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CFDBASED 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 CFDBased Optimization .............. .. .......... ......... ... ........2
1.3.1 Optimization Techniques for Computationally Expensive Simulations ................25
1.3.2 Design Space Refinement in CFDBased 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 MultiObj 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 SingleElement Injectors .............. ...............106....
4. 1.2 MultiElement 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 MULTIELEMENT INJECTOR FLOW MODELING AND ELEMENT SPACING
EFFECT S............... ...............19
7. 1 Introducti on ................. ...............191........... ...
7.2 Problem SetUp .................... ...............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 SingleElement Inj ector Flow Modeling .....__ ................ ............... 222 ...
8.4 MultiElement Inj ector Flow Modeling. ......... ........____ ......... ..........22
8.5 Future Work............... ...............224.
REFERENCE LIST .............. ...............225....
BIOGRAPHICAL SKETCH .............. ...............238....
LIST OF TABLES
Table page
11 Summary of CFDbased design optimization applications with highlighted surrogate
based optimization techniques. ............. ...............32.....
21 Summary of DOE and surrogate modeling references. ........._..__......_. ..............67
22 Design space refinement (DSR) techniques with their applications and key results.........68
31 Variable names and descriptions. ............. ...............92.....
32 Response surface fit statistics before (feasible DS) and after (reasonable DS) design
space reduction............... ...............9
33 Original and final design variable ranges after constraint application and design
space reduction............... ...............9
34 Baseline and optimum design comparison. ............. ...............93.....
41 Selected inj ector experimental studies ................. ...............128..............
42 Selected CFD and numerical studies for shear coaxial inj ectors. ........._..... .........._....129
43 Reduced reaction mechanisms for hydrogenoxygen combustion. ............. .................129
51 Number of grid points used in various grid resolutions. ........._._ ....... .... ............149
52 Data statistics in the grid comparison of the CFD data. ................ ................. .... 149
53 Comparison of cubic response surface coefficients and response surface statistics. .......150
54 Comparison of radialbasis neural network parameters and statistics. .................. .......... 150
55 RMS error comparison for response surface and radial basis neural network. ...............150
56 Total pressure loss coefficient and mixing index for extreme and two regular designs
for multiple grids............... ...............151.
57 Total pressure loss coefficient and mixing index for designs in the immediate
vicinity of Case 1 using Grid 3......... ........ ......... ................ ...............151
61 Flow regime description. ............. ...............174....
62 Effect of gri d re soluti on on wall heat flux and combustion l ength ................. ...............17 5
63 Flow conditions............... ..............17
71 Flow conditions and baseline combustor geometry for parametric evaluation. .............204
72 Kriging PRESSrms error statistics for each design space iteration. ................ ...............205
LIST OF FIGURES
Figure page
11 Rocket engine cycles............... ...............33.
12 SSME thrust chamber component diagram .............. ...............33....
13 SSME component reliability data ................. ...............34...............
14 Surrogatebased optimization using multifidelity data ................. ................. ...._.34
21 Optimization framework flowchart. ........_................. ...............69. ....
22 DOEs for noisereducing surrogate models. .............. ...............70....
23 Latin Hypercube Sampling. ............. ...............70.....
24 Design space windowing .............. ...............71....
25 Smart point selection............... ...............7
26 Depiction of the merit function rank assignment for a given cluster. .............. .... ..........._72
27 The effect of varying values of p on the loss function shape.............__ ........._ ......72
28 Variation in SSE with p for two different responses ........... _...... __ ........_......73
29 Pareto fronts for RSAs constructed with varying values of p............... ..................7
210 Ab solute percent difference in the area under the Pareto front curves ................... ...........74
31 Midheight static pressure (psi) contours at 122,000 rpm. ............. ......................9
32 Predicted Meanline and CFD totaltostatic efficiencies. ................. .................9
33 Predicted Meanline and CFD turbine work. ............. ...............95.....
34 Feasible region and location of three constraints ..........._._ ......_.._ .. ......_.._......9
35 Constraint surface for Cx2 Utip = 0.2............... ...............96..
36 Constraint surfaces for B1 = 0 and B1 = 40. ................ ............... ......... ....... ..96
37 Region of interest in function space ................. ...................... ..................97
38 Error between RSA and actual data point............... ...............97.
39 Pareto fronts for p = 1 through 5 for second data set............... ...............98..
310 Pareto fronts for p = 1 through 5 for third data set .............. ...............99....
311 Pareto Front with validation data ................. ...............99...............
312 Variation in design variables along Pareto Front ................. ............... ......... ...10
313 Global sensitivity analysis .............. ...............100....
314 Data points predicted by validated Pareto front compared with the predicted values
using six Kriging models based on 20 selected data points ................. ............. .......101
315 Absolute error distribution for points along Pareto front............... ...............101.
316 Average mean error distribution over 100 clusters ................. ................ ......... .102
317 Median mean error over 100 clusters............... ...............10
318 Average maximum error distribution over 100 clusters .............. ....................10
319 Median maximum error over 100 clusters ................. ...............104.............
41 Coaxial inj ector and combustion chamber flow zones. ................ ........................130O
42 Flame from gaseous hydrogen gaseous oxygen single element shear coaxial
inj ector. ............. ...............13 0....
43 Multielement inj ectors ................. ...............13. 1..............
44 Wall burnout in an uncooled combustion chamber. ............. ...............132....
45 Test case RCM1 inj ector. ................ ...............132........ ...
46 Temperature contours for a single element injector. ............. ...............133....
47 CFD heat flux results as compared to RCM1 experimental test case. ................... ........134
48 Multielement inj ector simulations ....__. ................. ........._.._ ....... 13
49 Fuel rich hydrogen and oxygen reaction with heat release..........._.._.. ............. .......136
51 M odified FCCD. ............. ...............15 1....
52 Bluff body geometry ................. ...............151......... .....
53 Computational domain for trapezoidal bluff body. ....._____ ..... ...___ ................. 152
54 Computational grid for trapezoidal bluff body. ................ .....___............... ....15
55 Bluff body streamlines and vorticity contours ................. ...............152........... ..
57 Comparison of response surface (top row) and radial basis neural network (bottom
row) prediction contours for total pressure loss coefficient............... ...............5
58 Comparison of response surface (top row) and radial basis neural network (bottom
row) prediction contours for mixing index (including extreme cases) ................... .........154
59 Comparison of response surface (top row) and radial basis neural network (bottom
row) prediction contours for mixing index (excluding extreme cases) ...........................155
510 Variation in obj ective variables with grid refinement ........._._. ......___ ............... 155
511 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...... ......
512 Comparison of response surface and radial basis neural network prediction contours
for mixing index at B* = 0 and h* = 0 ..........._ ....._.._ ...............15
61 Blanching and cracking of combustion chamber wall due to local heating near
inj ector el ements. ........... ..... ._ ............... 175...
62 Hydrogen flow geometry. ............. ...............176....
63 Hydrogen inlet mesh. .............. ...............176....
64 Zvorticity contours............... ...............17
65 Swirl number at each x location ................. ...............177........... ..
66 Average axial velocity u and average tangential velocity vs with increasing x...............178
67 Reynol ds numb er profile es ........._.._ ...... .___ ............... 178..
68 Nondimensional pressure as a function of x ................. ...............178.............
69 Hydrogen inlet flow profiles ................. ...............179........... ...
610 Combustion chamber crosssectional geometry and thermocouple locations. ................179
611 Estimated wall heat flux using linear steadystate and unsteady approximations. ..........180
612 1D axisymmetric assumption for heat conduction through combustion chamber wall .180
613 Estimated wall temperatures using linear and axisymmetric approximations .................18 1
614 Temperature (K) contours for 2D unsteady heat conduction calculations ................... ..181
615 Experimental heat flux values using unsteady assumptions ................. .....................182
616 Computational model for singleelement inj ector flow simulation .............. ..............182
617 Velocity contours vx(m/s) and streamlines............... ..............18
618 Temperature (K) contours............... ...............18
619 CFD heat flux values as compared to experimental heat flux approximations. ..............183
620 Wall heat transfer and eddy conductivity contour plots. ............. .....................8
621 Streamlines and temperature contours at plane := 0............... ...............184...
622 Heat flux and y+ profiles along combustion chamber wall. ........._.._.._ ....._.._.. ......185
623 Temperature and eddy conductivity profiles at various y locations on plane := 0.........185
624 Mass fraction contours for select species............... ...............186
625 Mole fractions for all species along combustion chamber centerline (y = 0, z = 0) ........187
626 Select species mole fraction profiles............... ...............18
627 Sample grid and boundary conditions............... ..............18
628 Computational grid along symmetric boundary. ............. ...............189....
629 Wall heat flux and y+ values for select grids ........... ......__ ... ..___.......18
630 Comparison of temperature (K) contours for grids with 23,907, 3 1,184, 72,239, and
103,628 points, top to bottom, respectively ................. ...............190........... ..
71 Injector element subsection............... ..............20
72 Design points selected for design space sensitivity study .............. ......................0
73 Effect of hydrogen mass flow rate on obj ectives ................. ...............206............
74 Oxygen isosurfaces and hydrogen contours. ............. ...............207....
75 Hydrogen contours and streamlines ................. ...............207........... ...
76 Maximum heat flux for a changing radial distance r* ................... ............... 20
77 Heat flux distribution. .............. ...............208....
78 Maximum heat flux as a function of aspect ratio ................. ...............209............
79 Design points in function and design space. .............. ...............209....
710 Merit function (M~Fz) contours for ........._ ....... ...............210.
711 Kriging surrogates based on initial 16 design points. ................... ................1
712 Kriging surrogates and merit function contours for 12 design points in Pattern 1 ..........211
713 Kriging surrogates and merit function contours for 12 design points in Pattern 2..........212
714 Kriging fits for all 15 design points from Pattern 1 .............. ...............213....
715 Kriging fits for all 12 design points from Pattern 2. ................................... 21
716 Pareto front based on original 16 data points (dotted line) and with newly added
points (solid line). ............. ...............214....
717 Approximate division in the design space between the two patterns based on A) peak
heat flux .............. ...............215....
718 Variation in flow streamlines and hydrogen contours in design space. ................... ........215
719 Location of best tradeoff points in each pattern group in design space. ................... .....216
720 Injector spacing for selected best tradeoff design point .............. ....................21
721 Predicted heat flux profies for baseline case (case 1), best tradeoff from Pattern 1
(case 19), and best tradeoff from Pattern 2 (case 18). ................ ............... ...._...217
722 Heat flux profies for different grid resolutions. The coarser grid was used to
construct the surrogate model.. ............ ...............218.....
723 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
CFDBASED 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 baseddesign optimization can be made computationally affordable
through the use of surrogate modeling which can then facilitate additional parameter sensitivity
assessments. The present study investigates surrogatebased 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 surrogatebased 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 multiobj ective design point selection reduced the number of design
points by an order of magnitude while maintaining good surrogate accuracy among the best
tradeoff points. Second, bluff bodyinduced 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 threedimensional 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 multielement 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 multiobjective merit function
formulation facilitated an efficient framework to investigate the multiple and competing
objectives. The analysis suggests that by adjusting the multielement 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 trialanderror exercises. Optimization tools can be used to guide the
selection of design parameters and highlight the predicted best designs and tradeoff 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 CFDbased 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 CFDbased design. In
summary, this research effort seeks to improve the design efficiency of complex rocket engine
components using improved CFDbased 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, CFDbased optimization and DSR are introduced as methods for improving
upon traditional experimentallybased 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.
Pressurefed engines are relatively simple;' two high pressure tanks are connected directly
to the combustion chamber, and valves are used to regulate the flow rate. Pressurefed engines
have low costs and good reliability because they are simple with few components. However,
pressurefed 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 gasgenerator cycle (Figure 11B) 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 pressurefed engine
cycle.ll
The expander cycle (Figure 11A) eliminates the need for a gas generator, thus resulting in
a simpler and more reliable system. The expander cycle is the simplest of the pumpfed engine
cycles,12 and it offers a high specific impulse.13 Current expander cycle rocket engines in use are
Pratt and Whitney's RL60 and the Ariane 5 ESCB. 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 12, 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 fuelrich 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 11C. 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 13. The components with the
lowest reliabilities are the highpressure fuel turbopump (HPFTP), the highpressure 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 gasgas 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 smallscale tests,
which are then scaled to a fullscale model.19 Many inj ectors are also designed based on cold
(nonreacting) 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 CFDbased 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 uncooled 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 computeraided designs. These early analyses
included onedimensional approximations such as equilibrium reactions. These equilibrium
approximations were based on the assumption of "fastchemistry," 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 8species, 18 reaction chemical
kinetics model with the kE 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 multielement 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 1D 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 finetune 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 CFDBased 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 777300,35 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 gradientbased methods,384 adjoint methods,44,45 and surrogate modelbased
optimization methods such as response surface approximations (RSA),46 Kriging models,474 Of
neural network models.'o Gradientbased 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 gradientbased and adj oint methods, the optimization is
serially linked with the computer simulation.
