, 96





0.2 04 0. 08 I 12 0 i: 0 08 JI 12






5 c12





on~




0.60 1.2.

A B r


Figure 7-18. Vapriton int flowo stemin h es and hydroen cowen tour ino desigrns space.






pek et lu.Diiio i ls hon o B cmusin ent. e215 er h












1.3
07i O
1.2 t------;--------

1.124

Pattern 2 .




0i13 '

06 02! 0
0.7-

017


0 14




O 8 1




' ..


attem 1



O 25


05




010

O 28

019

04

O 27


2


02 04 06 08


Figure 7-19. Location of best trade-off points in each pattern group in design space.


-100


-150 -100 -50 0
Z


50 100


Figure 7-20. Injector spacing for selected best trade-off design point for A) Pattern 1 (case 19
shown with 50 inj ector elements) and B) Pattern 2 (case 18 shown with 54 inj ector
elements).


216






































-150 -100 -50 0
Z


50 100


Figure 7-20. Continued.



45

40

35

30






15\


case 1
casel18
case 19


0.02 0.04 0.06 0.08
x (m)


0.1 0.12 0.14 0.16


Figure 7-21. Predicted heat flux profiles for baseline case (case 1), best trade-off from Pattern 1
(case 19), and best trade-off from Pattern 2 (case 18).


217














































































218


40


S30
E

Z. 20

10


0


2.75 x 105
6.03 x 105


0.05


0.1 0.15


x (m)


2.53 x 105
5.94 x 105


10


0


0.05 0.1 0.15
x (m)


Figure 7-22. Heat flux profiles for different grid resolutions. The coarser grid was used to
construct the surrogate model. A) Baseline (case 1). B) Case 19 of Pattern 1.


*2
100


95 0e8 11 1 &21
90 5;4 *1 1

**4 1 e 19 e 6


815

80


E1






E
m


50~ e12
*14 *16


e 17


e 22


E 40

~30

20


58

*7 e9


el 4


e 24 e 25 13 =
e 26 e 19 e 21


a 23


e 18 4


e 3
e 5


10
0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06
mdot H2 (kg/s)


75
0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06
mdot H2 (kg/s)


Figure 7-23. Peak heat flux and combustion length as a function of hydrogen mass flow rate.
Peak heat flux decreases and combustion length increases with an increase in mass
flow rate.










CHAPTER 8
CONCLUSIONS

CFD-based optimization can improve rocket engine component design by predicting the

effect of design changes on performance and life indicators. The development of a

comprehensive and efficient multi-obj ective optimization framework for CFD analysis is

necessary in this effort. Additionally, design space refinement techniques can help direct the

design optimization by guiding the process in meaningful directions. Design space refinement

techniques were used to reduce the cost of CFD-based optimization through the practice of

removing infeasible regions from the design space using low-fidelity analyses. Coupled with

tools including merit functions and Pareto fronts, design space refinement is a useful tool in

improving the accuracy and applicability of a surrogate model.

Design space refinement techniques were developed to account for optimization problems

with multiple obj ectives. This includes the use of Pareto fronts to identify favorable design

space regions in both objectives to allow for the removal of unnecessary regions. Additionally, a

merit function was selected based on its ability to provide a good accuracy surrogate with a

minimum number of points. A rank formulation was applied with data clustering to choose data

points that were simultaneously (a) in regions of high surrogate uncertainty, (b) have good values

in each function, and (c) are spread evenly across the design space. The design space

refinement techniques were used in conjunction with problems of interest to liquid rocket engine

design, looking at techniques to improve the mixing in flame-holding devices, improve turbine

efficiency, and improving performance and life characteristics of a rocket engine combustion

chamber. The results show that even simple analyses can provide significant insight into rocket

engine component flow dynamics that can supplement the existing experimental studies.


219









8.1 Radial Turbine Efficiency and Weight Optimization

The compact radial turbine preliminary optimization identified several factors that can

improve its efficiency. The low-cost surrogate model analysis revealed that reducing the radius

of the turbine while increasing blade size has a positive impact on efficiency without impacting

the turbine's weight. The most efficient designs have higher rotational speeds and isentropic

velocity ratios. A grid sensitivity analysis revealed that the rotational speed and the velocity

ratio accounted for 97% of the variability in the rotor weight, while the turbine efficiency was

sensitive to all variables included in the analysis.

The applicability of DSR techniques was successfully demonstrated in the preliminary

optimization of a radial turbine optimization.

1. In the case where the feasible design space was unknown, response surfaces proved
invaluable in determining design constraints.

2. After the feasible design space was determined, information from a Pareto front was used
to refine the design space to improve surrogate accuracy without compromising the
performance of either objective.

For the radial turbine optimization, the infeasible points were not thrown away. Instead,

they served a useful purpose by providing the location of the bounds of the feasible design space.

For a six-variable design space, surrogate models were necessary in extracting these bounds--a

task which would have been very difficult, otherwise. In addition, the study addressed the

problem of refining the design space in the presence of multiple obj ectives. To prevent missing

the region of interest in one obj ective or the other when the design space was refined, Pareto

fronts were used to simultaneously locate the regions of interest in both obj ectives.

Finally, the radial turbine case was analyzed again using merit functions to aid the

analysis. It was discovered that the use of merit functions could improve the efficiency of the

optimization process by improving point selection during design space refinement. Using merit


220









functions also means that fewer points are required during each stage in the analysis. Because

the radial turbine optimization was low cost, comparisons could be made of the accuracy

between the selection of a large number of points using standard DOE techniques of Latin

Hypercube Sampling and factorial designs and the selection of a small number of points using

merit functions. By using the merit function that maximized the probability of improvement of

the function values, the number of points in the final optimization cycle could be reduced by

94% while maintaining a Pareto front accuracy within 10%. This substantial reduction in the

number of points required meant that the use of merit functions would be a suitable method of

point selection in the optimization cycles of a computationally expensive problem, such as

inj ector flow.

8.2 Bluff Body Mixing Dynamics

A study of the sensitivity of flame-holder mixing to the bluff body geometry illustrated the

multiple roles of surrogate-based optimization.

1. In addition to facilitating the optimization, surrogates can be used to warn against an
inadequate CFD model. In this case, the large difference between an RSA and RBNN
approximation showed that a problem existed.

2. Complete improvement cannot always be gained using an alternative surrogate. A neural
network provided a better approximation in some regions, while an RSA performed better
in other regions.

3. It may be necessary to improve the accuracy of the CFD model or refine the design space
to obtain a better result. In this case, problems in the CFD model must be resolved before
the analysis can continue.

4. Exploration of the design space may be necessary to determine the deficiency in the CFD
model. Additional data points were added in suspect regions to reveal the true nature of
the CFD model response.

These conclusions are particularly important, as similar issues can arise in other CFD-based

design optimizations with complex flow situations. Without the use of surrogate models and










DSR, the problem of insufficient grid refinement in the CFD model might not have been

discovered.

Understanding the mixing dynamics and the effects of the geometry on mixing is

important in the design of combustion devices such as inj ector flow. In this case, the CFD-based

surrogate model was able to reveal important aspects on the sensitivity of the mixing to the bluff

body geometry. It was found that the mixing could be increased by changing the shape of the

trailing edge of the bluff body. This increased mixing often came at the expense of increasing

drag on the body, however, locations of extreme cases revealed designs that gave favorable

mixing characteristics and low drag.

8.3 Single-Element Injector Flow Modeling

The inj ector flow modeling was conducted as a preliminary exercise to the combustion

chamber flow optimization. Flow was analyzed within the inj ector itself to determine

combustion chamber inlet conditions. Model verification was provided by developing a

computational model of an experimental combustion chamber. Finally, a grid sensitivity

analysis was conducted. It was discovered that the combustion length was more sensitive to the

grid resolution than the wall heat flux. Grid independency is not reached in the present study.

The wall treatment of the turbulence model reduces the sensitivity of wall heat transfer

computations. A grid resolution was selected for the inj ector flow optimization that provided

good trade-off between heat flux profile accuracy and computational run time.

8.4 Multi-Element Injector Flow Modeling

The CFD model sought to approximate the effects near a single inj ector element in the

outer injector row of multi-element combustor flow. Geometric design variables were selected

to improve the performance and life of a rocket engine combustion chamber. The spacing


222









between inj ector elements and the distance to the wall were adjusted to determine the effect on

the peak heat flux profile and combustion length.

Determination of the feasible design space revealed interesting flow characteristics of the

initial data set. It was found that each point in the design space resulted in inj ector flow that

exhibited one of two characteristics:

1. The outer shear layer is directed towards the combustion chamber wall, resulting in a sharp
peak in heat flux near the inj ector face. This occurs due to a large distance to the wall as
compared to the inter-element distance. Although these designs, in general, had lower
mass flow rates per inj ector, overall, this scenario resulted in very high peak heat fluxes.

2. The outer shear layer is directed towards the neighboring element within the same row,
resulting in a low heat flux near the inj ector face that gradually increases to a peak
downstream. This occurs due to an injector-to-wall distance that is small compared to the
inter-element distance. These designs had higher mass flow rates per inj ector, but the heat
transfer was spread more evenly across the combustion chamber wall.

The behavior of the different flow patterns and trends dictated that separate analyses be

conducted based on the characteristic flow pattern.

The accuracy of the initial surrogate was low, indicating the need for a design space

refinement. In the radial turbine analysis, merit functions were shown to have the ability to

minimize the number of points needed to improve the surrogate model without significantly

compromising the future accuracy of the surrogate. For this reason the merit function that

maximizes the probability of improvement of the surrogate model was used to select additional

points for each region of the design space. The refinement process consisted of a change in the

design variable bounds as well as an addition of points based on the multi-obj ective merit

function-based point selection procedure. The DSR was applied to each of the two regions of the

design space based on the predicted flow characteristics. All of the new points selected using the

merit function-based DSR dominated all of the original design points. The multi-objective


223










surrogate-based DSR using merit functions for point selection was thus successful in providing

optimal design using a small number of design points.

8.5 Future Work

This study provides a starting point for CFD-based design of rocket engine components.

In particular, this study provides a new method of attacking complex rocket engine component

design. Future studies may include

1. Full 3-D simulations of the selected optimum radial turbine design and a comparison with
the 1-D Meanline code results to determine if the efficiency gains are comparable over the
baseline case.

2. A three-dimensional blade shape optimization of the selected optimum radial turbine
design.

3. Adding additional design points near the selected designs of the multi-element inj ector
analysis. Doing so would provide a more accurate representation of the design space near
the selected designs.

4. Improving the accuracy of the multi-element inj ector model by including the second row
of inj ector elements in the CFD model. In this case, the best designs as selected in the
current study can provide a starting point for the additional analysis. The effect of the
inj ector interactions can be quantified, and the applicability of the slip boundary and
symmetric computational domain can be determined.

5. Systematic experimental investigation of the effects of the wall distance and/or the inter-
element inj ector spacing on peak heat flux and combustion efficiency for a multi-element
injector. Experimental studies on the selected best, or similar, multi-element injector
designs and a comparison with the baseline case would provide experimental confirmation
of the results of the simple study.


224










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237









BIOGRAPHICAL SKETCH

Yolanda Mack was born in Willingboro, New Jersey, on January 2, 1980. Born to parents

who served in the U.S. Air Force, she has lived around the country and in Japan. In 2002, she

received her Bachelor of Science degree with honors in mechanical engineering from the

University of Florida. She also received her Master of Science degree in 2004 from the

University of Florida specializing in computational fluid dynamics(CFD)-based optimization. In

2004, she was awarded a fellowship by the South East Alliance for Graduate Education and the

Professoriate that seeks to increase diversity in the professoriate in the science, technology,

engineering and mathematics fields. She is also a 2006 recipient of the Amelia Earhart

Fellowship awarded by Zonta International to outstanding women Ph.D. students specializing in

aerospace-related sciences and engineering. She pursued her doctoral studies in CFD-based

optimization at the University of Florida in aerospace engineering under the guidance of Drs.

Wei Shyy and Raphael Haftka.

Yolanda currently lives in Gainesville and stays active by spending time with her son,

Trevor. After completing her doctoral studies, she will join Raytheon in Tucson, AZ as a Senior

Multidisciplined Engineer and plans on later pursuing an academic career.


238





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1 CFD-BASED SURROGATE MODELING OF LIQUID ROCKET ENGINE COMPONENTS VIA DESIGN SPACE REFINEMENT AND SENSITIVITY ASSESSMENT By YOLANDA MACK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Yolanda Mack

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3 To my son, Trevor.

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4 ACKNOWLEDGMENTS I would like to thank the peopl e in my life who have supported me during my graduate studies. In particular, I would like to thank my advisor, Dr. Wei Shyy, for his support and for pushing me to strive for excellence. I also thank Dr. Raphael Haftka for his guidance through the years on my research. I thank the member s of my supervisory committee members, Dr. Corin Segal, Dr. William Lear, Dr. Andreas Hase lbacher, and Dr. Don Slinn, for reviewing my work and offering suggestions during my studies. I thank Dr. Nestor Queipo for inviting me to explore new methods and new ideas that have been invaluable in my work. I thank Dr. Siddharth Thakur and Dr. Jeffrey Wright for their troubleshooting assistance over the years. I thank Mr. Kevin Tucker and the others that I ha ve collaborated with at NASA Marshall Space Flight Center along with the Institute for Future Space Transport under the Constellation University Institute Project for providing the motivations for my work as well as financial assistance. I would also like to thank Zonta International and the South East Alliance for Graduate Education and the Professoriate for their financial support and recognition. I thank Antoin Baker for his love and suppor t through any difficulty as we pursued our graduate degrees. I would like to thank my father, Calvin Mack, for his gentle guidance, and my mother, Jacqueline Mack, to whom I express my deepest gratitude for he r unconditional love and assistance over the years. I thank my sisters, Brandee and Cailah Mack, and other members of my family for their undying support and for forever believing in me.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .......10 ABSTRACT....................................................................................................................... ............15 CHAPTER 1 INTRODUCTION................................................................................................................. .17 1.1 Thrust Chamber Characteristics.......................................................................................18 1.1.1 Rocket Engine Cycles.............................................................................................19 1.1.2 Engine Reliability Issues........................................................................................20 1.2 Current and Past Design Practices....................................................................................21 1.2.1 Rocket Engine Injector Design...............................................................................22 1.2.2 Rocket Engine Turbine Design..............................................................................24 1.3 CFD-Based Optimization.................................................................................................25 1.3.1 Optimization Techniques for Comput ationally Expensive Simulations................25 1.3.2 Design Space Refinement in CFD-Ba sed Optimization Using Surrogate Models......................................................................................................................... .29 1.4 Summary.................................................................................................................... .......30 1.5 Goal and Scope............................................................................................................. ....31 2 OPTIMIZATION FRAMEWORK.........................................................................................35 2.1 Optimization Using Surrogate Models.............................................................................36 2.1.1 Design of Experiments...........................................................................................36 2.1.2 Surrogate Model Identif ication and Fitting............................................................38 2.1.2.1 Polynomial response surface approximation................................................39 2.1.2.2 Kriging.........................................................................................................41 2.1.2.3 Radial basis neural networks........................................................................44 2.1.3 Surrogate Model Accuracy Measures....................................................................47 2.1.3.1 Root mean square error................................................................................47 2.1.3.2 Coefficient of multiple determination..........................................................48 2.1.3.3 Prediction error sum of squares....................................................................49 2.2 Dimensionality Reduction Using Global Sensitivity Analysis.........................................49 2.3 Multi-Objective Optimization Using the Pareto Optimal Front.......................................52 2.4 Design Space Refinement Techniques.............................................................................54 2.4.1 Design Space Reduction for Surrogate Improvement............................................56 2.4.2 Smart Point Selection for Second Phase in Design Space Refinement..................59 2.4.3 Merit Functions for Data Selection and Reduction................................................61 2.4.4 Method of Alternative Loss Functions...................................................................65

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6 3 RADIAL TURBINE OPTIMIZATION.................................................................................75 3.1 Introduction............................................................................................................... ........76 3.2 Problem Description........................................................................................................ .79 3.2.1 Verification Study..................................................................................................79 3.2.2 Optimization Procedure..........................................................................................81 3.3 Results and Discussion..................................................................................................... 81 3.3.1 Phase 1: Initial Design of Experime nts and Construction of Constraint Surrogates....................................................................................................................8 1 3.3.2 Phase 2: Design Space Refinement........................................................................84 3.3.3 Phase 3: Construction of the Pareto Front and Validation of Response Surfaces....................................................................................................................... .86 3.3.4 Phase 4: Global Sensitivity Analysis and Dimensionality Reduction Check.........87 3.4 Merit Function Analysis...................................................................................................8 8 3.4.1 Data Point Selection and Analysis.........................................................................88 3.4.2 Merit Function Comparison Results.......................................................................89 3.5 Conclusion................................................................................................................. .......91 4 MODELING OF INJECTOR FLOWS................................................................................105 4.1 Literature Review.......................................................................................................... .106 4.1.1 Single-Element Injectors......................................................................................106 4.1.2 Multi-Element Injectors........................................................................................107 4.1.3 Combustion Chamber Effects and Considerations...............................................108 4.1.4 Review of Select CFD Mode ling and Validation Studies....................................112 4.2 Turbulent Combustion Model.........................................................................................117 4.2.1 Reacting Flow Equations......................................................................................117 4.2.2 Turbulent Flow Modeling.....................................................................................120 4.2.3 Chemical Kinetics................................................................................................124 4.2.4 Generation and Decay of Swirl............................................................................125 4.3 Simplified Analysis of GO2/GH2 Combusting Flow......................................................127 5 SURROGATE MODELING OF MIXING DYNAMICS....................................................137 5.1 Introduction............................................................................................................... ......137 5.2 Bluff Body Fl ow Analysis..............................................................................................138 5.2.1 Geometric Description and Computational Domain............................................139 5.2.2 Objective Functions and Design of Experiments.................................................140 5.3 Results and Discussion...................................................................................................14 2 5.3.1 CFD Solution Analysis.........................................................................................142 5.3.2 Surrogate Model Results......................................................................................143 5.3.3 Analysis of Extreme Designs...............................................................................145 5.3.4 Design Space Exploration....................................................................................147 5.4 Conclusions................................................................................................................ .....148 6 INJECTOR FLOW MODELING.........................................................................................157

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7 6.1 Introduction............................................................................................................... ......157 6.2 Experimental Setup......................................................................................................... 158 6.3 Upstream Injector Flow Analysis...................................................................................159 6.3.1 Problem Description.............................................................................................159 6.3.2 Results and Discussion.........................................................................................160 6.3.3 Conclusion............................................................................................................162 6.4 Experimental Results and Analysis................................................................................163 6.5 Injector Flow Modeling Investigation............................................................................166 6.5.1 CFD Model Setup.................................................................................................167 6.5.2 CFD Results and Experimental Comparison of Heat Flux..................................168 6.5.3 Heat Transfer Characterization.............................................................................169 6.5.4 Species Concentrations.........................................................................................170 6.6 Grid Sensitiv ity Study..................................................................................................... 171 6.7 Conclusion................................................................................................................. .....173 7 MULTI-ELEMENT INJECTOR FLOW MODELING AND ELEMENT SPACING EFFECTS........................................................................................................................ ......191 7.1 Introduction............................................................................................................... ......191 7.2 Problem Set-Up............................................................................................................. .193 7.3 Feasible Design Space Study..........................................................................................195 7.4 Design Space Refinement...............................................................................................199 7.5 Conclusion................................................................................................................. .....203 8 CONCLUSIONS.................................................................................................................. 219 8.1 Radial Turbine Efficiency and Weight Optimization.....................................................220 8.2 Bluff Body Mixing Dynamics........................................................................................221 8.3 Single-Element Injector Flow Modeling........................................................................222 8.4 Multi-Element Injector Flow Modeling..........................................................................222 8.5 Future Work................................................................................................................ ....224 REFERENCE LIST................................................................................................................. ....225 BIOGRAPHICAL SKETCH.......................................................................................................238

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8 LIST OF TABLES Table page 1-1 Summary of CFD-based desi gn optimization applications with highlighted surrogatebased optimization techniques...........................................................................................32 2-1 Summary of DOE and surr ogate modeling references......................................................67 2-2 Design space refinement (DSR) techniques with their applications and key results.........68 3-1 Variable names and descriptions.......................................................................................92 3-2 Response surface fit statistics before (f easible DS) and after (reasonable DS) design space reduction................................................................................................................ ...93 3-3 Original and final design variable ranges after cons traint application and design space reduction................................................................................................................ ...93 3-4 Baseline and optimum design comparison........................................................................93 4-1 Selected injector experimental studies.............................................................................128 4-2 Selected CFD and numerical stud ies for shear coaxial injectors.....................................129 4-3 Reduced reaction mechanisms for hydrogen-oxygen combustion..................................129 5-1 Number of grid points used in various grid resolutions...................................................149 5-2 Data statistics in the grid comparison of the CFD data...................................................149 5-3 Comparison of cubic response surface coe fficients and response surface statistics........150 5-4 Comparison of radial-basis neural network parameters and statistics.............................150 5-5 RMS error comparison for response surf ace and radial basis neural network................150 5-6 Total pressure loss coefficient and mixi ng index for extreme and two regular designs for multiple grids............................................................................................................. .151 5-7 Total pressure loss coefficient and mi xing index for designs in the immediate vicinity of Case 1 using Grid 3........................................................................................151 6-1 Flow regime description..................................................................................................17 4 6-2 Effect of grid resolution on wa ll heat flux and combustion length..................................175 6-3 Flow conditions............................................................................................................ ....175

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9 7-1 Flow conditions and baseline combustor geometry for parametric evaluation...............204 7-2 Kriging PRESSrms error statistics for each design space iteration...................................205

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10 LIST OF FIGURES Figure page 1-1 Rocket engine cycles....................................................................................................... ...33 1-2 SSME thrust chamber component diagram.......................................................................33 1-3 SSME component reliability data......................................................................................34 1-4 Surrogate-based optimizati on using multi-fidelity data.....................................................34 2-1 Optimization framework flowchart....................................................................................69 2-2 DOEs for noise-reducing surrogate models.......................................................................70 2-3 Latin Hypercube Sampling................................................................................................70 2-4 Design space windowing...................................................................................................71 2-5 Smart point selection...................................................................................................... ....71 2-6 Depiction of the merit function rank assignment for a given cluster.................................72 2-7 The effect of varying values of p on the loss function shape.............................................72 2-8 Variation in SSE with p for two different responses.........................................................73 2-9 Pareto fronts for RSAs constr ucted with varying values of p ............................................73 2-10 Absolute percent difference in th e area under the Pareto front curves..............................74 3-1 Mid-height static pressure (psi) contours at 122,000 rpm.................................................94 3-2 Predicted Meanline and CFD to tal-to-static efficiencies...................................................94 3-3 Predicted Meanline and CFD turbine work.......................................................................95 3-4 Feasible region and location of three constraints...............................................................95 3-5 Constraint surface for Cx2/Utip = 0.2................................................................................96 3-6 Constraint surfaces for 1 = 0 and 1 = 40.........................................................................96 3-7 Region of interest in function space...................................................................................97 3-8 Error between RSA and actual data point..........................................................................97 3-9 Pareto fronts for p = 1 through 5 for second data set.........................................................98

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11 3-10 Pareto fronts for p = 1 through 5 for third data set............................................................99 3-11 Pareto Front with validation data.......................................................................................99 3-12 Variation in design vari ables along Pareto Front.............................................................100 3-13 Global sensitivity analysis............................................................................................... 100 3-14 Data points predicted by validated Pareto front compared with the predicted values using six Kriging models based on 20 selected data points.............................................101 3-15 Absolute error distribution for points along Pareto front.................................................101 3-16 Average mean error distribution over 100 clusters..........................................................102 3-17 Median mean error over 100 clusters...............................................................................103 3-18 Average maximum error di stribution over 100 clusters..................................................103 3-19 Median maximum error over 100 clusters.......................................................................104 4-1 Coaxial injector and com bustion chamber flow zones....................................................130 4-2 Flame from gaseous hydrogen – gase ous oxygen single element shear coaxial injector....................................................................................................................... ......130 4-3 Multi-element injectors.................................................................................................... 131 4-4 Wall burnout in an uncooled combustion chamber.........................................................132 4-5 Test case RCM-1 injector................................................................................................132 4-6 Temperature contours for a single element injector........................................................133 4-7 CFD heat flux results as compared to RCM-1 experimental test case............................134 4-8 Multi-element injector simulations..................................................................................135 4-9 Fuel rich hydrogen and oxygen reaction with heat release..............................................136 5-1 Modified FCCD.............................................................................................................. .151 5-2 Bluff body geometry........................................................................................................ 151 5-3 Computational domain fo r trapezoidal bluff body...........................................................152 5-4 Computational grid for trapezoidal bluff body................................................................152 5-5 Bluff body streamlines and vorticity contours.................................................................152

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12 5-7 Comparison of response surface (top row) and radial basis neural network (bottom row) prediction contours for to tal pressure loss coefficient.............................................154 5-8 Comparison of response surface (top row) and radial basis neural network (bottom row) prediction contours for mixi ng index (including extreme cases)............................154 5-9 Comparison of response surface (top row) and radial basis neural network (bottom row) prediction contours for mixi ng index (excluding extreme cases)...........................155 5-10 Variation in objective vari ables with grid refinement.....................................................155 5-11 Difference in predicted mixing index va lues from response surface (top row) and radial basis neural network (bottom row) prediction cont ours constructed with and without extreme cases......................................................................................................156 5-12 Comparison of response surface and radial basis neural network prediction contours for mixing index at B = 0 and h = 0.............................................................................156 6-1 Blanching and cracking of combustion chamber wall due to local heating near injector elements.............................................................................................................. 175 6-2 Hydrogen flow geometry.................................................................................................176 6-3 Hydrogen inlet mesh........................................................................................................ 176 6-4 Z -vorticity contours..........................................................................................................17 7 6-5 Swirl number at each x location.......................................................................................177 6-6 Average axial velocity u and average tangential velocity v with increasing x ...............178 6-7 Reynolds number profiles................................................................................................178 6-8 Non-dimensional pressure as a function of x ...................................................................178 6-9 Hydrogen inlet flow profiles............................................................................................179 6-10 Combustion chamber cross-sectional geometry and thermocouple locations.................179 6-11 Estimated wall heat flux using linear steady-state and unsteady approximations...........180 6-12 1-D axisymmetric assumption for heat conduction through combustion chamber wall.180 6-13 Estimated wall temperatures using li near and axisymmetric approximations.................181 6-14 Temperature (K) contours for 2-D unsteady heat conduction calculations.....................181 6-15 Experimental heat flux valu es using unsteady assumptions............................................182

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13 6-16 Computational model for single-el ement injector flow simulation.................................182 6-17 Velocity contours vx(m/s) and streamlines.......................................................................183 6-18 Temperature (K) contours................................................................................................18 3 6-19 CFD heat flux values as compared to experimental heat flux approximations...............183 6-20 Wall heat transfer and e ddy conductivity contour plots..................................................184 6-21 Streamlines and temperature contours at plane z = 0.......................................................184 6-22 Heat flux and y+ profiles along combustion chamber wall.............................................185 6-23 Temperature and eddy conductivity profiles at various y locations on plane z = 0.........185 6-24 Mass fraction contours for select species.........................................................................186 6-25 Mole fractions for all species along combustion chamber centerline ( y = 0, z = 0)........187 6-26 Select species mo le fraction profiles................................................................................188 6-27 Sample grid a nd boundary conditions..............................................................................189 6-28 Computational grid along symmetric boundary..............................................................189 6-29 Wall heat flux and y+ values for select grids..................................................................189 6-30 Comparison of temperature (K) co ntours for grids with 23,907, 31,184, 72,239, and 103,628 points, top to bottom, respectively.....................................................................190 7-1 Injector element subsection..............................................................................................20 5 7-2 Design points selected for design space sensitivity study...............................................206 7-3 Effect of hydrogen mass flow rate on objectives.............................................................206 7-4 Oxygen iso-surfaces and hydrogen contours...................................................................207 7-5 Hydrogen contours and streamlines.................................................................................207 7-6 Maximum heat flux for a changing radial distance r *.....................................................208 7-7 Heat flux distribution..................................................................................................... ..208 7-8 Maximum heat flux as a function of aspect ratio.............................................................209 7-9 Design points in function and design space.....................................................................209 7-10 Merit function ( MF2) contours for...................................................................................210

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14 7-11 Kriging surrogates based on initial 16 design points.......................................................210 7-12 Kriging surrogates and merit function c ontours for 12 design points in Pattern 1..........211 7-13 Kriging surrogates and merit function c ontours for 12 design points in Pattern 2..........212 7-14 Kriging fits for all 15 design points from Pattern 1.........................................................213 7-15 Kriging fits for all 12 design points from Pattern 2.........................................................213 7-16 Pareto front based on original 16 data points (dotted line) an d with newly added points (solid line)............................................................................................................ .214 7-17 Approximate division in the design space between the two pattern s based on A) peak heat flux...................................................................................................................... .....215 7-18 Variation in flow streamlines a nd hydrogen contours in design space............................215 7-19 Location of best trade-off points in each pattern group in design space.........................216 7-20 Injector spacing for select ed best trade-off design point.................................................216 7-21 Predicted heat flux profiles for baseline case (case 1), best trade-off from Pattern 1 (case 19), and best trade-o ff from Pattern 2 (case 18).....................................................217 7-22 Heat flux profiles for different grid re solutions. The coarser grid was used to construct the surrogate model..........................................................................................218 7-23 Peak heat flux and combustion length as a function of hydrogen mass flow rate...........218

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15 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CFD-BASED SURROGATE MODELING OF LIQUID ROCKET ENGINE COMPONENTS VIA DESIGN SPACE REFINEMENT AND SENSITIVITY ASSESSMENT By Yolanda Mack August 2007 Chair: Wei Shyy Cochair: Raphael Haftka Major: Aerospace Engineering Computational fluid dynamics (CFD) can be us ed to improve the design and optimization of rocket engine components that traditi onally rely on empirical calculations and limited experimentation. CFD based-design optimizatio n can be made computationally affordable through the use of surrogate modeling which can th en facilitate additional parameter sensitivity assessments. The present study investigates su rrogate-based adaptive design space refinement (DSR) using estimates of surrogate uncertainty to probe the CFD analyses and to perform sensitivity assessments for complex fluid phys ics associated with liquid rocket engine components. Three studies were conducted. First, a surrogate-based pr eliminary design optimization was conducted to improve the efficiency of a comp act radial turbine for an expander cycle rocket engine while maintaining low weight. Design space refinement was used to identify function constraints and to obtain a high accuracy surrogate model in the region of interest. A merit function formulation for multi-objective design point selection reduced the number of design points by an order of magnitude while mainta ining good surrogate accuracy among the best trade-off points. Second, bluff body-induced flow was investigated to identify the physics and

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16 surrogate modeling issues relate d to the flow’s mixing dynamics. Multiple surrogates and DSR were instrumental in identifying designs for which the CFD model was deficient and to help to pinpoint the nature of the defi ciency. Next, a three-dimensional computational model was developed to explore the wa ll heat transfer of a GO2/GH2 shear coaxial single element injector. The interactions between turbulen t recirculating flow structures chemical kinetics, and heat transfer are highlighted. Finally, a simplified computati onal model of multi-element in jector flows was constructed to explore the sensitivity of wa ll heating and improve combustion e fficiency to injector element spacing. Design space refinement using surrogate models and a multi-objective merit function formulation facilitated an efficient framework to investigate the multiple and competing objectives. The analysis suggests that by adjus ting the multi-element injector spacing, the flow structures can be modified, resu lting in a better balance betw een wall heat flux and combustion length.

