Citation
Application of a three-dimensional Navier-Stokes model for a subsonic jet in a crossflow

Material Information

Title:
Application of a three-dimensional Navier-Stokes model for a subsonic jet in a crossflow
Creator:
Roth, Karlin Renée, 1961-
Publication Date:
Language:
English
Physical Description:
x, 145 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Boundary conditions ( jstor )
Boundary layers ( jstor )
Flat plates ( jstor )
Fluid jets ( jstor )
Jet aircraft ( jstor )
Navier Stokes equation ( jstor )
Symmetry ( jstor )
Trajectories ( jstor )
Turbulent jets ( jstor )
Velocity ( jstor )
Aerospace Engineering, Mechanics, and Engineering Science thesis Ph. D
Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF
Jets -- Fluid dynamics ( lcsh )
Navier-Stokes equations -- Mathematical models ( lcsh )
Short take-off and landing aircraft -- Jet propulsion ( lcsh )
Vertically rising aircraft -- Jet propulsion ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Karlin Renee Roth.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. §107) for non-profit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
Resource Identifier:
024836563 ( ALEPH )
20076866 ( OCLC )

Downloads

This item has the following downloads:


Full Text












APPLICATION OF
A THREE-DIMENSIONAL NAVIER-STOKES MODEL
FOR A SUBSONIC JET IN A CROSSFLOW




BY

KARLIN RENtE ROTH


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

1988


ITh~Q















ACKNOWLEDGMENTS


Without the continued guidance, encouragement and support of my advisor, Dr. Richard L. Fearn, this work would not have been possible. His suggestions gave me new ideas and new approaches which enabled me to understand the physical problem and to relate the computed results to the experimental data. I am especially grateful for the advantages that he gave me by arranging for me to complete my research at the NASA Ames Research Center.

Special thanks are extended to Mr. Richard J. Margason and to all the members of the Fixed Wing Aerodynamics Branch at the Ames Research Center for the opportunity to work with them and for their assistance during the past three years. The computer facilities that they made available to me were essential for calculating the flow and for visualizing the computed flowfield.

Appreciation is extended to Dr. James Ross and to Dr. William Van Dalsem for their many helpful discussions on both aerodynamics and computational fluid dynamics.

I also want to recognize the efforts of Mr. Siddharth Thakur who performed some of the data analysis required for making comparisons between the vortex properties for the computation and the experiment.

This project was supported by NASA Grant NCC 2-403.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS ................................................ ii

LIST OF SYMBOLS ................................................ v

ABSTRACT ....................................................... lx

CHAPTERS

1 INTRODUCTION ............................................ 1

2 BACKGROUND .............................................. 5

Experimental Studies .................................... 5
The Jet Plume........................................ 6
Conditions on the Flat Plate........................... 11
Near-Field Measurements .............................. 13
Prediction Techniques ................................... 13
Analytical, Empirical and Potential Flow Models...... 14 Navier-Stokes Methods ................................ 17

3 NUMERICAL APPROACH ...................................... 22

Thin-Layer Assumption ................................... 22
Finite Difference Equations ............................. 23
Time Differencing .................................... 27
Linearization of Nonlinear Terms...................... 28
Approximate Factorization ............................ 29
Flux Splitting ....................................... 31
Spatial Differencing ................................. 32
Numerical Dissipation ................................ 34
Code Implementation .................................. 35
Computational Grid ...................................... 36
Boundary Conditions ..................................... 40

4 RESULTS AND DISCUSSION .................................. 44

Grid Dependence ......................................... 48
Flow in the Symmetry Plane .............................. 51
Jet Centerline ....................................... 53
Decay of the Jet Velocity ............................ 55
Contrarotating Vortex Pair .............................. 57
Vortex Models ........................................ 59
Comparison with Experiment ........................... 61
Surface Pressure Distribution ........................... 63









Flow Visualization ...................................... 67
Streamlines .......................................... 67
Entrainment .......................................... 70
Topological Considerations ........................... 71
Observed Flow Patterns ............................... 74

5 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY........ 129 APPENDIX ....................................................... 136

REFERENCES ..................................................... 138

BIOGRAPHICAL SKETCH ............................................ 145














LIST OF SYMBOLS


A,B Arbitrary constants A,B,C,M Flux Jacobians resulting from the local linearization of
the fluxes about the previous time level

a,b,c Constants in the equation for the empirical vortex curve a,b,c,d Coefficients for first order Taylor series expansion of
F and G about P0

c Local speed of sound, (yp/p)0.5 C Curve or path in the phase plane Cp Pressure coefficient, (p-p.)/q, D Jet diameter De,Di Numerical dissipation operators for the central space
differenced directions e Total energy 6 Unit vector E Splitting error for the approximate factorization f Arbitrary function F,G Arbitrary functions F,G,H,Q Flux vectors for the thin-layer Navier-Stokes equations h Constant which has the value At or At/2 in the F3D
difference algorithm

h Effective vortex spacing I Identity matrix J Jacobian of the coordinate transformation L Differential operator M Mach number N Node