At times, gradientbased optimization is not feasible. For example, Burgee et al.5 found
that an extremely noisy obj ective function did not allow the use of gradientbased optimization,
as the optimizer would often become stuck at noiseinduced 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. Surrogatebased optimization (SBO) uses a
simplified, loworder 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 tradeoffs, 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 multiobj 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 centralcomposite designs (CCD) and face
centered cubic designs (FCCD), or they may be tested across the full range of design parameter
values as in multilevel 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 geometrybased
bounds on the design variables. Vaidyanathan et al.63 chose variable ranges based on variations
of an existing design in the CFDbased optimization of a single element rocket inj ector.
Commonly, the design space cannot be represented by a simple boxlike 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 CFDbased 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 lowfidelity model is a
qualitatively similar to the high fidelity model. In other instances, lowfidelity data may be
combined with highfidelity data to reduce the overall number of expensive highfidelity runs.
The lowfidelity data may employ a correction surrogate for improved accuracy, as shown in
Figure 14.
Han et al.64 perfOrmed a surrogatebased optimization on a lowfidelity 2D CFD model
and tested the results on a more expensive 3D model in the optimization of a multiblade fan
and scroll system and verified the 2D model using physical experiments as well as comparisons
to previous studies and the 3D model. Keane65 built an accurate surrogate model of a transonic
wing using highfidelity 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 highfidelity 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 lowfidelity model to better approximate a high fidelity model in
the prediction of crack propagation. Thus, the accuracy of the lowfidelity models can be
improved by combining the lowfidelity 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 lowfidelity model to aid in the optimization of shell structures for
buckling. Balabanov et al.70 applied a surrogate correction to a lowfidelity 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, computationalbased 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, CFDbased 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 modelperhaps one with more
flexibilityand 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 CFDBased 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 highdimensional 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 boxshaped 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 superdetonative 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 lowfidelity analysis prior to optimizing
using a highfidelity 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 CFDbased 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
lowfidelity 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 CFDbased 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 lowfidelity 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 1D 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, CFDbased optimization techniques are applied to improve the design process of a
multielement rocket inj ector. In this case, the optimization techniques must be applied in the
context of a CFDbased design process where a limited number of function evaluations are
available. Chapter 8 concludes the effort.
Table 11. Summary of CFDbased 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
Multiblade
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
Lowfidelity
optimization
Multifidelity
optimization
Multifidelity
Optimization
Multifidelity
optimization
Multifidelity 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 geometrybased
bounds on design variables
Design variables chosen from variations of
existing design
Optimized lowfidelity 2D model to predict for
highfidelity 3D model
Supplemented highfidelity CFD data with low
fidelity empirical model
Varied mesh density for variable fidelity models
Saved 255 CPU hours using lowfidelity
supplemental data
Used coarse and fine finite element models as
low and highfidelity 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 lowfidelity model to refine design space
before optimizing using highfidelity 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 11. Rocket engine cycles.
cycle.
A) Expander cycle, B) gas generator cycle, C) preburner
I POTP
Figure 12. SSME thrust chamber component diagram showing lowpressure fuel turbopump
(LPFTP), lowpressure oxidizer turbopump (LPOTP), highpressure fuel turbopump
(HPFTP), highpressure 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
Higfdeit i ;Eb mmBt~ P
model
Surrogate K~
Figure 13.S Asses surrogante refia ine design space rpodcdfrm[
Lwfidelitymoe
Figure 14. Surrogatebased op imizatio sn utieiydta utieiyaayi
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 21 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 multiobjective 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 1D 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 21.
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 facecentered cubic designs (FCCD) for boxshaped domains. Design
optimality designs such as D or Aoptimal83 (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, spacefi11ing designs such
as LatinHypercube Sampling (LHS)60 Or Orthogonal Arrays (OAs)46 are preferred to efficiently
cover the entire design space.
Central composite designs and facecentered 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 22. The center point detects curvature in the system while all other points are
pushed as far as possible from the center provide noisesmoothing 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 spacefi11ing 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 LatinHypercube 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 23A. 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 quasiuniform 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 23B).
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 nonparametric 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 nonparametric 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 rootmeansquare (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 crossvalidation 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 2ndorder model, but higher order models may also be used. A 2ndorder model requires a
minimum of(k +1I)(k + 2)/2 design points, where k is the number of design variables. Thus, a
2ndorder model with three variables would require at least 10 points. A 3rdorder model with
three variables would require twice that amount. A 2ndorder model provides good tradeoff 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. RaisRohani 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 multifidelity analyses in the
multidisciplinary optimization of aircraft. Additional references and their key results are given
in Table 21.
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 (21)
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 21 can be expressed in matrix form as
f = XP+ e E (E)=0 V (E) = 0 (22)
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 (23)
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 (24)
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 (25)
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 highorder 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
CFDbased 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, secondorder, 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 (26)
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') (27)
In this study, a Gaussian correlation structure'oo is used
R =ep k X/k X (28)
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 nonsampled points are
j)= + rTR' 1 ) (29)
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 Nsvector of ones. The mean is estimated by
p= 1 R'1) 1 I'R y (210)
and the standard deviation of the response is estimated as
<^r =(lLT 'yp (211)
On the other hand, the estimation variance at nonsampled design points is given by
11'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 overfitting 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 userdefined error criterion to filter noise to prevent data
overfitting. 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 twolayer networks consisting of a radialbasis function
(RBF) and a linear output layer. The radial basis function is given by
f= radbas c xb) (213)
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) (214)
where n is given by the term in parenthesis in Equation 213. Based on these equations, the
function prediction is given by
j) ,~=1 w, ep b (215)
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 spacefilling 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 overfitting 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 undertrained 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 overfitting 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 (216)
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= (217)
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 (218)
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 211i. 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= ,= (219)
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/ (220)
and PRESSms is given by
PRESS '=1 (221)
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,) (222)
where / = = of dx If the following condition
dxk = 0(223)
is imposed for k = ii, ..., i,, then the decomposition described in Equation 222 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 (224)
where V ( f) = EjT i f f, and each of the terms in Equation 224 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, (225)
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 (226)
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 (227)
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~' (228)
SV(f)
V1totax= V+Vz,z (229)
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 (230)
Sobolll2 prOposed a variancebased nonparametric 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 irregularshaped domains, so
the analytical treatment works best in a wellrefined 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 222 and the sensitivity indices can be
obtained.
2.3 MultiObjective 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
multiobj 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)) (232)
If two designs are compared, then the designs are said to be nondominated 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 nondominated with respect to a set A c S if Bae A: a < x
Such designs in function space are called nondominated solutions. All the designs x (x e S)
which are nondominated 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 hypersurface.
In this shtdy, an elitist nondominated sorting genetic algorithm NSGAIIll3 and a parallel
archiving strategy to overcome the Pareto drift problem114 are used as the multiobj 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 nondomination criteria. Many individuals can have the same
rank with the best individuals given the designation of rank1. Initialize an archive with all
the nondominated 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 operatorsselection, crossover, and mutationto 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 rank1 solutions in the combined population.
10. Remove all dominated solutions from the archive.
11. Add all rank1 solutions in the current population which are nondominated 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 boxlike domains. For noisedominated problems and RSAs, alphabet designs (A
optimal, Doptimal, Goptimal, 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 CFDbased 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 24, 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 boxshaped design space. The study by Balabanov et al. used this
method in the RSA of a highspeed civil transport model. The approach removes large areas of
the standard boxshaped 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 subregions. It was found that surrogate accuracy was improved more for carefully
selected points within a small subregion 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 loworder polynomial, the model can often be
inadequate. This is termed as "bias error," and can obviously be reduced by using higherorder
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. RaisRohani and Singhss used sequential response surfaces to increase the
structural efficiency in various problems. In the study, RaisRohani 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 multiobj 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 25 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
multiobjective optimization. Jeong and Obayashill9 USed EGO in the multiobjective
optimization the pressure distribution at two operating conditions of a twodimensional 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 multiobj ective
problems. The new algorithm called ParEGO by simultaneously using Pareto frontbased
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 NSGAIIll3
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.
FarhangMehr 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. FarhangMehr 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 noninterest.
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 multiobjective 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 multiobj 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)kV(x) (233)
where k is an adjustable scaling factor that is set to 1 in this study.. The variance Vis predicted as
given in Equation 212. 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~ (234)
where Tis a target value given by
T= f ,P ~f, (235)
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) (237)
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 multiobj 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 (238)
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 23 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, (239)
where k is the number of terms in the polynomial response surface and x,, are values from the
matrix X from Equation 23. 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 (240)
or simplified to
f = b + b,x, +b2x 2+b3x 3+b4x 4+b,x, (241)
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, (242)
=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 27 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, =
(243)
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 28 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 29 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 totalto
static efficiency, rls. Figure 29A 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 210 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 21. 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
RaisRohani Composite LHS, OA, RSA Accuracy and efficiency of RSA varies
and Singh"' structures random depending on choice of DOE
Kurtaran et al."" Crashworthiness Factorial Linear, Difference between RSAs reduces as
design elliptic, design space size reduces
with D
optimal
Doptimal
IMSEa
Doptimal
LHS
Full factorial
quadratic
RSA
RSA
1storder 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
Crashworthiness
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 21. 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,
Doptimal
NN, RSA
Shi and
Hagiwara 0
Crashworthiness
CCD with D RSA, NN
optimal
alMSE integrated mean square error92
bMARS multivariate adaptive regression splines'2
"MPD minimum point design96
Table 22. Design space refinement (DSR) techniques with their applications and key results.
Author
Balabanov et
al. "
Roux et al.7
Papila et al.8
RaisRohani and
Singh8
BosqueSendra
et al.'2
Papila et al.n
Jeong and
Obayashill9
Knowles "
FarhangMehr
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
BoxBehnken 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 21. Optimization framework flowchart.
69
MI I
b
.
*
,e
III
Figure 22. DOEs for noisereducing 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 23. 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 25. 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 24. 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~ Or01~
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 26. Depiction of the merit function rank assignment for a given cluster given by
Equation 238 for two objectives where an" = Fn" F"'
Figure 27. 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 28. 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 29. 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 210. 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 surfacebased dualobj 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 multiobj 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 multiobj 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
multiobj 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 multiobjective 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 1D radial turbine model adapted from the 1D 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 algorithmbased 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 1D Meanline code. Using a 1D
code allowed for the availability of relatively inexpensive computations. To determine whether
using the 1D code was feasible for optimization purposes, a 3D verification study was
conducted. Once the Meanline code was verified, the radial turbine optimization could proceed.
3.2.1 Verification Study
Threedimensional unsteady NavierStokes 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 highspeed of 140,300
RPM. A 2vane/1rotor 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 threedimensional, unsteady, NavierStokes 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 BaldwinLomax 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 31 shows static pressure contours (psi) at the midheight 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 3D simulations as
compared to 0.55 for the 1D simulation. Figure 32 contains the predicted totaltostatic
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 fourpoint 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 33 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 1D Meanline code and 3D CFD analyses 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 overpredict the totaltostatic efficiency by an
expected degree. The predicted optimum point based on the 1D Meanline code will likely yield
overly optimistic results, but the predicted degree of improvement should translate to the 3D
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 31.
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 facecentered 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 31.
AN 2 = AN 2 (AnsqrFrac)
Tip, Spd = Tip, Spd (U/C isen)
(31)
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 31, 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 34.
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 35. The B1 constraint was found to be the most
demanding and resulted in a feasible design space as shown in Figure 36. 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 3level 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 37. 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 37.