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17 CHAPTER 1 INTRODUCTION The Space Shuttle Main Engine (SSME) was de veloped as a reusable launch engine. However, even with noticeable success, the co mplexity of the SSME drove up the maintenance costs of the engine due to the harsh operating environment among the engine components. These additional maintenance costs were unforeseen in the original design and analysis efforts.1 For this reason, NASA is striving to design a simpler, more reliable engine.2,3 Today, there are more computational, diagnostic and experimental tool s available to assist in streamlining the design process than when the SSME was designed. For example, flow and combustion processes can now be modeled using a dvanced computational fl uid dynamics (CFD) tools.4,5 Using CFD tools can help narrow down the design of an engine co mponent before a prototype is built and tested. Often, designs are based on past experience, in tuition, and empirical calc ulations. Without a comprehensive and methodical process, the desi gns may not be satisfact orily refined without tedious and inefficient trial-anderror exercises. Optimization t ools can be used to guide the selection of design parameters a nd highlight the predicted best de signs and trade-off designs. In this study, CFD and optimization tools are used to develop further capabilities in the design and analysis of a radial turbine and an injector : critical liquid rocket engine components. Current design practices for inje ctor and turbine designs incl ude past experience, intuition, and empirical calculations. It is difficult to fully analyze, design, and optimize the combustion chamber using experimental information alone Because of measur ement difficulties in combustion chambers due to high temperatures and pressures, much is still too poorly understood of the flow dynamics to accurately pred ict the response of futu re injector designs. Using CFD can be used as a tool to help an alyze and understand the flow dynamics within the combustion chamber to fill in knowledge gaps left by the injector experiments. This research

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18 effort will use CFD as an analysis and design tool for rocket engine components to improve component design and provide information on the system processes that may be difficult to obtain experimentally. CFD valid ation efforts are ongoing to fine tune CFD codes and improve their predictive ability. To use CFD effectivel y as a design tool for ro cket engine component design, an efficient design methodology must be used. A CFD-based optimization methodology can be used to analyze, design, and optimi ze combustion chambers by using the increased knowledge provided by CFD simulation to pred ict better designs. Through successive refinement of the design process, or design sp ace refinement (DSR), the design optimization methodology is streamlined and improved to make it suitable for complex CFD-based design. In summary, this research effort seeks to improve the design efficiency of complex rocket engine components using improved CFD-based optimiza tion and design space refinement techniques. This chapter presents a general introduction to the issues that motivate the work, including background information of rocket engine cycles a nd individual component reliability. Past and current design practices for sele ct rocket engine components ar e described and their limitations are noted. Next, CFD-based optimization and DS R are introduced as methods for improving upon traditional experimentally-bas ed analysis and design. Fina lly, a roadmap of the various chapters is presented. 1.1 Thrust Chamber Characteristics A rocket engine thrust chamber is comprised of a main injector with hundreds of injector elements, a combustion chamber, and a nozzle. Supporting elements include turbopumps and possibly a preburner or gas generator. In addi tion to the fuel tanks, pi ping, and valve system, a rocket engine is complex, and all components must be designed fo r maximum efficiency.

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19 1.1.1 Rocket Engine Cycles Rocket engines carry both the fuel and the propellant with the vehicle. For chemical rockets, the propellant is a large part of the total mass of the vehi cle. The basic goal in rocket engine design is to obtain the highest thrust possible with the lowest total vehicle weight.6 As a general objective, it is desirable to have a specific impulse that is as larg e as possible. Hydrogen and oxygen are often chosen as the fuel and propellant because they have very high values of specific impulse. While the goal of maximizing the specific impulse is straightforward, many rocket engines designed to help achieve the goa l are inherently complex and expensive. The need for better performance drives the complexity. Pressure-fed engines are relatively simple;7 two high pressure tanks are connected directly to the combustion chamber, and valv es are used to regulate the fl ow rate. Pressure-fed engines have low cost8 and good reliability9 because they are simple with few components. However, pressure-fed engines require heavy, massive tanks to provide the required pressure. For this reason, turbopumps were used in other rocket en gine cycles to deliver the needed pressure, allowing for lighter fuel and propellant tanks. Ad ditionally, gas generators or preburners were required to supply hot gases to the turbines. This led to a need for additional components, where each component could compromise engine reliability. In a gas-generator cycle (Figur e 1-1B) a portion of the fuel is burned before it reaches the combustion chamber. The exhaust from the gas ge nerator is used to drive a turbine, which is used to drive the pumps. After the e xhaust leaves the turbine, it is dumped.10 Gas generator engine cycles provide improved th rust and specific impulse as compared to a pressure-fed engine cycle.11. The expander cycle (Figure 1-1A) eliminates the need for a gas generator, thus resulting in a simpler and more reliable syst em. The expander cycle is the simplest of the pump-fed engine

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20 cycles,12 and it offers a high specific impulse.13 Current expander cycle rocket engines in use are Pratt and Whitney’s RL-60 and the Ariane 5 ESC-B. In the expander cycle, the unburned fuel is preheated by being passed through tubes used to cool the combustion chamber. This heated gas is passed through the turbine used to drive the fuel and oxidizer pumps The fuel is then sent to the combustion chamber where it is combusted. Us ing preheated fuel instead of hot combustion exhaust leads to a low turbine failure rate.14 One disadvantage to the expander cycle is that the engine is limited by the amount of power that the turbine can deliver. In this case, power output can be increased by improving the turbine drive, or increasing the amount of heat transferred to the unburned fuel used to drive the turbine.15 In the Space Shuttle main engine (SSME) cycle shown in Figure 1-2, a portion of the fuel and oxygen is used to power the turbopumps. The SSME has two separate gas generators; the exhaust of one powers the turbopump that deliver s fuel, while the other powers the turbopump that delivers oxygen. Unlike the gas generator cy cle, the fuel-rich exhaust is not dumped. Instead, the remaining fuel and oxygen also en ter the combustion chamber and are combusted. For this reason, the gas generators in the SSME are referred to as preburners. A simple preburner cycle is shown in Figure 1-1C. Using preburners maximized the specific impulse of the engine.16 Thus, the SSME has very high performance, but is also highly complex. For this reason, it is very expensiv e to operate and maintain. 1.1.2 Engine Reliability Issues One of NASA’s goals in reusable launch vehi cles is to improve engine reliability. After each shuttle flight, the engines we re removed from the Space Shuttle and inspected. Parts of the engine were replaced as nece ssary. Improving engine reliabilit y and engine component life can help drive down the costs of reus able launch vehicles. Jue and Kuck1 examined the reliability of the various Space Shuttle Main Engine (SSME) components by determining the probability for

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21 catastrophic failure of various components as shown in Figure 1-3. The components with the lowest reliabilities are the highpressure fuel turbopump (HPFT P), the high-pressure oxidizer turbopump (HPOTP), and the large throat main combustion chamber (LTMCC). The turbopumps and the combustion chamber are com ponents common to all rocket engine cycles (with the exception of a pressure fed cycle in which turbopum ps are absent). Improving these components can therefore cause large improvement s in overall engine re liability. Using an expander cycle, for example, can improve th e turbopump reliability by providing a milder turbine environment, thus improving the life of the turbine blades. Improving the injector configuration can increase the reliability of the main combustion chamber (MCC) by helping to alleviate local hot spots within the co mbustion chamber and prevent wall burnout. To help address the issue of engine reliability, NASA is gath ering data for the combustion chamber design of the next generation of reus able launch vehicles. Specifically, NASA is overseeing tests on several gas-gas H2/O 2 rocket injectors by Marshall et al.17 Particularly, the injector tests are being run to supply data for the purpose of CFD validation. In this way, direct comparisons can be made between the experiment al and computational da ta. Also, by steering the experiments, NASA can ensure that any specifically needed da ta is extracted in sufficient amounts. Efforts are also in place to improve th e life and efficiency of additional thrust chamber components, such as the turbin e which drives the fuel pump. 1.2 Current and Past Design Practices Injectors of the past and even of today are designed primarily thr ough the use of empirical models.18 Injectors are also frequently designed based on the results of simple small-scale tests, which are then scaled to a full-scale model.19 Many injectors are also designed based on cold (non-reacting) flow tests.20 These cold flow tests are us ed primarily to test the mixing capabilities of an injector desi gn. Then correlations were de veloped to apply them to hot

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22 (reacting) flow tests.21 Various design parameters are test ed experimentally by making the thrust chamber apparatus of modular design. In this way, the injector can be analyzed by switching out injectors, changing combustion ch amber length, and changing nozzl es in addition to varying the chamber pressure, mass flow rates, etc.17,22 The prospects of CFD-based design have been explored only recently. This is due to mode ling difficulties as a result of the presence of turbulence in combination with reacting flow and possibly including multiphase and supersonic flow. CFD validation efforts are still ongoing. The design of turbines for rocket engine appl ications is more advanced than injector design. This is because turbines are designed straightforwardly for its aerodynamic and stress properties; the turbines for rocket engines have no added complications due to reacting and/or multiphase flow. Turbines are now routinely designed with the aid of CFD, allowing for sophisticated flow prediction and designs. 1.2.1 Rocket Engine Injector Design Osborn et al.23 reported on early water cooled co mbustion chambers that experienced significant wall erosion near th e injector. This erosion wa s common, and trial and error experimentation often resulted in the quick de struction of much of the thrust chamber. Peterson et al.24 constructed an injector with 137 el ements based on tests from a smaller injector with only 37 elements. It was noted th at due to improved coolin g in the larger scale injector, the number of injector elements, and hence, injector spacing, could be increased. This in turn meant that the mass flow rate per elemen t was increased, requiring an increase in the size of the injector orifices to regul ate pressure. Separate thrust ch ambers were built to distinguish differences in performance for cool ed and un-cooled thrust chambers. The early optimization efforts of an injector element may have consisted of a dimensional analysis of geometric design variables and flow properties, as in the Apollo Service Propulsion

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23 System Injector Orifice Study Program.25 In this study, a dimensional analysis along with Buckingham’s pi theory reduced the number of design variables from 14 to 8, which still made testing all possible combinations an intractable problem using e xperimental techniques alone. Thus, an experimental design analyses was applied to reduce the number of tests. Limited by the use of experiments only, six criti cal design variables were chosen and were tested at two levels per variable. After 32 cases had been built and tested, the sensitivity of the response to the design variables required a reevaluation of the optim ization process. However, the results of the analysis eventually yielded information about th e relationship between the flow behavior and the chosen geometry parameters. Curve fits were then developed based on the optimization results for later use. Gill26 described the movement from injector de signs based on histor ical data and the injector designer’s experience and intuition to co mputer-aided designs. These early analyses included one-dimensional approximations such as equilibrium reactions. These equilibrium approximations were based on the assumption of “fas t-chemistry,” and were used to characterize the combustion field to aid the design process. Gill’s research looked at the effects of the injector geometry and injection pattern on the co mbustion process. Performance of the injector had to be maintained while increasing the life of the injector by preventing burning through or erosion of the combustor components. Most experimental techniques were insufficient to predict for many conditions that could reduce injector or combustion chamber life. For this reason, researchers now look to employ CFD techniques to aid in injector design. For example, Liang et al.27 developed semiempirical calculations to aid in the computer modeling of liquid rocket combustion chambers. Sindir and Lynch28 simulated a single element GO2/GH2 injector using an 8-species, 18 reaction chemical

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24 kinetics model with the kturbulence model and compared the velocity profile to experimental measurements with good results. Tucker et al.5 stressed the importance of supplying experimental results to aid in the task of CFD validation. Shyy et al.29 and Vaidyanathan et al.30 identified several issues inherent in CFD modeling of injectors, including the n eed for rigorous validation of CF D models and the difficulty of simulating multi-element injector flows due to lengthy computational times. They chose to simulate single injector flow, as this allowed fo r the analysis of key co mbustion chamber life and performance indicators. The lif e indicators were the maximum temperatures on the oxidizer post tip, injector face, and combustion chamber wall, and the performance indicator was the length of the combustion zone. These conditions were expl ored by varying the impinging angle of the fuel into the oxidizer. The studies were successful in using CFD to specifica lly access the effects of small changes in injector geometry on combustion performance. 1.2.2 Rocket Engine Turbine Design Turbines are developed based on basic knowledge of the fuel characteristics, estimated stresses, and turbine pressure. The turbines are driven using hot exhaust gas from the combustion process. Turbines were and sti ll are designed by using em pirical relations to calculate a preliminary starting point for the desi gn. The designs rely both on past information from similar turbine designs and on experiment al correlations. Unlike for injector design, computers have been used in the design process fo r rocket engine turbines for over 40 years. Beer 1965 used the aid of computers to estimate the velocity distributi on across a turbine blade to help improve turbine design. Simple 1-D co des are routinely used to provide preliminary information on design temperatures, stresses, overa ll flow geometry, velocities, and pressures.31 The prediction of turbine perf ormance was later improved by th e consideration of aerodynamic effects that lead to pressure losses.32

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25 Current design methodologies now combine past design practices with CFD analyses. Experimental correlations and simple computer programs are used to determine the preliminary design. CFD can then be used to fine-tune or analyze the design.33 In particular, the CFD analyses are used to shape the blade such that problems such as large pressure losses or unsteadiness are minimized.34 1.3 CFD-Based Optimization The designs of many aerospace components are m oving from the experimental realm to the computational realm. Some products can now be completely designed on computers before they are built and tested, including aircraft such as the Boeing 777-300,35 Bombardier Learjet 45,36 and Falcon 7X.37 Using CFD in coordination with optim ization techniques provides a means of effectively handling complex design problem s such as rocket engine components. 1.3.1 Optimization Techniques for Com putationally Expensive Simulations Scientific optimization techni ques have evolved substantiall y and can now be effectively used in a practical environment. When appropriately combined with computer simulations, the entire design process can be streamlined, reduci ng design cost and effort. Popular optimization methods include gradient-based methods,38-43 adjoint methods,44,45 and surrogate model-based optimization methods such as res ponse surface approximations (RSA),46 Kriging models,47-49 or neural network models.50 Gradient-based methods rely on a step by step search for an optimum design using the method of steepest descent on th e objective function accord ing to a convergence criterion. Adjoint methods require formulations th at must be integrated into the computational simulation of the physical laws. In both gradie nt-based and adjoint methods, the optimization is serially linked with the computer simulation. At times, gradient-based optimization is not feasible. For example, Burgee et al.51 found that an extremely noisy objective function did no t allow the use of grad ient-based optimization,

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26 as the optimizer would often become stuck at noise-induced local minima. Also, for simulating designs with long run times, it may be more time effi cient to run simulations in parallel, as this will allow many simulations to run simultaneously. For these problems, optimization based on an inexpensive surrogate is a good choice. Surrogate-based optimiz ation (SBO) uses a simplified, low-order characterization of the design space instead of relying exclusively on expensive, individually conducted co mputational simulations or expe rimental testing. Surrogatebased optimization allows for the determination of an optimum design while exploring multiple design possibilities, thus providing insight into the workings of the design. A surrogate model can be used to help revise the problem defin ition of a design task by providing information on existing data trends. Furthermore, it can conveni ently handle the existence of multiple desirable design points and offer quantitative assessment of trade-offs, as well as facilitate global sensitivity evaluations of the desi gn variables to assess the effect of a change in a design variable on the system response.52,53 The SBO approach has been shown to be an effective approach for the design of computationally expensive m odels such as those found in aerodynamics,54 structures,55 and propulsion.56 The choice of surrogates may depend on the problem. For example, Shyy et al.57 compared quadratic and cubic polynomi al approximations to a radial basis neural network approximation in the multi-objective optimization of a rocket engine injector. Simpson et al.58 suggested using Kriging as an altern ative to traditiona l quadratic response surface methodology in the optimization of an aerospike nozzle. Dornberger et al.59 conducted a multidisciplinary optimization on turbomachinery designs using response surfaces and neural networks. To construct the surrogate model, a sufficient nu mber of different designs must be tested to capture how a system response varies with differe nt design parameters. Th e process of selecting

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27 different designs for testing is termed Design of Experiments (DOE). Designs may be tested mainly at the design parameter extremes as in central-composite designs (CCD) and facecentered cubic designs (FCCD), or they may be tested across the full range of design parameter values as in multi-level factorial designs46 or Latin hypercube sampling.60 The design space is defined by the physical range of parameters, or va riables, to be explored Variable ranges are often determined based on experience with an existing design, but in many cases, the choice of variable ranges is an educated guess. The simp lest choice is to set simple range limits on the design variables before proceeding with the opti mization, but this is not always possible. Simpson et al.61 used a hypercube design space with desi gn variables range selection based on geometry to prevent infeasible designs in the SBO of an aerospike nozzle. Rodriguez 62 successfully optimized a complex jet configura tion using CFD with simple geometry-based bounds on the design variables. Vaidyanathan et al.63 chose variable ranges based on variations of an existing design in the CFD-based optimizat ion of a single elemen t rocket injector. Commonly, the design space cannot be represented by a simple box-like domain. Constraints or regions of infeasibility may dict ate an irregular design. Designs that work best for irregular design spaces include Latin Hypercube Sampling60 and orthogonal arrays.46 Once the design space is bounded, the optimization commences. In or der to reduce computational expense, it is very important for a surrogate to be an accura te replacement for an expensive CFD simulation. The application of the SBO approach in th e context of CFD-based optimization with complex flow models can present a significant cha llenge, in that the data required to construct the surrogates is severely limited due to time and computational constraints of the high fidelity CFD runs. In some cases, the problem of high computational cost in SBO can be addressed by performing an optimization on a low fidelity model and translating the result to a high fidelity

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28 model. Doing so requires additional analyses to ensure that the low-fidelity model is a qualitatively similar to the high fidelity model. In other instances, low-fidelity data may be combined with high-fidelity data to reduce the overall number of expensive high-fidelity runs. The low-fidelity data may employ a correction surrogate for improved accuracy, as shown in Figure 1-4. Han et al.64 performed a surrogate-based optimization on a low-fidelity 2-D CFD model and tested the results on a more expensive 3-D model in the optimization of a multi-blade fan and scroll system and verified the 2-D model us ing physical experiments as well as comparisons to previous studies and the 3-D model. Keane65 built an accurate surrogate model of a transonic wing using high-fidelity CFD data supplemented with data from a low fidelity empirical model. Alexandrov et al.66 used variation in mesh refinements as variable fidelity models for a wing design. Knill et al.67 saved 255 CPU hours by supplementing high-fidelity data with a lowfidelity aerodynamic model in the optimization of a high speed civil transport wing. Vitali et al.68 used surrogates to correct a low-fidelity model to better approximate a high fidelity model in the prediction of crack propagation. Thus, the accuracy of the low-fi delity models can be improved by combining the low-fidelity mode l with the surrogate correction, and a new surrogate could be constructed based on th e composite model. Venkataraman et al.69 also used a surrogate correction on a low-fideli ty model to aid in the optimization of shell structures for buckling. Balabanov et al.70 applied a surrogate co rrection to a low-fide lity model of a high speed civil transport wing. Haftka71 and Chang et al.72 scaled surrogate models with local sensitivity information to improve model accuracy for analytical and structural optimization problems.

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29 With continuing progress in computational si mulations, computational-based optimization has proven to be a useful tool in reducing the design process duration and expense. However, because of the complexity of many design problems such as is involved with liquid rocket engine components, CFD-based optimization can be a daun ting task. In some cases, a surrogate model is fit to the points and the predictive ca pability for the unknown cases is found to be unacceptably low. One solution is to try an alte rnative surrogate model—perhaps one with more flexibility—and approximate the response again. Another remedy is to refine the design space based on the imperfect but useful insight gained in the process. By reducing the domain size, design space refinement often naturally leads to the improvement of a surrogate model. 1.3.2 Design Space Refinement in CFD-Based Optimization Using Surrogate Models The design space is the region in which objec tive functions are sampled in search of optimal designs, where the region is bounded by defi ned constraints. Design space refinement (DSR) is the process of narrowing down the sear ch by excluding regions as potential sites for optimal designs because (a) they obviously violat e the constraints, or (b) the objective function values in the region are poor. In the optimiza tion of a high speed civil transport wing, Balabanov et al.70 discovered that 83% of the points in th eir original design space violated geometric constraints, while many of the remaining point s were simply unreasonable. By reducing the design space, they eliminated their unreasonable designs and improved the accuracy of the surrogate model. Roux et al.73 found that the accuracy of a polyno mial RSA is sensitive to the size of the design space for structural optimi zation problems. They recommended the use of various measures to find a small “reasonable design space.” It is often advantageous to re fine the design space to meet th e needs of the design problem. For example, design constraints can prohibit the use of a simple hypercube design space.74,75 In other instances the fitted surrogate is simply inaccurate. For a high-dimensional design space,

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30 adequately filling a hypercube design space regi on with points becomes cost prohibitive. Balabanov et al.70 noted that to sample only at the vert ices of a box-shaped design requires 2 n points, where n is the number of design variables. In contrast, a simplex sampled at the vertices would require only n + 1 points. In particular, for n = 25, a box domain has a volume 1010 times larger than a corresponding ellipsoid. Several techniques are used to improve the optimization process and surrogate accuracy by reducing the domain over which the function is approximated. Madsen et al.74 refined the design space to an ellipsoid shape in the shape optimization of a diffuser to prevent the unnecessa ry CFD analysis of physically infeasible designs. In the optimization of a super-detonative ram accelerator, Jeon et al.76 found that a surrogate model that was a poor fit due to sharp curvature in the resp onse and was able to ach ieve a better result by performing a transformation on the de sign space, hence, smoothing out the curvature. Papila et al.77 explored the possibility of reducing errors in regions the design space by constructing a reduced design space on a small region of interest. Balabanov et al.70 eliminated suspected infeasible design regions from the design space usi ng a low-fidelity analysis prior to optimizing using a high-fidelity CFD analysis in the design of a high speed ci vil transport wing. Papila et al.78 refined the design space by expanding the va riable bounds to account for optimal designs that lay on the edge of the design space in th e shape optimization of a supersonic turbine. 1.4 Summary Rocket engine component design is a co mplex process that can be made more straightforward using CFD-based optimization tec hniques. The performance and efficiency of key components can be improved using the SBO me thods. The application of the SBO approach to CFD problems can be challenging due to time a nd computational constraints. Tools including low-fidelity analyses and design space refinement can help redu ce computational expense. In

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31 particular, design space refineme nt techniques can be employed to facilitate a more efficient optimization process by reducing th e number of necessa ry computationally expensive CFD runs. 1.5 Goal and Scope The research goal is to investigate ad aptive DSR using measures based on model uncertainty from varying surrogate models and focusing on regions of interest to improve the efficiency of the CFD-based design of rocket engine components. Chapter 2 reviews the proposed DSR techniques as well as the optimiza tion framework that is used to complete the optimizations. This includes the use of new op timization tools to accelerate the optimization process. In Chapter 3, DSR techniques are applie d in the low-fidelity optimization of a radial turbine for liquid rocket engines. The preliminary design of a ra dial turbine is a logical next step, because 1) it is a new design where th e optimal design space size and location is unknown, and 2) a 1-D solver is used to provide function evaluations allowi ng for the collection of a large number of data points. The DSR techniques are used to dete rmine both an applicable design space size by excluding infeasible regions a nd the region of interest by excluding poorly performing design points. The optimization framework technique s are used to complete the optimization process. The optimization techni ques then combined with CFD with a final objective of improving the rocket engine injector design process. The governing equations for combusting flow and a background on rocket engi ne injectors are given in Chapter 4. The optimization methods are demonstrated using an analysis of a trapezoidal bluff body to improve combustor mixing characteristics in Chapter 5. Th e bluff body analysis is used to show that DSR can identify regions of poor simulation accu racy. Chapter 6 presents the CFD modeling effort for a single element injector. The CFD resu lts are compared to an experimental work for accuracy. Then, the flow dynamics leading to wall heat transfer are analyzed, providing information that would be difficult to obtain using an experimental analysis alone. Finally, in

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32 Chapter 7, CFD-based optimization techniques are applied to improve the design process of a multi-element rocket injector. In this case, th e optimization techniques must be applied in the context of a CFD-based design process where a limited number of function evaluations are available. Chapter 8 concludes the effort. Table 1-1. Summary of CFD-based design optimiza tion applications with highlighted surrogatebased optimization techniques. Author Application Method highlighted Key results Shyy et al.79 Rocket engine injector Surrogate comparison Polynomial approximation compared to neural network surrogate model. Simpson et al.80 Aerospike nozzle Surrogate comparison, design space selection Kriging model compared to quadratic response surface. Simple design variable ranges selected using geometry. Dornberger et al.59 Turbomachinery designs Surrogate comparison Multidisciplinary optimization using response surfaces and neural networks. Rodriguez62 Jet configuration Design space selection Optimization using simple geometry-based bounds on design variables Vaidyanathan et al.63 Rocket engine injector Design space selection Design variab les chosen from variations of existing design Han et al.64 Multi-blade fan/scroll system Low-fidelity optimization Optimized low-fidelity 2-D model to predict for high-fidelity 3-D model Keane65 Transonic wing Multi-fidelity optimization Supplemented high-fidelity CFD data with lowfidelity empirical model Alexandrov et al.66 Wing design Multi-fidelity optimization Varied mesh density for variable fidelity models Knill et al.67 High speed civil transport wing Multi-fidelity optimization Saved 255 CPU hours using low-fidelity supplemental data Balabanov et al.70 High speed civil transport wing Multi-fidelity analysis with surrogate correction Used coarse and fine finite element models as lowand high-fidelity models Madsen et al.74 Diffuser Design space refinement Removed infeasible design space regions to prevent unnecessary CFD analyses Jeon et al.76 Ram accelerator Design space refinement Performed transformation on design space for better response surface fit Papila et al.77 Supersonic turbine Design space refinement Refine design space to improve regions with high error Balabanov et al.70 Civil transport wing design space refinement Used low-fidelity model to refine design space before optimizing using high-fidelity model Papila et al.78 Supersonic turbine Design space refinement Design space expansion to explore optimal designs at edge of design space

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33 A fuel p um p oxidizer p um p oxidizer fue l Thrust chamber turbine B fuel p um p oxidizer p um p o xidizer fue l Thrust chamber turbine gas generator exhaust C fuel p um p oxidizer p um p oxidizer fue l Thrust chamber turbine preburner Figure 1-1. Rocket engine cycles A) Expander cycle, B) gas ge nerator cycle, C) pre-burner cycle. Figure 1-2. SSME thrust chamber component di agram showing low-pressure fuel turbopump (LPFTP), low-pressure oxidi zer turbopump (LPOTP), high-pressure fuel turbopump (HPFTP), high-pressure oxidize r turbopump (HPOTP), main fuel valve (MFV), main oxidizer valve (MOV), combustion chamber valve (CCV), and the main combustion chamber (MCC). Picture reproduced from [81].

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34 Figure 1-3. SSME component reliability data. Chart reproduced from [1]. Figure 1-4. Surrogate-based optimization using mu lti-fidelity data. A multi-fidelity analysis may require a surrogate model for correc ting the low-fidelity model to better approximate the high-fidelity model. refine design space construct surrogate Verify optimum location with highfidelity model Low-fidelity model w/ correction Surrogate correction Low-fidelity model Assess surrogate High-fidelity model

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35 CHAPTER 2 OPTIMIZATION FRAMEWORK The optimization framework steps include 1) modeling of the objectiv es using surrogate models, 2) refining the design space, 3) reduc ing the problem dimensionality, and 4) handling multiple objectives with the aids of Pareto front and a global sensitivity evaluation method. Several procedures may require moving back and fo rth between steps. Fi gure 2-1 illustrates the process used to develop optimal designs fo r the problems involving multiple and possibly conflicting objectives. The steps of the framew ork are detailed in the following sections. The first step in any optimization problem is to identify the performance criteria, the design variables and their allowable ranges, and the design constraints. This critical step requires expertise about the physical process. A multi-ob jective optimization problem is formulated as (),:1,;:1,ji M inimizewherefjMxiN FxFx Subject to ()0,:1, ()0,:1,p kwherecpP wherehkK CxC HxH Once the problem is defined, the designs are evaluated through experiments or numerical simulations. For numerical simulations, the type of numerical model and design evaluation used varies with the goals of the study. For a simple preliminary design optimization, the use of an inexpensive 1-D solver may be su fficient. However, for the final detailed design, more complex solvers may be needed.82,78 The choice of a model has an important bearing on the computational expense of evaluating designs. When obtaining many design points is timeprohibitive, it is often more prudent to use an inexpensive surrogate model in place of the expensive numerical model.

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36 2.1 Optimization Using Surrogate Models This step in the framework involves deve loping alternate models based on a limited amount of data to analyze and optimize designs. The surrogates provide fast approximations of the system response making optimization and se nsitivity studies possible. Response surface approximations, neural network tec hniques, spline, and Kriging are examples of methods used to generate surrogates for simulations in the optimization of complex flows52 involving applications such as engine diffusers,74 rocket injectors,63 and supersonic turbines.78,82 The major benefit of surrogate models is th e ability to quickly obtain any number of additional function evaluations with out resorting to more expensive numerical models. In this aspect, surrogate models can be used for multiple purposes. Obviously, they are used to model the design objectives, but they can also be used to model the constraint s and help identify the feasible region in design space. Key stages in the construction of surrogate models are shown in Figure 2-1. 2.1.1 Design of Experiments The search space, or design space, is the set of all possible combinations of the design variables. If all design variables are real, the design space is given asN xR where N is the number of design variables. The feasible domain S is the region in design space where all constraints are satisfied. For adequate accuracy, the data points used in the surrogate mode l must be carefully selected. The proper data selec tion is facilitated by using Desi gn of Experiments (DOE). One challenge in design optimization despite the type of surrogate used is called the “curse of dimensionality.” As the number of design variables increase, the number of simulations or experiments necessary to build a surrogate incr eases exponentially. This must be taken into consideration in choosing a DOE. The key issues in the selection of an appropriate DOE include

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37 ( i ) the dimensionality of the problem, ( ii ) whether noise is important source of error, ( iii ) the number of simulations or experi ments that can be afforded, ( iv ) the type of surrogate used to model the problem, and ( v ) the shape of the design space. If noise is the dominant source of error, DOEs that reduce sensitivity to noise ar e commonly used. These include central composite designs (CCD) or face-centered cubic designs (FCCD) for box-shaped domains. Design optimality designs such as Dor A-optimal83 (Myers and Montgomery 2002, pp. 393 – 395) designs are useful for irregular shaped domains and high dimens ional domains when minimizing noise is important. Specificall y, these designs can be used to re duce the number of points in an experimental design for a given accuracy. When noi se is not an issue, sp ace-filling designs such as Latin-Hypercube Sampling (LHS)60 or Orthogonal Arrays (OAs)46 are preferred to efficiently cover the entire design space. Central composite designs and face-centered cubic designs ar e intended to minimize the presence of noise in the response. They are co mprised of a single point at the center of the design space with the remaining points situated along the periphery of the design space, as shown in Figure 2-2. The center po int detects curvature in the system while all other points are pushed as far as possible from the center provid e noise-smoothing character istics. The FCCD is actually a modified version of th e CCD with the axial points move d onto the edges of a square. An FCCD is required when the limits of the de sign space are strictly set, requiring a square design space. The CCD and FCCD are most popular for re sponse surface approximations. They are commonly used to reduce noise in the response by placing most of the po ints on the boundary of the design space. For responses that do not c ontain noise, such as data obtained using deterministic computational simulations, a spacefilling design may be required for accuracy.