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 242 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 38, 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 39. 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 33. Quadratic response surfaces were constructed for
the turbine weight, Wrotor, and the turbine totaltostatic 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 37, 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 5level factorial design over these three variables. The remaining variables were
held constant according the values observed in the best tradeoff 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 5level factorial design used over RPM, Tip Flw, and U/C isen
(1 19 / 125 feasible points) to improve resolution among the best tradeoff 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 310. 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 311. Within the Pareto front, a region was identified that
would provide the best value in terms of maximizing efficiency and minimizing weight. This
tradeoff 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 34.
Within the best tradeoff region, only RPM~ and Tip Flw vary along the Pareto front as seen
in Figure 312. The other variables are constant within the tradeoff 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 tradeoff 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 highperformance 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 312. 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 31) 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 34. The results are shown in Figure 313. 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. (233) through (237).
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 23 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 311 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 314 and Figure 315 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 314. 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 =(34)
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 315. 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 316 and Figure 318, respectively. Figure 317 and Figure 319
provide direct comparisons of all selection criteria using the median values from Figure 316 and
Figure 318, 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 liquidrocket 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.
* Multiobjective Optimization using Pareto Front. Using the Pareto front information
constructed using genetic algorithms, a best tradeoff 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 multiobjective optimization. Using
merit functions gives the ability of dramatically reducing the number of points required in
the final cycle of multiobj 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 31. Variable names and descriptions.
Objective Description Baseline design
variables
Wrotor Relative measure of "goodness" for overall weight 1.147
Y7ts Totaltostatic 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 34. Baseline and optimum design comparison.
Objectives Description Baseline Optimum
Wrotor Relative measure of "goodness" for overall 1.147 1.147
weight
Grls Totaltostatic efficiency 85.0% 89.7%
Design Variables Baseline Optimum
Table 32. 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 33. 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 31. Midheight 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 32. Predicted Meanline and CFD totaltostatic efficiencies.
tllc. I!!!!!!c
E
100 110 120
103 RPM
130 140 150
Figure 33. 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 34. 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 35. 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 36. 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 37. 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 38. 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 38. Continued.
0. 18
0. 16
0. 14 
0. 12
0.08
0
0.5 1 1 .5 2 2.5
W
Figure 39. 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 310. 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 311. 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 radeoff
01 region
0.5 1 1.5 2
0.5 1 1.5 2
Figure 312. Variation in design variables along Pareto Front.
Dhex %
3%
AnsqrFrac
5%
React*RPM
2%
Other
53%
Figure 313. 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 314. 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 315. 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 316. 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 317. 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 318. 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 319. 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 CFDbased design.
This chapter outlines current and proposed methodologies in the CFDbased 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 onedimensional 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 41.
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 coldfire and hotfire testing to study general injector
characteristics. The results of these tests were used along with a number of additional multi
element coldfire 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 SingleElement 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 42.
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.
Singleelement 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 wellmixed 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 MultiElement Injectors
A multielement 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 43. 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 nearwall 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 44 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 gasgas 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 gasgas 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 offaxis 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 7element gasgas 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 gasgas single element shear coaxial
inj ector element. A singleelement inj ector was used rather than a multielement 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
stagedcombustion 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 OHPlanar Laser Induced Fluorescence (OHPLIF) 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 multiphase modeling that included liquid,
gas, and liquid droplets. A 2D axisymmetric model was used along with a chemical model
consisting of a 9equation 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 gasgas
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 2D model to simulate LOX/GH2 multielement inj ector flow by changing the walls from a
noslip 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 ratecontrolling 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 finiterate chemical
model. The chemistry parameters were determined based on the temperature field. The kE
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 2D 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 NavierStokes (FDNS) solver used a kE model
with wall functions, and the real fluids models were used for the multiphase 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. Finiterate and equilibrium chemistry models were used. The GO2/GH2
computation used the 4equation equilibrium chemistry model for hydrogenoxygen 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 crosssection with rounded corners, while all three research groups chose to treat it as an
axisymmetric computational domain. None expected the nonasymmetry 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 kE 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 2D simulation of 3D
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 RCM1 of the 3rd Rocket Combustion Modeling (RCM) Workshop held in
Paris, France in March, 2006. The experimental setup of the RCM1 test case is shown in Figure
45. Lin et al. uses the FDNS code introduced by Cheng et al.15 The model was simulated
using two codes: FDNS, a Einitevolume pressurebased solver for structured grids, and loci
CHEM, a densitybased 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, lociCHEM is convenient for complex geometries. The code
used a 7species, 9reaction Einite rate hydrogenoxygen combustion model. The computational
grid encompassed the preinj ector flow, combustion chamber, and nozzle. Computations are 2D
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
LociCHEM code, an ignition point was specified. The temperature contours for each code by
integrating the turbulence equations to the wall are given in Figure 46. The best matching of the
heat flux values came by using the lociCHEM 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 47A. 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 RCM1
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 nonboundary 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 kmo model. The basic kmo 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 lociSTREAM against the same RCM1 test case
used by Lin et al.15 A 2D axisymmetric model was used for the computational domain. A 7
species, 9reaction finite rate hydrogenoxygen 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 46. 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 47. 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 threedimensional CFD models and simulation of multielement inj ector flow
simulations. Tucker et al.15 reports on several worksinprogress at Marshall Space Flight
center. This includes a simplified CFD model that uses a 7element inj ector element flow along
with bulk equilibrium assumption in the remainder of the 17 degree subsection to help
approximate full multielement combustor dynamics of the Integrated Powerhead
Demonstratorl59 IPD inj ector. Sample results are given in Figure 48A and Figure 48B.
Additional simulations are being conducted on the Modular Combustor Test Article (MCTA) to
supplement the IPD inj ector studies as shown in Figure 48C.
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 (41)
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 (42)
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 (43)
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 (44)
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 (45)
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 ] (47)
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' (48)
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 (49)
at 8x,
S( pk ) C3P122 ap dI
+ + (411)
8 t "x Dx D
(pH p)+d(puH)= d4 '+ 2ZJ (412)
at 8x D D
where p is given by Equation 42, Vk is the diffusion velocity of species k, and
H = h +_uru, (413)
k=1 (414)
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 (415)
SPrL a ] PrL 8 ] PrL a ]
4.2.2 Turbulent Flow Modeling
The conservation equations given in equations (49) (412) are applicable for laminar
flow. For turbulent flow, the Reynolds Averaged Navier Stokes methods (RANS) use ensemble
averaging to obtain the timeaveraged 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 (416)
at ax,
S(y~ t drx (i> k k k) dk,,)~~ k =1,N (417)
+~~,~ i l=, + r p1"u (418)
at Dx, Dx
St~ P H+pu"u,"P) + pl 2H +&  pu,"u,
(419)
C + pu,"h") + dr u(? ,, pu,"u") + ,r, u"(1uu
where a tilde and double prime denote a Favreaveraged mean and its fluctuation, respectively,
and a bar and a prime denote a Reynoldsaveraged 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, (420)
where k is the kinetic energy per unit volume of the turbulent velocity fluctuations, and the
Reynoldsstress tensor pu,"ul" that is denoted by
po, = puu,"u (421)
These terms can be modeled using the Boussinesq eddy viscosity models (EVM). The EVM
models the Reynoldsstresses by
du SH 2 85", 2
p5", = pu,"ul" = pu, i + p, 2 pk 3, (422)
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 (423)
P r, Dx,
where Prt is the turbulent Prandtl number, and the eddy conductivity is given by
p, cp
kt (424)
Pr,
In this study, the Prandtl number is kept constant at Prt = 0.9. The term iiEz, pu j~ul"u," from
equation (419) 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 (426)
The final forms of the conservation equations for mass, species, momentum and energy are given
respectively as follows:
+ = 0 (427)
at ax
+ pH, +fk k, k=l 1N (428)
+t + (pcp, '+: 3 (429)
Dx S, x,3 dx, B xL~, crdk 1
where
& h"+ 0,0 +k (431)
and the turbulent eddy viscosity Prt is modeled as a product of the velocity scale 5and a length
scale emultiplied by a proportionality constant C
pu, = cy( (432)
The velocity and length scales can be determined using the ke 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 (433)
8t 8r 8 366
(pe+pue)= p++C~k C~p (434)
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 : 2 435
and the velocity and length scales are given respectively by
r=J (436)
= "' (43 7)
and the constants are given by
C, = 0.09; C, = 1.44; C2 = 1.92; ok = 1.0; o, = 1.3 (438)
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 ke model and the use of wall functions. Examples of low
Reynolds number ke 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 loglaw 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 probabilitydensity functions. Further details on using the conservedscalar approach to
account for turbulencecombustion interactions can be found in Libby and Williams.169
The method used in this research is the finite rate chemistry assumption. In particular, a
kineticcontrolled 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 hydrogenoxygen combustion reaction. Within
the global reaction
2H2, +0 2H2,0 (43 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) (440)
where Rz, is the universal gas constant and A, b, and EA are experimentally determined empirical
parameters given in Table 43.
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 = ryd (441)
R uv dA
where R is the hydraulic radius, and a ~is the angular velocity. The swirl number can be
approximated as
S =4 (442)
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" (443)
where : is the nondimensional 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) =1ep (444)
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 49). The reaction is given as
20H2 + 02 4 2H2 + 2(#1)H2 (445)
and the following assumptions are made:
* Constant pressure
* Complete combustion with no dissociation
* Fuelrich combustion ($> 1)
* Steadystate
* All combustion products exit at the same temperature
For this system the heat output in watts is given by
+1 a ,
(446)
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 41. Selected injector experimental studies.
Author Inj ector type Propellants Key results
Hutt and Single element LOX/GH2 IHROVative measurements of LOX swirl
Cramer oxidizerrich spray
(1996)135 COaxial swirl
inj ector
Preclik Multipleelement 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 gasgas
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
Multipleelement
coaxial for FFSC
Nonreacting j et
Singleelement
shear coaxial
Singleelement
Shear coaxial
GO2/GH2
LN2
GO2/GH2
GO2/GH2
q = lilH, hH,
1 1
MHoO + 2 AhH,
Table 43. Reduced reaction mechanisms for hydrogenoxygen combustion.
Reaction A ((m3/gmOl)n1/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 42. 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 threephase
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.
kE Good species matching with
inhouse 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 inhouse
experiment.
Menter Good peak heat flux
BSL prediction. Overpredicts
downstream heat flux. Low
grid dependence.
GO2/
GH 2
2D
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
(RCM1)
5.2 767/798 Heat flux
comparison
(RCM1)
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
(RCM1)
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 OXygenrich core
recirculation
heat transfer
recirculation
I
n/ shear layer
reattachmentpm
homogenous products
shear layer
Figure 42. Flame from gaseous hydrogen gaseous oxygen single element shear coaxial
inj ector.
Figure 41. Coaxial injector and combustion chamber flow zones.
`1.
:v NAS
j ~
Figure 43. Multielement injectors. A) Sevenelement 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 44. 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 45. Test case RCM1 injector. Test rig A) schematic and B) photo.
Figure 45. Continued.
C i;
Figure 46. Temperature contours for a single element injector. A) FDNS and lociCHEM,15 B)
lociSTREAM, and C) lociCHEM156 COdes. Plots are not drawn to same scale.
O Test Dtar
Chem3 BSL
 Chem3SST T) :r
I:  Chemr3 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 47. CFD heat flux results as compared to RCM1 experimental test case. A) FDNS and
lociCHEM codes," B) lociSTREAM code," and C) lociCHEM code using
hybrid grids and various turbulent models.156
134
Figure 48. Multielement injector simulations. Reproduced from [158]. A) Integrated
Powerhead Demonstrator (IPD) injector showing computational domain and B) a
closeup 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 49. Fuel rich hydrogen and oxygen reaction with heat release.
136
CHAPTER 5
SURROGATE MODELING OF MIXING DYNAMIC S
A bluff bodyinduced 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 CFDbased surrogate modeling.
Plausible alternative surrogate models can lead to different results in surrogatebased
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 highmixing
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 2D 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 nearwake region that decays
to form a welldeveloped 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 timeaveraged 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 52. 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) (51)
The computational domain consists of a trapezoidal bluff body within a rectangular channel.
The flow area is illustrated in Figure 53. 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 2D, unsteady NavierStokes 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 secondorder central
difference schemes. The grid was constructed using ICEMCFDin7 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 54. 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 51. 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 tradeoff 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 (53)
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 (54)
where pu is the fluid viscosity, Uis the timeaveraged velocity, and u' and v' are the fluctuation
velocities (u' = u U, v' = v v). Equation 54 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 timeaveraged quantity represents the mixing index value. The
constraints are given as
B+b<1
0.5
(55)
0.0 < b < 0.5
0.0
The first constraint in Equation 55 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 facecentered composite design
(FCCD) is used to select 27 of the 52 data points. The facecentered 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 55. 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 Latinhypercube 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 55A 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 timeaveraged flow is a more concise
quantity. Figure 55C shows the timeaveraged 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 52 provides the range of the total pressure loss coefficient and mixing index for the
52 design points. From Table 52, 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 56. 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 53.
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 53, 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 crossvalidation. The values of SPREAD and GOAL that minimize the
error in the test cases are given in Table 54. 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 57. 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 58. 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 59. Yet,
large differences still exist in the overall shape of the prediction model surfaces.
The RMS error values shown in Table 54 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 55.
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 56. 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 56) 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 timeaveraged 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 57.
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 510. 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
58, 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 (57)
which leads to the contour plots in Figure 511. 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 512 plots the response surface and radial basis
neural network predicted values as compared to the actual CFD data from
Table 57 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 underpredicts 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 51. 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 52. Data statistics in the grid comparison of the CFD data.
CD .
Max 2.21 857
Min 1.79 515
Range (maxmin) 0.42 342
Mean 2.03 617
Standard deviation 0.11 66
Table 53. 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 54. Comparison of radialbasis 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 55. 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 56. 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 57. 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 51. Modified FCCD.
Figure 52. 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 53. Computational domain for trapezoidal bluff body.
>0
0 10 20 30
Figure 54. Computational grid for trapezoidal bluff body.
Figure 55. Bluff body streamlines and vorticity contours. A) Typical instantaneous vorticity
contours, B) typical instantaneous streamlines, and C) timeaveraged streamlines for
flow past a trapezoidal bluff body (B = 0.5, b = 0.25, h = 0.25) at Re = 250.
 ~
//
///// =
C
Figure 55. 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
: H1UB
~ IO)..
''
Ofi O~
Figure 56. Flow characterization in design space on the Bb axis. At b = 0, the angle of the
lower bluffbody 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 59. 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 510. 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 511. 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 512. 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 gasgas 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) twodimensional 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 gasgas
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 preinjector 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 62. 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 63. The computation was run using the lociSTREAMm5
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 64 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 xplane 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 nondimensional 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 xplane vorticity,
thus reducing the overall swirl value to negligible levels. The nondimensional 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 fullydeveloped flow conditions at the exit
of the pipe.
Figure 65 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 66 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 67A. 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 103 MPa across the
inj ector, nearly all of which occurs in the converging region and final pipe section, as shown in
Figure 68. The hydrogen exit velocity profile, k, and co, are shown in Figure 69. 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 xdirection 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 610. The
heat flux equation was calculated based on the timedependent 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 steadystate (SS) heat
flux was then calculated using
qi\,,moss = y T (61)
where conduction coefficient k of Copper 1 10, the material used for the combustion chamber
wall, is k = 388 W/mK Ay is the distance between the thermocouple holes as shown in Figure
610, 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/kgK. 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 61 and 62, respectively, and are shown in Figure 611.
Equations 61 through 63 assume that the heat flow is onedimensional.179 For a square
duct, the onedimensional linear assumption may not be correct. The actual equation would need
to be determined through a numerical solution of the twodimensional 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 twodimensional 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 onedimensional axisymmetric
approximation was included in addition to the onedimensional linear approximation.
If it is assumed that the isotherms within the combustion chamber wall are axisymmetric, a
second onedimensional assumption can be conducted. The geometry of the isotherms is given
in Figure 612. 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 1D axisymmetric heat conduction, the steady state heat flux approximation is given by
kAT
q",,~s (64)
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 612. Using this equation, the heat flux at
the wall centerline can be determined based on the 1D heat conduction equation, giving
4,,a~,,,, rwlannIn (ro /r,) (65
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 (66)
Taxlsymm ( o)
An unsteady approximation is determined by adding the second term from equation (62) to the
steady state heat flux approximations in Equation 65. Figure 613 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 crosssections 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 104 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 610), respectively, are matched. The resulting temperature contours are given
in Figure 614 for the prescribed boundary conditions for the combustion chamber crosssection
at x = 84 mm.
Significant differences are seen in the comparisons of the estimated wall heat fluxes shown
in Figure 615. 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 2D unsteady conduction analysis. The location of the peak heat flux from the conduction
analysis is shifted from that of the 1D 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 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 based on the experimental results presented
by Conley et al.22 Of a singleelement 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 threedimensional flow effects.
The CFD simulations were conducted using lociSTREAMm which is a pressurebased,
Einiterate chemistry solver for combustion flows for arbitrary grids developed by Streamline
Numerics, Inc. at the University of Florida. The code numerically solves the 3D unsteady,
compressible, NavierStokes equations. Menter' s SST model is used for turbulence closure. A
sevenspecies, nineequation 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 616. The
values for specific flow conditions are given in Table 61. The unstructured grid contains
1.7 x106 elements and 3.7 x105 nodes. The computational domain represents a threedimensional,
oneeighthsection of the full combustion chamber. Noslip 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 selfsustaining, and the
computation was run until a steadystate 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 (67)
The calculated heat flux based on Equation 67 was calculated in the CFD simulation and then
compared to the experimental values.
Figure 617 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 618 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, onedimensional 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, twodimensional representation
between the heat flux and the temperature distribution as described in Section 6.4. These three
cases, labeled as (i) linear onedimensional, (ii) axisymmetric, and (iii) 2D unsteady adiabatic
are shown in Figure 619 along with the computed result. Figure 620 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 2D conduction
analysis are similar, but shifted. The shift in the CFD data as compared to the results of the 2D
conduction analysis indicates a faster heat release in the CFD simulation than may actually occur
in the experiments. In all cases, the CFD data underpredicts 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 threedimensional simulation allows for a view of the heat flux characteristics across
the flat wall. Figure 620 shows the twodimensional nature of the predicted heat flux along the
wall. Also shown in Figure 620 is the eddy conductivity. The location of peak heat flux can be
clearly seen in Figure 620A, 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 twodimensional heat flux and temperature.
Across x = 24 mm mark, there is a sudden jump in heat flux magnitude that corresponds to
the change in xvelocity from positive to negative as shown in Figure 621. The sudden jump in
heat flux is shown in Figure 622A. 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 622B. The effect of the velocity change is echoed in the temperature
values near the wall (y+ ~ 2.5) as seen in Figure 623A.
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 623B. 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 618 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 624.
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 625.
Figure 625 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 626 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 nearwall inj ector element. The subdomain is approximated
using a rectangular shaped computational domain with width and height of 4.03 mm, and a
length of 100 mm. Only onehalf 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 627. 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 62, and each grid had a grid distribution as shown in Figure 628. 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 62.
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 629. 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 630. 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 630. 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 3D 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 2D
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 61. 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 62. 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 63. 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 61. Blanching and cracking of combustion chamber wall due to local heating near
inj ector elements.
NI 11 11
Figure 62. Hydrogen flow geometry.
Figure 63. 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 65. Swirl number at each x location.
NO
5 0 Y L~~
5 10 15 100
10 15 100
B Y
Figure 64. Zvorticity 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 66. 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 67. Reynolds number profiles. A) axial velocity and B) tangential velocity.
2.5E03
2.0E03
g 1.5E03
8 1.0E03
turbulent, 1 = Oo
5.0E04 
turbulent, = 5.7o
0.0E+00
20 0 20 40 60 80 100 120
Figure 68. Nondimensional 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 69. Hydrogen inlet flow profiles. A) Axial velocity, B) turbulent production k, and C)
turbulent dissipation co.
Figure 610. Combustion chamber crosssectional geometry and thermocouple locations. Units
are given in millimeters.
Thermocouples
I I wall
 Steadystate 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 611. Estimated wall heat flux using linear steadystate and unsteady approximations.
Figure 612. 1D 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 613. 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 614. Temperature (K) contours for 2D 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
2D 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 615. 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 616. Computational model for singleelement injector flow simulation. A) Combustion
chamber boundary conditions, B) grid closeup, and C) representative combustion
chamber section.
182
Figure 617. 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 618. 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 619. 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 620. Wall heat transfer and eddy conductivity contour plots. A) Heat flux along
combustion chamber wall (xzplane 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 xaxis.
(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 621. 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 622. 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~ 63 eprtr n dycnutvt rflsa aiu oain npae=0
Figur 6A) Temperature. B) Eddy conductivity.poie tvrosyloain npaez=0
185
Unsteady approximation
p vx = 0
Ifreattachment point
vx = 0
IIIIIIIT
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 624. 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 625. 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 626.
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 35d
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 629. 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 630. 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 timedependent
oscillations.
O I
CHAPTER 7
MULTIELEMENT 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 multielement 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 tradeoffs 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 multielement inj ector
model is constructed. An effort is made to capture the 3D effects of a full multielement
injector face in a simplified 3D 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 SetUp
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 multielement inj ector to make it suitable for a parametric study. The
inj ector face simplification of the full inj ector shown in Figure 71 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 multielement 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 lociSTREAMm (Thakur and Wright 2006).
The CFD model attempts to approximate the singleelement section near the chamber' s outer
wall given in Figure 71. The singleelement section was approximated using a rectangular
shaped computational domain. Only onehalf 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 runtime. The solutions are
expected to give good heat flux accuracy, but the combustion lengths are expected to be
somewhat underpredicted.
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 (71)
N' < Nmx /Nbaseline
where
N* = N/Nbaseline T lbaseline (72)
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 71. 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 71B is given by
w (N = Rtan(73)
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 RCM1 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 71. 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 72. 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 73 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 73B, 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 73 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 crosssectional 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 74. 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 75A.
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 76 in the plot of maximum heat flux
versus r*. Figure 77 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 73. It can be seen in Figure 78 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 ectorinj 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 78. 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 73D). 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 tradeoff points are located in one or the other region of the design space, as
shown in Figure 79. 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 79B. The design space will
also be expanded slightly to
0.60 < N* <1.25
(74)
0.20 <; r* <1.25
due to the best tradeoff 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 multielement 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 multiobj ective problems introduced in Equation
238, 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 functionbased 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 72.
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 710. 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 711. While
Kriging can provide an adequate surrogate model for the peak heat flux, Figure 711B 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 712 and
Figure 713, respectively. Figure 712 indicates the locations of the three points added to the
design space in the region of Pattemn 1. Figure 713 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 tradeoff
200
designs. The resulting Kriging fits including the new data points are shown for Pattern 1 and 2
in Figure 714 and Figure 715, respectively. The response surface for peak heat flux of Pattern
1 in Figure 714 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
716, 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 tradeoff 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 tradeoff 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 717 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 718. The longest combustion lengths, or worst combustion efficiencies, occur
in the points in Pattern 2. This is likely due to the smaller crosssectional 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 716, cases 18 and 19 were chosen as the best tradeoff designs. These
designs exist on opposite sides of the design space and in different pattern regions, as shown in
Figure 719, but have similar number of inj ectors in the outer row of the multielement inj ector,
as shown in Figure 720. 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
721 shows the predicted heat flux profile for the selected tradeoff cases as compared to the
baseline cases. The grid resolutions of the baseline case and one tradeoff 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 722A. 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 tradeoff 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 723. 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 tradeoff 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 multielement 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 multielement 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 71. 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 72. 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 211
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
~ "
noslip
H2 inlet
'02 pOst tip
slip
Figure 71. Inj ector element subsection. A) Outer row representation of baseline inj ector with
computational subsection shown at top and B) in closeup. 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 72. 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 73. 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 74. Oxygen isosurfaces 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 75. 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 76. 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 77. Heat flux distribution. A) Center of combustion chamber wall for case 6 and case 12.
B) Heat flux distribution along first onethird length of entire wall for case 12 (top)
and case 6 (bottom). Horizontal axis is xaxis.
208
A Imax\"'' B
Figure 79. 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 tradeoff
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 78. 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 CtI0 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 710. 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 tradeoff
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 711. 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 tradeoff 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, *PjiO Q 8
0.4
0 .6 0. 0 00. 0 .81 1.