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38 While CCD and FCCD only contain points at th e center and edges of the design space, spacefilling designs distribute points throughout the design space. The most common type of spacefilling design is Latin-Hypercube Sampling (LHS). The LHS technique consists of dividing the design space N times along each variable axis creating Nk subsections where N is the number of data points and k is the number of design variables. A total of N data points are randomly placed such that only one point exists in each subsect ion interval. This procedure can possibly leave holes in the design space as shown in Figur e 2-3A. A criterion to maximize the minimum distance between points can help spread points more evenly and prevent bunching. Orthogonal array LHS84 is another method of improving the distri bution of data points within the design space. It insures a quasi-uniform distributi on of points across the design space. The design space is uniformly divided into larger subsectio ns and LHS is applied while ensuring one point lies in each large subsection (Figure 2-3B). 2.1.2 Surrogate Model Identification and Fitting There are many types of surrogate models to choose from. There are parametric models that include polynomial response surfaces approxim ations (RSA) and Kriging models, and there are non-parametric models such as radial ba sis neural networks ( RBNN). The parametric approaches assume the global functional form of the relationship between the response variable and the design variables is known, while the non-parametric ones us e different types of simple, local models in different regions of the data to build up an overall model. Response surface approximations assume that the data is noisy a nd thus it fits the data with a simple, smooth global function to minimize the root-mean-square (rms) error. Kriging and RBNN use functions localized to the data points. Or dinary Kriging, the type of Krigi ng used in this study, uses all of the data points for that purpose and interpolates over all of them, while RBNN uses a subset of the points and minimizes the rms error. The subs et is determined based on the desired minimum

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39 size of the error. RSA requires a solution of linear equations, as does Kriging for a given correlation function. In Krig ing, the process of finding th e correlation function is an optimization problem that is done either with cr oss-validation or minimum likelihood estimates. On the other hand, RBNNs let the user experi ment with one parame ter that defines the correlation (SPREAD). However, the process of minimizing the rms is an optimization problem. 2.1.2.1 Polynomial response surface approximation Response surface approximations (RSA) we re developed for experimental design processes. They consist of a polynomial curve fit to a set of da ta points. Usually the polynomial is a 2nd-order model, but higher order models may also be used. A 2nd-order model requires a minimum of 122 kk design points, where k is the number of design variables. Thus, a 2nd-order model with three variables woul d require at least 10 points. A 3rd-order model with three variables would require twice that amount. A 2nd-order model provides good trade-off in terms of accuracy and computational expense. Fo r RSAs to be effective in design optimization, the sampled points must be near the optimum desi gn. In this case, it is often necessary to perform a screening exercise on th e data to ensure that the de sign space is selected in the appropriate region. Once the pr oper design region has been narrowed down, the region can be populated with data points using a DOE such as CCD or FCCD. The popular polynomial RSAs have been used extensively in design optimization for numerous applications using varying DOEs with varying re sults. Rais-Rohani and Singh85 used polynomial RSAs to increase the structural effici ency of composite stru ctures while maximizing reliability. They found that the accuracy and e fficiency of RSAs can vary depending on the choice of DOE. Polynomial RSAs have been su ccessful in smoothing re sponses that contain noise. In the aerodynamic design of a subsonic wing, Sevant et al.86 encountered numerical

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40 noise resulting in local maxima. Polynomial RSAs were used to smooth the noise. Sturdza et al.87 used a response surface to optim ize the shape of a fuselage to retain laminar flow across a wing. Clues to the accuracy of RSAs can be found through the simultaneous use of multiple RSAs. Kurtaran et al.88 used linear, elliptic, and quadratic RSAs to optimize for crashworthiness. Kurtaran et al. found that the differences between the pr edictive capabi lities of the various RSAs decreased as the desi gn space was reduced. Hosder et al.89 used RSAs in conjunction with techniques to reduce the desi gn space and perform multi-fidelity analyses in the multidisciplinary optimization of aircraft. Additi onal references and their key results are given in Table 2-1. The polynomial RSA assumes that the function of interest f can be represented as a linear combination of Nc basis functions zj and an error term For a typical observation i a response can be given in the form of a linear equation as ()2 1()0 zcN i ijjiii jfzEV (2-1) where the errors i are considered independent with an expected value equal to zero and a variance equal to 2. The coefficients j represent the quantitative re lation among basis functions zj. Monomials are the preferred basis functions. The relationship between the coefficients j and the basis functions zj is obtained using Ns sample values of the response fi for a set of basis functions zj ( i ) such that the error in the prediction is minimized in a least squares sense. For Ns sample points, the set of equations specified in Equation 2-1 can be expressed in matrix form as 20 fX E VI (2-2)

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41 where X is a s cNN matrix of basis functions, also known as a Gramian design matrix, with the design variable values as the sampled points. A Gramian design matrix for a quadratic polynomial in two variables ( Ns = 2; Nc = 6) is given by 22 112111112121 22 122212122222 22 121122 22 1211221 1 1 1ssssssiiiiii NNNNNNxxxxxx xxxxxx x xxxxx xxxxxx X (2-3) The vector b of the estimated coefficients, which is an unbiased estimate of the coefficient vector and has minimum variance, can then be found by 1 TTbXXXf (2-4) At a new set of basis function vector z for design point P, the predic ted response and the variance of the estimation are given by 1 ()2 1ˆˆ ()()TzzzXXzcN iT jj PP jfbzandVf (2-5) 2.1.2.2 Kriging In cases where a given surrogate performs poorly, it is necessary to use a different type of surrogate model, because the high ly nonlinear nature of some pr ocesses may not be captured by surrogates such as RSAs. Kriging is a popular ge ostatistics technique named after the pioneering work of D.G. Krige47 and was formally developed by Matheron.90 The Kriging method in its basic formulation estimates the value of a functi on or response at some location not sampled as the sum of two components: the linear model (e .g., polynomial trend) and a systematic departure representing low (large scale) and high fr equency (small scale) variation components, respectively. Kriging has the added flexibility of being able to eith er provide an exact data fit or

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42 to act as a smoothing function. It can be used to approximate a highly nonlinea r response with fewer points than a comparable high-ord er response surface. Koch et al.91 asserts that using Kriging may be preferable over RSAs for de sign problems with a large number of design variables. Mahadevan et al.92 found that Kriging was more accura te than an RSA and used fewer function evaluations for engineering reliability estimates. Kriging has the capability of overcoming the limit of relatively small design space that is inherent in RSAs, and can be used effectively fo r larger design spaces. It is gaining popularity in CFD-based optimization as an alterna tive to RSAs. Forsberg and Nilsson93 found that Kriging provided a better approximation as compared to an RSA in the structural optimization of crashworthiness. However, Jin et al.94 found that Kriging was very accurate, but also very sensitive to noise as compared to other surrogate s in the data fitting of several analytical test problems. Rijpkema et al.95 found that Kriging was better able to capture local details in an analytical test function than an RSA, but also cautioned against the unintentional fitt ing of noise. Chung and Alonso96 compared Kriging to RSAs and found that while both can be accurate, Kriging is better at fitting functions with several local optima. Simpson et al.97 found that Kriging and response surface performed comparably in the optimization of an aerospike nozzle. Kanazaki et al.98 used Kriging to reduce the computati onal cost involved in optimizing a threeelement airfoil using genetic al gorithms by using the Kriging surr ogate for function evaluations. Jouhaud, J. –C. et al.99 used Kriging to adaptively refine the design space in the shape optimization of an airfoil. Kriging is represented by a trend perturbed by a “systematic departure.” In this study, ordinary Kriging is used, where the constant trend is represented by the sample mean. Other types of Kriging include simple Kriging that uses a constant trend of zer o, and universal Kriging

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43 uses a linear trend. The systematic departure component represents the fluctuations around the trend with the basic assumption being that thes e are correlated, and the correlation is a function of distance between the locations under consideration. More precise ly, it is represented by a zero mean, second-order, stationary process (mean and variance constant with a correlation depending on a distance) as described by a correlation mode l. Hence, these models suggest estimating deterministic functions as ()()cov((),())()()00,ijyEij xxxx (2-6) where is the mean of the response at sampled design points, and is the error with zero expected value and with a correlation structur e that is a function of a generalized distance between the sample data points. ()()2cov(),(),ijijR xxxx (2-7) In this study, a Gaussi an correlation structure100 is used 2 1,expijij kkk kvNR xxxx (2-8) where R is the correlation matr ix among the sample points, Nv denotes the number of dimensions in the set of design variables x identifies the standard devia tion of the response at sampled design points, and k is a parameter which is a measure of the degree of correlation among the data along the kth direction. Specifically, the parameters and are estimated using a set of N samples ( x y ) such that a likelihood function is maximized.100 Given a probability distribution and the corresponding parameters, the likelihood f unction is a measure of the probability of the sample data being drawn from it. The model estimates at non-sampled points are 1ˆˆˆ ()Ty rR y 1 (2-9)

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44 where ˆ is the estimated mean, r identifies the correlation vect or between the set of prediction points and the points used to construct the model, y a vector of the response at the sampled points, and 1 denotes an Ns-vector of ones. The mean is estimated by 1 11ˆTT 1R11R y (2-10) and the standard deviation of the response is estimated as 1 2ˆˆ ˆT sN y 1R y 1 (2-11) On the other hand, the estimation variance at non-sampled design points is given by 1 21 11 (())1T T TR VyR R 1r xrr 11 (2-12) In this study, the model estimates, estimation va riance, and standard deviation of the response are calculated using the Matlab toolbox DACE.101 2.1.2.3 Radial basis neural networks Neural networks are anothe r alternative to traditiona l response surface methods. Specifically, neural networks are able to fit da ta with a highly complex and nonlinear response. Artificial neural networks are made of inte rconnected local models called neurons. A high number of neurons can improve the accuracy of the fit, while using fewer neurons improves the smoothing qualities of the model. Mahajan et al.102 used neural networks (NN) to optimize a mechanically aspirated radiation shield for a meteorological temperat ure sensor to gain information on the workings of the temperatur e sensor system and the performance of the radiation shield. Papila et al.78 used radial basis neur al networks to supplement the data used to construct an RSA in the optimization of a supersonic turbine for rocket propulsion. Charalambous et al.103 used neural networks in the predic tion of bankruptcy and stressed that the

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45 prediction ability of a neural network is problem dependent. Hsken et al.104 used neural networks in turbine blade op timizations. Shi and Hagiwara105 used neural networks to maximize the energy dissipation of the crashworthiness of a vehicle component. Pidaparti and Palakal106 compared the prediction capability of neural networ ks with experimental results in the prediction of crack propagation in agi ng aircraft. Chan and Zhu107 demonstrated the ability of neural networks in modeling highly non linear aerodynamic characteristi cs with many variables. Brigham and Aquino108 used neural networks to accelerat e their optimization search algorithm by providing inexpensive function evaluations. Neural networks are made up of a distributi on of artificial neurons, whose function is modeled after biological neurons. A shape f unction is applied at each neuron with the combination being a complete function curve. Each neuron is given a weight in the network that defines that individual neuron’s strength. Neural networks are often diffi cult to train due to the required specification of several parameters that ca n affect the fit of the model. Care must also be taken, as there is sometimes danger in ove r-fitting the data, resul ting in poor prediction capability away from the known data points. None theless, the models are extremely flexible and can provide an exact data fit, or meet a user-defin ed error criterion to filter noise to prevent data over-fitting. Neural networks us e only local models, and the de gree of effective influence of each local model, or neuron, is a selection made by the user. Radial basis neural networks are two-layer networks consistin g of a radial-basis function (RBF) and a linear output layer. The radial basis function is given by bfradbas cx (2-13) where b is the bias, c is the center vector asso ciated with each neuron, x is the input design vector, and the function radbas is the Matlab function name for an RBF, or a local Gaussian

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46 function with centers at individual locations from the matrix x. The radial basis function is given by 2expradbasnn (2-14) where n is given by the term in parenthesis in Equation 2-13. Based on these equations, the function prediction is given by 2 1ˆ expN ii iywb cx (2-15) where wi is the weight vector for each neuron i, and N is the number of neurons. The Gaussian RBF decreases monotonically with the distance from its center. The bias b is set to 0.8326/SPREAD where SPREAD is the spread constant, a user defined value that specifies to the radius of influence for each neuron where the ra dius is equal to SPREAD/2. Specifically, the radius of influence is the dist ance at which the influence reaches a certain small value. If SPREAD is too small, then prediction is poor in regions that are not near the position of a neuron. If SPREAD is too large, then the sensi tivity of the neurons will be small. Because RBNNs are based on the combination of multiple local approximations, a space-filling DOE is necessary. If large spaces exist in the DOE, it may be difficult to ove rlap information from nearby points. The capability of the RBNN to f ill the information gaps in the design space is based on the size SPREAD. If SPREAD is too sm all, the RBNN surrogate may not provide any predictive ability in some regi ons within the large space. For this region, there must be coordination between the spacing of data point s within the design sp ace and the size of SPREAD. Increasing the number of neurons will improve the accuracy of the surrogate. Neurons are added to the network one by one until the sum of the squares of the errors (SSE) is reduced enough to reach a specified error goa l. If the error goal is set to zero, neurons will be added until

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47 the network exactly predicts the input data. Howe ver, this can lead to over-fitting of the data which may result in poor prediction between data point s. On the other hand, if error goal is large, the network will be under-trained and predictions even on data points will be poor. For this reason, an error goal is judicious ly selected to provide early st opping of the network training to prevent over-fitting and increase the overall prediction accuracy. The purpose of the user is obviously to de termine good values of SPREAD and the error goal, GOAL. This can be done by reserving a sma ll number of data points as test data. The remaining data is designated as the training data. First, the network is trained using the training data and the Matlab function newrb. Then, the network is simulated at the test points and the error is calculated between the actual and predicte d response. The process is repeated for a small number of values with selected range of SPR EAD and a small set of values for GOAL. The values of SPREAD and GOAL that produce the smalle st errors in the test data are identified, then the ranges of SPREAD and GOAL are reduced and new values are selected. The process iteration continues until the SPR EAD and GOAL combination is identified that minimizes the error in the test data. The idea l combination can usually be identified within three iterations. In this way, the training data is used to train the neural network, and th e test data is used to tune it. 2.1.3 Surrogate Model Accuracy Measures Once the surrogate models are available, it is imperative to establish the predictive capabilities of the surrogate mode l away from the available data. Several measures of predictive capability are given below. 2.1.3.1 Root mean square error The error i at any design point i is given by ˆiii f f (2-16)

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48 where fi is the actual value and ˆi f is the predicted value. For Ns design points, the root mean square (rms) error is given by 2 1 i i ssN N (2-17) A small rms error indicates a good fit. The rms e rror is a standard measur e of fit that may be used to compare the performances of surrogates of different types. The adjusted rms error a is used to adjust the rms error based on the number of parameters in a polynomial RSA and is given by 2 1sN i i sa N q (2-18) where Ns is the number of data points, and q is the number of terms in the polynomial approximation. A polynomial with a large num ber of terms is penalized and deemed less desirable for a polynomial approximation with fewer terms for the same number of data points. For a good fit, a should be small compared to the data value ranges. For Kriging, the standard error measurement is given by the estimated standard deviation of the response given by Equation 2-11. The squa re root of the estimated standard deviation of the response can be compared to the rms error of polynomial RSAs. 2.1.3.2 Coefficient of multiple determination The adjusted coefficient of multiple determination 2 adj R defines the prediction capability of the polynomial RSA as 2 1 2 1211;wheres sN as i i N s i iadjN f f N ffR (2-19)

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49 For a good fit, 2 adj R should be close to 1. 2.1.3.3 Prediction error sum of squares When there are an insufficient number of data points available to test the RSA, the prediction error sum of squares (PRESS) statisti c is used to estimate the performance of the surrogate. A residual is obtained by fitting a su rrogate over the design sp ace after dropping one design point from the training set. The value pr edicted by the surrogate at that point is then compared with the expected value. PRESS is given by 2 1ˆPRESSsN ii i f f (2-20) and PRESSrms is given by 2* 1ˆ PRESSsrmsN ii i s f f N (2-21) where *ˆi f is the value predicted by the RSA for the ith point which is excluded while generating the RSA. If the PRESSrms value is close to a, this indicates that the RSA performs well. 2.2 Dimensionality Reduction Using Global Sensitivity Analysis At the beginning of a design optimization, seve ral variables are chosen as design variables with the assumption that they are important to the optimization. However, having large numbers of design variables can greatly increase the cost of the optimization. Section 2.1.2 introduced the concept of the “curse of dimensionality,” meaning that as the number of design variables increase, the number of data points required to obtain a good approximation of the response can increase exponentially. It is of great benef it, therefore, to simplify the design problem by identifying variables that are unimportant and removing them fr om the analysis. The most efficient way of doing this is to perform a sensiti vity analysis. A global sensitivity analysis can

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50 provide essential information on the sensitivity of a design object ive to individua l variables and variable interactions. By rem oving the variables that have ne gligible influence on the design objective, the dimensionality of the problem can be reduced.109,110,111 Global sensitivity analyses enable the study of the behavior of differe nt design variables. This information can be used to identify the variables which are the least important and thereby can reduce the number of vari ables. A surrogate model f (x) of a square integrable objective as a function of a vector of in dependent input variables x ( 0,11,i x iN ) is assumed and modeled as uniformly distributed random variables. The surrogate model can be decomposed as the sum of functions of increas ing dimensionality as 01212,,,,xiiijijNN iij f ffxfxxfxxx (2-22) where0d1 x0x f f. If the following condition 11 ... 00siikfdx (2-23) is imposed for k = i1, …, is, then the decomposition describe d in Equation 2-22 is unique. In the context of a global sensitivity analysis, the total variance denoted as V ( f ) can be shown to be equal to 1 11()n iijN iijNVfVVV (2-24) where 2 0VfEff and each of the terms in Equation 2-24 represent the partial variance of the independent variables ( Vi) or set of variables to the total variance. This provides an indication of their relative importance. The partial variances can be calculated using the following expressions:

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51 ([|]) ([|,]) ([|,,])ii ijijij ijkijjijikjkijkVVEfx VVEfxxVV VVEfxxxVVVVVV (2-25) and so on, where V and E denote variance and the expected value respectively. Note that 1 0|iii E fxfdx and 1 2 0([|])iiiVEfxfdx Now the sensitivity i ndices can be computed corresponding to the independent variables and set of variables. For example, the first and second order sensitivity indi ces can be computed as ()()ij i iijV V SS VfVf (2-26) Under the independent model inputs assumption, the sum of all the sensitivity indices is equal to one. The first order sensitivity index for a given va riable represents the main effect of the variable but it does not take into account the effect of interaction of the va riables. The total contribution of a variable on the total variance is given as the sum of all the interactions and the main effect of the variable. The total sensit ivity index of a variable is then defined as ,,,... ()iijijk jjijjikki total iVVV S Vf (2-27) To calculate the total sensitivity of any design variable xi, the design variable set is divided into two complementary subsets of xi and Z ,1,;j Z xjNji The purpose of using these subsets is to isolate the influence of xi from the influence of the remaining design variables included in Z The total sensitivity index for xi is then defined as total totali iV S Vf (2-28)

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52 ,total Z iiiVVV (2-29) where Vi is the partial variance of the objective with respect to xi and Vi,Z is the measure of the objective variance that is depe ndent on interactions between xi and Z. Similarly, the partial variance for Z can be defined as VZ. Therefore, the total objective variability can be written as Z ZiiVVVV (2-30) Sobol112 proposed a variance-based non-parametric a pproach to estimate the global sensitivity for any combination of design variables using M onte Carlo methods that is also amenable. While Sobol used Monte Carlo simulations to conduct the global sensitivity analysis, the expressions given above can be eas ily computed analytically once th e RSA is available. In the present study, the above referenced expression s are evaluated analytic ally using polynomial RSAs of the objective functions. No accommodations are made for irregular-shaped domains, so the analytical treatment works best in a wellrefined design space where the assumption of a boxlike domain will not introduce si gnificant errors. Using a polynomial RSA as the function f(x), the approximation can be decomposed as in Equation 2-22 and the sensitivity indices can be obtained. 2.3 Multi-Objective Optimization Using the Pareto Optimal Front After developing a computationally inexpensiv e way of evaluating di fferent designs, the final step is to perform the actual optimization. In the case of a single objective, this requires a simple search of design space for the minimum value of the objective. For two or more objectives, additional treatment is needed. Highl y correlated objectives can be combined into a single objective function. When the objectives are conflicting in nature, there may be an infinite number of possible solutions th at will provide possi ble good combinations of objectives. These solutions are known as Pareto optimal solutions. While there are numerous methods of solving

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53 multi-objective optimization problems, the use of evolutionary algorithms (EAs) is a natural choice to get many Pareto optimal solutions in a single simulation due to its population based approach and ability to converg e to global optimal solutions. A feasible design x(1) dominates another feasible design x(2) (denoted by x(1) < x(2)), if both of the following conditions are true: 1. The design x(1) is no worse than x(2) in all objectives, i.e., (1)(2) jjffxx for all j = 1,2,…, M objectives. (1) x (2)(1)(2)(1)(2) jjjjjMfforjMff xxxxx (2-31) 2. The design x(1) is strictly better than x(2) in at least one objective, or (1)(2) jjffxxfor at least one {1,2,...,} j M (1)(2)(1)(2)()()jjjMff xxxx (2-32) If two designs are compared, then the designs are said to be non-dominated with respect to each other if neither design dominates the other. A design xS, where S is the set of all feasible designs, is said to be non-dominated with respect to a set AS, if : aAax. Such designs in function space are called non-dominated solutions. All the designs x ( xS) which are non-dominated with respect to any othe r design in set S, comprise the Pareto optimal set. The function space representation of the Pa reto optimal set is the Pareto optimal front. When there are two objectives, the Pareto optim al front is a curve, when there are three objectives, the Pareto optimal front is represente d by a surface and if there are more than three objectives, it is represented by a hyper-surface. In this study, an elitist non-dominat ed sorting genetic algorithm NSGA-II113 and a parallel archiving strategy to overcome the Pareto drift problem114 are used as the multi-objective

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54 optimizer to generate Pareto optimal solutions. The description of the algorithm is given as follows: 1. Randomly initialize a popul ation (designs in the design space) of size npop. 2. Compute objectives and constraints for each design. 3. Rank the population using non-dom ination criteria. Many indi viduals can have the same rank with the best individua ls given the designation of rank-1. Initialize an archive with all the non-dominated solutions. 4. Compute the crowding distance. This distance finds the relative closeness of a solution to other solutions in the function space and is us ed to differentiate between the solutions on same rank. 5. Employ genetic operators—selection, crossover, and mutation—to create intermediate population of size npop. 6. Evaluate objectives and constraint s for this intermediate population. 7. Combine the two (parent and intermediate ) populations, rank them, and compute the crowding distance. 8. Update the archive: 9. Compare archive solutions with rank-1 solutions in the combined population. 10. Remove all dominated solutions from the archive. 11. Add all rank-1 solutions in the current population whic h are non-dominated with respect to the archive. 12. Select a new population npop from the best individuals based on the ranks and the crowding distances. 13. Go to step 3 and repeat until the termination criteria is reached, whic h in the current study is chosen to be the number of generations 2.4 Design Space Refinement Techniques Given knowledge about a problem, it is desira ble to perform CFD si mulations with (a) design parameters set in a region that is known to provide good results in order to refine the design (exploitation) or (b) to set the design parameters in an unexplored region and use

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55 additional simulations to decide whether thes e designs have potentia l (exploration). The approaches of exploitation versus exploration are the basis of most global search algorithms. In the context of design space refinement, a mo re modest goal is set by using exploitation to identify the bad regions of a design space: A “reasonable design space” approach uses simple analysis models or inexpensive constraints to crop areas of the design space that give blatantly poor results. A surrogate based on a small number of points can be used in the identification of “bad” regions. In this case, the surrogate may not be accurate enough to identify the good regions, but is sufficien t to identify regions th at are obviously bad. Exploration is needed to identify poorly repr esented areas of the design space by using error indicators to identify regions where the surrogate model fits poorly. E xploration is necessary even in regions with apparent moderate performa nce, as the improved surrogate may reveal that the performance in the region is better than expe cted. The design space refinement process using the ideas of exploitation and exploration shoul d provide more accurate surrogate models and reduce the simulation of poorly performing designs. Design space refinement often leads to irre gular domains. Most DOE techniques are intended for box-like domains. For noise-dominat ed problems and RSAs alphabet designs (Aoptimal, D-optimal, G-optimal, etc.) are available.46 For problems dominated by bias errors, space filling designs such as LHS are normally used, generating them first in an enclosing box and throwing out point s outside the domain. Surrogate models have been shown to be an effective means of reducing the number of computationally and time intensive function evalua tions required in CFDbased optimization. However, to use surrogates effectively, it must fi rst be ensured that the surrogate models are high accuracy representations of the system response. An inaccurate surrogate model can prevent further analysis and can cast doubt on the validity of the data. Accuracy can suffer if the choice

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56 of DOE was inappropriate for the ty pe of surrogate chosen or if the design space was too large. The advantages of design space refinement offe r promising ways to deal with an initial inaccurate surrogate model. Current techniques often include forms of design space reduction or windowing to obtain a reasonable design space. 2.4.1 Design Space Reduction for Surrogate Improvement Reducing the design space by reduc ing the ranges of the desi gn variables often leads to better accuracy in the surrogate model. This is often due to the fact that the response in the reduced design space has less curvature. This f eature makes the response easier to approximate. Papila et al.78 reduced the ranges of the design variables to perform a preliminary optimization on a supersonic turbine for the rocket engines of a reusable launch vehicle using RSAs. The goal of the study was to maximize the turbine effici ency and vehicle payload while minimizing the turbine weight. The preliminary design used a si mplified model to provide the system response. A FCCD was applied for each turbine stage. A composite response surface was constructed using desirability functions, and an optimum design was located. The error between the actual response and the predicted response at the lo cated optimum design wa s high, indicating poor RSA accuracies. As a result, Papila et al. re duced the size of the design space by about 80%. The smaller design space resulted in substantial improvement in the RSA prediction accuracies. Reducing the size of the design space in this manner is depicted in Figure 2-4, and is an example of design space exploitation, and can lead to a good accuracy surrogate in a small region of interest. Shaping the design space based on feedback fr om the objective function values allows the designer to concentrate solely on the region of interest. Balabanov et al.70 pointed out that effort could be saved if the design space was shaped su ch that it only considered reasonable designs rather than using the standard box-shaped desi gn space. The study by Balabanov et al. used this

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57 method in the RSA of a high-speed civil transport model. The a pproach removes large areas of the standard box-shaped design leav ing more of a simplex or ellips oid design. Inexpensive, lowfidelity models are used to predict the system response, and the result is used to shape the design space into a reasonable design space. Points that would lie on the edge of the standard box in a traditional DOE procedure are moved inward to the boundary of the identified reasonable design space. By concentrating the data points on the region of interest, the accuracy of the response surface within that region is improved. The t echnique was used to separately estimate the structural bending material weight and the lift and drag coeffici ents. In these cases, the design space was reduced at least by half, resulting in improvements in RSA accuracy. Roux et al.73 employed techniques such as intermediate response surfaces and identification of the region of interest to improve surrogate accuracy in structural optimization problems. Linear and quadratic response surfaces were used to fit the data. The design space was iteratively reduced around the suspected optimum by fitting respons e surfaces on smaller and smaller sub-regions. It was found that surroga te accuracy was improv ed more for carefully selected points within a small s ub-region than for a large number of design points within a large design space. Both Balabanov et al.70 and Papila et al.82 combine multiple objectives into a composite objective function before optimizing. However, this procedure can limit design selection to one or a few designs, when mu ltiple other desirable designs might exist. Papila et al.77 pointed out that the inaccuracy in th e initial response surf ace may result in windowing to a region that does no t actually contain the optimum design. Papila et al. argued that particular attention must be paid to the regions of higher e rror in an RSA. Papila et al. explains that because a quadratic RSA is a low-order polynomial, the model can often be inadequate. This is termed as “bias error,” a nd can obviously be reduced by using higher-order

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58 approximating polynomials. The study by Papila et al. seeks to improve the accuracy of RSAs by using a design space windowing technique to focus only on the region of interest. At the same time, it seeks to exclude regions of the de sign space with high bias errors. Papila studied the effects of fitting three different RSAs: a standard quadratic RSA, a RSA within the region of interest using the design space windowing t echnique, and an RSA using the design space windowing technique while excluding regions with high error. Papila found that the design space windowing technique improved the accuracy of the RSA over the global RSA. However, using the design space windowing technique excl uding regions of high error did not improve accuracy over a standard design space windowing technique. Sequential response surfaces115,116 involve several design space windowing steps. This method is used in the search of a single optimum by coupling surroga te modeling with the method of steepest descent. After a surrogate is fitted, a new design space box is constructed at a new location by stepping in the direction of the steepest descent. A new DOE is applied, and a new surrogate is constructed for the new design space region. The obvious disadvantage to this method is that it could potentiall y require a large number of data points for several different local optimizations. Rais-Rohani and Singh85 used sequential response surfaces to increase the structural efficiency in various problems. In the study, Rais-Rohani and Singh employ global and local response surface models. They noted that the global response surface model required a large number of data points for sufficient accu racy. For the local model, a linear response surface model was fit to the local regions. A tota l of ten design subspaces were needed before the optimum was reached. The two methods obtained similar optimum points, indicating comparable accuracy. However, the global response surface required 1000 to 3000 design points, while the local response surface method re quired from 69 to 406 data points. The study

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59 shows that using a large overall design space region over a small, local design space is not always the best choice. Goel et al.117 reached a similar conclusion in the optimization of a diffuser vane shape. Goel et al. found that su rrogates constructed with in a smaller, refined design space were significantly more accurate th an surrogates constructed over the unrefined design space. It has been shown that design space refine ment techniques can increase the accuracy of RSAs. One obvious omission in current design space refinement techniques is a method of refining the design space in the presence of mu ltiple objectives when a satisfactory weighting criterion is not known. In the use of RSAs fo r multi-objective optimizat ion, it is difficult to locate a design space that can simultaneously represent the multiple objectives while retaining accuracy for each objective. One solution is to ensure that the design space is large enough to encompass all of the effects the design. Th is course of action, as Papila et al.78 noted, can introduce substantial error into RSA. Using an al ternative surrogate such as Kriging or neural networks can alleviate this problem. However, doing so may require begi nning the analysis from scratch to obtain a data set using a DOE that is amenable to the desired surrogate model. 2.4.2 Smart Point Selection for Second Phase in Design Space Refinement A popular technique in optimization is the us e of the Efficient Global Optimization (EGO) algorithm proposed by Jones et al.118 This technique uses the variance prediction provided by Kriging to predict the location of the optimum poin t. The procedure used to predict the optimum point is shown briefly in Figure 2-5 and is described in detail in Section 2.4.3, and involves maximizes the expected improvement of a sample d function. The optimization process usually proceeds by sampling the function at a limited number of points. Kriging used as the surrogate model fit to the data points, and the variance and expected improvement across the design space is calculated. The optimization th en proceeds in a linear fashion, using the value of the expected

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60 improvement to select the optimum. The predicted optimum is sampled, and then added to the data set. The iteration continue s until convergence is reached. Various attempts have been made to impr ove upon the process to make it suitable for multi-objective optimization. Jeong and Obayashi119 used EGO in the multi-objective optimization the pressure distribution at two operating conditions of a two-dimensional airfoil shape. Genetic algorithms were used to develop the predicted Pareto front using the value of the expected improvement as the f itness criteria. The Pareto fr ont identifies points that are nondominated in the expected improvement of two objectives. Jeong and Obayashi choose three new data points based on the original Kriging mode l: 1) the predicted point with the maximum expected improvement in the first objective whic h lay at one end of the Pareto Front, 2) the predicted point with the maximu m expected improvement in the second objective at the other end of the Pareto front, and 3) a point with an expected improvement that lies in the middle of the Pareto front. The process was iterated multiple times until a converged optimum was achieved in both objectives. Th is straightforward optimizati on was concerned with and was successful in finding a single optimum, but did not address competing objectives. Knowles120 sought modify EGO for the use of co mputationally expensive multi-objective problems. The new algorithm called ParEGO by simultaneously using Pareto front-based optimization with the EGO algorithm, similar to the procedure used by Jeong and Obayashi.119 In this case, the goal was to develop an accurate Pareto front using a smaller number of function evaluations than methods typically used to generate Pareto fronts such as the NSGA-II113 algorithm. Again, EGO was used to select the points. However, this procedure relies on a few hundred function evaluations. This type of evaluation is only possible with relatively inexpensive function evaluations or a surrogate model that is already sufficiently accurate.

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61 Farhang-Mehr and Azarm121 used a different selection tech nique for constructing surrogate models for crash analyses. The technique also required a standard DOE and Kriging fit to initialize the response surface. The method then added points based on the predicted irregularities in the response surface. This method allowed for the response surface to be improved while requiring potentially fewer points overall. Thus, the design space was adaptively refined using points th at were selected based on th e predicted objective function, itself. Farhang-Mehr and Azarm cited the benefits of their approach in reducing the total number of points needed for optimizations based on computationally expensive simulations. The procedure is good for improving surr ogate accuracy, but does not ta ke into account the value of the function, meaning that points may be added in regions of non-interest. 2.4.3 Merit Functions for Data Selection and Reduction Using the concepts such as design space reduction (Section 2.4.1) along with the benefits of innovative point selection as in the previous section, merit f unctions, facilitated by Kriging, can be smoothly integrated into the multi-objective DSR process. Merit functions are statistical measures of merit that use information of th e function values and model uncertainty in the surrogate model to indicate the locations wh ere the function values can be improved and uncertainty reduced. In this research effort, mer it functions are used to select data points such that the accuracy of the surrogate model can be improved using a minimal number of data points. The merit functions rely on pred ictions of the function values and function uncertainty and attempt to balance the effects of the two. Krig ing is selected as the surrogate model for the optimization due to its inherent ability to provi de estimates of both the function value and the prediction variance. Although several merit functi ons are available, the strengths of a given merit function over another for the purpose of multi-objective optimization with an extremely limited number of function evaluatio ns has not been determined.

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62 Four different merit functions are considered. Because ordinary Kriging assumes that the response data is normally distribut ed, a statistical lowe r bound can be calculated. The first merit functions seeks to minimizi ng the statistical lower bound (MF1) given by 1ˆMFykVxxx (2-33) where k is an adjustable scaling factor that is set to 1 in this study.. The variance V is predicted as given in Equation 2-12. The statistical lower bou nd is simply the predicted function value minus the estimated standard error in the surrogate pr ediction at a given location, where the estimated error is a function of the distan ce between data points. Locations far away from any data points will have higher estimated error, or variance. The second merit function is maximizing the probability of improvement122 (MF2). The function is used to calculate the probability of improving the function value beyond a value T at any location. A point is selected where the probability of improvement is the highest. 2ˆ Ty MF V x x x (2-34) where T is a target value given by minminTfPf (2-35) where fmin is the current predicted optimum, is the normal cumulative distribution function, and P is an adjustable scaling factor set to 0.1 in this study, which corresponds to an improvement of 10% over the current best f unction value. Depending on the value of P using the criterion of maximizing the probability of improvement ( MF2) can results in a highly localized search. Jones123 suggests that selecting several target s can help the search proceed in a more global manner. The third merit func tion is maximizing the expected improvement118 ( MF3). Maximizing the e xpected improvement ( MF3) is the criterion used in the Efficient Global

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63 Optimization (EGO) algorithm developed by Jones et al.118 and has been shown to be effective in searching globally for the optimum It calculates the amount of improvement that can be expected at a given location. minmin 3minˆˆ ˆ fyfy MFfyV VV xx xx x xx (2-36) where is the normal density function. The fourth merit function to be considered is minimizing the expected value of th e minimum function value122 ( MF4). The expected value is simply the predicted value at a given lo cation minus the expected im provement. The criterion MF4 is similar to MF3, but the search tends to be more local. Sasena et al.122 suggests that MF4 should be used only when there is confidence that the optimum region has been found. minmin min 4ˆˆ ˆˆ ,0 00 fyfy yfyVV MF VV V xx xxx x xx (2-37) The procedure is developed such that it acc ounts for the presence of multiple objectives. In this way, a single point can be selected to improve the response surface across multiple objectives. The procedure involves applying a DOE with a large number of points across the design space. The points are then divided into cl usters such that the number of clusters equals the number of desired new points. Clustering the data prior to selection ensures that the data are spread sufficiently across the design space, reduci ng the possibility of choosing two points that are very near each other. For the purpose of mu lti-objective optimization, the characteristics for each objective need to be combined before the best points can be selected. The points are selected based on a simple weighting function. In this study, the data is clustered based on proximity using the Matlab function kmeans For a given cluster m a rank R is given to each point n

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64 ,,min 1 ,max,minmm k knk m n i kkFF R FF (2-38) where F is the value of the selection criterion (merit function, function value, etc.), ,minm kF is the point with the smallest value of F within the cluster m n is a point from the cluster m and ,max kF and ,min kF are the maximum and minimum values F respectively, within the entire data set for objective k For MF1 and MF4, the point in each cluste r with the minimum value R is selected. For MF2 and MF3, the point in each cluster with the maximum value R is selected. The full procedure for using merit functions to select points is as follows: 1. Construct a response surface using Kriging fo r each objective using an initial sampling. 2. Determine if the design space size should be reduced. Construct a DOE using a large number of points M within the new design space. 3. Choose a merit function and calculate the merit function value for each objective k at each point M in the new DOE using the predicte d function value and variance. 4. Calculate R at each point M as determined by Equation 2-38. 5. Choose the desired number of points m to be sampled from the new DOE. The number of points m is selected by the user according to th e number of sample points that can be afforded by the simulation or experiment. The quantity M should be much larger than m 6. Group the points M into m number of clusters. 7. Select the point that has the best value of R from each cluster. The final set of m design points will satisfy three basic criteria: 1. The new points will be evenly spread across the new design space 2. The selected points will be selected such that they are predicted to improve the function values in each objective 3. The selected points are those in each cluster predicted to best reduce the errors in the surrogate models for each objective.

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65 2.4.4 Method of Alternative Loss Functions By modifying the least squares loss function, it is possible to fit several similar, but slightly different polynomial response surfac es to the same set of data. For adequately refined design spaces, it can be shown that the difference among these polynomials is negligible. For design spaces that require some type of refinement, larg e differences can exist. This property of using the differences in different surrogates highlight potential problems in the surrogate model or in the data is not new.75 However, the proposed method has an advantage in that surrogates of the same type can be compared requiring no change in the data set. This may enable a quantitative measurement in the degree of refi nement needed in a problem. For a polynomial response surface the prediction at any point i is given as 1ˆk ijij jybx (2-39) where k is the number of terms in the polynomial response surface and xij are values from the matrix X from Equation 2-3. For example, at any point i a quadratic response surface with two variables can be given as 22 011213241252ˆybbxbxbxbxxbx (2-40) or simplified to 01122334455ˆybbxbxbxbxbx (2-41) The notation used for these and the followi ng equations are adapted from Myers and Montgomery.46 To determine the coefficients bj, the loss function to be minimized can be defined as 0 1111ˆp nnnk pp iiiijij iiijLyyybbx (2-42)

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66 where is the error given by the differe nce between the actual response y and the predicted response and k is the number of terms in the response surface equation. Fi gure 2-7 illustrates the effect of changing the value of p in the loss function formulation. For a traditional least squares loss function, p = 2. The coefficient vector bj must be determined such that the loss function is minimized. To do so, the partial derivatives must be set equal to zero 1 1 0 1 1sgn0 sgn0,1,2,,n p ii i n p iiij i jL p b L p xjk b (2-43) resulting in a set of simultaneous equa tions that must be solved where sgn(i) = i /| i | The coefficient vector b can be determined using any met hod for solving simultaneous nonlinear equations. In this study, the Matlab function fsolve can be used to obtain the coefficients. Preliminary tests were conducted on data from the preliminary optimization of a radial turbine. Details of this optimi zation are given in Chapter 3. This optimization problem involved the refinement of the design space for a problem with two objectives and involved screening data (Data Set 1) to determine the feasible design space (Data Set 2). Figure 2-8 shows the results of response surfaces fit to Data Set 2 and illustrate s how the sums of the squares of the errors (2 1sN i iSSE ) of a quadratic res ponse surface vary with p for two objectives. In general, SSE increases as p increases. The amount of change in SSE depends on the size of the design space. This can be better illustrated through use of Pareto fronts. Figure 2-9 shows the Pareto fronts construc ted using response surfaces from the two objectives for coordinating values of p. In this case, data is used from the preliminary optimization of a radial turbine, and the objectives are the turbine weight, Wrotor, and the total-tostatic efficiency, ts. Figure 2-9A shows a large difference in the Pareto fronts for an identical

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67 data set and identical forms of the RSAs The only difference was the value of p in the loss function. After the design space was refined (Data Set 3) to improve the accuracies of the RSAs, the Pareto fronts based on varying values of p are quite similar. To study the quantitative differences, the curves were integrated to obtai n the area under the Pareto front curves, and the results were compared to the Pareto front usi ng RSAs obtained from a standard least squares regression. Figure 2-10 shows that the differences in the Pareto fronts from the unrefined design space were very large. In contrast, the Pare to fronts from the refined design space differed by less that five percent. This may be a useful tool in determining wh ether a design space is adequately refined. Table 2-1. Summary of DOE and surrogate modeling references. Author Application DOE Surrogate(s) Key results Han et al.64 Multiblade fan/scroll CCD Quadratic RSA RSA using CCD required no special treatment, in this case Sevant et al.86 Subsonic wing CCD RSA CCD used to ensure equal variance from center of design sp ace. RSA used to smooth numerical noise Rais-Rohani and Singh85 Composite structures LHS, OA, random RSA Accuracy and efficiency of RSA varies depending on choice of DOE Kurtaran et al.88 Crash-worthiness Factorial design with Doptimal Linear, elliptic, quadratic RSA Difference between RSAs reduces as design space size reduces Hosder et al.89 Aircraft configuration D-optimal RSA A 30 variable optimization required design space refinement and multifidelity analysis Mahadevan et al.92 Engineering reliability analysis IMSEa 1st-order RSA, Kriging For very small number of points, Kriging performed better than 1storder RSA Forsberg and Nilsson93 Crash-worthiness D-optimal RSA, Kriging Kr iging performs better than RSA in this case Jin et al.94 Analytic test functions LHS RSA, MARSb, RBF*, Kriging Kriging more susceptible to noise than other surrogates Rijpkema et al.95 Analytic test functions Full factorial RSA, Kriging Kriging better captures local perturbations, but causes problems when strong noise exists

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68 Table 2-1. Continued. Author Application DOE Surrogate(s) Key results Chung and Alonso96 Supersonic business jet CCD, MPDc RSA, Kriging Kriging better at fitting when there are several local optima Simpson et al.97 Aerospike nozzle OA RSA, Kriging Kriging and RSA produce comparable results Mahajan et al.102 Radiation shield Full factorial NN Gain information on system trends and shield performance Papila et al.78 Supersonic turbine FCCD, OA, D-optimal NN, RSA NN data used to supplement CFD data for RSA Shi and Hagiwara105 Crash-worthiness CCD with Doptimal RSA, NN Maximized vehicle energy dissipation aIMSE – integrated mean square error92 bMARS – multivariate adaptive regression splines124 cMPD – minimum point design96 Table 2-2. Design space refineme nt (DSR) techniques w ith their applications and key results. Author Application DSR Technique Key Results Balabanov et al.70 Transport wing Reasonable design space Unreasonable designs eliminated and surrogate accuracy improved Roux et al.73 Structural optimization Design space windowing Polynomial RSA accuracy sensitive to design space size Papila et al.82 Supersonic turbine Design space reduction Design space reduction substantially improved RSA accuracy Rais-Rohani and Singh85 Structural optimization Sequential response surface Successive small design spaces more efficient than large design space Bosque-Sendra et al.125 Chemical applications Sequential response surface Box-Behnken design used to move and resize design space while including previously used points Papila et al.77 Preliminary turbine design Design space windowing Windowing improved RSA accuracy over global RSA Jeong and Obayashi119 2D airfoil shape Smart point selection using EGO Used expected improvement uncertainty parameter for Pareto front fitness to select new design points Knowles120 Arbitrary functions Pareto front refinement using EGO (ParEGO) ParEGO more efficient than random search for Pareto front construction Farhang-Mehr and Azarm121 Crash analysis Smart point selection based on function irregularities Increasing point density in irregular function regions improves surrogate accuracy using minimal points

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69 Figure 2-1. Optimization framework flowchart. Problem definition and optimization setup 2. Design space refinement Refinement necessary? 3. Dimensionality reduction check 4. Multi-objective optimization using Pareto fron t Two or more confliction objectives? Find/choose optimum Design of Experiments Numerical simulations at selected locations Construction of surrogate models (model Selection and identification ) Model validation 1. Surrogate modeling yes no no yes Iterate, if necessary

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70 A x 1 x2 B x 1 x 2 Figure 2-2. DOEs for noise-reduc ing surrogate models. A) Ce ntral composite design and B) face centered cubic design for two design vari ables. The extreme points are selected around a circle for CCD and a square for FCCD. A x1 x2 B x1 x2 Figure 2-3. Latin Hypercube Sa mpling. A) LHS with holes in the design space. B) Orthogonal array LHS can help address this issue by filling the design space more evenly.

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71 Original design space x 1 x 2 Refined design space Figure 2-4. Design space window ing showing optimum based on original design space and the final optimum based on the refined design space. actual function kriging model merit function original data points new data points Figure 2-5. Smart point selecti on. The final Kriging approxim ation of the actual function is shown. The new point added in each cycle is shown along with the three points at the function ends and center that comprised the original Kriging appr oximation. In this case, points are selected based on the locat ion of the minimum merit function value.

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72 Figure 2-6. Depiction of the merit functi on rank assignment for a given cluster given by Equation 2-38 for two objectives where min mmm nnaFF Figure 2-7. The effect of varying values of p on the loss function shape. A traditional least squares loss function is shown by p = 2.

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73 1 2 3 4 5 1.7 1.8 1.9 2 2.1 2.2 SSE for objective 1 py1 1 2 3 4 5 0.06 0.08 0.1 0.12 0.14 0.16 SSE for objective 2 py2 Figure 2-8. Variat ion in SSE with p for two different responses in the preliminary optimization of a radial turbine for the unref ined design space (Data Set 2). A B Figure 2-9. Pareto fronts for RSAs constructed with varying values of p for A) the original design space and B) the refined design space. The blue dots represent actual validation data along the Pareto front cons tructed on the refined design space using standard least squares regression.

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74 Figure 2-10. Absolute percent difference in the area under the Pareto front curves for the original feasible design space (RS 2) and the refined design space (RS 3) for various values of p as compared to a traditional least squares loss function (p = 2).

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75 CHAPTER 3 RADIAL TURBINE OPTIMIZATION Surrogates can be used to obtain necessary information about the design space. This information can be used to help refine the de sign space to improve the acc uracy of the surrogate model. The following case study demonstrates how response surfaces can be used to provide constraint information necessary in refining the design space to pr event infeasible CFD runs. It also shows ways in which the design space can be refined to a small region of interest by using Pareto fronts to simultaneously satisfy two objectives. A response surface-based dual-objective de sign optimization was conducted in the preliminary design of a compact radial turbin e for an expander cycle rocket engine. The optimization objective was to increase the effi ciency of the turbine while maintaining low turbine weight. Polynomial response surface ap proximations were used as surrogates and the accuracy of such approximations improves by limiting the size of the domain and the number of variables of each response of interest. This wa s done in three stages using an approximate, onedimensional model. In the first stage, a relativel y small number of points were used to identify approximate constraint boundaries of the feasib le domain and reduce th e number of variables used to approximate each one of the constraints. In the second stage, a small number of points in this approximate feasible domain were used to identify the domain wh ere both objectives had reasonable values. The last stage focused on obtaining high accuracy approximation in the region of interest with a large number of points. The approximations were used to identify the Pareto front and perform a global sensitivity an alysis. Substantial improvement was achieved compared to a baseline design. A second study was conducted on the radial turb ine data after the first optimization study was completed. The research effort was also used to test strategies of improving the efficiency

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76 of the DSR process by reducing the necessary nu mber of points for use in the multi-objective optimization of computationally expensive pr oblems. These improved DSR strategies were applied to the optimization of a compact radial turbine, which was originally optimized using traditional response surface methodology and Pareto fr onts. Merit functions were used to reduce the number of points in the analysis, while ensu ring that the accuracy of the surrogate was maintained. In particular, the use of merit f unctions in a multi-objective optimization process was explored. This research seeks to explore th e benefits of using merit functions to select points for computationally expens ive problems. The purpose of th e proposed analysis procedure is twofold. First, it must be en sured that using a given selection cr iterion, such as merit functions to select the data points, result s in better surrogate accuracy th an using a seemingly more obvious method, such as simply choosing points with the best function values. Second, it must be determined whether the use of one merit function for the purpose of computationally expensive multi-objective optimization is superi or over the other merit functions. 3.1 Introduction In rocket engines, a tu rbine is used to drive the pumps that deliver fuel and oxygen to the combustion chamber. Cryogenic fuel is heated, resu lting in a phase change to a gaseous state. The increased pressure drives the turbine. Howeve r, the survivability of turbine blades limits the degree to which the fuel can be heated. There is a limited amount of h eat available from the combustion process with which to preheat the fuel, resulting in low chamber pressure and temperature. This can be an advantage because lower fuel temperatures can improve turbine reliability. However, the low chamber pressure means that the turbine work output is also limited. Turbine work can be increased in two ways: increasing the available energy in the drive gas or improving the efficiency by which the turbin e can extract the available energy. Increasing

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77 the available energy for a given fluid is accomplishe d by increasing the turbine inlet temperature. To increase the turbine inlet temp erature in an expander cycle, a higher heat flux from the thrust chamber to the cooling fuel is needed. Obtaining this higher heat flux is problematic in several ways. First, materials and manuf acturing development is necessary to produce a thrust chamber with high heat flux capability. This work is an ongoing area of tec hnology development. Second, to enable a higher heat exchange, in creased surface area and co ntact time between the thrust chamber and cooling fuel is needed. Th ese requirements lead to a larger, and heavier, thrust chamber. In addition, significantly raising the turbin e inlet temperature defeats the expander cycle’s advantage of maintaining a be nign turbine environment. Immich et al.126 reviewed methods of enhancing heat transfer to the combustion chamber wall in an expander cycle. The authors tried three methods of enhanc ing heat flux to the unburned fuel: 1) Increase the length of the combustion chamber, 2) in crease the combustion chamber wall surface area by adding ribs, and 3) increase the combusti on chamber wall roughness. The authors also mentioned that in the future they would expe rimentally investigate the influence of the distribution of the injector elem ent distribution and the effect of the distance of the injector element to the wall on the heat transfer to the wall. However, results of the future analysis are not available. The second approach to increasing turbine work is to improve turbine efficiency. If the turbine inlet temperature is held constant, an increase in turbine work is directly proportional to efficiency increase. If the required work can be achieved with moderate efficiency, an improvement in that efficiency can be traded for reduced inlet temper atures, providing better design environment margins. One way to improve tu rbine efficiency is to use a radial turbine. Radial inflow turbines perform better than axial turbines at high velocity ratios, exhibit better

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78 tolerance to blade incide nce changes, and have lo wer stresses than axial de signs. Radial turbines have been used successfully in automotive applications, but are not often used in rocket engines due to their relatively large size and weight. The size of the comp act radial turbines makes them applicable to rocket engine cycles wh en a high velocity ratio is involved. This research will focus on improving turbine efficiency. The radial turbine design must provide maximum efficiency while keeping the overall weight of the turbine low. This necessitates a multi-objective optimization. A response surface analysis46 provides an efficient means of tackling the optimization problem. The research presented here represents the preliminary optimization of a radial turbine using a simplified 1-D radial turbine model adapted from the 1-D Meanline127 code utilized by Papila et al.82 The Meanline code provides performance and geometry predictions based on selected input conditions. It is an approximate and inexpensive model of the actual processes. Because it is an approximate model, there is some degree of uncertainty involved, but it can provide a good starting poin t in the design process. Using response surface analysis, an accurate surrogate model was constructed to predict the radial turbine weight and the efficiency ac ross the selected design space. The surrogate model was combined with a genetic algorithm-bas ed Pareto front construction and facilitates global sensitivity evaluations. Because the radial turbine represented a new design, the feasible design space was initially unknown. Techni ques including design constraint boundary identification and design space reduction were nece ssary to obtain an acc urate response surface approximation (RSA). The analysis used the optimization framewor k outlined in Chapter 2. The framework steps included in the analysis are 1) modeling of the objec tives using surrogate models, 2) refining the design space, 3) reduc ing the problem dimensionality, and 4) handling

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79 multiple objectives with the aids of Pareto fr ont and a global sensitivity evaluation method. Finally, the ability of Merit Func tions to improve the efficiency of the optimization process was demonstrated. 3.2 Problem Description The radial turbine performance was simulate d using the 1-D Meanli ne code. Using a 1-D code allowed for the availability of relatively inexpensive computations To determine whether using the 1-D code was feasible for optimi zation purposes, a 3-D ve rification study was conducted. Once the Meanline code was verified, th e radial turbine optimi zation could proceed. 3.2.1 Verification Study Three-dimensional unsteady Navi er-Stokes simulations were performed for the baseline radial turbine design, and the pr edicted performance parameters were compared with the results of the Meanline analysis. Simulations were perf ormed at three rotational speeds: the baseline rotational speed of 122,000 RPM, a low speed of 103,700 RPM and a high-speed of 140,300 RPM. A 2-vane/1-rotor model was used. The simulations were run with and without tip clearance, and the computational grids contained approximately 1.1 million grid points. The PHANTOM code was used to pe rform the numerical simulations.128 The governing equations in the PHANTOM code are the three-di mensional, unsteady, Navier-Stokes equations. The equations have been written in th e Generalized Equatio n Set (GES) format,129 enabling it to be used for both liquids and gases at operating conditions ra nging from incompressible to supersonic flow. A modified Baldwin-Lomax tur bulence model is used for turbulence closure.130 In addition to the perfect gas approximation, the code contains two options for the fluid properties. The first option is based on the equations of state, thermodynamic departure functions, and corresponding state pr inciples constructed by J. C. Oefelein at Sandia Corporation in Livermore, California. The second option is based on splines generated from the NIST

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80 Tables.131 A detailed description of th e code/algorithm development, as well as its application to several turbine and pump te st cases, is presented in Venkateswaran and Merkle129 and Dorney et al.132 Figure 3-1 shows static pressu re contours (psi) at the mid-he ight of the turbine for the baseline rotational speed of 122,000 RPM. This figur e illustrates the geometry of the turbine, and the contours indicate that the pressure decrease is nearly evenly divided between the vane and the rotor. In fact, the reaction was approximately 0.60 for each of the th ree 3-D simulations as compared to 0.55 for the 1-D simulation. Figur e 3-2 contains the predicted total-to-static efficiencies from the Meanline and CFD analys es. The CFD results include values with and without tip clearance. In genera l, fair agreement is observe d between the Meanline and CFD results. The trends are qualitatively similar, but the Meanline analysis predicts higher efficiencies. There is approxima tely a four-point difference in the quantitative values. The quantitative differences in the results are not surp rising considering the lack of experimental data available to anchor the Meanline code. The differences in the pr edictions with and without tip clearance decrease with increasing rotational spee d. Figure 3-3 shows the predicted work from the Meanline and CFD analyses. The trends are again similar between the Meanline and CFD analyses, but the Meanline values are consiste ntly 5 – 6% higher than the CFD values. The similar trends between the 1-D Meanline code and 3-D CFD analys es indicate that the optimization can be confidently performed on the 1D Meanline code. It can be expected that for a given turbine speed the Meanline code will over-predict the total-to-static efficiency by an expected degree. The predicted optimum point ba sed on the 1-D Meanline code will likely yield overly optimistic results, but the predicted degr ee of improvement should translate to the 3-D CFD analysis.

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81 3.2.2 Optimization Procedure The purpose of the design optimization is to maximize the turbine efficiency while minimizing the turbine weight. A total of six desi gn variables were identifie d. The ranges of the design variables were set based on current design practices. Additionally, five constraints were identified. Two of the five cons traints are structural constraints, two are geometric constraints, and one is an aerodynamic constraint. The aerodynamic constraints are based on general guidelines. The descriptions of a ll objectives, variables, and constr aints are given in Table 3-1. It is unknown in what way the constraints depend on the design variable s. It is possible that certain combinations of desi gn variables will cause a constraint violation. It is also unknown whether the selected ranges resu lt in feasible designs. The res ponse surface analysis can help clarify these unknown factors. The design of experiments (DOE) procedure was used to select the location of the data points that minimize the effect of noise on the fitted polynomial in a response surface analysis. A face-centered cubic design was used to generate a total of 77 data points within the selected ranges. 3.3 Results and Discussion 3.3.1 Phase 1: Initial Design of Experiments a nd Construction of Constraint Surrogates The values of the objective functions were obt ained using the Meanline code. Of the 77 solutions from the initial DOE, seven cases failed and 60 cases violated one or more of the five constraints, resulting in only 10 successful cases. Before the op timization could be conducted, a feasible design space needed to be identified. Because there was limited information on the dependencies of the output constraints on the de sign variables, response surface analyses were used to determine these dependencies. Response su rfaces were used to properly scale the design variable ranges and identify ir regular constraint boundaries.

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82 Quadratic response surfaces were fit to the output constraint s. The change in each response surface with respect to changes in some of design variables was small, indicating that the response surfaces were largely insensitive to ce rtain design variables. The accuracy of the response surfaces was not compromised when the identified design variables were neglected. Therefore, for each response surf ace, the effects of va riables that contribu ted little to the response surface were removed. In this wa y, each response surface was simplified. The simplified dependencies are shown in Equation 3-1. 22 11,, ,, ,, ANANAnsqrFrac TipSpdTipSpdU/Cisen Cx2/UtipCx2/UtipRPMU/CisenAnsqrFrac ReactU/CisenTipFlw Rsex/RsinRsex/RsinAnsqrFracU/CisenDhex% (3-1) The information obtained from the response surfaces about the output constraint dependences were further used to develop c onstraints on the design va riables. A quadratic response surface was constructed for each design va riable as a function of the output constraint and the remaining design variables, for example, 1,, R eactReactU/CisenTipFlw (3-2) The most accurate response surfaces (R2 adj 0.99) were used to dete rmine the design variable constraints. The output c onstraints were in turn set to th e constraint limits. For example, a constraint on React was applied to coincide with the constraint 1 0: 11 1,,0 0,, ReactU/CisenTipFlw R eactReactU/CisenTipFlw (3-3) Constraint boundary approximati ons were developed in this manner for each constraint. As can be seen from Equation 3-1, two of the five constraints (AN2 and Tip Spd) were simple limits on a single variable. For the single variable constraints, the variable ranges were simply

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83 reduced to match the constraint boundaries. Th e remaining constraints were more complex. However, one of the complex constraints (Rsex/Rsin) was automatically satisfied by the reduction of the variable ranges for the single vari able constraints. The f easible region is shown in Figure 3-4. The remaining constraints involved three vari ables each. It was discovered that many low values of RPM violated the Cx2/Utip constraint. The region of violation was a function of RPM, U/C isen, and AnsqrFrac as shown in Figure 3-5. The 1 constraint was found to be the most demanding and resulted in a feasib le design space as shown in Figur e 3-6. Much of the original design space violated this cons traint. It was also discovere d that the constraint surface representing the bounds for 1 40 lay outside of the origin al design variable range for React. In this case, the lower bound for React was sufficient to satisfy this constraint. The predictive capability of the constraints wa s tested using the available data set. The design variable values were input into the RSAs for the constraints. Using the RSAs, all points that violated the output constraints were correc tly identified. Now that the feasible design space was accurately identified, data points could be placed in the feasible data region. The results and summary of the prediction of constr aint violations are as follows: 1. Quadratic response surfaces we re constructed to determine the relationship between the output constraints and the design variables. 2. The variable ranges were adjusted based on information from the constraint surfaces. 3. A 3-level factorial design (729 points) wa s applied within new variable ranges. 4. Points that violated any cons traints (498 / 729 points) were eliminated based on RSAs of constraints. Using the response surface constraint appr oximations, 97% of the 231 new data points predicted to be feasible lay w ithin the actual feasib le design space region after simulation using the Meanline code. The point s that were predicted to be feas ible, but were actually found to be

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84 infeasible, often violated the 1 0 requirement by a slight amount. The resulting data set contained 224 data points. 3.3.2 Phase 2: Design Space Refinement Plotting the data points in function space re vealed additional information about the location of the design space as shown in Figure 3-7. A large area of function space contained data points with a lower efficiency than was de sired. There also existed areas of high weight without improvement in efficiency. Additionally the fidelity of the response surface for ts was apparently compromised by the existen ce of a design space that was too large. These undesirable areas could be eliminated, and the response surf ace fidelity could be improved by refining the design space. The density of points could then be in creased within the region of interest, eliminating the possibility of unnecessary invest igation of undesirable points. The region of interest is shown in Figure 3-7. To further test the necessity of a reduced desi gn space, five RSAs each were used to fit the data for Wrotor and ts in the original feasible design space. These response surfaces were constructed using the general loss function given in Equation 2-42 for p = 1…5. The RSA constructed using the least square loss function (p = 2) was used as a reference point. As seen in Figure 3-8, regardless of the RSA used, the error in the RSA at the data points is high at high Wrotor and low ts. To improve RSA performance, additiona l data points could be added in these regions, or these data regi ons could be eliminated. Pareto fronts were constructed for each RSA se t. The results are show n in Figure 3-9. In this case, the Pareto fronts differ by as much as 20%. Because the results differ significantly depending on which RSA is used, this also indi cates that further design space refinement is necessary.

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85 The design variable bounds were further redu ced to match the new design space. The new design variable ranges are given in Table 3-3. Quadratic response surfaces were constructed for the turbine weight, Wrotor, and the turbine total-to-static efficiency, ts, using the original feasible design space to screen data points. Points pred icted to lie outside of the refined design space would be omitted. For the refined design space, a third set of data was required. As seen in Figure 3-7, using a factorial desi gn tended to leave holes in functi on space. It was possible that this could hamper construction of an accurate Pareto front. To prevent this, Latin Hypercube Sampling was used to help close possible holes for the third data set. The points were efficiently distributed by maximizing the mini mum distance between any points. Of the best points from the second data set, it only RPM, Tip Flw, and U/C isen varied, while React and Dhex% remained constant at their lowest values, and AnsqrFrac remained constant at its maximum value among these points. To ensure that this effect was captu red in the third data set, additional points were added using a 5-level factorial de sign over these three variables. The remaining variables were held constant according the values observed in the best trade-off poi nts. Quadratic response surfaces previously constructed for the turbine weight, Wrotor, and the turbine total to static efficiency, ts, were used to screen the potential data poi nts. Points predicted to lie outside of the newly refined design space would be omitted from the analysis. In summary, 1. Only the portion of design space with best performance was reserved to allow for a concentrated effort on the region of interest and to increase response surface fidelity. 2. Latin Hypercube Sampling (181 / 300 feasible points) was used over all six variables and was supplemented by a 5-level factorial design used over RPM, Tip Flw, and U/C isen (119 / 125 feasible points) to improve re solution among the best trade-off designs. 3. Points that were predicted to violate constrai nts or lie outside of re gion of interest were omitted. The combination of the DOEs resulted in a total of 323 feasible design points.

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86 3.3.3 Phase 3: Construction of the Pareto Fr ont and Validation of Response Surfaces As in the second data set, five RSAs each with p = 1…5 were used to fit the data for Wrotor and ts for the third data set. Pareto fronts were constructed for each RSA set and are shown in Figure 3-10. In this case, the Pareto fronts differed by a maximum of only 5%. Because the difference in the response surfaces for varying values of p is small for the third data set, the design space was determined to be adequately refined. Function evaluations from the quadratic response surfaces (p = 2) were used to construct the Pareto Front shown in Figure 3-11. Within the Pareto front, a region was identified that would provide the best value in terms of maxi mizing efficiency and minimizing weight. This trade-off region was selected for the validation of the Pareto front. The results of the subsequent validation simulations indicated that the respon se surfaces and corresponding Pareto front were very accurate. A notable improvement was attain ed compared to the baseline radial turbine design. The design selected optimum design had the same weight (Wrotor) as the baseline case with approximately 5% improvement in effici ency. The specifications for the optimum design are given in Table 3-4. Within the best trade-off region, only RPM and Tip Flw vary along the Pareto front as seen in Figure 3-12. The other variables are constant within the trade-off region and are set to their maximum or minimum value. This indicates that increasing the range of one of these variables might result in an increase in performan ce. The minimum value of the variable React was chosen as the only variable range that could reas onably be adjusted. The validation points were simulated again using a reduced React value. Reducing the minimum value of React from 0.45 to 0.40 increased the maximum efficiency only for Wrotor > 1. The maximum increase in efficiency improved from 4.7% to 6.5%, but this increase occu rred outside of the preferred trade-off region.