0.4 0. 0.8 1 1.2
C '
Figure~.4 712 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 1 440 495
1 ** 0 4
0 496
".~ 046
Z p 0 497
g, 10 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 713. 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 714. 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 715. 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 716. 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 718. 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 719. Location of best tradeoff points in each pattern group in design space.
100
150 100 50 0
Z
50 100
Figure 720. Injector spacing for selected best tradeoff 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 720. 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 721. Predicted heat flux profiles for baseline case (case 1), best tradeoff from Pattern 1
(case 19), and best tradeoff 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 722. 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 723. 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
CFDbased 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 multiobj 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 CFDbased optimization through the practice of
removing infeasible regions from the design space using lowfidelity 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 flameholding 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 lowcost 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 sixvariable design space, surrogate models were necessary in extracting these boundsa
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 flameholder mixing to the bluff body geometry illustrated the
multiple roles of surrogatebased 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 CFDbased
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 CFDbased
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 SingleElement 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 tradeoff between heat flux profile accuracy and computational run time.
8.4 MultiElement Injector Flow Modeling
The CFD model sought to approximate the effects near a single inj ector element in the
outer injector row of multielement 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 interelement 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 injectortowall distance that is small compared to the
interelement 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 multiobj ective merit
functionbased 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 functionbased DSR dominated all of the original design points. The multiobjective
223
surrogatebased 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 CFDbased 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 3D simulations of the selected optimum radial turbine design and a comparison with
the 1D Meanline code results to determine if the efficiency gains are comparable over the
baseline case.
2. A threedimensional blade shape optimization of the selected optimum radial turbine
design.
3. Adding additional design points near the selected designs of the multielement 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 multielement 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 multielement
injector. Experimental studies on the selected best, or similar, multielement 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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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
aerospacerelated sciences and engineering. She pursued her doctoral studies in CFDbased
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 CFDBASED 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
PAGE 3
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 CFDBased Optimization.................................................................................................25 1.3.1 Optimization Techniques for Comput ationally Expensive Simulations................25 1.3.2 Design Space Refinement in CFDBa 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 MultiObjective 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 SingleElement Injectors......................................................................................106 4.1.2 MultiElement 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 MULTIELEMENT INJECTOR FLOW MODELING AND ELEMENT SPACING EFFECTS........................................................................................................................ ......191 7.1 Introduction............................................................................................................... ......191 7.2 Problem SetUp............................................................................................................. .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 SingleElement Injector Flow Modeling........................................................................222 8.4 MultiElement 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 11 Summary of CFDbased desi gn optimization applications with highlighted surrogatebased optimization techniques...........................................................................................32 21 Summary of DOE and surr ogate modeling references......................................................67 22 Design space refinement (DSR) techniques with their applications and key results.........68 31 Variable names and descriptions.......................................................................................92 32 Response surface fit statistics before (f easible DS) and after (reasonable DS) design space reduction................................................................................................................ ...93 33 Original and final design variable ranges after cons traint application and design space reduction................................................................................................................ ...93 34 Baseline and optimum design comparison........................................................................93 41 Selected injector experimental studies.............................................................................128 42 Selected CFD and numerical stud ies for shear coaxial injectors.....................................129 43 Reduced reaction mechanisms for hydrogenoxygen combustion..................................129 51 Number of grid points used in various grid resolutions...................................................149 52 Data statistics in the grid comparison of the CFD data...................................................149 53 Comparison of cubic response surface coe fficients and response surface statistics........150 54 Comparison of radialbasis neural network parameters and statistics.............................150 55 RMS error comparison for response surf ace and radial basis neural network................150 56 Total pressure loss coefficient and mixi ng index for extreme and two regular designs for multiple grids............................................................................................................. .151 57 Total pressure loss coefficient and mi xing index for designs in the immediate vicinity of Case 1 using Grid 3........................................................................................151 61 Flow regime description..................................................................................................17 4 62 Effect of grid resolution on wa ll heat flux and combustion length..................................175 63 Flow conditions............................................................................................................ ....175
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9 71 Flow conditions and baseline combustor geometry for parametric evaluation...............204 72 Kriging PRESSrms error statistics for each design space iteration...................................205
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10 LIST OF FIGURES Figure page 11 Rocket engine cycles....................................................................................................... ...33 12 SSME thrust chamber component diagram.......................................................................33 13 SSME component reliability data......................................................................................34 14 Surrogatebased optimizati on using multifidelity data.....................................................34 21 Optimization framework flowchart....................................................................................69 22 DOEs for noisereducing surrogate models.......................................................................70 23 Latin Hypercube Sampling................................................................................................70 24 Design space windowing...................................................................................................71 25 Smart point selection...................................................................................................... ....71 26 Depiction of the merit function rank assignment for a given cluster.................................72 27 The effect of varying values of p on the loss function shape.............................................72 28 Variation in SSE with p for two different responses.........................................................73 29 Pareto fronts for RSAs constr ucted with varying values of p ............................................73 210 Absolute percent difference in th e area under the Pareto front curves..............................74 31 Midheight static pressure (psi) contours at 122,000 rpm.................................................94 32 Predicted Meanline and CFD to taltostatic efficiencies...................................................94 33 Predicted Meanline and CFD turbine work.......................................................................95 34 Feasible region and location of three constraints...............................................................95 35 Constraint surface for Cx2/Utip = 0.2................................................................................96 36 Constraint surfaces for 1 = 0 and 1 = 40.........................................................................96 37 Region of interest in function space...................................................................................97 38 Error between RSA and actual data point..........................................................................97 39 Pareto fronts for p = 1 through 5 for second data set.........................................................98
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11 310 Pareto fronts for p = 1 through 5 for third data set............................................................99 311 Pareto Front with validation data.......................................................................................99 312 Variation in design vari ables along Pareto Front.............................................................100 313 Global sensitivity analysis............................................................................................... 100 314 Data points predicted by validated Pareto front compared with the predicted values using six Kriging models based on 20 selected data points.............................................101 315 Absolute error distribution for points along Pareto front.................................................101 316 Average mean error distribution over 100 clusters..........................................................102 317 Median mean error over 100 clusters...............................................................................103 318 Average maximum error di stribution over 100 clusters..................................................103 319 Median maximum error over 100 clusters.......................................................................104 41 Coaxial injector and com bustion chamber flow zones....................................................130 42 Flame from gaseous hydrogen gase ous oxygen single element shear coaxial injector....................................................................................................................... ......130 43 Multielement injectors.................................................................................................... 131 44 Wall burnout in an uncooled combustion chamber.........................................................132 45 Test case RCM1 injector................................................................................................132 46 Temperature contours for a single element injector........................................................133 47 CFD heat flux results as compared to RCM1 experimental test case............................134 48 Multielement injector simulations..................................................................................135 49 Fuel rich hydrogen and oxygen reaction with heat release..............................................136 51 Modified FCCD.............................................................................................................. .151 52 Bluff body geometry........................................................................................................ 151 53 Computational domain fo r trapezoidal bluff body...........................................................152 54 Computational grid for trapezoidal bluff body................................................................152 55 Bluff body streamlines and vorticity contours.................................................................152
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12 57 Comparison of response surface (top row) and radial basis neural network (bottom row) prediction contours for to tal pressure loss coefficient.............................................154 58 Comparison of response surface (top row) and radial basis neural network (bottom row) prediction contours for mixi ng index (including extreme cases)............................154 59 Comparison of response surface (top row) and radial basis neural network (bottom row) prediction contours for mixi ng index (excluding extreme cases)...........................155 510 Variation in objective vari ables with grid refinement.....................................................155 511 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......................................................................................................156 512 Comparison of response surface and radial basis neural network prediction contours for mixing index at B = 0 and h = 0.............................................................................156 61 Blanching and cracking of combustion chamber wall due to local heating near injector elements.............................................................................................................. 175 62 Hydrogen flow geometry.................................................................................................176 63 Hydrogen inlet mesh........................................................................................................ 176 64 Z vorticity contours..........................................................................................................17 7 65 Swirl number at each x location.......................................................................................177 66 Average axial velocity u and average tangential velocity v with increasing x ...............178 67 Reynolds number profiles................................................................................................178 68 Nondimensional pressure as a function of x ...................................................................178 69 Hydrogen inlet flow profiles............................................................................................179 610 Combustion chamber crosssectional geometry and thermocouple locations.................179 611 Estimated wall heat flux using linear steadystate and unsteady approximations...........180 612 1D axisymmetric assumption for heat conduction through combustion chamber wall.180 613 Estimated wall temperatures using li near and axisymmetric approximations.................181 614 Temperature (K) contours for 2D unsteady heat conduction calculations.....................181 615 Experimental heat flux valu es using unsteady assumptions............................................182
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13 616 Computational model for singleel ement injector flow simulation.................................182 617 Velocity contours vx(m/s) and streamlines.......................................................................183 618 Temperature (K) contours................................................................................................18 3 619 CFD heat flux values as compared to experimental heat flux approximations...............183 620 Wall heat transfer and e ddy conductivity contour plots..................................................184 621 Streamlines and temperature contours at plane z = 0.......................................................184 622 Heat flux and y+ profiles along combustion chamber wall.............................................185 623 Temperature and eddy conductivity profiles at various y locations on plane z = 0.........185 624 Mass fraction contours for select species.........................................................................186 625 Mole fractions for all species along combustion chamber centerline ( y = 0, z = 0)........187 626 Select species mo le fraction profiles................................................................................188 627 Sample grid a nd boundary conditions..............................................................................189 628 Computational grid along symmetric boundary..............................................................189 629 Wall heat flux and y+ values for select grids..................................................................189 630 Comparison of temperature (K) co ntours for grids with 23,907, 31,184, 72,239, and 103,628 points, top to bottom, respectively.....................................................................190 71 Injector element subsection..............................................................................................20 5 72 Design points selected for design space sensitivity study...............................................206 73 Effect of hydrogen mass flow rate on objectives.............................................................206 74 Oxygen isosurfaces and hydrogen contours...................................................................207 75 Hydrogen contours and streamlines.................................................................................207 76 Maximum heat flux for a changing radial distance r *.....................................................208 77 Heat flux distribution..................................................................................................... ..208 78 Maximum heat flux as a function of aspect ratio.............................................................209 79 Design points in function and design space.....................................................................209 710 Merit function ( MF2) contours for...................................................................................210
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14 711 Kriging surrogates based on initial 16 design points.......................................................210 712 Kriging surrogates and merit function c ontours for 12 design points in Pattern 1..........211 713 Kriging surrogates and merit function c ontours for 12 design points in Pattern 2..........212 714 Kriging fits for all 15 design points from Pattern 1.........................................................213 715 Kriging fits for all 12 design points from Pattern 2.........................................................213 716 Pareto front based on original 16 data points (dotted line) an d with newly added points (solid line)............................................................................................................ .214 717 Approximate division in the design space between the two pattern s based on A) peak heat flux...................................................................................................................... .....215 718 Variation in flow streamlines a nd hydrogen contours in design space............................215 719 Location of best tradeoff points in each pattern group in design space.........................216 720 Injector spacing for select ed best tradeoff design point.................................................216 721 Predicted heat flux profiles for baseline case (case 1), best tradeoff from Pattern 1 (case 19), and best tradeo ff from Pattern 2 (case 18).....................................................217 722 Heat flux profiles for different grid re solutions. The coarser grid was used to construct the surrogate model..........................................................................................218 723 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 CFDBASED 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 baseddesign 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 rrogatebased 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 surrogatebased 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 multiobjective design point selection reduced the number of design points by an order of magnitude while mainta ining good surrogate accuracy among the best tradeoff points. Second, bluff bodyinduced flow was investigated to identify the physics and