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87 It was further determined that using a React value of 0.40 resulted in a turbine design that was too extreme. Based on the results of the current study, seve ral observations can be made concerning the design of high-performance radial turbines. 1. The more efficient designs have a higher velo city ratio based on the rotational speed as shown by U/C isen in Figure 3-12. This is a re sult of the inlet blad e angle being fixed at 0.0 degrees due to structural considerations. Classic radial turbine designs have velocity ratios of approximately 0.70. 2. The more efficient designs have a smaller tip radius and larger bl ades (AnsqrFrac) as compared to designs with a larger radius and smaller blades. Incr easing the annulus area leads to higher efficiencies. In this case, th e annulus area (AnsqrFrac) is set to the highest value allowable while respecting stress limitations. 3. The higher performing designs have higher rota tional speeds (as compared to the original minimum RPM value of 80,000 given in Table 31) as a result of th e smaller radius and large blades. The higher rotational speeds also lead to more efficient pump operation. 3.3.4 Phase 4: Global Sensitivity Analysis and Dimensionality Reduction Check A global sensitivity analysis was conducted using the response surface approximations for the final design variable ranges gi ven in Table 3-4. The results ar e shown in Figure 3-13. It was discovered that the tu rbine rotational speed RPM had the largest impact on the variability of the resulting turbine weight Wrotor. The effects of th e rotational speed RPM along with the isentropic velocity ratio U/C isen make up 97% of the variability in Wrotor. All other variables and variable interactions have minimal effect on Wrotor. For the total to static efficiency, ts, the effects of the design variables are more evenly distributed. The reaction variable, React, has the highest overall impact on ts at 28%. This information is useful for fu ture designers. For future designs, it may be possible to eliminat e all variables except RPM and U/C isen when evaluating Wrotor, whereas for ts, it may be necessary to keep all variables.

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88 It must be noted that the local variability at any given point can vary significantly from the global sensitivity values due to the nonlinear natu re of the RSAs. However, the global sensitivity analysis can be scaled to the region of in terest to explore more localized effects. 3.4 Merit Function Analysis Merit functions were used in an attempt to reduce the number of points in the DSR analysis as compared to the orig inal radial turbine optimization st udy. They were used to select data points such that the accura cy of the surrogate model can be improved using a minimal number of data points. The merit functions rely on predictio ns of the function values and variance and attempt to balance the effects of th e two. Kriging was selected as the surrogate model for the optimization. The merit function analysis demonstrates how the merit functions, facilitated by Kriging, can be integrated into the DSR process. 3.4.1 Data Point Selection and Analysis First, a Kriging model is fit to the 231 points that comprised the feasible design space constructed as the result of the first cycle in th e optimization process. This feasible design space is designated DS1. Next, the 323 points from the refined design space constructed during the second cycle of the optimization process are used as a databank. This reasonable design space is designated DS2, but no function valu es are initially made at these points. Out of the databank of data point locations from DS2, points are clus tered based on proximity, and the point from each cluster that is predicted to have the best charac teristics based on merit func tion values is selected for function evaluation. It is a ssumed that the points in DS2 lay very near the suspected optima. The procedure used in the anal ysis is given as follows: 1. Fit Kriging model to points in the feasib le design space (DS1) for each objective. 2. Using Kriging model from DS1, predict function valueˆ yand prediction variance V(x) at each point in reasonable design space (DS2) for each objective.

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89 3. Calculate merit function values at each point in DS2 based on Eqs. (2-33) through (2-37). 4. Normalize variables by the design variable range and cl uster the data points from DS2 into m groups, based on proximity. 5. Select 6 sets of m points each: a. Choose the data points with the best rank R from each cluster and predict objective value at those points for each of the four merit functions. b. Choose the points with the best rank R based on the minimum function value fmin from each cluster using Equation 2-38. c. Choose the data points that lie cl osest to the cluster centers. 6. Evaluate objectives at each point in each of the six data sets. 7. Fit Kriging model to the points in each of the six data sets. 8. Use Kriging model to predict objective values along Pareto front for each set. 9. Calculate the errors in the objec tive values for each scenario. 10. Repeat steps 1 through 9 for 100 times such th at the random variati on that occurs when clusters are selected sets is minimal. 11. Compare accuracy of different selection criteria among the 6 selection criteria. 12. Repeat steps 1 through 11 using m = 20, 30, 40, 50 clusters (with one data point selected per cluster) to check sensitivity to number of data points. 13. Compare accuracy among the 6 selection criteria for different numbers of data points. 3.4.2 Merit Function Comparison Results The points along the validated Pareto front gi ven by the blue curve in Figure 3-11 were predicted by each of the six new Kriging approximations based on MF1, MF2, MF3, MF4, fmin, or the cluster centers. The differences were small for data sets containing 50 data points, so the results shown in Figure 3-14 and Fi gure 3-15 are illustrated using the results of the smallest data set. The results for a selected cluster set a nd 20 data points are shown in Figure 3-14. Although only 20 points were selected by maximizing the probability of improvement (MF2) and used to generate the Kriging model, the predicted points only vary sligh tly from the actual values along

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90 most of the Pareto front. This indicates that the 20 data points selected by MF2 actually lie on or very near the Pareto front. Because Kriging inter polates the data points, the prediction of points near the 20 data points is very good which leads to the construction of an accurate Pareto front. It is likely that the pr ediction of points away from the Pare to front would be very poor. Other selection criteria were not as successful in predicting points along the Pareto front. The absolute scaled error at each point is calculated as maxminˆscaledyy e yy (3-4) and the absolute combined error is given as ,1 2rotortscombinedscaled Wee (3-5) The error distribution for data points along the Pare to front for each selection criterion is given in Figure 3-15. The use of MF2 results in the lowest combined e rror. Thus, it is demonstrated qualitatively that se lecting points using MF2 results in the highest accuracy for data sets with a low number of points. Due to the random nature of the Matlab function kmeans that is used to generate the clusters, the cluster sets can be slightly different depending on the starting point of the search, especially if the data set being clustered is well distributed. For this r eason, 100 different cluster sets were used to reduce effects due to cl uster selection. The mean and maximum error distributions were compared for various numbers of data points and th e different selection criteria and are shown in Figure 3-16 and Figure 3-18, respectivel y. Figure 3-17 and Figure 3-19 provide direct comparisons of a ll selection criteria us ing the median values from Figure 3-16 and Figure 3-18, respectively. By usi ng the probability of improvement MF2 as a selection criterion, the overall accuracy was maintained to a surprising degree. Except for MF1, using merit

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91 functions as selection criteria a ppeared to result in better accuracy than simply selecting the point with the minimum function value or centers from each cluster. Using MF1 gave results that were very similar to using fmin. When the maximum error was considered, however, all merit functions except MF2 resulted in higher maximum errors as the number of points decreased. Selecting points based on maxi mizing the probability of impr ovement resulted in the best performance given a small number of data points as compared to simply choosing the minimum function value or cluster centers. The results in the use of the remaining merit functions were mixed. Maximizing the probability of improveme nt, maximizing the expected improvement, or minimizing the expected value of the minimum f unction value resulted in lower average errors than selection based on minimi zing the statistical lower bound, minimum function value, or cluster centers. However, simply selecting th e cluster centers seemed to limit the maximum error better than any merit function except maxi mizing the probability of improvement. In this analysis, it was possible to reduce the number of points in the final op timization cycle by 94% while keeping the accuracy of points along the Pare to optimal front within 10% with an average error of only 3% or less. 3.5 Conclusion An optimization framework can be used to f acilitate the optimization of a wide variety design problems. The liquid-rocket compact radial turbine analysis demonstrated the applicability of the framework: Surrogate Modeling. The radial turbine optimization process began without a clear idea of the location of the feasible design region. RSAs of output constraints were successfully used to identify the feasible design space. Design Space Refinement. The feasible design space wa s still too large to accommodate the construction of an accurate RSA for the prediction of turbine efficiency. A reasonable design space was defined by eliminating poor ly performing areas thus improving RSA fidelity.

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92 Dimensionality Reduction. A global sensitivity analysis provided a summary of the effects of design variables on objective variab les, and it was determined that no variable could be eliminated from the analysis. Multi-objective Optimiza tion using Pareto Front. Using the Pareto front information constructed using genetic algor ithms, a best trade-off region was identified within which the Pareto front and the response surfaces used to create the Pareto front were validated. At the same weight, the RS optimization result ed in a 5% improvement in efficiency over the baseline case. Merit Functions. A methodology was presented to use merit functions as a selection criteria to reduce the number of points requ ired for multi-objective optimization. Using merit functions gives the ability of dramatical ly reducing the number of points required in the final cycle of multi-objective optimization. Through this study, a number of aspects from the framework were demonstrated, and the benefits of the various steps were made a pparent. The framework provides an organized methodology for attacking several issues that arise in design optimization. Table 3-1. Variable names and descriptions. Objective variables Description Baseline design Wrotor Relative measure of “goodness” for overall weight 1.147 ts Total-to-static efficiency 85% Design variables MIN Baseline MAX RPM Rotational Speed 80,000 122,000 150,000 React Percentage of stage pressure drop across rotor 0.45 0.55 0.70 U/C isen Isentropic velocity ratio 0.50 0.61 0.65 Tip Flw Ratio of flow parameter to a choked flow parameter 0.30 0.25 0.48 Dhex % Exit hub diameter as a percent of inlet diameter 0.10 0.58 0.40 AnsqrFrac Used to calculate annulus area (stress indicator) 0.50 0.83 1.0 Constraints Desired range Tip Spd Tip speed (ft/sec) (stress indicator) 2500 AN2 Annulus area speed2 (stress indicator) 850 1 Blade inlet flow angle 0 1 40 2C/Utip Recirculation flow coefficient (indication of pumping upstream) 0.20 Rsex/Rsin Ratio of the shroud radius at the exit to the shroud radius at the inlet 0.85

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93 Table 3-2. Response surface fit statistics before (feasible DS) and afte r (reasonable DS) design space reduction. Feasible DS Reasonable DS Wrotor ts Wrotor ts R2 0.987 0.9170.9960.995 R2 adj 0.985 0.9050.9960.994 Root Mean Square Error 0.094 0.0200.02350.00170 Mean of Response 1.04 0.7711.040.844 Observations 224 224310310 Table 3-3. Original and final de sign variable ranges after constr aint application and design space reduction. Original ranges Final ranges Design variable Description MIN MAX MIN MAX RPM Rotational Speed 80,000150,000 100000 150,000 React Percentage of stage pressure drop across rotor 0.450.68 0.45 0.57U/C isen Isentropic velocity ratio 0.50.63 0.56 0.63Tip Flw Ratio of flow parameter to a choked flow parameter 0.30.65 0.3 0.53Dhex% Exit hub diameter as a % of inlet diameter 0.10.4 0.1 0.4AnsqrFrac Used to calculate annulus area (stress indicator) 0.50.85 0.68 0.85 Table 3-4. Baseline and optimum design comparison. Objectives Description Baseline Optimum Wrotor Relative measure of “goodness” for overall weight 1.147 1.147 ts Total-to-static efficiency 85.0% 89.7% Design Variables Baseline Optimum RPM Rotational Speed 122,000 124,500 React Percentage of stage pressu re drop across rotor 0.55 0.45U/C isen Isentropic velocity ratio 0.61 0.63Tip Flw Ratio of flow parameter to a choked flow parameter 0.25 0.30Dhex % Exit hub diameter as a % of inlet diameter 0.58 0.10AnsqrFrac Used to calculate annulus area (stress indicator) 0.83 0.85

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94 Figure 3-1. Mid-height static pressure (psi ) contours at 122,000 rpm. 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 100000105000110000115000120000125000130000135000140000145000150000RPMTotal-to-Static Efficiency Meanline CFD No tip clearance CFD W/tip clearance Figure 3-2. Predicted Meanline and CFD total-to-static efficiencies. 1300.0 3200.0 100 110 120 130 140 150 0.92 0.9 0.88 0.86 0.84 0.82 0.8 0.78 0.76 0.74 Total-to-Static Efficiency Meanline CFD – No tip clearance CFD – W/ti p clearance 103 RPM

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95 220 230 240 250 260 270 280 290 100000105000110000115000120000125000130000135000140000145000150000RPMWork (BTU/lbm) Meanline CFD No tip clearance CFD W/tip clearance Figure 3-3. Predicted Mean line and CFD turbine work. Feasible AN2 > 850 Tip Spd < 2500 Rsex/Rsin < 0.85 Infeasible AN 2 < 850 R se x /Rsin > 0.85 Tip Spd > 2500 U/C isen A ns q rFrac Figure 3-4. Feasible region and location of three constraints. Three of five constraints are automatically satisfied by the rang e reduction of two design variables. 290 280 270 260 250 240 230 220 100 110 120 130 140 150 Work ( BTU/lbm ) Meanline CFD – No tip clearance CFD – W/tip clearance 103 RPM

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96 Infeasible Cx2/Utip < 0.2 Feasible Cx2/Utip > 0.2 U/C isen AnsqrFrac RPM 105 Figure 3-5. Constraint surface for Cx2/Utip = 0.2. At higher values of AnsqrFrac and U/C isen, lower values of RPM are infeasible. Figure 3-6. Constraint surfaces for 1 = 0 and 1 = 40. Values of 1 > 40 lay outside of design variable ranges.

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97 Approximate region of interestNote: Maximum Etats 90% Approximate region of interestNote: Maximum Etats 90% Approximate region of interestNote: Maximum Etats 90% Figure 3-7. Region of interest in function space. (The quantity 1 – ts is used for improved plot readability.) A 0 0.2 0.4 0.6 0.8 1 1.2 012345 Wrotorerror p = 2 p = 5 p = 1 Figure 3-8. Error between RSA and actual data point. A) Wrotor and B) ts at p = 1, 2, and 5. Wroto r vs. ts Wroto r 1 – ts

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98 B 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.550.60.650.70.750.80.850.90.95ts error p = 2 p = 5 p = 1 Figure 3-8. Continued. Figure 3-9. Pareto fronts for p = 1 through 5 for second data set. (The quantity 1 – ts is used for improved plot readability.) Pareto fronts differ by as much as 20%.

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99 Figure 3-10. Pareto fronts for p = 1 through 5 for third data set. (The quantity 1 – ts is used for improved plot readability.) Pareto fronts differ by no more than 5%. Figure 3-11. Pareto Front with validation data. Deviations fr om the predictions are due to rounded values of the design variables (predi ction uses more signi ficant digits). The quantity 1 – ts is used for improved plot readability.

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100 Figure 3-12. Variation in desi gn variables along Pareto Front. A Other < 1%Dhex % < 1% Tip Flw 1% U/C isen*RPM 1%U/C isen 7% RPM 90%B Dhex % 3% AnsqrFrac 5% RPM 15% React 28% U/C isen 21% Tip Flw 23% React*RPM 2% Other 3% Figure 3-13. Global sensitivity analysis. Effect of design variables on A) Wrotor and B) ts.

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101 Figure 3-14. Data points predicte d by validated Pareto front compared with the predicted values using six Kriging models based on 20 selected data points. The validated Pareto front is labeled “Actual.” Figure 3-15. Absolute error distribution for poi nts along Pareto front using 20 selected data points each where the points were selected using (1) MF1, (2) MF2, (3) MF3, (4) MF4, (5) fmin, and (6) cluster centers.

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102 Figure 3-16. Average mean error distribution over 100 cl usters for (1) 50 point s, (2) 40 points, (3) 30 points, and (4) 20 points. As number of points decreases, merit functions 2, 3, and 4 perform better than MF1, fmin, or cluster centers with MF2 showing the best performance.

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103 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 10 20 30 40 50 Number of PointsMedian Average Mean MF 1 MF 2 MF 3 MF 4 fmin centers Figure 3-17. Median mean error over 100 clusters Merit function 2 shows the best performance as the number of points decreases. Figure 3-18. Average maximum error distribu tion over 100 clusters for (1) 50 points, (2) 40 points, (3) 30 points, and (4) 20 points.

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104 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 10 20 30 40 50 Number of PointsMedian Average Max MF 1 MF 2 MF 3 MF 4 fmin centers Figure 3-19. Median maximum error over 100 cl usters. Merit function 2 shows the best performance as the number of points decreases.

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105 CHAPTER 4 MODELING OF INJECTOR FLOWS The combustion chamber of a liqui d rocket engine is cooled by transferring heat to the unburned fuel that circulates ar ound the outside of the combustion chamber via a series of tubes or coolant channels. Rocket injectors deliver th e fuel and oxidizer to the combustion chamber. Commonly, the injector face is made up of a series of injector elements arranged in concentric rows. The row of injectors near the chamber wall can cause considerable local heating that can reduce the life of the combustion chamber. The local heating near each inj ector element takes the form of a sinusoidal wall heat flux profile cause d by the interactions of the outer row elements. Often, this effect is not consid ered during the design process. Reduc ing the intensity of this local heating is of prime importance. For example, in the design of the Space Shuttle Main Engine (SSME), the effect of the local heating in the combustion chamber was not considered during the design process. After construc tion, it was discovered that hotspots along the chamber wall severely reduced the expected chamber life. This resulted in an unforeseen increase in the reusability operating costs of the engine. Local hotspots can even cause actual burnout of the chamber wall. By accurately predicting potentially detrimental phenomena in advance, it may be possible that issues such as wa ll burnout can be avoided. CFD m odeling and validation efforts, in conjunction with the experimental data, can a ssist in the understanding of combustor flow dynamics, eventually leading the way to efficient CFD-based design. This chapter outlines current and proposed methodologies in the CFD-based optimization of liquid rocket engine components. First, a summary is presente d of past and present injector analysis techniques. Then, the basic governing equa tions are presented that are applicable in the simulation of turbulent reacting flow. Fina lly, a simplified one-dimensional analysis is

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106 demonstrated to help define the relationship betw een flow parameters and heat transfer due to reacting flow. 4.1 Literature Review Injector research has been ongoing for over 40 year s. Injector inlet fl ow has a significant influence on combustor performance. The understandi ng of the injector charac teristics is critical in determining the nature of the flow within the combustion chamber. In particular, the inlet flow geometry and injector outlet diameter ha ve large influences on flow in the combustion chamber.26 However, at times, the research has raised more questions than answers. Work must still be done to explain certain elements of the flow within the combustion chamber. A general diagram of injector flow is given in Figure 4-1. Historically, injectors have been de signed using experi mental techniques133 and empirical calculations. A design was built and tested, and then improvements were made based on the results. For example, Calhoon et al.134 extensively reviewed standa rd techniques for injector design, including the analysis of cold-fire and hot-fire test ing to study general injector characteristics. The results of these tests we re used along with a number of additional multielement cold-fire tests to design th e full injector. The full injector was then fabricated and tested for performance and heat flux characteristic s and combustion stability. However, the experimental design techniques were insufficient to predict for many conditions that could reduce injector or combustion chamber life. 4.1.1 Single-Element Injectors A significant portion of injector research has been conducted using experiments consisting of a single injector element. The single elemen t analysis is often used as a starting point in modeling full combustor flow. A test firing of a single element injector is shown in Figure 4-2. Hutt and Cramer135 found that if it is assumed that all of the injectors in the core are identical,

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107 then the measure of energy release from a single element’s flow field is a good approximation of the efficiency of the entire core. In particular the mixing characteristics of a single injector can be used to determine the element pattern and spacing needed fo r good mixing efficiency. Single-element injectors are often used in in jector experimentation to approximate the mixing effects of the full injector. Calhoon et al.134 tested a multiple element injector chamber such that the radial spacing of the elements coul d be varied. In cold flow testing, it was found that mixing increased dramatically when multiple elements were used. However, it was determined that the element spacing only had a sli ght influence on the mixing efficiency. Yet, as the distance between the elements increased, the mixing efficiency increased. This was due to large recirculation regions that br ought flow from the well-mixed fa r region to the near field. When recirculation effects are compensated for, Calhoon et al. found that there is no significant effect of multiple element interactions compared to single element mixing. This, in theory, indicates that a satisfactory combustion analysis can be done using only a single injector. 4.1.2 Multi-Element Injectors A multi-element injector face is made up of an array of injector elements, and usually contains from seven to hundreds of injector elements as shown in Figure 4-3. Small changes in the design of the injector and the pattern of el ements on the injector f ace can significantly alter the performance of the combustor. Elements mu st be arranged to maximize mixing and ensure even fuel and oxidize r distribution. Gill26 found that the element diam eters and diameter ratios largely influence mixing in the combustion chambe r, and that small diameters lead to overall better performance. The type of elements need not be consistent across the entire injector face. The outer elements must be chosen to help provide some wall cooling in the combustion chamber. Gill suggests that using a coaxial type injector for the outer row of injectors provides an ideal near-wa ll environment.

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108 Heating effects on the combustion chamber wa ll due to the arrangement of injector elements is of prime importance. Injector placement can result in high local heating on the combustion chamber wall. Figure 4-4 shows the e ffect of local heating that resulted in the burnout of an uncooled combusti on chamber. Rupe and Jaivin136 found a positive correlation between the temperature profile al ong the wall and the placement of injector elements. Farhangi et al.137 investigated a gas-gas injector and measur ed heat flux to the combustion chamber wall and injector face. It was found that the mixing of the propellants controlled the rate of reaction and heat release. Farhangi et al. suggested that the injector element patte rn could be arranged in a way that moved heating away from the injector face by delaying the mixing of the propellants. 4.1.3 Combustion Chamber Effects and Considerations Several studies have suggested that pressu re has little influence on the combustion dynamics. Branam and Mayer138 studied turbulent length scales by injecting cr yogenic nitrogen through a single injector to help in understanding cryoge nic rocket propellant mixing efficiency. It was found that changes in pres sure and injection velocity ha d very little effect on flow dynamics. Similarly, Calhoon et al. 134 found that the pressure effects on chamber heat flux were very small. Quentmeyer and Roncace139 found no change in the heat flux data with a change in pressure when calculating the heat flux to the wall of a plug flow calor imeter chamber. In addition, Wanhainen et al.140 found that changing the chamber pressure had no effect on combustion stability. While, Mayer et al.141 found that pressure is re latively constant throughout the combustion chamber, while Moon142 asserts that even the very small pressure gradients that exist near the injector exist can significantly al ter velocity profiles and mixing in the combustion chamber. The relatively constant pressure indi cates that density gradients in the combustion chamber must be due to temperature, rather than pressure. Thus, the flow can be considered incompressible. In addition to pressure independence, Preclik et al.143 experimentally measured

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109 heat fluxes alon the combustion chamber wall. It was found that the wa ll heat fluxes were largely independent of the mixtur e ratio in the analysis of inj ection patterns and velocities on wall heat flux when the oxygen mass flow ra te was held constant. Conley et al.144 also discovered a negligible influence on wall heat flux due to the mixture ratio. Several analyses consistently found that there is a sharp jump in the heat flux a certain ways down the combustion chamber wall. The expl anations for the jump have been varied. Quentmeyer and Roncace139 found a sharp jump in heat flux 3 cm from the injector face. They took this to mean that at this point, the reactants had mixed su fficiently for rapid combustion to begin. Calhoon et al.134 found a sharp increase in heat flux at 4.44 cm from the injector face. It was believed that this was due to the deterioration of the cooling effect of the fuel near the wall. Reed145 sought to understand mixing and heat transfer characteristics in small film cooled rockets using a gas-gas hydrogen/oxygen combusto r. Reed also found a local maximum heat flux in the flow. Similar to Calhoon et al., Reed proposed that the loca l maximum heat flux was due to a breakdown of cooling due to increased mixi ng. Reed also suggeste d that the peak heat flux has to do with the mixture ratio and the amou nt of available oxygen pres ent to react with the cooling flow. One characteristic of reacting flow in a ro cket engine combustion chamber is that the reaction happens very quickly, and is limite d by the amount of tur bulent mixing in the combustion chamber. The combusting shear area is highly turbulent, so mixing, and thus reaction, can occur very quickly. Foust et al.146 investigated the use of gaseous propellants for the main combustion chamber by exploring the mixing characteris tics, combustion length, flame holding, injector face heating, and the potential for combustion chamber wall cooling. Foust found that reaction occurs almost immediately af ter injection, and the high velocity gaseous

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110 hydrogen jet decelerates rapidly after leaving the injector, then reaccel erates due to heat addition. Morren et al.147 found that the temperature profiles along the combustion chamber wall were caused exclusively by mixing and shear layer effect s, and not by pressure gradients. de Groot148 found that for gaseous injection, a high degree of mixing is cr itical. Additi onally, combustion must occur away from injector face to reduce injector face wall heating. Several efforts have been made to study inject or dynamics in detail, such that the results could be used to improve combustion chamber perf ormance. Experiments have only been able to provide a limited picture due to measurement difficulties posed by the high temperature, high pressure, and multiple species flow. Rupe and Jaivin136 studied the effect of injector proximity to the wall on combustion chamber heat flux. However, the available experimental techniques were not enough to fully understanding the bounda ry layer phenomena and were insufficient to allow accurate future predictions on heat flux based on the results. Strakey et al.149 experimentally examined the eff ects of moving the inner oxygen post off-axis and away from the combustion chamber wall to study the effects on wall heat transfer. While an attempt was made to balance between wall heating and combustion performance, the experimental results were not enough to accurately predict whether the wall heat transfer would be sufficiently low for the optimized designs. Smith et al.150 conducted hot fire tests of a 7element gas-gas coaxial injector for different pressures to assess the effect on inj ector face heating, but we re unable to accurately evaluate chamber performance due to the pres ence on interior boundary layer cooling. CFD promises to fill in the gaps of knowledge left by experimental evaluations. The experimental procedures are currently di rected at providing va lidation data for CFD simulations. Experimental measurements are inva luable in the ongoing task of CFD validation. A benchmark injector experiment was run by Marshall et al.17 for the purpose of CFD validation.

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111 In this study, wall heat flux measurements were taken for a gas-gas single element shear coaxial injector element. A single-element injector was used rather than a multi-element injector because the experimental set up is simpler and cheaper. The study was conducted using hot and ambient temperature gaseous oxygen and gaseous hydrogen at four chamber pressures. Preburners were used to supply th e hot gases. The preburners were used to simulate flow in a staged-combustion cycle, such as the SSME. Th ey found that for a given geometry and injection velocity, the heat flux along the wall is pressure independent. Conley et al.22 investigated a GO2/GH2 coaxial shear injector to examine the heat flux to the combustion chamber wall for the use of CFD validation. The experi mental setup included optical access and the ability to measure heat fl ux, temperature, and pressure in the combustion chamber. In the experiment, Conl ey et al. investigated the eff ects of combustion chamber length, pressure, and mass flow rate on th e heat flux to the wall. A square chamber was used to allow for a window to be placed on th e combustion chamber. The optical access was in place for the purpose of providing information about combusti on dynamics by viewing the flame at various pressures. This was accomplished using OH-Pl anar Laser Induced Fluorescence (OH-PLIF) to show the primary reaction region, or flame. The edges of the square combustion chamber were rounded to reduce stress on the chamber due to pr essure. Conley et al. found no dependence of the heat flux on pressure when the heat flux was scaled by th e mass flow rate. Specifically, Conley found that when the heat flux could be scaled by the inlet hydrogen mass flow as 2, s caledHinqqm the heat flux profile along the wall co llapsed to a single curve for a given combustion geometry. Conley concluded from th e experiments that th e amount of heat flux along the combustion chamber wa ll was primarily a function of the combustion chamber geometry.

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112 4.1.4 Review of Select CFD Modeling and Validation Studies The majority of CFD validation efforts to date have concentrated on single element coaxial injectors. Work has focused on LOX/GH2 or GO2/GH2 combustion. Most efforts focused on matching species or velocity profiles. Only recen tly have efforts been in place to work towards using CFD as a combustor design tool. These recent efforts focus on accurately modeling the heat characteristics of combustors. Liang et al.27 sought to improve the e ffects of multi-phase mode ling that included liquid, gas, and liquid droplets. A 2-D axisymmetric model was used along with a chemical model consisting of a 9-equation kinetic model along wi th 4 equilibrium equations. The equilibrium equations were included to help anchor the flame during the comput ation by allowing for instantaneous reactions. In this case, equilibrium reactions were necessary due to the coarseness of the grid used. The turbulence model used is th e eddy viscosity model. In this case, no effort is made to couple combustion with the turbulen ce equations. For the simulation of a gas-gas injector, a mixture fraction of unity was used for the propellants, and the combustion chamber was originally filled with oxygen. An artificia l ignition region was plac ed near the oxygen post tip. The computation was run for a physical equiva lent of 10ms. An attempt was made to use the 2-D model to simulate LOX/GH2 multi-elemen t injector flow by changing the walls from a no-slip to a slip boundary conditi on. Local peak temperature is around 2000K, and the average temperature is around 1500K. Liang et al. mentio ned that, based on the CFD results, atomization might be a rate-controlling factor, but there was insufficient experimental data to confirm this fact. Liang et al. notes that grid resolution has a significant e ffect on flame ignition and flame steadiness, but no grid sensitivity study is conducted to explore the effects. Temperature, velocity, and mass fraction contours were obtained, but no comp arison was made to experimental results, so the accur acy of the simulation is unknown.