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16 surrogate modeling issues relate d to the flows 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 threedimensional 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 multielement 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 multiobjective merit function formulation facilitated an efficient framework to investigate the multiple and competing objectives. The analysis suggests that by adjus ting the multielement 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 trialanderror exercises. Optimization t ools can be used to guide the selection of design parameters a nd highlight the predicted best de signs and tradeoff 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 CFDbased 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 CFDbased design. In summary, this research effort seeks to improve the design efficiency of complex rocket engine components using improved CFDbased 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, CFDbased optimization and DS R are introduced as methods for improving upon traditional experimentallybas 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. Pressurefed 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. Pressurefed engines have low cost8 and good reliability9 because they are simple with few components. However, pressurefed 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 gasgenerator cycle (Figur e 11B) 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 pressurefed engine cycle.11. The expander cycle (Figure 11A) 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 pumpfed 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 Whitneys RL60 and the Ariane 5 ESCB. 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 12, 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 fuelrich 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 11C. 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 NASAs 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 13. The components with the lowest reliabilities are the highpressure fuel turbopump (HPFT P), the highpressure 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 gasgas 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 smallscale tests, which are then scaled to a fullscale model.19 Many injectors are also designed based on cold (nonreacting) 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 CFDbased 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 uncooled 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 Buckinghams 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 designers experience and intuition to co mputeraided designs. These early analyses included onedimensional approximations such as equilibrium reactions. These equilibrium approximations were based on the assumption of fas tchemistry, and were used to characterize the combustion field to aid the design process. Gills 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 8species, 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 multielement 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 1D 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 finetune 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 CFDBased 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 777300,35 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 gradientbased methods,3843 adjoint methods,44,45 and surrogate modelbased optimization methods such as res ponse surface approximations (RSA),46 Kriging models,4749 or neural network models.50 Gradientbased 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 ntbased and adjoint methods, the optimization is serially linked with the computer simulation. At times, gradientbased 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 ientbased optimization,
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26 as the optimizer would often become stuck at noiseinduced 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. Surrogatebased optimiz ation (SBO) uses a simplified, loworder 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 tradeoffs, 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 multiobjective 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 centralcomposite designs (CCD) and facecentered cubic designs (FCCD), or they may be tested across the full range of design parameter values as in multilevel 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 geometrybased bounds on the design variables. Vaidyanathan et al.63 chose variable ranges based on variations of an existing design in the CFDbased optimizat ion of a single elemen t rocket injector. Commonly, the design space cannot be represented by a simple boxlike 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 CFDbased 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 lowfidelity model is a qualitatively similar to the high fidelity model. In other instances, lowfidelity data may be combined with highfidelity data to reduce the overall number of expensive highfidelity runs. The lowfidelity data may employ a correction surrogate for improved accuracy, as shown in Figure 14. Han et al.64 performed a surrogatebased optimization on a lowfidelity 2D CFD model and tested the results on a more expensive 3D model in the optimization of a multiblade fan and scroll system and verified the 2D model us ing physical experiments as well as comparisons to previous studies and the 3D model. Keane65 built an accurate surrogate model of a transonic wing using highfidelity 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 highfidelity 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 lowfidelity model to better approximate a high fidelity model in the prediction of crack propagation. Thus, the accuracy of the lowfi delity models can be improved by combining the lowfidelity 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 lowfideli ty model to aid in the optimization of shell structures for buckling. Balabanov et al.70 applied a surrogate co rrection to a lowfide 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, computationalbased 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, CFDbased 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 modelperhaps one with more flexibilityand 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 CFDBased 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 highdimensional 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 boxshaped 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 superdetonative 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 lowfidelity analysis prior to optimizing using a highfidelity 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 CFDbased 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 lowfidelity 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 CFDbased 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 lowfidelity 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 1D 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, CFDbased optimization techniques are applied to improve the design process of a multielement rocket injector. In this case, th e optimization techniques must be applied in the context of a CFDbased design process where a limited number of function evaluations are available. Chapter 8 concludes the effort. Table 11. Summary of CFDbased 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 geometrybased 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 Multiblade fan/scroll system Lowfidelity optimization Optimized lowfidelity 2D model to predict for highfidelity 3D model Keane65 Transonic wing Multifidelity optimization Supplemented highfidelity CFD data with lowfidelity empirical model Alexandrov et al.66 Wing design Multifidelity optimization Varied mesh density for variable fidelity models Knill et al.67 High speed civil transport wing Multifidelity optimization Saved 255 CPU hours using lowfidelity supplemental data Balabanov et al.70 High speed civil transport wing Multifidelity analysis with surrogate correction Used coarse and fine finite element models as lowand highfidelity 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 lowfidelity model to refine design space before optimizing using highfidelity 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 11. Rocket engine cycles A) Expander cycle, B) gas ge nerator cycle, C) preburner cycle. Figure 12. SSME thrust chamber component di agram showing lowpressure fuel turbopump (LPFTP), lowpressure oxidi zer turbopump (LPOTP), highpressure fuel turbopump (HPFTP), highpressure 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 13. SSME component reliability data. Chart reproduced from [1]. Figure 14. Surrogatebased optimization using mu ltifidelity data. A multifidelity analysis may require a surrogate model for correc ting the lowfidelity model to better approximate the highfidelity model. refine design space construct surrogate Verify optimum location with highfidelity model Lowfidelity model w/ correction Surrogate correction Lowfidelity model Assess surrogate Highfidelity 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 21 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 multiob 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 1D 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 21. 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 facecentered cubic designs (FCCD) for boxshaped domains. Design optimality designs such as Dor Aoptimal83 (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 acefilling designs such as LatinHypercube Sampling (LHS)60 or Orthogonal Arrays (OAs)46 are preferred to efficiently cover the entire design space. Central composite designs and facecentered 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 22. The center po int detects curvature in the system while all other points are pushed as far as possible from the center provid e noisesmoothing 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 LatinHypercube 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 23A. 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 quasiuniform 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 23B). 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 nonparametric 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 nonparametric 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 rootmeansquare (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 ossvalidation 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 2ndorder model, but higher order models may also be used. A 2ndorder model requires a minimum of 122 kk design points, where k is the number of design variables. Thus, a 2ndorder model with three variables woul d require at least 10 points. A 3rdorder model with three variables would require twice that amount. A 2ndorder model provides good tradeoff 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. RaisRohani 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 multifidelity analyses in the multidisciplinary optimization of aircraft. Additi onal references and their key results are given in Table 21. 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 (21) 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 21 can be expressed in matrix form as 20 fX E VI (22)
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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 (23) 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 (24) 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 1 ()()TzzzXXzcN iT jj PP jfbzandVf (25) 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 highord 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 CFDbased 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, secondorder, 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 (26) 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 (27) In this study, a Gaussi an correlation structure100 is used 2 1,expijij kkk kvNR xxxx (28) 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 nonsampled points are 1 ()Ty rR y 1 (29)
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44 where 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 Nsvector of ones. The mean is estimated by 1 11TT 1R11R y (210) and the standard deviation of the response is estimated as 1 2 T sN y 1R y 1 (211) On the other hand, the estimation variance at nonsampled design points is given by 1 21 11 (())1T T TR VyR R 1r xrr 11 (212) 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 neurons 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 rfitting 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 userdefin ed error criterion to filter noise to prevent data overfitting. 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 twolayer networks consistin g of a radialbasis function (RBF) and a linear output layer. The radial basis function is given by bfradbas cx (213) 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 (214) where n is given by the term in parenthesis in Equation 213. Based on these equations, the function prediction is given by 2 1 expN ii iywb cx (215) 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 spacefilling 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 overfitting 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 undertrained 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 overfitting 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 iii f f (216)
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48 where fi is the actual value and i f is the predicted value. For Ns design points, the root mean square (rms) error is given by 2 1 i i ssN N (217) 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 (218) 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 211. 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 (219)
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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 1PRESSsN ii i f f (220) and PRESSrms is given by 2* 1 PRESSsrmsN ii i s f f N (221) where *i 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 (222) where0d1 x0x f f. If the following condition 11 ... 00siikfdx (223) is imposed for k = i1,
, is, then the decomposition describe d in Equation 222 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 (224) where 2 0VfEff and each of the terms in Equation 224 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 (225) and so on, where V and E denote variance and the expected value respectively. Note that 1 0iii 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 (226) 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 (227) 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 (228)
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52 ,total Z iiiVVV (229) 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 (230) Sobol112 proposed a variancebased nonparametric 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 irregularshaped 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 222 and the sensitivity indices can be obtained. 2.3 MultiObjective 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 multiobjective 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 (231) 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 (232) If two designs are compared, then the designs are said to be nondominated 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 nondominated with respect to a set AS, if : aAax. Such designs in function space are called nondominated solutions. All the designs x ( xS) which are nondominated 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 hypersurface. In this study, an elitist nondominat ed sorting genetic algorithm NSGAII113 and a parallel archiving strategy to overcome the Pareto drift problem114 are used as the multiobjective
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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 nondom ination criteria. Many indi viduals can have the same rank with the best individua ls given the designation of rank1. Initialize an archive with all the nondominated 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 operatorsselection, crossover, and mutationto 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 rank1 solutions in the combined population. 10. Remove all dominated solutions from the archive. 11. Add all rank1 solutions in the current population whic h are nondominated 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 boxlike domains. For noisedominat ed problems and RSAs alphabet designs (Aoptimal, Doptimal, Goptimal, 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 24, 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 boxshaped desi gn space. The study by Balabanov et al. used this
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57 method in the RSA of a highspeed civil transport model. The a pproach removes large areas of the standard boxshaped 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 subregions. It was found that surroga te accuracy was improv ed more for carefully selected points within a small s ubregion 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 loworder polynomial, the model can often be inadequate. This is termed as bias error, a nd can obviously be reduced by using higherorder
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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. RaisRohani and Singh85 used sequential response surfaces to increase the structural efficiency in various problems. In the study, RaisRohani 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 multiobjective 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 25 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 multiobjective optimization. Jeong and Obayashi119 used EGO in the multiobjective optimization the pressure distribution at two operating conditions of a twodimensional 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 multiobjective problems. The new algorithm called ParEGO by simultaneously using Pareto frontbased 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 NSGAII113 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 FarhangMehr 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. FarhangMehr 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 noninterest. 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 multiobjective 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 multiobjective 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 1MFykVxxx (233) where k is an adjustable scaling factor that is set to 1 in this study.. The variance V is predicted as given in Equation 212. 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. 2 Ty MF V x x x (234) where T is a target value given by minminTfPf (235) 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 3min fyfy MFfyV VV xx xx x xx (236) 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 4 ,0 00 fyfy yfyVV MF VV V xx xxx x xx (237) 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 ltiobjective 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 (238) 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 238. 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 1k ijij jybx (239) where k is the number of terms in the polynomial response surface and xij are values from the matrix X from Equation 23. For example, at any point i a quadratic response surface with two variables can be given as 22 011213241252ybbxbxbxbxxbx (240) or simplified to 01122334455ybbxbxbxbxbx (241) 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 1111p nnnk pp iiiijij iiijLyyybbx (242)
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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 27 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 (243) 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 28 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 29 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 totaltostatic efficiency, ts. Figure 29A 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 210 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 21. 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 RaisRohani and Singh85 Composite structures LHS, OA, random RSA Accuracy and efficiency of RSA varies depending on choice of DOE Kurtaran et al.88 Crashworthiness Factorial design with Doptimal Linear, elliptic, quadratic RSA Difference between RSAs reduces as design space size reduces Hosder et al.89 Aircraft configuration Doptimal RSA A 30 variable optimization required design space refinement and multifidelity analysis Mahadevan et al.92 Engineering reliability analysis IMSEa 1storder RSA, Kriging For very small number of points, Kriging performed better than 1storder RSA Forsberg and Nilsson93 Crashworthiness Doptimal 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 21. 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, Doptimal NN, RSA NN data used to supplement CFD data for RSA Shi and Hagiwara105 Crashworthiness CCD with Doptimal RSA, NN Maximized vehicle energy dissipation aIMSE integrated mean square error92 bMARS multivariate adaptive regression splines124 cMPD minimum point design96 Table 22. 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 RaisRohani and Singh85 Structural optimization Sequential response surface Successive small design spaces more efficient than large design space BosqueSendra et al.125 Chemical applications Sequential response surface BoxBehnken 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 FarhangMehr 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 21. Optimization framework flowchart. Problem definition and optimization setup 2. Design space refinement Refinement necessary? 3. Dimensionality reduction check 4. Multiobjective 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 22. DOEs for noisereduc 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 23. 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 24. 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 25. 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 26. Depiction of the merit functi on rank assignment for a given cluster given by Equation 238 for two objectives where min mmm nnaFF Figure 27. 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 28. 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 29. 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 210. 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 surfacebased dualobjective 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 multiobjective 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 multiobjective 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 multiobjective 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 cycles 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 multiobjective 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 1D radial turbine model adapted from the 1D 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 algorithmbas 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 1D Meanli ne code. Using a 1D code allowed for the availability of relatively inexpensive computations To determine whether using the 1D code was feasible for optimi zation purposes, a 3D ve rification study was conducted. Once the Meanline code was verified, th e radial turbine optimi zation could proceed. 3.2.1 Verification Study Threedimensional unsteady Navi erStokes 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 highspeed of 140,300 RPM. A 2vane/1rotor 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 threedi mensional, unsteady, NavierStokes 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 BaldwinLomax 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 31 shows static pressu re contours (psi) at the midhe 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 3D simulations as compared to 0.55 for the 1D simulation. Figur e 32 contains the predicted totaltostatic 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 fourpoint 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 33 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 1D Meanline code and 3D 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 overpredict the totaltostatic efficiency by an expected degree. The predicted optimum point ba sed on the 1D Meanline code will likely yield overly optimistic results, but the predicted degr ee of improvement should translate to the 3D 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 31. 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 facecentered 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 31. 22 11,, ,, ,, ANANAnsqrFrac TipSpdTipSpdU/Cisen Cx2/UtipCx2/UtipRPMU/CisenAnsqrFrac ReactU/CisenTipFlw Rsex/RsinRsex/RsinAnsqrFracU/CisenDhex% (31) 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 (32) 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 (33) Constraint boundary approximati ons were developed in this manner for each constraint. As can be seen from Equation 31, 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 34. 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 35. The 1 constraint was found to be the most demanding and resulted in a feasib le design space as shown in Figur e 36. 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 3level 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 37. 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 37. 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 242 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 38, 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 39. 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 33. Quadratic response surfaces were constructed for the turbine weight, Wrotor, and the turbine totaltostatic 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 37, 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 5level factorial de sign over these three variables. The remaining variables were held constant according the values observed in the best tradeoff 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 5level factorial design used over RPM, Tip Flw, and U/C isen (119 / 125 feasible points) to improve re solution among the best tradeoff 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 310. 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 311. Within the Pareto front, a region was identified that would provide the best value in terms of maxi mizing efficiency and minimizing weight. This tradeoff 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 34. Within the best tradeoff region, only RPM and Tip Flw vary along the Pareto front as seen in Figure 312. The other variables are constant within the tradeoff 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 tradeoff 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 highperformance 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 312. 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 34. The results ar e shown in Figure 313. 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 value 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. (233) through (237). 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 238. 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 311 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 314 and Fi gure 315 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 314. 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 maxminscaledyy e yy (34) and the absolute combined error is given as ,1 2rotortscombinedscaled Wee (35) The error distribution for data points along the Pare to front for each selection criterion is given in Figure 315. 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 316 and Figure 318, respectivel y. Figure 317 and Figure 319 provide direct comparisons of a ll selection criteria us ing the median values from Figure 316 and Figure 318, 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 liquidrocket 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. Multiobjective Optimiza tion using Pareto Front. Using the Pareto front information constructed using genetic algor ithms, a best tradeoff 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 multiobjective optimization. Using merit functions gives the ability of dramatical ly reducing the number of points required in the final cycle of multiobjective 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 31. Variable names and descriptions. Objective variables Description Baseline design Wrotor Relative measure of goodness for overall weight 1.147 ts Totaltostatic 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 32. 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 33. 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 34. Baseline and optimum design comparison. Objectives Description Baseline Optimum Wrotor Relative measure of goodness for overall weight 1.147 1.147 ts Totaltostatic 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 31. Midheight 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 100000105000110000115000120000125000130000135000140000145000150000RPMTotaltoStatic Efficiency Meanline CFD No tip clearance CFD W/tip clearance Figure 32. Predicted Meanline and CFD totaltostatic 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 TotaltoStatic 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 33. 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 34. 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 35. Constraint surface for Cx2/Utip = 0.2. At higher values of AnsqrFrac and U/C isen, lower values of RPM are infeasible. Figure 36. 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 37. 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 38. 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 38. Continued. Figure 39. 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 310. 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 311. 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 312. 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 313. Global sensitivity analysis. Effect of design variables on A) Wrotor and B) ts.
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101 Figure 314. 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 315. 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 316. 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 317. Median mean error over 100 clusters Merit function 2 shows the best performance as the number of points decreases. Figure 318. 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 319. 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 CFDbased design. This chapter outlines current and proposed methodologies in the CFDbased 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 onedimensional 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 41. 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 coldfire and hotfire test ing to study general injector characteristics. The results of these tests we re used along with a number of additional multielement coldfire 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 SingleElement 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 42. 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 elements 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. Singleelement 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 wellmixed 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 MultiElement Injectors A multielement 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 43. 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 nearwa 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 44 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 gasgas 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 gasgas 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 offaxis 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 gasgas 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 gasgas single element shear coaxial injector element. A singleelement injector was used rather than a multielement 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 stagedcombustion 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 OHPl anar Laser Induced Fluorescence (OHPLIF) 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 multiphase mode ling that included liquid, gas, and liquid droplets. A 2D axisymmetric model was used along with a chemical model consisting of a 9equation 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 gasgas 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 2D model to simulate LOX/GH2 multielemen t injector flow by changing the walls from a noslip 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 ratecontrolling 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 finiterate 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 2D 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 NavierStokes (FDNS) solver used a kmodel with wall functions, and the real fluids models were used for the multiphase 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. Finiterate and equilibrium chemistry models were used. The GO2/GH2 computation used the 4equation equilibrium ch emistry model for hydrogenoxygen 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 crosssection with rounded corners, while all three resear ch groups chose to treat it as an axisymmetric computational domain. None expected the nonasymmetry 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 2D simulation of 3D 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 RCM1 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 45. 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 finitevolume pressureb ased solver for structured grids, and lociCHEM, a densitybased 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, lociCHEM is convenient for complex geometries. The code used a 7species, 9reaction finite rate hydroge noxygen combustion model. The computational grid encompassed the preinjector flow, combusti on chamber, and nozzle. Computations are 2D 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 Menters 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 LociCHEM 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 46. The best matching of the heat flux values came by using the lociCHEM code integrated to the wall using Menters 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 47A. 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 RCM1 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 nonboundary 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 Menters SST model provided slightly better results than Menters 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 lociSTREAM against the same RCM1 test case used by Lin et al.155 A 2D axisymmetric model was used for the computational domain. A 7species, 9reaction finite rate hydrogenoxygen 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 46. 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 47. 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 threedimensional CFD models and simulation of multielement injector flow simulations. Tucker et al.158 reports on several worksinpr ogress at Marshall Space Flight center. This includes a simplifie d CFD model that uses a 7elem ent injector element flow along with bulk equilibrium assumption in the remainder of the 17 degree subsection to help approximate full multielement combustor dynamics of the Integrated Powerhead Demonstrator159 IPD injector. Sample results are gi ven in Figure 48A and Figure 48B. Additional simulations are being conducted on th e Modular Combustor Test Article (MCTA) to supplement the IPD injector studies as shown in Figure 48C. 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 (41) 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 (42) 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 (43) 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 (44) 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 Ficks law: k kkkdY mYmD dx (45) 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 (46) The mass rate of reaction is given by: 1 M kkkjj jWQ (47) 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 (48) 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 (49) ,,1,k jkkjkk jjY uYVYkN txx (410) ji ij i jjiuu u p txxx (411) iij j j jjju q HpuH txxx (412) where is given by Equation 42, Vk is the diffusion velocity of species k, and 1 2ii H huu (413)
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120 0 1 0 111 N Tfkk k NNN Tkkskkfkk kkkh hhh (414) where hs is the sensible enthalpy. The heat flux qj can be written as 1 2PrPrPrii j LjLjLjuu hH q xxx (415) 4.2.2 Turbulent Flow Modeling The conservation equations given in equations (49) (412) are applicable for laminar flow. For turbulent flow, the Reynolds Averaged Navier Stokes methods (RANS) use ensemble averaging to obtain the timeaveraged 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 (416) ,,1,k jkkjkjkk jjY uYVYuYkN txx (417) ji i jiij jijuu u p uu txxx (418) 11 22 1 2 iijjii j jjiijijiijjii jjjHuupuHuuu tx quhuuuuuuu xxx (419)
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121 where a tilde and double prime denote a Favreaverag ed mean and its fluctuation, respectively, and a bar and a prime denote a Reynoldsaverage 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 (420) where k is the kinetic energy per uni t volume of the turbulent velo city fluctuations, and the Reynoldsstress tensor ijuu that is denoted by ijijuu (421) These terms can be modeled using the Boussine sq eddy viscosity models (EVM). The EVM models the Reynoldsstresses by 22 33j il ijijttijij jilu uu uuk xxx (422) where t is the turbulent, or eddy, viscosity. The turbulent heat transport term juh is modeled as Prt j tjh uh x (423) where Prt is the turbulent Pra ndtl number, and the eddy conductivity is given by Prtp t tc k (424) In this study, the Prandtl numb er is kept constant at Prt = 0.9. The term 1 2 iijjiiuuuu from equation (419) is modeled as
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122 1 2 t iijjii kjk uuuu x (425) In the species equation, the term jkuY is modeled as tk jk ktjY uY Scx (426) The final forms of the conservation equations fo r mass, species, momentum and energy are given respectively as follows: 0j ju tx (427) ,1,k tk jkk jjktktjY Y uYkN txxScScx (428) 2 3ji j i il tij jijjiluu u u uu p txxxxxx (429) PrPr 2 3j jjLtj j ilt itij jjiljkjh HpuH txxx u uu k u x xxxxx (430) where 1 2 iiHhuuk (431) 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 (432)
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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 (433) 2 12 t ik iikiuCPC txxxkk (434) where the production Pk of k from the mean flow shear stresses is defined as j ii kt jiju uu P x xx (435) and the velocity and length sc ales are given respectively by k (436) 33 42Ck (437) and the constants are given by 120.09;1.44;1.92;1.0;1.3kCCC (438) 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 loglaw 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 probabilitydensity functions Further details on using the conservedscalar approach to account for turbulencecombustion 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 kineticcontrolled 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 hydrogenoxyge n combustion reaction. Within the global reaction 2222HO2HO (439)
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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 (440) where Ru is the universal gas constant and A b and EA are experimentally determined empirical parameters given in Table 43. 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 (441) where R is the hydraulic radius, and is the angular velocity. The swirl number can be approximated as W S U (442) 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 (443) where z is the nondimensional 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 (444) 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 49). The reaction is given as 22222HO2HO21H (445) and the following assumptions are made: Constant pressure Complete combustion with no dissociation Fuelrich combustion ( > 1) Steadystate 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 (446) 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 41. Selected inj ector experimental studies. Author Injector type Propellants Key results Hutt and Cramer (1996)135 Single element oxidizerrich coaxial swirl injector LOX/GH2 Innovative measurements of LOX swirl spray Preclik (1998)143 Multipleelement 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 Multipleelement coaxial for FFSC GO2/GH2 Evaluate performance of gasgas injector. Mayer (2002)141 Nonreacting jet LN2 Characterization of liquid jet flow Marshall et al. (2005)17 Singleelement shear coaxial GO2/GH2 Wall heat flux measurements Conley et al. (2007)22 Singleelement shear coaxial GO2/GH2 Wall heat flux independent of pressure, injection velocity
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129 Table 42. 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 threephase combustion model N/R Eddyviscosity Unknown. Cited inadequate experimental knowledge. Foust et al. (1996)146 GO2/ GH2 1.3 297 Velocity and species comparison 2D Cartesian 10151 kGood species matching with inhouse 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 inhouse experiment. Lin et al. (2005)155 GO2/GH2 5.2 767/798 Heat flux comparison (RCM1) 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 (RCM1) 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. Menters SST model provides superior peak heat flux prediction. Thakur and Wright (2006)157 GO2/GH2 5.2 767/798 Heat flux comparison (RCM1) Axisym. 26,000 104,000 points Menter SST Good peak heat flux prediction. Overpredicts downstream heat flux. Table 43. Reduced reaction mechan isms for hydrogenoxygen 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 1.2210140.91 8.37103HOMOHM 1.001010 0.00 0.02OOMOM 2.5510121.00 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 oxygenrich core heat transfer reattachment p oint heat transfer Figure 41. Coaxial injector a nd combustion chambe r flow zones. Figure 42. Flame from ga seous hydrogen gaseous oxygen single element shear coaxial injector.