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113 Foust et al.146 modeled GO2/GH2 combustion using an 18 reac tion finite-rate chemical model. The chemistry parameters were determ ined based on the temperature field. The kturbulence model was used. Preliminary co mputations were done to obtain the boundary conditions upstream of the inject or. The final computa tional model extends s lightly upstream of the injector exit. The model is 2-D Cartesia n, and the grid was coarse at 10151. The computation found good agreement in the specie s concentration profiles and the velocity profiles. Using CFD, Foust confirmed the fl ame holding ability of th e oxidizer post tip over wide ranges of equivalence ratios for a given injector. Cheng et al.151 looked to develop a CFD spray comb ustion model to help understand the effects on wall erosion. The Finite Differe nce Navier-Stokes (FDNS) solver used a kmodel with wall functions, and the real fluids models were used for the multi-phase flow. Heat and mass transfer between phases was neglected. The velocity and species concentrations were solved based on a constant pressure assumpti on, and then the density and temperature ere determined based on the real fluids model. The pressure was then corrected based on the newly determined density. Finite-rate and equilibrium chemistry models were used. The GO2/GH2 computation used the 4-equation equilibrium ch emistry model for hydrogen-oxygen combustion. The shear layer growth was well predicted by the model. Good co mparisons were made between the CFD and experiment for velocity and species profiles. Disagreement in the H2O species profile was attributed to experime ntal measurement error. It is assumed that the grid was twodimensional, however, no grid information is provided. Schley et al.152 compared the Penn State University (PSU) code used by Foust et al.146 and the NASA FDNS code used by Cheng et al.151 with a third computat ional code by called Aeroshape 3D153 (AS3D). The purpose in the CFD development was to reduce combustor

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114 development time and costs. The CFD comput ations were all based on the experiment performed by Foust et al.146 Schley noted that the experimental combustion chamber had a square cross-section with rounded corners, while all three resear ch groups chose to treat it as an axisymmetric computational domain. None expected the non-asymmetry to impact the comparison, as other errors were thought to be more significant. The FDNS and AS3D codes included the nozzle in their computations, while the PSU code did not. The FDNS code did not include any upstream injector analysis, while the ot hers did at least some simulation of the flow within the injector. All three codes used the kturbulence model. Coarse grids were used, as the researchers found that the solution increased in unsteadin ess with increasing grid resolution. Despite the different codes used and the di fferent treatments in boundary conditions, all computations agreed reasonably wi th each other and the experiment. Ivancic et al.154 used the AS3D code to do simulate LOX/GH2 combustion using a coaxial shear injector. Ivancic et al. assumed chemical equilibrium for the combustion modeling with no coupling between turbulence and chemistry. A real gas model was used for the oxygen. An adaptive grid was used with 150,000 grid elem ents and was grid independent based on finer meshes. The simulated shear layer thickness and th e flame location were not consistent with the experimental results. It is s uggested that the discrepancy is due to the 2-D simulation of 3-D effects. Another reason for the discrepanc y was the overly simplifie d combustion model. Lin et al.155 specifically investigates the ability of CFD to predict wall heat flux in a combustion chamber for a shear coaxial single elem ent injector based on th e experimental results from the test case RCM-1 of the 3rd Rocket Combustion Modeling (RCM) Workshop held in Paris, France in March, 2006. The experimental setup of the RCM -1 test case is shown in Figure 4-5. Lin et al. uses the FDNS code introduced by Cheng et al.151 The model was simulated

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115 using two codes: FDNS, a finite-volume pressure-b ased solver for structured grids, and lociCHEM, a density-based solver for generalized gr ids. The FDNS code has long been used by NASA for reacting flow simulation, but is cumbersome to use for complex geometries. LociCHEM, on the other hand, is a new code that is still in development. Because it is applicable to structured or unstructured grid s, loci-CHEM is convenient for complex geometries. The code used a 7-species, 9-reaction finite rate hydroge n-oxygen combustion model. The computational grid encompassed the pre-injector flow, combusti on chamber, and nozzle. Computations are 2-D axisymmetric, and consisted of coarse grids with 61,243 points using wall f unctions, or fine grids with 117,648 points where the turbul ence equations were integrated to the wall using Menter’s baseline model. The mass flow rates are fixed at the inlet to the computational zone with a uniform velocity profile. The experimental temp erature profile is appl ied along the wall of the computational domain. The computational domains were initially filled w ith steam, and for the Loci-CHEM code, an ignition point was specified The temperature contours for each code by integrating the turbulence equations to the wall ar e given in Figure 4-6. The best matching of the heat flux values came by using the loci-CHEM code integrated to the wall using Menter’s SST model. Prediction of the h eat flux was good in the recircul ation zone and poor beyond the reattachment point, as shown in Figure 4-7A. Lin et al. cited th e need for additional validation and improved turbulence m odels to improve results. West et al.156 expanded on the work by Lin et al.155 using an updated version of lociCHEM. West et al. performed a detailed grid refinement study on the CFD model of the RCM-1 test case. In particular, West et al. found that a structured grid was prohibitive, in that a large number of grid points were requi red for adequate reso lution near the combus tion chamber wall. For this reason, West et al. us ed hybrid grids consisting of a st ructured grid near the wall and

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116 triangle elements away from the wall. The st ructured grid had 51,000 points, while the hybrid grids had between 500,000 and 1,000,000 points. West et al. separately expl ored the effects of grid resolution in the boundary layer region, the non-boundary layer region, and the region in the vicinity of the flame. The effect s of grid resolution were small, and the coarsest hybrid grid was found to be sufficient to accurate ly predict the peak heat flux. The study also compared three turbulence models. The use of Menter’s SST model provided slightly better results than Menter’s BSL model and Wilcox kmodel. The basic kwas found to provide the worst approximation, as the prediction of the peak heat flux was in the wrong lo cation, indicating that the model would not be appropria te for a design problem. All tu rbulence models were found to overpredict the heat flux downstr eam of the peak heat flux. Thakur and Wright157 tested the code loci-STREAM against the same RCM-1 test case used by Lin et al.155 A 2-D axisymmetric model was used for the computational domain. A 7species, 9-reaction finite rate hydrogen-oxygen combus tion model was used. The CFD model was initiated with steam filling the combustion cham ber. Similarly to Lin et al., different grids were investigated: a fine grid with 104,000 grid points and a co arse grid with 26,000 points for integrating the turbulence equations to the wa ll, and a grid with 104, 000 points that remains coarse near the combustion chamber wall to use with wall function formul ations. The maximum chamber temperature was 3565 K, and the temperat ure contours are shown in Figure 4-6. The pressure was found to be approximately constant from the injector inlet to the nozzle throat. Thakur and Wright found that th e use of wall functions lowered th e predicted heat flux at the wall, regardless of whether the grid was fine or co arse. It is suspected that this may be due to errors in the computational code, as this same phenomenon was corrected in Lin et al. 155 The heat flux profiles obtained when in tegrating to the wall are similar to the results by Lin et al., as

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117 shown in Figure 4-7. It was also determined that the wall heat flux was not sensitive to the grid distribution far from the combustion chamber wall, as long as the flame re gion is well resolved. Improving the fidelity of CFD computations is ongoing. Cu rrently, work is advancing with the use of three-dimensional CFD models and simulation of multi-element injector flow simulations. Tucker et al.158 reports on several works-in-pr ogress at Marshall Space Flight center. This includes a simplifie d CFD model that uses a 7-elem ent injector element flow along with bulk equilibrium assumption in the remainder of the 17 degree subsection to help approximate full multi-element combustor dynamics of the Integrated Powerhead Demonstrator159 IPD injector. Sample results are gi ven in Figure 4-8A and Figure 4-8B. Additional simulations are being conducted on th e Modular Combustor Test Article (MCTA) to supplement the IPD injector studies as shown in Figure 4-8C. 4.2 Turbulent Combustion Model Combusting flows are highly complex, simultane ously involving chemical kinetics, fluid dynamics, and thermodynamics.160 For example, in a rocket combustion chamber, heat transfer to the walls depends on the boundary layer characteristics.161 Turbulence enhanced mixing along with the chemical kinetics can affect the rate of combustion.162 This section re views the basic governing equations, chemical ki netics, and equilibrium equations needed for analyzing combusting flow. 4.2.1 Reacting Flow Equations Because several species exist in combusting flow, the properties of each fluid must be accounted for. Mass fr actions are defined by k km Y m (4-1) where mk is the mass of species k in a given volume, and m is the total mass of gas.

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118 The total pressure and density are 11,whereandNN kkkk kk k R R p ppTpT WW (4-2) where Wk is the molecular weight of species k, R is the ideal gas constant, T is the temperature, and W is the mean molecular weight given by 1 1 N k k kY W W (4-3) The mass enthalpy of formation of species k at temperature T0 (the enthalpies needed to form 1 kg of species k at the reference temperature) is a property of the substance and is designated by 0 f kh (usually T0 = 298.15K). The enthalpy of species k is then given by 00 chemical sensible T kpkfk ThCdTh (4-4) where Cpk is the specific heat of species k. The heat capacities at constant pressure of species k are usually not constant with temperature in combusting flows. In fact, large changes are possible. They are usually tabulated as polynomial functions of temperature. The diffusion coefficient of species k in the rest of the mixture is Dk. Different gases diffuse at different rates. In CFD codes, this is usually resolved by using simplified diffusion laws such as Fick’s law: k kkkdY mYmD dx (4-5) In any chemical reaction, species must be c onserved. This conser vation is expressed in terms of the stoichiometric coefficients where kj and kj designate reactants and products, respectively:

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119 11 NN kjkkjk kkWW (4-6) The mass rate of reaction is given by: 1 M kkkjj jWQ (4-7) where N is the number of species, M is the number of reactions, and Qj is the rate of progress of reaction j which can be written as 11kjkjNN kk jfjrj kk kkYY QKK WW (4-8) where Kf and Kr are the experimentally dete rmined forward and reverse reaction rate coefficients. To solve for reacting flow, the conservation of mass, species, momentum, and energy is required. In this case, the governing equations in Cartesian coordinates are the conservation of mass, species, momentum, and energy, respectively. 0j ju tx (4-9) ,,1,k jkkjkk jjY uYVYkN txx (4-10) ji ij i jjiuu u p txxx (4-11) iij j j jjju q HpuH txxx (4-12) where is given by Equation 4-2, Vk is the diffusion velocity of species k, and 1 2ii H huu (4-13)

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120 0 1 0 111 N Tfkk k NNN Tkkskkfkk kkkh hhh (4-14) where hs is the sensible enthalpy. The heat flux qj can be written as 1 2PrPrPrii j LjLjLjuu hH q xxx (4-15) 4.2.2 Turbulent Flow Modeling The conservation equations given in equations (4-9) – (4-12) are applicable for laminar flow. For turbulent flow, the Reynolds Averaged Navier Stokes methods (RANS) use ensemble averaging to obtain the time-averaged form of the conservation equations The flow properties are decomposed into their mean and fluctuati ng components where Reynolds averaging is used for the pressure and density components, while Fa vre averaging is used for the velocity, shear, temperature, enthalpy, and heat flux components. The mean fl ow governing equations are given as the conservation of mass, sp ecies, momentum, and energy: 0j ju tx (4-16) ,,1,k jkkjkjkk jjY uYVYuYkN txx (4-17) ji i jiij jijuu u p uu txxx (4-18) 11 22 1 2 iijjii j jjiijijiijjii jjjHuupuHuuu tx quhuuuuuuu xxx (4-19)

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121 where a tilde and double prime denote a Favre-averag ed mean and its fluctuation, respectively, and a bar and a prime denote a Reynolds-average d mean and its fluctu ation, respectively. The values of several terms in the mean averaged conservation equations are unknown, and must be modeled instead. Specifically, the terms are 1 2iikuu (4-20) where k is the kinetic energy per uni t volume of the turbulent velo city fluctuations, and the Reynolds-stress tensor ijuu that is denoted by ijijuu (4-21) These terms can be modeled using the Boussine sq eddy viscosity models (EVM). The EVM models the Reynolds-stresses by 22 33j il ijijttijij jilu uu uuk xxx (4-22) where t is the turbulent, or eddy, viscosity. The turbulent heat transport term juh is modeled as Prt j tjh uh x (4-23) where Prt is the turbulent Pra ndtl number, and the eddy conductivity is given by Prtp t tc k (4-24) In this study, the Prandtl numb er is kept constant at Prt = 0.9. The term 1 2 iijjiiuuuu from equation (4-19) is modeled as

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122 1 2 t iijjii kjk uuuu x (4-25) In the species equation, the term jkuY is modeled as tk jk ktjY uY Scx (4-26) The final forms of the conservation equations fo r mass, species, momentum and energy are given respectively as follows: 0j ju tx (4-27) ,1,k tk jkk jjktktjY Y uYkN txxScScx (4-28) 2 3ji j i il tij jijjiluu u u uu p txxxxxx (4-29) PrPr 2 3j jjLtj j ilt itij jjiljkjh HpuH txxx u uu k u x xxxxx (4-30) where 1 2 iiHhuuk (4-31) and the turbulent eddy viscosity t is modeled as a product of the velocity scale and a length scale multiplied by a proportionality constant C tC (4-32)

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123 The velocity and length scales can be determined using the kmodel163 for turbulence modeling. This model is based on idealized as sumption of homogenous is otropic turbulence and consists of a transport equation of kinetic energy of turbulence k and second for rate of dissipation of turbulent energy Note that in all subsequent e quations, bars and tildes have been dropped for convenience. t ik iikik kukP txxx (4-33) 2 12 t ik iikiuCPC txxxkk (4-34) where the production Pk of k from the mean flow shear stresses is defined as j ii kt jiju uu P x xx (4-35) and the velocity and length sc ales are given respectively by k (4-36) 33 42Ck (4-37) and the constants are given by 120.09;1.44;1.92;1.0;1.3kCCC (4-38) For turbulent flows, the sharp va riations in velocity and temperature profiles in the near wall region require special treatments for the wall boundary conditions. Two common methods are the use of a low Reynolds number k model and the use of wall functions. Examples of low Reynolds number k models can be found in Patel et al.,164 Avva et al.,165 and Chien.166 Wall functions assume that the turbulence calculations stop at some distance away from the wall. It

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124 applies log-law relations near the wall.163 Further descripti on can be found in Pope167 and Tennekes and Lumley.168 4.2.3 Chemical Kinetics In rocket combustion, the fuel and oxid izer enter the combustion chamber in a nonpremixed fashion. The resulting diffusion flam es must form in the midst of turbulence fluctuations, leading to a highl y complex flow. One way of handling this is by making the assumption of “fast chemistry.” This assumption assumes that the reacti on rates of the species are very fast compared to the mixing time. This means that the instantaneous species concentrations and temperatures are functions of conserved scalar equations, such as the mixture fraction. The various average thermodynamic variables can be obt ained from scalar statistics using probability-density functions Further details on using the conserved-scalar approach to account for turbulence-combustion interactions can be found in Libby and Williams.169 The method used in this research is the finite rate chemistry assumpti on. In particular, a kinetic-controlled model is used that depends primarily on chemical kinetics as described previously. This method involves solving the co nservation equations desc ribing the convection, diffusion, and reaction sources. Exampl es can be found in Thakur et al.170 Turbulencecombustion models struggle in forms of accuracy and level of descriptio n, so new models are continually being developed. Chemical reaction rates control the rate of co mbustion and are related to flame ignition and extinction. Chemical kinetics ma p the pathways and rates of reaction from reactants to products. The global reaction of interest in this effort is the hydrogen-oxyge n combustion reaction. Within the global reaction 2222HO2HO (4-39)

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125 there are several elementary reactions. A globa l reaction can possibly be made up of dozens of elementary reactions. However, the effects of some mechanisms are small at certain temperatures and pressures. In this case, the nu mber of reaction mechanisms used to represent a global reaction can be reduced. If necessary, the number of r eactions can be reduced as to contain only the key reactions. Often, the reaction mechanisms ar e “optimized” for this purpose. In this work, nine elementary reactions are considered for the reac tion of hydrogen and oxygen to water. The reaction mechanisms were s upplied by NASA Marshall. Each of the nine reactions has a rate coefficient K given by expb AuKTATERT (4-40) where Ru is the universal gas constant and A b and EA are experimentally determined empirical parameters given in Table 4-3. 4.2.4 Generation and Decay of Swirl Swirl number is the ratio of axial flux of angular momentum to the axial flux of axial momentum rd S R ud vA vA (4-41) where R is the hydraulic radius, and is the angular velocity. The swirl number can be approximated as W S U (4-42) where W is the mean angular flow velocity and U is the mean axial velocity. A swirl number of S < 0.5 indicates weak to moderate swirl.

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126 For the case of annular flow, it can be expected that the rate of sw irl decay will be much greater than that of pipe flow due to the increased fluid drag at the center core. If the difference between the diameters becomes suffic iently small, the flow is approximately equal duct flow. In the case of the hydrogen inlet flow, the outer to in let diameter ratio is 1.22. Any swirl should quickly be eliminated for di ameter ratios close to unity. For confined swirling flows, Weske and Sturove171 examined the decay of swirl and turbulence in a pipe flow. They found that the tu rbulence field decayed qui ckly, and that the rate of decay was a strong function of swirl number. Yajnik and Subbaiah172 found that swirl decay, in general, was not significant for pi pes less than 5 pipe diameters long. In general, it has been found that swirl deca ys exponentially. For laminar swirling pipe flow (Re < 4000), Talbot173 determined a theoretical formulatio n for decay of swirl in a pipe. The experimental values were found to fall within the theoretical values for Re from 100 to 4000. The solution to the swir l equation for pipe flow is ,,0zvrzvre (4-43) where z is the non-dimensional length given by hzLD and is not a function of the initial flow distribution ,0 vr and is given by two theoretical limits: 1Re22.2 and 2Re74.3 The decay of swirl Ds can therefore be given by 11 ,0z svrz De vr (4-44) If it is simply assumed that the pipe flow example can be extended to annular flow, it can be estimated that the swirl decay in the hydroge n pipe will be between 47% and 88% without considering the effects of a fuel baffle. In actu ality, this amount woul d be greater for annular flow due to the growth of the boundary layer on the inner and outer surfaces of the annular pipe.

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127 If the amount of swirl generated in the pipe is small, it can be assumed that there will be negligible swirl at the exit of the hydrogen inlet. 4.3 Simplified Analysis of GO2/GH2 Combusting Flow As shown earlier, several factors are involved in injector and injecti on pattern design that are of interest in this research effort: A single injector can be analyzed as an approximation to full injector analysis The outer injector elements must be chose such that cooling is enhanced at the combustion chamber wall Injector placement can result in high local heating Pressure effects on combustion cham ber wall heat flux are very small Wall heat fluxes are largely inde pendent of the mixture ratio A sharp increase in heat flux occurs consiste ntly at a certain distance away from the injector face Analysis of the flow upstream of the injecti on may be necessary to determine the effect on combusting flow A preliminary combustion analysis can give some insight into the nature of H2/O2 reacting flow. In particular, the relationship between temperatur e, mass flow rate, pressure, heat flow rate, and equivalence ratio can be expl ored using a simplified versi on of the combustion chamber. A system can be considered wherein hydrogen and oxygen are injected such that the molar flow rate of hydrogen exceeds that of oxygen (Figure 4-9). The reaction is given as 22222HO2HO21H (4-45) and the following assumptions are made: Constant pressure Complete combustion with no dissociation Fuel-rich combustion ( > 1) Steady-state All combustion products exit at the same temperature

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128 For this system the heat output in watts is given by 22222 ,, 22111 2 2HinOin outHHOHOH TT Tqmhhhh (4-46) where the species enthalpies h are equal to the molar enthalpies in J/mol divided by the molecular weight of each species. Using this equa tion, several observations can be made. First, the heat flow rate is dependent only on the specie s enthalpies at the inlet and outlet temperatures, the mass flow rates, and the equivalence ratio. Second, it must be noted that enthalpies are largely independent of pressure. This would indicat e that the heat flow ra te is also independent of pressure when the inlet temperatures, outle t temperatures, and mass flow rates are held constant. Table 4-1. Selected inj ector experimental studies. Author Injector type Propellants Key results Hutt and Cramer (1996)135 Single element oxidizer-rich coaxial swirl injector LOX/GH2 Innovative measurements of LOX swirl spray Preclik (1998)143 Multiple-element coaxial injector LOX/LH2 Wall heat flux measurements Farhangi et al. (1999)137 GO2/GH2 Design, develop, and demonstrate a subscale hydrogen/oxygen gaseous injector Smith et al. (2002)150 Multiple-element coaxial for FFSC GO2/GH2 Evaluate performance of gas-gas injector. Mayer (2002)141 Non-reacting jet LN2 Characterization of liquid jet flow Marshall et al. (2005)17 Single-element shear coaxial GO2/GH2 Wall heat flux measurements Conley et al. (2007)22 Single-element shear coaxial GO2/GH2 Wall heat flux independent of pressure, injection velocity

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129 Table 4-2. Selected CFD and numerical studies for shear coaxial injectors. Author Propellants P (MPa) T(K) Summary/ parameters compared Grid spec. Turbulence model Accuracy Liang et al. (1986)27 LOX/GH2 37.9 94.4/298, 118/ 154 Developed three-phase combustion model N/R Eddyviscosity Unknown. Cited inadequate experimental knowledge. Foust et al. (1996)146 GO2/ GH2 1.3 297 Velocity and species comparison 2-D Cartesian 10151 kGood species matching with in-house experiment. Velocity matching degrades downstream. Cheng et al. (1997)151 LOX/GH2, GO2/GH2 3.1, 1.3 117/309, 290/298 Velocity and species comparison Axisym. 15181 kGood species matching to experiment by Foust et al. 146 Poor H2O species matching. Ivancic et al. (1999)154 LOX/GH2 6.0 127/125 Time/length scale investigation Axisym. 1.5105 elements kFair matching of radial OH distribution to in-house experiment. Lin et al. (2005)155 GO2/GH2 5.2 767/798 Heat flux comparison (RCM-1) Axisym. 61,243– 117,648 points Menter BSL Good peak heat flux prediction. Overpredicts downstream heat flux. Low grid dependence. West et al. (2006)156 GO2/GH2 5.2 767/798 Heat flux comparison (RCM-1) Axisym structured/ hybrid 50,000 – 1,000,000 points Wilcox k, Menter BSL and SST Good peak heat flux prediction when adequate grid resolution in high gradient areas. Menter’s SST model provides superior peak heat flux prediction. Thakur and Wright (2006)157 GO2/GH2 5.2 767/798 Heat flux comparison (RCM-1) Axisym. 26,000– 104,000 points Menter SST Good peak heat flux prediction. Overpredicts downstream heat flux. Table 4-3. Reduced reaction mechan isms for hydrogen-oxygen combustion. Reaction A ((m3/gmol)n -1/s) B EA (kJ/gmol) 22HOOHOH 1.7010100.00 2.41104 22HOHHOH 2.1910100.00 2.59103OHOHOHH 6.021090.00 5.501022HOOHH 1.80 1071.00 4.481032HOOHO 1.221014-0.91 8.37103HOMOHM 1.001010 0.00 0.02OOMOM 2.551012-1.00 5.941042HHMHM 5.00109 0.00 0.02OHHMHOM 8.401015 -2.00 0.0

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130 H 2 O2 shear layer shear layer recirculation recirculation flame homogenous products oxygen-rich core heat transfer reattachment p oint heat transfer Figure 4-1. Coaxial injector a nd combustion chambe r flow zones. Figure 4-2. Flame from ga seous hydrogen – gaseous oxygen single element shear coaxial injector.

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131 A B Figure 4-3. Multi-element inject ors. A) Seven-element shear coaxial injector and B) multielement IPD159 injector. Both injectors contain smaller holes at the outer edge that inject unburned fuel for wall cooling.

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132 Figure 4-4. Wall burnout in an uncooled combustion chamber. A Figure 4-5. Test case RCM-1 injector. Test rig A) schematic and B) photo.

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133 B Figure 4-5. Continued. A B C Figure 4-6. Temperature cont ours for a single element injector. A) FDNS and loci-CHEM,155 B) loci-STREAM,157 and C) loci-CHEM156 codes. Plots are not drawn to same scale.

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134 A B C Figure 4-7. CFD heat flux results as compared to RCM-1 experimental test case. A) FDNS and loci-CHEM codes,155 B) loci-STREAM code,157 and C) loci-CHEM code using hybrid grids and various turbulent models.156

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135 A B C Figure 4-8. Multi-element injector simula tions. Reproduced from [158]. A) Integrated Powerhead Demonstrator (IPD) injector showing computational domain and B) a close-up of temperature contours. C) Norm alized temperature contours are shown for Modular Combustor Test Ar ticle (MCTA) injector.

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136 22,,,HinHinmT 22,,,OinOinmT q2,,HoutoutmT 2,,HOoutoutmT 22,,,HinHinmT 22,,,OinOinmT q2,,HoutoutmT 2,,HOoutoutmT Figure 4-9. Fuel rich hydrogen a nd oxygen reaction with heat release.

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137 CHAPTER 5 SURROGATE MODELING OF MIXING DYNAMICS A bluff body-induced flow is used as a m odel problem to help probe the physics and surrogate modeling issues rela ted to the mixing dynamics. Understanding of the mixing characteristics in the wake of the bluff body has im portant implications to reacting flow, such as in the shear reacting layer of in jector flows. The sensitivity of the mixing dynamics to the trailing edge geometry of the bluff body can be explored using CFD-based surrogate modeling. Plausible alternative surrogate models can le ad to different results in surrogate-based optimization. The current study demonstrates the ability of using multiple surrogate models to discover the inadequacy of the CF D model. Since the cost of constructing surrogates is small compared to the cost of the simulations, using multiple surrogates may offer advantages compared to the use of a single surrogate. Th is idea is explored for a complex design space encountered in a trapezoidal bl uff body. Via exploration of local regions within the design space, it is shown that the design space has small islands where mixing is very effective compared to the rest of the design space. Both polynomial response surfaces and radial basis neural networks are used as surrogates, as it is difficult to use a single surrogate model to capture such local but critical features The former are more accura te away from the high-mixing regions, while the latter are mo re accurate near these regions. Thus, surrogate models can provide benefits in addition to simple model approximation. 5.1 Introduction The case study focuses on the mixing and tota l pressure loss characteristics of time dependent flows over a 2-D blu ff body. Bluff body devices are often used as flameholding devices such as in afte rburner and ramjet systems. Blu ff body devices shoul d have good mixing capability and low pressure loss across them. Ch allenges exist in that the bluff body flow is

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138 unsteady and difficult to predict. There is recirculating flow in the near-wake region that decays to form a well-developed vortex street in the wa ke region. The instantaneous loss and degree of mixing changes over time and must be well resolved for accurate solutions. An earlier investiga tion of the optimization of a tr apezoidal bluff body was conducted by Burman et al.138 The study uses a relatively coarse me sh (9272 computational cells) that was selected by performing a grid sens itivity study on a single case ba sed on the drag coefficient. The effects of grid resolution on the measure of mixing were not inves tigated, so it is possible that not all of the flow effects were captured. Fo r higher Reynolds number flow, as observed by Morton et al.,174 grid resolution can have a marked eff ect on the prediction of unsteady flows and can substantially affect the fidelity of the surrogate model. Surrogate models are used to approximate the effect of bluff body geometry changes on total pressure loss and mixing effectiveness. The time-averaged flow field solutions are compared by looking for common trends and co rrelations in the flow structures. 5.2 Bluff Body Flow Analysis In practical considerat ions, such as the aforementioned af terburner and ramjet combustor, the flow around many bluff body devi ces is turbulent. To simplif y the analysis, an effective viscosity is often estimated ba sed on engineering turbulence cl osures. The effective Reynolds number is therefore considerab ly lower than the nominal Reynolds number. For the purpose of this study, the flow is modeled using a Reynolds number in the range of the effective Reynolds number without resorting to turbulence models. Furthermore, without e ither enforcing the wall function or suppressing high gradie nt regions, both typically obse rved in engineering turbulent flow computations, the fluid fl ow tends to exhibit more unste ady behavior, rendering the flow computations more interesting and challenging. The bluff body geomet ry along with the flow

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139 area is described in detail. The computationa l domain and bluff body geometry specifications are based on the study performed by Burman et al.175 5.2.1 Geometric Description a nd Computational Domain The dimensions of the bluff body are given in Fi gure 5-2. The variables of interest are B b and h Altering the variables changes the slant a ngles of the upper and lower surfaces. The area, A of the bluff body is held cons tant and equal to unity. The fr ontal height is kept constant at D = 1. The value of H can be calculated from the afor ementioned variables and constants using the equation 2HAhbDDB (5-1) The computational domain consists of a trapezoid al bluff body within a rectangular channel. The flow area is illustrated in Figure 5-3. The fluid is incompressible, and the flow is laminar with a Reynolds number of 250 given by UD Re (5-2) where D is the frontal height of the bluff body, is the kinematic viscosity, and U is the freestream velocity. The upper and lower boundaries have “slip”, i.e., zero gradient, boundary conditions. The left boundary is the inlet, and the right boundary is the outlet. The 2-D, unsteady Navier-Stokes equations are solved using a CFD code called STREAM.176 The time dependent calculations were solved using the PISO (Pressure Implicit with Splitting of Operators) algorithm, and co nvective terms were calculated using the secondorder upwind scheme. Other spatial derivati ves are treated with th e second-order central difference schemes. The grid was constructed using ICEM-CFD177 software. The grid is nonuniform, with a higher density of grid points n ear the body and in the wake area as shown in Figure 5-4. Grids of varying resolution were used to determine the grid that provided the best

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140 balance in terms of solution run time and soluti on convergence in terms of grid resolution as detailed in Goel et al.178 Multiple grid refinements were investigated to identify the adequate resolution for this problem. The number of grid po ints in each grid is given in Table 5-1. While the difference in the total number of grid points between Grid 1 and Grid 2 is not large, the grid density in the near field and wake region for Grid 2 is higher than Grid 1. In Grid 3 and Grid 4, both the near field grid density and far field grid density are high, making them more refined. Overall, Grid 1 has the poorest resolution and Grid 4 has the best resolution. Grid 2 was selected for the analysis as it appeared to offer the best trade-off in terms of grid resolution and computational run time. The run time for a si ngle case was approximately eight hours on a 16 CPU cluster with Intel Itanium proce ssors (1.3 GHz) and 16GB of RAM. 5.2.2 Objective Functions and Design of Experiments The two objectives are the tota l pressure loss coefficient, CD, and the mixing index, M.I.. The total pressure loss coefficient at any time in stant is equal to the su m of the pressure and shear forces on the body divide d by the drag force. The total pressure loss coefficient is averaged between time t*, and time t* + T. It is assumed that T is large compared to the oscillatory time period. The total pressure loss coefficient is given by 2 1 2 *11tT Dxixi tCpnndSdt TUD (5-3) where p is the pressure, ix is the viscous stress tensor, is the fluid density, and D is the frontal height of the trapezoidal bluff body. The measure of mixing efficacy is given by the laminar shear stress plus the unsteady stress. The mixing index, M.I., is chosen as the integral of the mixing efficacy over the entire computational domain averaged over a time range T. It is given by

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141 *11 ..tT V tU M IuvdVdt TVy (5-4) where is the fluid viscosity, U is the time-averaged velocity, and u and v are the fluctuation velocities (, uuUvvV ). Equation 5-4 is related to the full Reynolds momentum conservation equation. The mixing is thus described as the combination of the momentum transfer due to viscosity and the momentum transfer by the fluctuation velocity field. The magnitude of the product of the fluctuation velocities is used to keep the sign of this term consistent throughout the data points. Only th e magnitude of the mixing index is important, so the absolute value of the time-averaged quant ity represents the mixing index value. The constraints are given as 1 0.51.0 0.00.5 0.00.5 Bb B b h (5-5) The first constraint in Equation 5-5 is incorporated to keep the frontal he ight equal to unity so that the Reynolds number remains constant. The remaining constraints maintain convexity of the geometry. The constraints are identical to those used in Burman et al.175 A total of 52 data points were used in the analysis. The design of experiments (DOE) procedure was used to select the location of the data points that minimize the effect of noise on the fitted polynomial in a response surface anal ysis. A modified face-centered composite design (FCCD) is used to select 27 of the 52 data po ints. The face-centered composite design usually uses the corners and center faces of a cube for poin t selection, but in this case, half of the cube would violate the first constraint in Equation 5-5. This means that four points from the FCCD are infeasible and must be removed from the design. Additional points were added on the

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142 interior of remaining triangular section to compensate for the missing points. The remaining 25 data points were selected using Latin-hyp ercube sampling to fill the design space. 5.3 Results and Discussion The CFD solutions were obtained for the 52 da ta points. These solutions were compared and categorized based on the mean flow field. Response surface and radial basis neural network prediction models were constructed from the data. Comparisons of fit were made between the surrogates. 5.3.1 CFD Solution Analysis Figure 5-5A shows typical instantaneous vort icity characteristics for a given trapezoidal bluff body geometry. The flow exhibits characte ristic vortex pair shedding. This oscillatory behavior is reflected in the instantaneous total pressure loss coefficient and mixing index values. The vortex pair shedding is asymmetric due to the asymmetry of the bluff body. However, in order to evaluate the overall device performan ce, the time-averaged flow is a more concise quantity. Figure 5-5C shows the time-averaged solution for the same geometry. In the time averaged solution it can be seen that the oscillatory behavior in the wake has been completely averaged out. Table 5-2 provides the range of the total pressure loss coefficient and mixing index for the 52 design points. From Table 5-2, it is ev ident the maximum mixing index value lies well outside of the range indicated by the standard deviation. This indicates evidence of possible outliers. Along closer inspection, it was discov ered that three of the 52 cases had mixing index values that lay far beyond the range bounded by the standard deviation. These were termed “extreme cases.” A complete analysis of th e extreme cases is given in Goel et al.178 The flow fields from the CFD runs for geom etries representing the full expanse of the design space can be divided into several major groups as illustrated in Figure 5-6. The flow

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143 fields are categorized based on the location and si ze of mean recirculation regions near the bluff body. These groups will provide a means of re ference for describing general trends in the sample data. Geometries in Group F had the lo west total pressure loss coefficient with an average value of 1.85. Group B had the highest mi xing index with an average value of 655. It can be predicted that the best designs in terms of high mixing and low pressure drop, and thus the focus of a possible future design refinement may lie in the region including and between Group B and Group F. 5.3.2 Surrogate Model Results When constructing the response surface models of the two objectives, a cubic response surface model was found to perform better than a simpler quadratic model. The full cubic model is given by ****2***2** 01234567 ***2*3*2***2*3*2* 891011121314 ****2***2**2**2*3 141516171819ˆ yBbhBBbbBh hbhBBbBbbBh BbhbhBhbhbhh (5-6) where a starred value indicates the variable is no rmalized between 0 and 1. Insignificant terms were then removed from some cubic response surfaces, resulting in reduced cubic models. The models used in this analysis are given in Table 5-3. Problems were discovered in the mixing index response surface model fit: The response surface has a low R2 adj value and a high RMS error, so the fit is very poor. Roughly half of the coefficients in th e response surface are very large (> 1000). Three of the large coefficients are positive and four are negative. The last two symptoms indicate a polynomial with sharp gradients and oscillatory behavior. It was suspected that these symptoms were due to behavior exhibited by the extreme cases. The extreme cases were removed in an attempt to improve the response su rface fit for the mixing index. As seen from Table 5-3, after remova l of the extreme cases the response surface fit

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144 improved to 0.921, and the magnitude of the highest term dropped from 4780 to 1550. The number of terms needed for the response surface fit also reduced from 16 to 14 and the RMS error reduced from 39 to 12. Radial basis neural networks we re also constructed to model the data to determine whether it was a better surrogate model for the bluff bod y. For the analysis, 10 of the 52 points were reserved as test cases for cross-validation. The values of SPREAD and GOAL that minimize the error in the test cases are given in Table 5-4. The neural network also had problems in fitting the data when all 52 data points were used. Indi cations of those problems are given as follows. The optimum value of SPREAD for the mixing index is quite low. A large number of neurons are needed to fit the data. This behavior is indicative of large local variations in the data. When the same three points that were removed in the response surface analysis we re removed in the neural network analysis, the number of neurons needed to fit the data re duced from 35 to 21. The RMS error was also reduced and the optimum spread consta nt increased from 0.14 to 0.35. The response surface and neural network predic tion surfaces for the total pressure loss coefficient are very similar. The most noticeable difference in the total pressure loss coefficient predictions occurs at high h values. This behavior can be observed in Figure 5-7. For example, at h* = 1, B* = 0, and b* = 0.5, the response surface predicts a relatively high value for the total pressure loss coefficient, while the neural network predicts a relatively low total pressure loss coefficient value. The differences between the prediction models fo r the mixing index are striking, as seen in Figure 5-8. The neural network clearly shows three points with mixing index values that are considerably higher than that of the surrounding area. The respon se surface attempts to fit the same three points, but the effects are wider spread. The result is that prediction is poor in areas

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145 near these points. The large differences be tween the response surface and neural network surrogates are useful in that they can alert us to the fact that both models may have problems in some parts of the design space, so that at a minimum they serve as a warning signal. The three points with disproportionately high values were the ones identified as extreme cases in the previous analysis. When these ex treme cases are removed, the response surface and neural network prediction surface are more consistent with each other as seen in Figure 5-9. Yet, large differences still exist in the over all shape of the prediction model surfaces. The RMS error values shown in Table 5-4 indicat e that, overall, the neural network is able to fit the data better than the response surface. The points reserved as test points in the construction of the neural network should have an RMS error that is near the RMS error for all 52 points. For the mixing index, the RMS errors for the test points in the neural network prediction surface are lower than the RMS errors of the response surfaces as seen in Table 5-5. The fit of the neural network to the data that includes extreme cases is only fair, but is considerably better than that of the response surface. Removal of the extreme cases further improves the fit. The error in the test data is co nsiderably reduced and is, in fact, lower than the total RMS error. The RMS error for the total pressure loss coefficient approximation in the test data is also less than the tota l RMS error and is considerably smaller than the response surface RMS error. This indicates that better overall pred iction ability is gained for using the radial basis neural network. 5.3.3 Analysis of Extreme Designs A grid resolution study was conducted by Goel et al.178 Because the quality of the response surface for the mixing index from Grid 2 was poor an outlier analysis was conducted to identify possible outliers. Three cases were detected as potential outliers. The suspect outlier designs were analyzed to identify possible issues. The results for the grids used had adequate time

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146 convergence, and these three potential outlier cases did not indicate any abnormal behavior. In addition to the three potential outliers, two other cases were also analyzed to observe the trends of grid refinement on other solutions. The total pressure loss coefficient and mixing index for these cases are given in Table 5-6. The to tal pressure loss coeffi cient did not change significantly but the mixing index further increased. Also the contrast of the outlier designs with the other designs became even sharper. This indicated that the cases detected as potential outliers might represent a true phenomenon. Thes e designs are therefore designated as extreme designs rather than outliers. Six additional designs between extreme Case 1 and Case 3 (see Table 5-6) were analyzed using Grid 2. Designs for Case 1 and Case 3 were mirror symmetric. Since B and h were at the lower extremes, b = 0.25 was the axis of mirror symmetry. This means that the designs with b = 0.05 and b = 0.45 were mirror images and so on. The to tal pressure loss coefficient and mixing index for mirror symmetric designs were expected to be the same and observed to be almost the same. The mixing indices for these intermediate designs did not exhibit the trend of high mixing indices. The time-averaged flow fields also di d not reveal any abnormality. The sudden change in behavior near the extreme designs was not captured, and higher resolution was required. To investigate this sharp change in the mi xing index, additional simulations were conducted in the vicinity of the extreme design Case 1. Grid 3 was used to conduct the simulations to get a more accurate estimate of the total pressure loss coefficient and mixing index. In these designs, h was set at the lower limit and B wa s also fixed at the lower limit for most cases. Only b was varied to differentiate between desi gns. It was observed that the mixing index was very high for the designs near the extreme case up to b = 0.02 and there was a sudden drop in the mixing index for design with b = 0.03. The results are shown in Table 5-7.

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147 Extreme design cases 1, 3, 52 and regular ca ses 7 and 17 were analyzed using Grid 4 to assess the convergence trends with respect to grid size. The results for these representative cases for different grids are shown in Figure 5-10. The mixing indices for different cases showed trends of achieving asymptotic behavior with im provements in grid resolutions. However, it is apparent that the Grid 2 solution has not reache d grid convergence, and th is may be contributing to the problem with the response surface fit. 5.3.4 Design Space Exploration The extreme cases cannot be dismissed as outliers, as Goel et al.178 determined, as they were found to be properties of the simulated fl ow. Thus the behavior of the prediction surfaces constructed using the extreme cases needed to be further investigated. The RMS errors indicate that when the extreme cases are included in th e surrogate model construc tion, the radial basis neural network fits the data better than the respon se surface. In reality, by inspection of Figure 5-8, the neural network appears quite deficient in areas away from the outlier. The fits of the response surface and radial basis neural networ ks constructed without the extreme cases were determined to be quite good. Thus, these surfaces could be compared to the prediction surfaces constructed with the ex treme cases to determine areas where the prediction was poor by using the equation with extremeswithout extremes...... MIMIMI (5-7) which leads to the contour plots in Figure 5-11. In areas away from the extreme cases, the change in the mixing index M.I. for the response surface is small. In contrast, even away from the extreme cases, there are areas of relatively high M.I.. This suggests that the response surface is better than the neural network at predicting the response away from the extreme cases.

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148 To test if the response surf ace fit was better away from the extreme cases, an exploration of the design space was conducted. Figure 512 plots the response su rface and radial basis neural network predicted values as co mpared to the actual CFD data from Table 5-7 along the line M.I. = f(B* = 0, b*, h* = 0). Of the cases used to construct the surrogate models, only the extreme cases (B,b,h) = (0,0,0) and (B,b,h) = (0,0,1) fall on this line. The radial basis neural network predicts both of these points very well, while the response surface under-predicts the mixing index at b = 1. The variation in the data is very high, with a sharp drop in the mixing index value over a sh ort distance near the extreme cases and very smooth behavior away from the extreme cases. Neither surface captures the complete behavior. In this case, the response surf ace predicted the interior points very well, as expected, but performed poorly near the extreme cases. The neur al network, on the other hand, better predicts the trends in the data near the extreme cases. Areas of large differences within the desi gn space between the tw o surrogates correspond to regions where the accuracy in the CFD model is compromised. Particularly, these regions are concentrated around the extreme designs. These designs were found by Goel et al.178 to have not reached grid convergence. The results indicate that the extreme designs may require a greater degree of grid refinement than other design points. 5.4 Conclusions Polynomial response surface and radial basis ne ural networks were used as surrogates for flow over a trapezoidal bluff body. Both the resp onse surface and the radial basis neural network approximations adequately predicted the tota l pressure loss coefficient. However, both surrogates poorly fit the mixing index and were substantially different from each other. The large differences in the surrogate approximations served as a warning signal and prompted further investigation. Inspection of the surroga te models’ statistics re vealed that the fitting

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149 problems were due to high variations in the da ta in localized regions. Removing three extreme cases greatly improved the fits. Further investig ation revealed that the CFD simulations were not converged, and may have been contributing to the inability of the surrogates to properly approximate the model. Simulations of complex flows can sometimes re sult give unexpected results. Here, three cases had considerably higher mixi ng indices than the other cases. This behavior could not be accurately captured by a single surrogate model al one. The radial basis neural network was found to better approximat e the response near the extreme cases due to its local behavior, while the response surface provided be tter prediction of the response away from the extreme cases. Refinement of the design space near the extreme designs revealed that the differences in the surrogate models corresponded to the locations of the low accuracy CFD designs. Thus, surrogate models and design space refinement tech niques can be used to identify the presence and location of inaccurate CFD models. Table 5-1. Number of grid points used in various grid resolutions. Grid # Number of grid points 1 37,320 2 44,193 3 74,808 4 147,528 Table 5-2. Data statistics in the grid comparison of the CFD data. CD M.I. Max 2.21857 Min 1.79515 Range (max-min) 0.42342 Mean 2.03617 Standard deviation 0.1166

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150 Table 5-3. Comparison of cubic response surf ace coefficients and response surface statistics. Terms CD M.I. M.I.a Intercept 1.95 801576 B* -0.368 -1490-60.6B* 0.672 -562795H* 0.337 -186830B*2 1.24 2900394B*b* 3.29 47800B*2 0 515-1550B*h* 0.507 8420H*b* -0.633 494-556H*2 -0.700 0-39.9B*3 -0.844 1660-360B*2b* -4.01 -4620-902B*b*2 -4.08 -36600B*3 -0.686 0765B*2h* -0.597 -715-80.7B*b*h* -0.740 -1000319B*2h* 0 -769484B*h*2 0.517 00B*h*2 0 2680H*3 0.340 00 Response Surface Statistics R2 adj 0.963 0.6540.921RMS error 0.021 3912 # of terms 17 1614 Mean response 2.03 617604a extreme cases omitted normalized variable Table 5-4. Comparison of radial-basis neural network parame ters and statistics. Parameters CD M.I. M.I.a SPREAD 0.65 0.140.35 GOAL 0.0055 30005500 # of neurons 29 3521rms error 0.0109 15.611.6a extreme cases omitted Table 5-5. RMS error comparison for respons e surface and radial basis neural network. Objective function Quadratic RS rms error Cubic RS rms error RBNN rms error at test points RBNN rms error at all points CD 0.035 0.0220.0096 0.014 M.I. 54 4129 16M.I.a 16 1310 12a extreme cases omitted

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151 Table 5-6. Total pressure loss coefficient and mi xing index for extreme and two regular designs for multiple grids. CD M.I. Case # B b H Grid 1 Grid 2 Grid 3 Grid 1 Grid 2 Grid 3 1a 0.500 0.000 0.000 2.001.951.95477801 850 3a 0.500 0.500 0.000 2.001.961.96477805 853 7 0.500 0.250 0.250 2.092.172.14538635 643 17 0.875 0.125 0.375 1.931.921.91434541 579 52a 0.690 0.200 0.460 1.982.152.14502857 977a suspected outliers Table 5-7. Total pressure loss coefficient and mixi ng index for designs in the immediate vicinity of Case 1 using Grid 3. Case B b h CD M.I. 1 0.50 0.00 0.00 1.95850 I 0.50 0.01 0.00 1.95858 II 0.50 0.02 0.00 1.94886 III 0.50 0.03 0.00 1.96639 IV 0.50 0.04 0.00 1.98650 Figure 5-1. Modified FCCD. y x B b D =1 H h Figure 5-2. Bluff body geometry. The three design variables are B, b, and h. The parameter D is the frontal area which is kept constant at 1.

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152 6.5 D=1 32 5 14 Figure 5-3. Computational doma in for trapezoidal bluff body. X Y0102030 -5 0 5 Figure 5-4. Computational grid for trapezoidal bluff body. A Figure 5-5. Bluff body streamlines and vorticity contours. A) Typical instantaneous vorticity contours, B) typical instantaneous streamlines, and C) time-averaged streamlines for flow past a trapezoidal bluff body (B = 0.5, b = 0.25, h = 0.25) at Re = 250.

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153 B C Figure 5-5. Continued. Figure 5-6. Flow characterizat ion in design space on the B-b axis. At b = 0, the angle of the lower bluff-body surface is zero. At B = 0, small angles exist on lower and upper bluff body surfaces and the sum of the magn itude of these angles is at a maximum. Along the line B + b = 1, the angle of the upper surface is equal to zero.

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154 Figure 5-7. Comparison of res ponse surface (top row) and radial basis neural network (bottom row) prediction contours for total pressure loss coefficient at h* = 0 (left), 0.5 (center), and 1 (right). Normalized variables are shown. Figure 5-8. Comparison of res ponse surface (top row) and radial basis neural network (bottom row) prediction contours for mixing index (including extreme cases) at h* = 0 (left), 0.5 (center), and 1 (right). Norm alized variables are shown.

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155 Figure 5-9. Comparison of res ponse surface (top row) and radial basis neural network (bottom row) prediction contours for mixing index (excluding extreme cases) at h* = 0 (left), 0.5 (center), and 1 (right). No rmalized variables are shown. A B Figure 5-10. Variation in objective variables with grid refinement. A) CD and B) M.I.

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156 Figure 5-11. Difference in predicted mixing i ndex values from response surface (top row) and radial basis neural network (bottom row) prediction cont ours constructed with and without extreme cases at h* = 0 (left), 0.5 (center), and 1 (right). Normalized variables are shown. Figure 5-12. Comparison of response surface and radial basis neural network prediction contours for mixing index at B* = 0 and h* = 0. The CFD data is additional data and is not used to construct the surrogate models.

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157 CHAPTER 6 INJECTOR FLOW MODELING 6.1 Introduction Heating effects due to the arrangement of in jector elements is of prime importance. Injector placement can result in high local heating on the combustion chamber wall. A multielement injector face is made up of an array of injector elements. The type of elements need not be consistent across the entire injector face. The outer elements must be chosen to help provide some wall cooling in the combustion chamber. Sm all changes in the design of the injector and the pattern of elements on the injector face can significantly alter th e performance of the combustor. The elements must be arranged to maximize mixing and ensure even fuel and oxidizer distribution. For example, Gill26 found that the element diameters and diameter ratios largely influence mixing in the combustion chambe r, and that small diameters lead to overall better performance. Gill suggests that using a coax ial type injector for the outer row of injectors provides an ideal environment due to the outer flow being fuel. Rupe and Jaivin136 found a positive correlation between the temperature profil e along the wall and the placement of injector elements. Farhangi et al.137 investigated a gas-gas injector and measured heat flux to the combustion chamber wall and injector face. It was found that the mixing of the propellants controlled the rate of reaction and heat release. Fa rhangi et al. suggested th at the injector element pattern could be arranged in a way that moved heating away from the injector face by delaying the mixing of the propellants. A CFD model is constructed based on experi mental results provide d by Conley et al.22 The CFD model is used to help understand the flow dynamics within the GO2/GH2 liquid rocket combustion chamber. In particular, the CFD model attempts to match the experimental heat flux data at the wall. The CFD simulation is also used to help provide answers to questions that could

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158 not be easily answered using an experiment alone. These issues include (1) the ability to show flow streamlines, (2) two-dimensional wall heat fl ux effects, and (3) the relationships between flow dynamics and heat flux to the combustion ch amber wall. Finally, a grid sensitivity analysis is conducted to determine the effects of grid resolution and density on wall heat flux and combustion length. 6.2 Experimental Setup The validation exercise is based on an experimental setup by Conley et al.22 The injector itself was constructed of an outer hydrogen tube w ith an oxygen tube within. The oxygen tube is concentric to the hydrogen tube and is stabilized by a baffle. This baffle keeps the oxygen tube centered within the hydrogen tube and helps to evenly distribute th e hydrogen flow. The temperature and pressure of the propellants when they are injected is not known from the experiments. In modeling flow in the combustion chamber, Schley et al.152 compared the results from three different codes with experimental results with good agreement for gas-gas combustion. One primary concern for all simulati ons was the lack of inlet turbulent conditions for the combusting flow. All contributing parties simply made an approximation of suitable inlet conditions. By analyzing the pr e-injector flow, some of the gu esswork can be removed. A preinjector analysis is used to determine the flow conditions within the injector itself, as well as to determine how the flow conditions affect the flow entering the combustion chamber. The combustion chamber is allowed to co ol between tests. Conley et al.22 determined in a preliminary analysis that the temperature along the combustion chamber wall would reach a maximum of 450 K after 10 seconds of combustio n. It was estimated that the actual wall temperature reached up to 600 K. For a simila r CFD simulation, this would mean that the combustion chamber wall would be considerably cooler than the bulk flow, where the flame

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159 temperature can reach up to 3000 K. Thus, the th ermal boundary layer plays a significant part in the estimation of the heat flux via CFD. 6.3 Upstream Injector Flow Analysis Because it was determined that the geometry of the injector could have a large effect on combustion dynamics, the flow upstream of the injector must be characterized for flow properties and effects. A CFD st udy was conducted based on expe rimental results of a coaxial injector built and test ed by Conley et al.22 In particular, the CFD si mulation will seek to glean information from the injector that is difficu lt or impossible to meas ure using conventional experimental methods. The simulation will in vestigate the possibility of swirl development upstream of the injector, as well as determine the pressure drop across the injector. Finally, the shape of the velocity profile at the injector exit is investigated. 6.3.1 Problem Description The geometry consists primarily of an annula r flow region with the hydrogen entering at a direction perpendicular to the primary flow. A flow baffle, that exists in the experimental setup and is meant to center the oxygen tube within the hydrogen tube, has been omitted in this analysis. The flow undergoes a sharp reduction in area near the exit of the geometry. The flow parameters of interest are those of the exiting flow. For the hydrogen inlet flow, two separate cas es were examined. The first case assumes that the inlet flow is parallel to the walls, or = 0. This assumes that the hydrogen is being drawn from quiescent flow such as a tank, for example. The second case assumes that the flow enters at a slight angle, or = 5.7. This case represents a pe rturbation that may have existed in the original flow. The two cases are compared and analyzed for similarities and differences as well as for the potential for swirl development and subsequent decay.

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160 The flow regime was assumed to be turb ulent, compressible, and unsteady. The computational domain and boundary conditions are given in Figure 6-2. The computational mesh consisted of 2.36105 nodes and the mesh density was increas ed near the no slip walls and areas of flow curvature as shown in Figure 6-3. The computation was run using the loci-STREAM157 code using six 1.3GHz processors. Each computation was run long enough to correspond to a physical time of 1.6 seconds. 6.3.2 Results and Discussion Figure 6-4 shows the vorticity contours for the hydrogen flow given both inlet conditions. At x = 0, the flow makes a 90 turn while simultan eously flowing around the inner pipe. This results in the development of two vorticies in the x-plane on either side of the y axis. These vorticies are convected down stream while they gradually decay. Th is decay is due to a relatively low Reynolds number combined with a long non-dimensional chamber length of 10.5. However, in the converging section of the annular pipe, the remaining weak vorticies strengthen due to a sharp increase in the local Reynolds nu mber. Due to the small tangential velocity and small hydraulic diameter, the viscous effects result in a dramatic decrease in x-plane vorticity, thus reducing the overall swirl value to neglig ible levels. The non-di mensional length of the portion of the pipe with a small hydraulic diameter is comparable to that of the larger portion of the pipe at 10.7. This allows the flow to reach almost fully-developed flow conditions at the exit of the pipe. Figure 6-5 shows the overall swirl numbers that arise as a result of the individual vorticies. The overall swirl is very weak (< 0. 1) outside the entrance region at the x = 0 plane. The magnitude increases momentarily at the convergin g section of the pipe, then peaks as viscous effects overcome the accelerating effect cause by the convergence and retard the flow in the tangential direction. The swirl number then re duces to a very low, and negligible, amount.

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161 Figure 6-6 shows the average velocity for increasing x for each inlet condition. The average mean velocity in the x direction, u, is nearly identical for each of the two inlet conditions. However, the inlet conditions result in very different values for the velocity in the direction, v. While the = 0 inlet condition results in negligible flow rotation, the = 5.7 causes a noticeable, but still small maximum v velocity of 0.6m/s. The tangential velocity v continues to decrease until the converging porti on of the pipe where there is a sharp increase. As the pipe area again becomes constant, the tangential velocity decreases again to a value near 0.3. An examination of the Reynolds number reveals th at practically no difference occurs in the Reynolds numbers for the two inlet conditions, as shown in Figure 6-7A. The Reynolds number in the tangential direction for the = 0 inlet condition is nearly zero along the length of the pipe. For the the = 5.7 inlet condition, the Reynold s number in the tangential direction continually decreases until its value nearly reaches that of the the = 0 inlet condition. Based on these results, it can be concluded that perturbi ng the inlet flow ultimately has little effect on the resulting exiting flow. Because little difference exists in the flow ex iting the injector, the efforts can now focus on the single case based on uniform inlet fl ow. There is a pressure drop of 5.510-3 MPa across the injector, nearly all of which occurs in the conve rging region and final pipe section, as shown in Figure 6-8. The hydrogen exit velocity profile, k, and are shown in Figure 6-9. The hydrogen exit velocity profile is nearly symmetric with a maximum value of 60.6 m/s and a Reynolds number of 2.8103. A more distinct asymmetry can be seen in the turbulent kinetic energy k profile. Peaks in k occur near the pipe walls. Near the walls, the velocity gradients are high, leading to a large production of k. At the walls, the velocity is zero, so the turbulent kinetic energy is also set to zero. There is a decay of k away from the walls as the turbulent production

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162 of k again becomes zero at the centerline where the u velocity gradient is equal to zero. The production of turbulent dissipation is also dependent on the velo city gradients. However, at the wall, the value of turbulent dissipation is un known, and must be guessed. The value of turbulent dissipation at the wall is therefore set based on the cell distance to the wall. For an infinitely small cell, the turbulen t dissipation would be set to an infinite value. The free stream turbulent dissipation boundary condition is based on the free stream velocity and characteristic length. Therefore, there is a gradual reduction of from the walls to th e free stream value. Similar simulations were run for the oxygen flow. Full details are not given, as the geometry was very simple. The results are thos e of standard turbulent pipe flow. The oxygen flow Reynolds number is an orde r of magnitude higher at 3.7104 than the hydrogen flow due largely to a larger hydraulic diameter. The ma ximum velocity for the oxygen flow was 25.4 m/s. The shapes of the k and profiles are similar to those of the hydrogen flow, but are of lower magnitude than the hydrogen flow. 6.3.3 Conclusion Regardless of the characteristics of the flow entering the hydrogen inlet, the swirl at the outlet was found to be negligible. It was found that by biasing the inlet flow, a small amount of swirl developed, but it was not enough to sustai n through the final flow section. This flow reduction had two effects: (1) the flow accelerated in the x-direction to such a degree such that the swirl number became very small, and (2) the flow profile at th e outlet was slightly asymmetric. This asymmetry may or may not have a noticeable effect on the combustion chamber dynamics.

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163 6.4 Experimental Results and Analysis No temperature measurements are taken direc tly at the combustion chamber wall. Instead, equations were used to extrapolate temperature and heat flux values to the combustion chamber wall. The heat flux is measured along one of the combustion chamber walls and is determined using sensors that measure the temperature at tw o nearby locations, as shown in Figure 6-10. The heat flux equation was calculated based on the time-dependent experimental results to account for the unsteady effects of heat transfer. The heat flux measured along one of the combustion chamber walls and is determined using sensors that measure the temperature at tw o nearby locations. The steady-state (SS) heat flux was then calculated using ,linearSSk qT y (6-1) where conduction coefficient k of Copper 110, the material used for the combustion chamber wall, is k = 388 W/m-K y is the distance between the ther mocouple holes as shown in Figure 6-10, and T is the difference in temperature between the two thermocouple holes. The heat flux profile using this definition was thought to be inaccurate. Thus, the equation was altered to include a heat absorption correc tion which accounts for unsteady (U S) effects of heat transfer. ,2,1 ,,2,22oo linearUSioTT kcy qTT yt (6-2) where the subscript i denotes the thermocouple location cl osest to the combustion chamber wall and o denotes the outer thermocouple location. The subscripts 1 and 2 represents an initial and final time, respectively. The the density and heat capacity c of Copper 110 are given by = 8700 kg/m3 and c = 385 J/kg-K. The chamber wall temperat ures were determined using the heat flux data and the thermal conductivity of the wall.

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164 linearSS walllineariqy TT k (6-3) In the experimental documentation, the steady and unsteady heat fluxe s are calculated using Equations 6-1 and 6-2, respectively, and are shown in Figure 6-11. Equations 6-1 through 6-3 assume that the heat flow is one-dimensional.179 For a square duct, the one-dimensional linear assumption may not be correct. The actual equation would need to be determined through a numerical solution of the two-dimensional heat conduction equation. The correct heat flux equation would change the experimentally documented values of heat flux at the wall. However, because at each cross section only two temperature measurements are made, there is not enough data to do a full two-di mensional analysis of heat transfer through the combustion chamber wall. Additional temperature m easurements taken within the wall would be required for an accurate estimation of wall heat flux. Therefore, a one-dimensional axisymmetric approximation was included in addition to the one-dimensional linear approximation. If it is assumed that the isotherms within the combustion chamber wall are axisymmetric, a second one-dimensional assumption can be conducted. The geometry of the isotherms is given in Figure 6-12. This is reasonable due to th e approximate axisymmetr y of the temperature contours within the combustion chamber wall, a nd the high thermal conductivity of the copper. For 1-D axisymmetric heat c onduction, the steady state heat flux approximation is given by ,lnaxisymmSS walloikT q rrr (6-4) where rwall is the distance from the center of the combustion chamber to the combustion chamber wall, and ri and ro are the distances from the center of the combustion chamber to the inner and outer thermocouples, respectively, as shown in Figur e 6-12. Using this equa tion, the heat flux at the wall centerline can be determined based on the 1-D heat conduction equation, giving

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165 ,,lnwalloi wallaxisymmlinearSS iwallrrr qq rr (6-5) For the given combustion chamber diameter, this assumption would result in a wall heat flux value that is everywhere 28% be low the linear heat flux assumption. This is reasonable, as the assumption of axisymmetric isotherms would resu lt in the lowest possible heat flux given the temperatures at the thermocouple locations. The temperature at the wall would be given by ln lnaxisymmo iooTr TrT rrr (6-6) An unsteady approximation is determined by adding the second term from equation (6-2) to the steady state heat flux approxima tions in Equation 6-5. Figure 6-13 shows the estimated wall temperature profiles based on the experimental thermocouple readings. The estimated wall temperatures are very similar between the line ar and axisymmetric approximations, indicating that the linear assumption used in the experime ntal documentation was reasonable in the case of the wall temperature. Finally, an approximation of the wall heat flux was conducted using a numerical twodimensional unsteady heat conduction analysis. Vaidyanathan et al.180 completed a similar analysis on a different injector and provided the conduction analys is results for the injector by Conley et al.22 The cross-sections at each thermo couple pair location were analyzed independently. Because the experiment lasted only seven to eight seconds it was estimated that only negligible heat escaped through the outer combustion chamber wall. Therefore, the outer wall was assumed to be adiabatic. The thermal propagation time scale for the combustion chamber wall is equal to tP = 2/ = 3.13 s where is the thickness of the combustion chamber wall equal to 0.019 m, and is the thermal diffusivity and is equal to 1.158 10-4 m2/s. The

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166 thermal propagation time scale is on the same orde r of magnitude as the e xperimental run time of 7.75 s, indicating that the adiaba tic assumption is reasonable. Th e inner wall heat flux, taken to be uniform along the inner combustion chamber wa ll, was adjusted until the computational and experimentally measured temperatures obtaine d at 3.15 and 9.54 mm fr om the inner chamber walls (see Figure 6-10), respectively, are matched. The resulting temperature contours are given in Figure 6-14 for the prescribed boundary cond itions for the combustion chamber cross-section at x = 84 mm. Significant differences are seen in the comparis ons of the estimated wall heat fluxes shown in Figure 6-15. The shapes of the profiles given by the linear and axisymmetric assumptions are the same, albeit the axisymmetric assumption gives a proportionally lower heat flux value. The peak heat flux value predicted using the axisymmetric assumption is similar to that predicted by the 2-D unsteady conduction analysis. The locati on of the peak heat flux from the conduction analysis is shifted from that of the 1-D approximations. Conley et al.22 found that the point of maximu m heat release using the linear approximation is at 58 mm from the injector fa ce, whereas the maximum temperature occurs further downstream at 84 mm from the injector face. On the other hand, the point of maximum heat release using the unsteady 2D analysis gives a maximum heat flux at the same location as that of the maximum temperature. 6.5 Injector Flow Modeling Investigation A CFD modeling investigation was conducted ba sed on the experimental results presented by Conley et al.22 of a single-element shear coaxial GO2/GH2 injector. The CFD model is used to supplement the experimental results, and to ga in insight into releva nt flow dynamics. The modeling effort is particularly useful in capturing the variatio n of heat flux along the combustion chamber wall due to three-dimensional flow effects.