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131 A B Figure 43. Multielement inject ors. A) Sevenelement 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 44. Wall burnout in an uncooled combustion chamber. A Figure 45. Test case RCM1 injector. Test rig A) schematic and B) photo.
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133 B Figure 45. Continued. A B C Figure 46. Temperature cont ours for a single element injector. A) FDNS and lociCHEM,155 B) lociSTREAM,157 and C) lociCHEM156 codes. Plots are not drawn to same scale.
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134 A B C Figure 47. CFD heat flux results as compared to RCM1 experimental test case. A) FDNS and lociCHEM codes,155 B) lociSTREAM code,157 and C) lociCHEM code using hybrid grids and various turbulent models.156
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135 A B C Figure 48. Multielement injector simula tions. Reproduced from [158]. A) Integrated Powerhead Demonstrator (IPD) injector showing computational domain and B) a closeup 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 49. Fuel rich hydrogen a nd oxygen reaction with heat release.
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137 CHAPTER 5 SURROGATE MODELING OF MIXING DYNAMICS A bluff bodyinduced 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 CFDbased surrogate modeling. Plausible alternative surrogate models can le ad to different results in surrogatebased 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 highmixing 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 2D 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 nearwake region that decays to form a welldeveloped 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 timeaveraged 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 52. 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 (51) The computational domain consists of a trapezoid al bluff body within a rectangular channel. The flow area is illustrated in Figure 53. The fluid is incompressible, and the flow is laminar with a Reynolds number of 250 given by UD Re (52) 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 2D, unsteady NavierStokes 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 secondorder central difference schemes. The grid was constructed using ICEMCFD177 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 54. 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 51. 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 tradeoff 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 (53) 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 (54) where is the fluid viscosity, U is the timeaveraged velocity, and u and v are the fluctuation velocities (, uuUvvV ). Equation 54 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 timeaveraged 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 (55) The first constraint in Equation 55 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 facecentered composite design (FCCD) is used to select 27 of the 52 data po ints. The facecentered 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 55. 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 Latinhyp 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 55A 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 timeaveraged flow is a more concise quantity. Figure 55C shows the timeaveraged 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 52 provides the range of the total pressure loss coefficient and mixing index for the 52 design points. From Table 52, 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 56. 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 141516171819 yBbhBBbbBh hbhBBbBbbBh BbhbhBhbhbhh (56) 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 53. 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 53, 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 crossvalidation. The values of SPREAD and GOAL that minimize the error in the test cases are given in Table 54. 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 57. 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 58. 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 59. Yet, large differences still exist in the over all shape of the prediction model surfaces. The RMS error values shown in Table 54 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 55. 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 56. 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 56) 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 timeaveraged 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 57.
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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 510. 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 58, 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 (57) which leads to the contour plots in Figure 511. 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 57 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 underpredicts 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 51. 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 52. Data statistics in the grid comparison of the CFD data. CD M.I. Max 2.21857 Min 1.79515 Range (maxmin) 0.42342 Mean 2.03617 Standard deviation 0.1166
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150 Table 53. 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 149060.6B* 0.672 562795H* 0.337 186830B*2 1.24 2900394B*b* 3.29 47800B*2 0 5151550B*h* 0.507 8420H*b* 0.633 494556H*2 0.700 039.9B*3 0.844 1660360B*2b* 4.01 4620902B*b*2 4.08 36600B*3 0.686 0765B*2h* 0.597 71580.7B*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 54. Comparison of radialbasis 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 55. 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 56. 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 57. 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 51. Modified FCCD. y x B b D =1 H h Figure 52. 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 53. Computational doma in for trapezoidal bluff body. X Y0102030 5 0 5 Figure 54. Computational grid for trapezoidal bluff body. A Figure 55. Bluff body streamlines and vorticity contours. A) Typical instantaneous vorticity contours, B) typical instantaneous streamlines, and C) timeaveraged 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 55. Continued. Figure 56. Flow characterizat ion in design space on the Bb axis. At b = 0, the angle of the lower bluffbody 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 57. 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 58. 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 59. 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 510. Variation in objective variables with grid refinement. A) CD and B) M.I.
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156 Figure 511. 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 512. 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 gasgas 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) twodimensional 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 gasgas 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 einjector 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 62. 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 63. The computation was run using the lociSTREAM157 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 64 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 xplane 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 nondimensional 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 xplane vorticity, thus reducing the overall swirl value to neglig ible levels. The nondi 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 fullydeveloped flow conditions at the exit of the pipe. Figure 65 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 66 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 67A. 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 5.5103 MPa across the injector, nearly all of which occurs in the conve rging region and final pipe section, as shown in Figure 68. The hydrogen exit velocity profile, k, and are shown in Figure 69. 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 xdirection 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 610. The heat flux equation was calculated based on the timedependent 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 steadystate (SS) heat flux was then calculated using ,linearSSk qT y (61) where conduction coefficient k of Copper 110, the material used for the combustion chamber wall, is k = 388 W/mK y is the distance between the ther mocouple holes as shown in Figure 610, 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 (62) 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/kgK. 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 (63) In the experimental documentation, the steady and unsteady heat fluxe s are calculated using Equations 61 and 62, respectively, and are shown in Figure 611. Equations 61 through 63 assume that the heat flow is onedimensional.179 For a square duct, the onedimensional linear assumption may not be correct. The actual equation would need to be determined through a numerical solution of the twodimensional 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 twodi 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 onedimensional axisymmetric approximation was included in addition to the onedimensional linear approximation. If it is assumed that the isotherms within the combustion chamber wall are axisymmetric, a second onedimensional assumption can be conducted. The geometry of the isotherms is given in Figure 612. 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 1D axisymmetric heat c onduction, the steady state heat flux approximation is given by ,lnaxisymmSS walloikT q rrr (64) 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 612. Using this equa tion, the heat flux at the wall centerline can be determined based on the 1D heat conduction equation, giving
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165 ,,lnwalloi wallaxisymmlinearSS iwallrrr qq rr (65) 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 (66) An unsteady approximation is determined by adding the second term from equation (62) to the steady state heat flux approxima tions in Equation 65. Figure 613 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 crosssections 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 1.158 104 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 610), respectively, are matched. The resulting temperature contours are given in Figure 614 for the prescribed boundary cond itions for the combustion chamber crosssection at x = 84 mm. Significant differences are seen in the comparis ons of the estimated wall heat fluxes shown in Figure 615. 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 2D unsteady conduction analysis. The locati on of the peak heat flux from the conduction analysis is shifted from that of the 1D 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 singleelement 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 threedimensional flow effects.
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167 The CFD simulations were conducted using lociSTREAM157 which is a pressurebased, finiterate chemistry solver for combustion fl ows for arbitrary grids developed by Streamline Numerics, Inc. at the University of Florida. The code numerically solves the 3D unsteady, compressible, NavierStokes equations. Menters SST model is used for turbulence closure. A sevenspecies, nineequation 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 616. The values for specific flow conditi ons are given in Table 61. The unstructured grid contains 1.7106 elements and 3.7105 nodes. The computational domain represents a threedimensional, oneeighthsection of the full combustion chambe r. Noslip 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 selfsustaining, and the computation was run until a steadystate 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 (67) The calculated heat flux based on Equation 67 wa s calculated in the CFD simulation and then compared to the experimental values. Figure 617 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 618 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, onedimensional 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, twodimensional representation between the heat flux and the temperature distribu tion as described in Section 6.4. These three cases, labeled as (i) linear onedimensional, (ii) axisymmetric, and (iii) 2D unsteady adiabatic are shown in Figure 619 along with the co mputed result. Figure 620 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 2D conduction analysis are similar, but shifted. The shift in th e CFD data as compared to the results of the 2D 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 underpredicts 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 threedimensional simulation allows for a view of the heat flux characteristics across the flat wall. Figure 620 shows the twodimensi onal nature of the predicted heat flux along the wall. Also shown in Figure 620 is the eddy con ductivity. The location of peak heat flux can be clearly seen in Figure 620A, 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 twodimensional heat flux and temperature. Across x = 24 mm mark, there is a sudden jump in heat flux magnitude that corresponds to the change in xvelocity from positive to negative as shown in Figure 621. The sudden jump in heat flux is shown in Figure 622A. 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 622B. The effect of the velocity change is echoed in the temperature values near the wall (y+ ~ 2.5) as seen in Figure 623A. 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 623B. 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 618 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 624. 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 625. Figure 625 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 626 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 nearwall inj ector element. The subdomain is approximated using a rectangular shaped computational domai n with width and height of 4.03 mm, and a length of 100 mm. Only onehalf 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 627. 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 62, and each grid had a grid distribution as shown in Figure 628. 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 62. 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 629. 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 630. 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 630. 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 3D 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 2D 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 61. 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 62. 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 63. 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 61. 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 62. Hydrogen flow geometry. Figure 63. Hydrogen inlet mesh.
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177 A B Figure 64. Zvorticity 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 65. Swirl number at each x location.
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178 0 10 20 30 40 50 60 20020406080100 xu0.10 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 66. 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 67. Reynolds number profiles. A) ax ial velocity and B) tangential velocity. 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 20020406080100120 x(ppo)/po turbulent, = 0 turbulent, = 5.7 Figure 68. Nondimensional 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 69. 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 610. Combustion chamber crosssectiona 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) Steadystate approximation Unsteady approximation Figure 611. Estimated wall heat flux using li near steadystate and unsteady approximations. To Twall T i rwall r i ro Thermocou p les Figure 612. 1D 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 613. 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 614. Temperature (K) contours for 2D 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 2D unsteady adiabatic Figure 615. Experimental heat flux va lues using unsteady assumptions. The x values are given in meters. A B C Figure 616. Computational model for singleelem ent injector flow simulation. A) Combustion chamber boundary conditions, B) grid closeup, and C) representative combustion chamber section.
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183 Figure 617. 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 618. 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) 1D linear axisymmetric unsteady adiabatic CFD Figure 619. 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 620. Wall heat transf er and eddy conductivity contour plots. A) Heat flux along combustion chamber wall (xzplane 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 xaxis. (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 621. 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 622. 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 623. 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 624. Mass fraction c ontours for select species. H2 O2 OH H2O
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187 A B Figure 625. 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 626. Select species mole fraction profiles. A) H2 and B) H2O at plane z = 0.
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189 Figure 627. Sample grid and boundary conditions. Figure 628. 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 629. 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 630. 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 timedependent oscillations.
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191 CHAPTER 7 MULTIELEMENT 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 multielement 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 tradeoffs 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 multielement injector model is constructed. An effort is made to capture the 3D effects of a full multielement injector face in a simplified 3D 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 SetUp 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 multielement injector to make it suitable for a parametric study. The injector face simplification of the full inj ector shown in Figure 71 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 multielement 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 lociSTREAM157 (Thakur and Wright 2006). The CFD model attempts to approximate the sing leelement section near the chambers outer wall given in Figure 71. The singleelement section was approximated using a rectangular shaped computational domain. Only onehalf 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 runtime. The solutions are expected to give good heat flux accuracy, but the combustion lengths are expected to be somewhat underpredicted. 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 (71) where **,baselinebaselineNNNrrr (72) 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 71. 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 71B is given by
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195 tan 2 D wNR N (73) 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 RCM1 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 71. 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 72. 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 73 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 73B, 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 73 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 crosssectional 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 74. 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 75A. 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 76 in the plot of maximum heat flux versus r* Figure 77 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 domains aspect ratio, rather than separate ly being a function of radial or circumferential spacing, alone. The domains aspect ratio is given by AR = r* / w* where w* = w / wbaseline, and wbaseline is determined based on N = Nbaseline in Equation 73. It can be seen in Figure 78 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 injectorinjector 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 78. 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 73D). 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 tradeoff points are located in one or the other region of the design space, as shown in Figure 79. 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 79B. The design space will also be expanded slightly to *0.601.25 0.201.25 N r (74) due to the best tradeoff 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 multielement 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 multiobjective problems introduced in Equation 238, 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 functionbased 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 72. 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 710. 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 711. While Kriging can provide an adequate surrogate model for the peak heat flux, Figure 711B 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, while eight of the 20 design points were located in the region for Patte rn 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 eac