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167 The CFD simulations were conducted using loci-STREAM157 which is a pressure-based, finite-rate chemistry solver for combustion fl ows for arbitrary grids developed by Streamline Numerics, Inc. at the University of Florida. The code numerically solves the 3-D unsteady, compressible, Navier-Stokes equations. Menter’s SST model is used for turbulence closure. A seven-species, nine-equation finite rate chemistry model181 was used for gaseous hydrogen and gaseous oxygen. 6.5.1 CFD Model Setup The computational domain, grid, and boundary conditions are shown in Figure 6-16. The values for specific flow conditi ons are given in Table 6-1. The unstructured grid contains 1.7106 elements and 3.7105 nodes. The computational domain represents a three-dimensional, one-eighth-section of the full combustion chambe r. No-slip conditions are specified at the combustion chamber wall, the injector face, and the oxygen post tip. Symmetry conditions are specified along the planes of symmetry and along the centerline of the combustion chamber. The nozzle is not included in the analysis; instead, a constant pressure outlet condition is specified, as the pressure is nearly constant throughout the combus tion chamber. The initial temperature was set to 2000 K to place initial heat flux value within the range of the experimental measurements. A short ignition region with T = 3000 K was used in the recirculation region of the shear layer to start combustion. The ignition was turned off after the flame was self-sustaining, and the computation was run until a steady-state condition was reached based on the boundary conditions given. Applying an adiabatic c ondition to the combustion chamber wall was considered, but this condition would erase th e thermal boundary layer along the wall. The thermal boundary layer is essential to determining the temperature gradient at the wall, and thus, the heat flux. A constant temper ature condition was considered a su itable alternative, and would represent the case wherein the temperature along the wall was unknow n. A constant temperature

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168 of T = 500 K was specified along the combustion chamber wall. This temperature value was arbitrarily selected as it is w ithin the approximate range of the experimental value, but is not equal to the average experimental temperature at the wall. Results when the experimental temperature profile is used along the combustion chamber wall are not included, as there are no obvious differences in the flow contours whethe r the experimental temp erature profile or a constant temperature is applied at the wall. The injector face and the oxidizer post tip region between the hydrogen and oxygen inlets are set to adiabatic conditions. 6.5.2 CFD Results and Experimental Comparison of Heat Flux In the computation, the heat flux at a location j is calculate d using the following equation: jwallfluid jT qk y (6-7) The calculated heat flux based on Equation 6-7 wa s calculated in the CFD simulation and then compared to the experimental values. Figure 6-17 shows the presence of a strong recirculation region with relatively slow fluid flow. The hydrogen flow enters at a greater velo city than the oxygen flow, but quickly slows due to mixing. Due to the low Mach number of 0.04 and the relativ ely constant pressure throughout the combustion chamber, the density changes th at occur within the combustion chamber are essentially due to changes in temperature. Figu re 6-18 shows the temper ature contours along the z = 0 plane. The combustion length, or the dist ance from the injector face where 99% of the combustion is complete, is at 38 mm. The steady state, one-dimensional heat transf er approximation results in a linear heat flux and temperature relationship. If one accounts for the area variations via a steady state, axisymmetric treatment, then there is a different heat flux and temperatur e profile relationship. Finally, if one chooses to treat the outer wall boundary as adiabatic, to reflect the short

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169 experimental run time, then one can develop an unsteady, two-dimensional representation between the heat flux and the temperature distribu tion as described in Section 6.4. These three cases, labeled as (i) linear one-dimensional, (ii) axisymmetric, and (iii) 2-D unsteady adiabatic are shown in Figure 6-19 along with the co mputed result. Figure 6-20 shows the twodimensional nature of the heat flux along the wa ll. Overall, the agreement between the CFD results and the experiment fair The overall trends from the CFD data and the 2-D conduction analysis are similar, but shifted. The shift in th e CFD data as compared to the results of the 2-D conduction analysis indicates a fast er heat release in the CFD simulation than may actually occur in the experiments. In all cases, the CFD data under-predicts the experimental data. Clearly, there are substantial uncertainties in converting limite d temperature measurements within the wall to heat flux profiles on the wall. There is also uncertainty in the CFD results that may be due to the resolution of the computational domain. The eff ects of the grid resolu tion on the experimental injector element are expl ored in Section 6.6. 6.5.3 Heat Transfer Characterization The three-dimensional simulation allows for a view of the heat flux characteristics across the flat wall. Figure 6-20 shows the two-dimensi onal nature of the predicted heat flux along the wall. Also shown in Figure 6-20 is the eddy con ductivity. The location of peak heat flux can be clearly seen in Figure 6-20A, and corresponds to the maximum eddy conductivity value along the combustion chamber wall. The heat flux contour s show where the heated gases are circulated back towards the injector face located at the left of the plot. Additional measurements would be required for more accurate comparisons of the two-dimensional heat flux and temperature. Across x = 24 mm mark, there is a sudden jump in heat flux magnitude that corresponds to the change in x-velocity from positive to negative as shown in Figure 6-21. The sudden jump in heat flux is shown in Figure 6-22A. The locat ions of stagnation region resulting from the x-

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170 velocity change and the streamlin e reattachment points are clearly apparent in the dips in the y+ values shown in Figure 6-22B. The effect of the velocity change is echoed in the temperature values near the wall (y+ ~ 2.5) as seen in Figure 6-23A. The mixing action of the shear layer transports heat from the flame to the wall. The point of maximum heat flux occurs at 63 mm. This corresponds to the point of maximum eddy conductivity that also occurs at 63 mm from the injector face, shown in Figure 6-23B. Similar to the experimental results, the peak wall temperat ure does not occur in the same location as the maximum heat flux. On the other hand, the ma ximum temperature along the wall occurs at 72 mm, but is essentially constant between 64 mm a nd 79 mm. This is immediately upstream of the reattachment point which is 80 mm from the inje ctor face. The disconnection between the peak heat flux and the peak wall temperature can be re lated to the ratio of eddy viscosity to laminar viscosity. The wall heat flux prof ile corresponds to the eddy viscos ity, and similarly the thermal conductivity. The temperature profile on the other hand corresponds to the laminar, or molecular viscosity. 6.5.4 Species Concentrations From the temperature profile given in Figure 6-18 at y = 0, it can be seen that much of the reaction occurs before 72 mm. This is also evident in the ma ss fraction contours in Figure 6-24. The reaction proceeds beyond that at a much sl ower rate as the system moves toward equilibrium. The species concentration thr ough the flame center is shown in Figure 6-25. Figure 6-25 shows the diffusion of hydrogen towa rds the centerline between 0 and 22mm. The mole fraction of H2 increases at y = 0 due to the consumption of O2. The y = 2 profile is located just beyond the H2 inlet, so the increase in H2 reflects the spreading of the H2 stream. Because the reaction is fuel rich, the hydrogen is not comp letely consumed. There is almost the almost complete disappearance of gaseous oxygen by approxima tely 53mm. This is also consistent with

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171 the rise in temperature that o ccurs at this location. This consumption of oxygen and the high temperatures gives rise to intermediate species. Species concentrations and temperatures near the wall (y+ ~ 2.5) show a jump near 24mm where vx = 0 similar to what was seen in the temper ature and enthalpy profiles. This effect is still found somewhat away from the wall. Fi gure 6-26 shows the homogeneity of the combustion products beyond approximately 72 mm. By this point, the products are comprised almost completely of steam and unburned hydrogen. 6.6 Grid Sensitivity Study For the grid sensitivity study, the injector element used is the same as that from the modeling study along with boundary conditions that simulate a single injector element near a combustion chamber wall. In preparation for th e later parameter study in Chapter 8, the CFD model attempts to approximate a near-wall inj ector element. The sub-domain is approximated using a rectangular shaped computational domai n with width and height of 4.03 mm, and a length of 100 mm. Only one-half of the square cross section is si mulated, with a symmetry plane defined along the symmetric boundary. The boundary conditions for all cases are give n as shown in Figure 6-27. In the CFD model, the distance to the wall is held equal to the distance to the inner slip. The purpose of the inner slip boundary is to keep the computational doma in approximately symmetric. A constant temperature of T = 500 K was specified along the combustion chamber wall. The remaining boundary conditions are the same as those used in Section 6.5.1. The injector face and the oxidizer post tip region between the hydrogen and oxygen inlets were set to adiabatic conditions. The nozzle is not included in the CFD model, and a constant pressure condition is specified at the outlet. A constant mass flow rate for hydroge n and a constant mass flow rate for oxygen are

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172 specified at the inlets, with gaseous oxygen ente ring through the center core of the injector and hydrogen entering through th e outer annular region. The effects of modifying the grid resolution we re investigated via a grid sensitivity study. The computational grids investigated containe d from 23,907 points for the coarse grid 103,628 points for the finest grid. The number of points an d smallest grid density used for each grid is given in Table 6-2, and each grid had a grid distribution as shown in Figure 6-28. A layer of prism elements were used near the wall so that the turbulence equations could be integrated to the wall without the use of wall functions which might reduce accuracy. The prism layers were constructed with an exponential distribution with an increasing number of prism layers used as the initial wall layer height was decreased. The specifications for the initial height of the prism layers are given in Table 6-2. The coarsest grid solution (grid 6) predicts a mu ch sharper rise in the wall heat flux that of the fine grid solution (grid 1), a nd underpredicts the peak heat flux as seen in Figure 6-29. This indicates that performance could suffer when usi ng a very coarse grid. The largest difference in the results can be seen in the length of the co mbustion zone in Figure 6-30. The coarsest grid solution (grid 6) predicts a much shorter combustion zone than the finest grid solution. Using the coarse grid leads to much more smearing along the reaction zone than the two finest grid solutions, grid 1 and grid 2. The species diffuse fast er in grid 6 as compared to the grids 1 or 2. However, the velocity field is less affected by th e grid density than the species diffusion. Thus, the mixing that is responsible for convecting heat to the combustion chamber wall leads to heat flux profiles that are less sensitive to grid de nsity. Significant differences are seen in the CFD solutions as the grid resolution is increased to its finest level. Numerical unsteadiness was seen in

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173 the finest grid solution (grid 1) shown in Figure 6-30. For this reason, the peak heat flux recorded for grid 1 is not considered the “true” peak heat flux. In this case, true grid independency cannot be confirmed. Howeve r, there is relative consistency between the values of the peak heat flux, mean heat flux, and combustion length for the intermediate grids. Over all, the intermediate grid s with 31,184 nodes and 43,680 nodes provide arguably the best tradeoffs between r un time, numerical steadiness, and accuracy as compared with other grid resolutions. A grid with a resolution between that of grids 1 and 2 was used for the CFD modeling study in the Section 6. 5. This indicates that significant improvement in the heat flux profile would not necessarily be obtained by increasing the grid resolution. For the same injector element and mass flow rates used for the grid sensitivity study and for the modeling study, the reduction in the com bustion chamber area appears to result in a reduction of the combustion length. However, because of the numerical unsteadiness of the finest grid, it is difficult to determine whether this is due to the combustion chamber geometry, or due to inadequate grid resolution. It is possible th at the combustion length may be, for the most part, only a function of the injector geometry and mass flow rate, and not a function of the combustion chamber geometry. The parameter study in Chapter 8 tries to determine some of the geometric effects on injector flow. 6.7 Conclusion The CFD model was able to qualitatively ma tch the experimental results. Uncertainty exists in the prediction of the peak heat flux and heat rise. However, the predicted location of the peak heat flux was consistent with the experime nt. The heat flux downstream of the reattachment point was well predicted. In addition, the CFD resu lts were able to provide additional insight into the combusting flow. In particular, the relationshi p between the flow dynamics and heat transfer could be characterized. It was found that the e ffect of the 3-D square geometry shaped the

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174 recirculation in a way that noticeably affected the heat transfer profile. The heat flux along the wall is further shaped due to the recirculati on of fluid that has had time to cool along the combustion chamber wall, resulting in a relatively cool region of fluid n ear the injector face. Beyond the reattachment point, the ho mogenous fluid cools at a relati vely constant rate. The heat transfer within the combustion chamber can be characterized via the eddy conductivity which further relates to the peak heat flux at the wall. Additional experimental temperature measurements could improve the CFD modeling process by providing some measure of the 2-D wall heat transfer effects. A dditional measurements could also improve the accuracy of the estimated wall heat flux and temperatures. A grid sensitivity study on the injector el ement used for the CFD modeling remained inconclusive. A grid independent solution could not be obtained by refining the grid. Numerical unsteadiness resulted with the finest grid. The co arsest grid predicted a sharper heat flux rise, a lower peak heat flux, and a shorter combustion le ngth than the finer grids. With the exception of the coarsest grid and the finest grids, the predicted heat flux remained largely unaffected. This indicates that a finer or coarser grid would likely have little eff ect of the heat flux predicted in the modeling study. However, the predicted com bustion length varied si gnificantly based on the grid resolution. Additional studies would be requ ired to determine the reason for the numerical unsteadiness. Table 6-1. Flow regime description. Inlet temperature, T 300 K Exit pressure, pe 2.75 MPa Mass flow rate, m 0.187 g/s Max Reynolds number, Remax 3000 Min Reynolds number, Remin 960

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175 Table 6-2. Effect of grid resolution on wall heat flux and combustion length. Grid # Grid points Minimum element size in shear layer/near wall (mm) Run time (min) qmax (MW/m2) qmean (MW/m2) Combustion length (mm) 1 103628 0.15 / 0.03109205.412.11 31.1 2 72239 0.2 / 0.0331505.082.03 34.7 3 43680 0.5/ 0.017444.802.09 30.7 4 31184 0.5 / 0.035884.932.06 30.7 5 25325 0.5/ 0.053904.732.07 30.6 6 23907 1 / 0.033924.472.03 23.2 Table 6-3. Flow conditions. Inlet hydrogen mass flow rate, 2 H m 0.396 g/s Inlet oxygen mass flow rate, 2Om 1.565 g/s Equivalence ratio, 2.0 Chamber pressure, p 2.75 MPa Initial combustion chamber temperature, T 2000 K Average Mach number, M 0.04 Figure 6-1. Blanching and cr acking of combustion chamber wa ll due to local heating near injector elements.

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176 y INLET OUTLET x N OSLIP N OSLIP N OSLIP z Figure 6-2. Hydrogen flow geometry. Figure 6-3. Hydrogen inlet mesh.

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177 A B Figure 6-4. Z-vorticity contours. A) = 0 and B) = 5.7. -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 -61434547494 xSwirl Number, S0 50 100 150 200 250area (mm2) = 0 = 5.7 area Figure 6-5. Swirl number at each x location.

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178 0 10 20 30 40 50 60 -20020406080100 xu-0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70v u, = 0 u, = 5.7 v = 0 v = 5.7 Figure 6-6. Average axial velocity u and average tangential velocity v with increasing x. The axial velocity is two orders of magnitude higher than the tangential velocity. A 0 500 1000 1500 2000 2500 3000 -20020406080100120 xReu Reu, = 0 Reu, = 5.7 B 0 100 200 300 400 500 600 700 -20020406080100120 xRev Rev, = 0 Rev, = 5.7 Figure 6-7. Reynolds number profiles. A) ax ial velocity and B) tangential velocity. 0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 -20020406080100120 x(p-po)/po turbulent, = 0 turbulent, = 5.7 Figure 6-8. Non-dimensional pressure as a function of x.

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179 A 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.0020.0040.0060.0080.0 0 vXr B 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0510152025 kr C 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.E+041.E+051.E+061.E+071.E+081.E+09 omega r Figure 6-9. Hydrogen inlet flow profiles. A) Axial velocity, B) turbulent production k, and C) turbulent dissipation 25.4 63.5 + 3.2 19.1 19.1 9.5 15.9 1.6 1.6 Thermocou p les Figure 6-10. Combustion chamber cross-sectiona l geometry and thermocouple locations. Units are given in millimeters.

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180 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 1 1.5 2 2.5 3 3.5 4 x(m)q (MW/m2) Steady-state approximation Unsteady approximation Figure 6-11. Estimated wall heat flux using li near steady-state and unsteady approximations. To Twall T i rwall r i ro Thermocou p les Figure 6-12. 1-D axisymmetric assumption for heat conduction through combustion chamber wall.

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181 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 400 450 500 550 600 xTwall(K) linear axisymmetric Figure 6-13. Estimated wall temperatures using linear and axisymmetric approximations. Linear and axisymmetric assumptions give nearly equal wall temperature estimations. Temperature values at x = 0m and x = 0.16m are estimated based on nearest temperature values. A 0 0.005 0.01 0.015 0.02 0.025 0.03 -0.01 -0.005 0 0.005 0.01 0.015 0.02 B 0 0.005 0.01 0.015 0.02 0.025 0.03 -0.01 -0.005 0 0.005 0.01 0.015 0.02 C 0 0.005 0.01 0.015 0.02 0.025 0.03 -0.01 -0.005 0 0.005 0.01 0.015 0.02 D 0 0.005 0.01 0.015 0.02 0.025 0.03 -0.01 -0.005 0 0.005 0.01 0.015 0.02 550 300 Figure 6-14. Temperature (K) contours for 2-D unsteady heat conduc tion calculations at x = 0.084 m at A) 1 s, B) 2 s, C) 6.5 s, and D) 7.75 s. Units are given in meters. Onequarter of the combustion chamber cross sec tion is shown. The temperature scale is consistent across plots.

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182 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 2.5 3 3.5 4 x(m)q (MW/m2)Unsteady approximation unsteady linear unsteady axisymmetric 2-D unsteady adiabatic Figure 6-15. Experimental heat flux va lues using unsteady assumptions. The x values are given in meters. A B C Figure 6-16. Computational model for single-elem ent injector flow simulation. A) Combustion chamber boundary conditions, B) grid closeup, and C) representative combustion chamber section.

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183 Figure 6-17. Velocity contours vx(m/s) and streamlines. The injector is located in the lower right corner of the combustion chambe r. The reattachment point is at x = 77 mm. Figure 6-18. Temperature (K) cont ours. Injector center is at x,y = (0,0). Distances are given in millimeters. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 2.5 3 3.5 4 x(m)q (MW/m2) 1-D linear axisymmetric unsteady adiabatic CFD Figure 6-19. CFD heat flux values as compar ed to experimental heat flux approximations.

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184 A zx q (MW/m2) 0 20 40 60 80 100 120 140 160 -10 0 10 0 0.5 1 1.5 2 2.5 B C Figure 6-20. Wall heat transf er and eddy conductivity contour plots. A) Heat flux along combustion chamber wall (xz-plane at y = 12.7) B) eddy conductivity at y = 12.3, and C) eddy conductivity at z = 0 with wall at y = 12.7 mm. Eddy conductivity contours are oriented perpendicular to wall he at flux contours. Horizontal axis is x-axis. (Solutions mirrored across x = 0). Length units are in mm. The injector is centered at x,y,z = (0,0,0), and the flow proceeds from left to right. Figure 6-21. Streamlines and temperature contours at plane z = 0. All lengths are in millimeters.

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185 A 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 2.5 3 x(m)q (MW/m2)Unsteady approximation vx = 0B 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 2.5 3 3.5 4 x (m)y+ vx = 0 reattachment point Figure 6-22. Heat flux and y+ profiles along combustion chambe r wall. A) Heat flux profile along the wall at z = 0 showing sharp increase in heat flux corresponding to the location of velocity vx = 0. B) The y+ values along the near wall cell boundary dip at stagnation regions. A B Figure 6-23. Temperature and e ddy conductivity profiles at various y locations on plane z = 0. A) Temperature. B) Eddy conductivity.

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186 Figure 6-24. Mass fraction c ontours for select species. H2 O2 OH H2O

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187 A B Figure 6-25. Mole fractions for all speci es along combustion chamber centerline (y = 0, z = 0). A) uniform and B) log scaling.

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188 A B Figure 6-26. Select species mole fraction profiles. A) H2 and B) H2O at plane z = 0.

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189 Figure 6-27. Sample grid and boundary conditions. Figure 6-28. Computational gr id along symmetric boundary. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 1 2 3 4 5 6 7 xq (MW/m2) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 xy+ 23907 31184 72239 103628 23907 31184 72239 103628 Figure 6-29. Wall heat flux and y+ values for select grids. The distance from the injector face x is given in meters.

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190 Figure 6-30. Comparison of temp erature (K) contours for grids with 23,907, 31,184, 72,239, and 103,628 points, top to bottom, respectively. The finest grid shows time-dependent oscillations.

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191 CHAPTER 7 MULTI-ELEMENT INJECTOR FLOW MODE LING AND ELEMENT SPACING EFFECTS 7.1 Introduction Because of the arrangement of injector elemen ts near the wall of the combustion chamber, the heat flux peaks near each injector element. Chamber design and potential material failure is based on peak heat flux, so the peak heat flux mu st be minimized to increase component life and reliability. By controlling the injection pattern, a layer of cool gas near the wall can be created. One method of controlling the inj ection pattern is to offset the oxidizer post tip away from the wall. This results in a higher percentage of unburned fuel near the wall, and this technique was used in the SSME. The cooler layer near the wa ll can possibly further minimize the peak heat flux. Another method of controlling the heat flux is to change the spacing of injector elements near the combustion chamber wall. This research effort looks to quantify selected geometry effects by directly exploring the sensitivity of wall heating and injector performance on the injector spacing. Shyy et al.29 and Vaidyanathan et al.63 identified several issues inherent in CFD modeling of injectors, including the n eed for rigorous validation of CF D models and the difficulty of simulating multi-element injector flows due to lengthy computational times. They chose to simulate single injector flow, as this allowed fo r the analysis of key co mbustion chamber life and performance indicators. The lif e indicators were the maximum temperatures on the oxidizer post tip, injector face, and combustion chamber wall, and the performance indicator was the length of the combustion zone. These conditions were expl ored by varying the impinging angle of the fuel into the oxidizer. In other words, the injector type was gradually varied from a shear coaxial injector to an impinging injector element. Re sponse surface approximations were used to model the response of the injector system, and a co mposite response surface was built to determine the

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192 injector geometries that provide d the best trade-offs between performance and life. It was determined that a shear coaxial injector provi des the lowest maximum temperatures on the injector face and combustion chamber wall, but had the lowest performance. An impinging element provided the lowest ma ximum oxidizer post tip temperat ure and had high performance due to the short combustion length, but because the maximum injector face and combustion chamber wall temperatures were high, the life of the injector was compromised. The study confirmed that small changes in the injector geom etry can have a large impact on performance. The optimization procedure was successful in spec ifically accessing the effects of a small change in the injector geometry. This is important, as it allows for greater confidence in choosing the next design. This information can be coupled with the experi ence of injector designers to accelerate the design process. The purpose of the proposed research is to im prove the performance of upper stage rocket engines by raising the temperature of the fuel en tering the combustor. A multi-element injector model is constructed. An effort is made to capture the 3-D effects of a full multi-element injector face in a simplified 3-D model. The step s to improve the injector design are twofold: 1) the mean heat flux to the combustion chamber wall must be increased to increase fuel temperature and subsequently incr ease engine performance, and 2) the peak heat flux to the wall must be decreased to diminish the effects of lo cal heating on the chamber wall. The injector is optimized to best satisfy the objectives. The feasible design space is determined th rough exploration of the design space using available design space refinement and selection techniques. Afte r the feasible design space is determined, the optimization framework described in Chapter 2 is used to guide the injector

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193 element spacing study. The design space is refine d as needed to obtain an accurate surrogate model. 7.2 Problem Set-Up This study explores the effects of geometry on the peak heat flux and combustor length. Approximately 90% of the heat flux to the combusto r wall is due to heat transfer from the outer row of injector elements. The second row of inj ector elements is respon sible for approximately 10% of heat flux to the wall. This information wa s used to simplify the computational model of a GO2/GH2 shear coaxial multi-element injector to make it suitable for a parametric study. The injector face simplification of the full inj ector shown in Figure 7-1 is outlined below: 1. For the CFD model, only the outer row of elemen ts is considered. Currently, the effects and interactions of the outer el ement row with interior elements are simplified by imposing symmetric conditions. 2. The multi-element injector effects are modeled by imposing symmetric conditions between injectors with the computational domain si zes corresponding to the actual radial and circumferential spacing of the full injector. Thus, the CFD model is a scaled down version of what would occur in an actual combustion chamber. The CFD model attempts to capture e ffects near the outer edge of the injector. The variables are the circumfere ntial spacing (number of injector elements in outer row), wherein the total mass flow rate through the outer row injector elem ents is held constant, and the radial spacing, such that the distance to the wall is held equal to the distance to the inner slip boundary. By this assumption, the mass flow rate through a single injector is equal to the total mass flow rate through the outer ro w of injectors divided by the num ber of injectors in the outer row. The purpose of the inner slip boundary is to keep the computationa l domain approximately symmetric, as the contributions and interactions from inner elements are neglected in the analysis. It is desired that the CFD model be ab le agree qualitatively with related experiments as well as establish the sensitivity of certain geometry changes on wall heating.

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194 The CFD simulations were conducted using loci-STREAM157 (Thakur and Wright 2006). The CFD model attempts to approximate the sing le-element section near the chamber’s outer wall given in Figure 7-1. The single-element section was approximated using a rectangular shaped computational domain. Only one-half of the domain is simulated. The grid resolution used is equivalent to the intermediate grid 3 fr om the grid sensitivity study of Section 6.6. This grid was chosen for its balance of accuracy a nd relatively short run-time. The solutions are expected to give good heat flux accuracy, but the combustion lengths are expected to be somewhat under-predicted. In summary, the objectives are to minimize the peak heat flux qmax(N*,r*) and minimize the combustion length LC(N*,r*) where the independent variables are explored within the range max0.751.25 0.251.25baselineN r NNN (7-1) where **,baselinebaselineNNNrrr (7-2) where N is the number of injector elements, Nmax is the maximum possible number of injector elements in the outer row, and r is given as shown in Figure 7-1. The combustion length is the length at which combustion is 99% complete and a shorter combustion length indicates increased performance. A low peak heat flux would reduce the risk of wall burnout. Although the number of injector elements in the outer row changes, the total combined mass flow rate to the outer elements also remains constant, so the mass flow rates for the in dividual injectors are a function of the number of injector elements N. The distance w of the computational domain in Figure 7-1B is given by

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195 tan 2 D wNR N (7-3) where R is the distance from the center of the injector to the center of an outer element injector element and D is the injector element diameter. The baseline injector element geometry is ba sed on the experimental test case RCM-1 from Section 4.1.4, while the overall injector a nd element spacing is based on the Integrated Powerhead Demonstrator159 (IPD) main injector. For this injector, the maximum number or injector elements Nmax allowable in the outer row is 101. The flow conditions and injector geometry for the baseline injector are given in Table 7-1. The inlet flow represents incompletely burned flow from upstream preburners, so both th e fuel and oxidizer contain some water as a constituent. 7.3 Feasible Design Space Study The design points based on r* and N* are shown in Figure 7-2. The first seven design points were selected based on a pr eliminary design sensitivity study. Point 16 lies at the center of the design space. Latin Hypercube Sampling was then used to select the seven remaining design points within the variable ranges. Figure 7-3 shows the values of the objectives at each design point. It can be seen that the mean heat flux across the combustion chamber wall stays relatively constant across all of the design points. The maximum heat flux, however varies depending on the combustor geometry. The design points have an averag e mean heat flux of 8.2 MW/m2 and an average maximum heat flux of 37.3 MW/m2. In general, as the maximum heat flux becomes lower, the combustion length is longer, indicating slower combustion. For design purposes, there would need to be some balance between the maximum heat flux an d combustion length. From Figure 7-3B, it can be seen that case 6 shows good balance between minimizing the peak heat flux and minimizing

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196 the combustion length, while case 12 gives argu ably the worst characteristics due to its high maximum heat flux and long comb ustion length. Figure 7-3 also shows the maximum heat flux and combustion length as a function of the hydroge n mass flow rate. It is apparent that the maximum heat flux is not a function of the mass fl ow rate alone, but that there must be other influencing factors. There does seem to be an ove rall decrease in the maximum heat flux with the mass flow rate, as well as an overall increase in combustion length with increasing mass flow rate, which is a function of N* No obvious relationship was seen between the maximum heat flux or combustion length and th e cross-sectional area. This i ndicates that, a smaller crosssectional area does not automatically translate into a longer combustion length, as one might expect. This was also demonstrated by the grid sensitivity study. The CFD results for four cases, represen ting the baseline case along with cases demonstrating good or poor heat flux characteristics, are shown in Figure 7-4. It can be seen that the flame is flattened somewhat based on the sh ape of the computational domain. Another major observation is of the location of the recirculation region. Instead of hot gases being directed at the wall, as in the baseline case and case 12, th e recirculation region of case 2 and case 6 is located between injector elements, and the str eamlines are largely parallel to the combustion chamber wall. This led to an observation that a ll of the CFD solutions resulted in one of two scenarios. Eleven of the cases showed a sharp pe ak in the wall heat flux near the injector face with the downstream heat flux considerably lowe r than the peak heat flux. This is termed “Pattern 1,” and corresponds to cases that have ho t gases directed at the wall as in Figure 7-5A. The remaining five cases had a relatively low he at flux near the combustion chamber wall that grew gradually to a slight p eak further downstream. This gr oup is termed “Pattern 2,” and corresponds to the cases that have a recircula tion zone that would exist between injector

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197 elements. The two groups can be clearly seen in Figure 7-6 in the plot of maximum heat flux versus r* Figure 7-7 shows the heat flux profile and contours for case 12 of Pattern 1, and case 6 of Pattern 2. In Pattern 2, the heat flux is sp read across the combustion wall, rather than being localized as were the cases from Pattern 1. The best performing designs w ith regards to heat flux appear to be those with small distances between the injector and the wall. However, of the cases in Pattern 2, the worst cases are cases 7 and 9 th at have smaller domain widths as compared to the other cases. This would suggest that the peak heat flux might be a function of the domain’s aspect ratio, rather than separate ly being a function of radial or circumferential spacing, alone. The domain’s aspect ratio is given by AR = r* / w* where w* = w / wbaseline, and wbaseline is determined based on N = Nbaseline in Equation 7-3. It can be seen in Figure 7-8 that the value of the peak heat flux appears to be strongly depe ndent on the aspect ratio. The designs with the lowest peak heat flux also have th e lowest aspect ratio. This indi cates that it may be desirable to have a low ratio of wall spacing (radial spacing) to injector-injector sp acing (circumferential spacing) to reduce the peak heat flux. Due to the presence of case 5, it is difficult to determine whether the cases approximately follow a linear tre nd and case 5 is an ou tlier, or if the actual trend is quadratic. It appears that the peak heat flux is highly sensitive to the combus tion chamber geometry as evidenced by Figure 7-8. The re lationship of the combustion length to geometry is less clear. It appears that there is some sensitivity to the ge ometry due to the fact that equivalent injector elements with equal mass flow rates can result in combustion lengths that can differ significantly (see points 2, 4, and 6 in Figure 7-3D). In gene ral, the cases in the region defined by Pattern 1 have shorter combustion lengths than those in Pattern 2. This indi cates that increasing the size of the recirculation zone, and hence the mixing, may increase the combustion efficiency. An

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198 individual analysis on a single injector element of the eff ect of the combustion chamber geometry on combustion length with constant mass flow rate may be necessary to determine the relationship. The best trade-off points are located in one or the other region of the design space, as shown in Figure 7-9. Because the behavior of th e design points in the different regions are very different, this indicates the need for two sepa rate design spaces. The dividing line for the new design space is given approximately by the dash ed line in Figure 7-9B. The design space will also be expanded slightly to *0.601.25 0.201.25 N r (7-4) due to the best trade-off poi nts of pattern 2 having low N* and r* values. While it does appear that the combustion length is definitely affected by the geometry, the results of the feasible design space study were not able to reveal the direct relationship between combustion length and geometry. Based on a comput ational model for an injector element near the combustion chamber wall of a multi-element in jector, it was found that an injector element that is located far from the combustion chamber wall does not always result in the best heat transfer characteristics. The resu lts of the feasible design spa ce study suggest that by increasing the spacing between injector elem ents of the outer row, while re ducing the distance of the outer row to the wall, that the heat tr ansfer results could be better c ontrolled. This configuration helps to direct heat away from the wall, rather than towards it, and results in an even distribution of heat across the combustion chamber wall. To a lesser degree, increasing the distance to the combustion chamber wall can also result in reduced peak heat fl ux, but the required spacing may possibly be prohibitively large.

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199 7.4 Design Space Refinement Using the merit function smart point selec tion procedure introduced in Section 2.4.3, along with an assigned rank using the formulation for multi-objective problems introduced in Equation 2-38, additional points were added to the newly expanded design space. The merit function that maximizes the probability of improvements MF2 was chose for point selection, as it resulted in the least error for a small number of sampling point s as was shown the radial turbine analysis in Section 3.4. The design space refinement proceed ed over several steps. The steps are listed briefly, then descri bed in more detail: 1. Kriging RS was constructed for each objective over 16 initial design points. Based on the results initial Kriging RS, the design space vari able ranges were expanded due to favorable function values on the edge of design space. Four points were selected via the multiobjective merit function-based point selection crit eria to add to the original 16 points for a total of 20 designs. 2. The design space was separated based on flow patterns into two regions called Pattern 1 and Pattern 2. A Kriging RS was construc ted for each objective for the 12 out of 20 designs in Pattern 1. Three new points were selected using MF2 for a total of 15 designs in Pattern 1. 3. A Kriging RS was constructed for each objective for the 8 out of 20 designs in Pattern 2. Four new points were selected using MF2 for a total of 12 designs in Pattern 2. 4. A Kriging RS was constructed for each obj ective for the 15 desi gns in Pattern 1. 5. A Kriging RS was constructed for each obj ective for the 12 desi gns in Pattern 2. 6. For visualization of the full design space, a Kriging response surfaces were fit for each objective to all 27 design points. The error statistics for all six steps ar e given for each objective in Table 7-2. A Kriging surrogate was fit to the entire de sign space before the design space separation. The merit function value was calculated for point s sampled across the design space as shown in Figure 7-10. The densely sampled poi nts were divided into eight clusters based on location. The point with the best rank in each cluster was select ed. Out of the eight resulting points, four were

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200 chosen based on their favorable objective function values predicted by th e surrogate model. The Kriging surrogate results along w ith the four selected points ar e shown in Figure 7-11. While Kriging can provide an adequate surrogate model for the peak heat flux, Figure 7-11B shows that the design space is too large for an appropriate su rrogate fit for the combustion length. When the PRESSrms is calculated as a percent of the objectives sampling range or the standard deviation of the objectives, it can be seen that the PRESSrms error for the LC surrogate fit is about twice that of the qmax surrogate. The design space was thus divited into two smaller regions to try to improve the surrogate fits. Fifteen of the 20 design points were located in the region for Pattern 1, whi