0 Order operator p Fluid pressure Po Critical point q Fluid dynamic pressure Q Vector containing the fluid density, three components of
momentum and total energy in F3D Pr Prandtl number r Radius of jet exit r Distance to an arbitrary point from a vortex center R Velocity ratio, M /M rc Vortex core radius Re Reynolds number based on freestream velocity and jet
diameter, U D/V s Arclength S Viscous flux vector for the thin-layer Navier-Stokes
equations

S Saddle point S' Half-saddle t Time T Similarity transformation natrix U Velocity U,V,W Contravarient velocity components u,v,w Cartesian velocity components V Velocity vector x,y,z Cartesian coordinates

)Parameter in the numerical dissipation based on the
fluid pressure

)Diffusion constant a Differential









6 Finite difference operator 5* Boundary layer displacement thickness A Denotes an incremental change in a variable, often a
first order forward difference At Time step 2' E4 Constants in the numerical dissipation operator

8,, Arbtrary constants in the time differencing formula y Ratio of specific heat F Vortex strength F Strength of an isolated diffuse vortex
o

9Rotation angle for a cross-section perpendicular to the
vortex curve

K Coefficient of thermal conductivity

1 Eigenvalues of the flux Jacobian, A

1Parameter in phase plane solution

A Diagonal matrix containing the eigenvalues of the flux
Jacobian, A

pFluid viscosity

0 Angle defined for the diffuse vortex model

0 Angle measured in degrees from the negative x-axis v Fluid kinematic viscosity , 14 Generalized coordinates x Pi

p Fluid density C Time variable in generalized coordinate system








Vorticity distribution for a pair of Gaussian vortices


00


Infinity


Subscripts

c Jet centerline i,j,k Specify a location in the computational plane j Jet max Maximum min Minimum o Base solution v Vortex curve 1,2,3 Denote unique values of a variable

0 Freestream condition Superscripts

b Backward difference f Forward difference n Time level +,- Denote variables associated with positive or negative
eigenvalues after flux splitting


viii


CO(rl 0)














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 APPLICATION OF
A THREE-DIMENSIONAL NAVIER-STOKES MODEL
FOR A SUBSONIC JET IN A CROSSFLOW

By

Karlin Renbe Roth

December, 1988


Chairman: Richard L. Fearn
Major Department: Aerospace Engineering, Mechanics and Engineering Science

The aerodynamic/propulsive interaction between a subsonic jet exhausting perpendicularly through a flat plate into a crossflow is investigated numerically. An approximately factored, partially flux split, implicit solver for the three-dimensional thin-layer Navier-Stokes equations is used for the computations. This algorithm is applied to flows with a range of jet to crossflow velocity ratios between 4 and 8.

Both qualitative and quantitative agreement between the computed results and the existing experimental database are demonstrated. Qualitatively, all of the pertinent flow features, including the jet trajectory, the contrarotating vortex pair and the wake region near the flat plate, are captured numerically. Flow visualization of the computed results on a graphics workstation is instructive for understanding the complex









interaction between the jet and the crossflowing fluid in the region near the jet exit.

The computed jet centerlines deviate less than one jet

diameter from the empirically predicted and experimentally located centerlines for all cases tested, including an inviscid computation. This indicates that the location of the jet centerline is relatively insensitive to grid structure and is determined primarily by pressure effects rather than viscous effects.

The computed velocity field is analyzed so that a direct comparison can be made with the experimentally determined properties of the contrarotating vortex pair. The upwash velocities in vortex cross sections are used in a vortex model to infer the vortex strength, location and core radius. Good agreement between the vortex properties calculated for the simulation and the experiment is demonstrated for the most refined computations.














CHAPTER 1
INTRODUCTION


During the past 50 years, a considerable number of

theoretical, experimental and computational investigations have been directed toward the flowfield induced by the injection of a turbulent jet of fluid into a crossflow. Perhaps the simplest illustration of this flow phenomenon is the emission from a chimney on a windy day (1). In gas turbine combustors, the mixing of jets in a crossflow plays a dominant role in determining the temperature profile of gas leaving the combustor, and hence, the performance and durability of the turbine (2). Yet another example of a jet in a crossflow is given by the discharge of effluent into a waterway

(3). The present study is motivated by the aerodynamic/propulsive interaction problem associated with Short Take-Off and Vertical Landing (STOVL) aircraft.

In the transition between hover and wingborne flight, STOVL aircraft rely on the direct thrust of lift jets to supplement the aerodynamic lift. As a result, a relatively high velocity jet is injected into a crosswind created by the forward motion of the aircraft during transition. The interaction of the lifting jets with the flow over the aerodynamic lifting surface produces a complex flowfield around the aircraft. This interaction usually results in a loss of lift and an increase in nose-up pitching moment with increasing forward flight velocity when compared with









the results of a linear model for the lift generated by the forward flight of the aircraft added to the direct lift of the jets without interference (4). Thus, the aerodynamic/propulsive interference effect poses a significant problem for STOVL aerodynamics.

Traditionally, the development of STOVL aircraft has been guided primarily by experimental investigations. In order to reduce the time and expense of the design and testing of these vehicles, researchers have developed prediction techniques for the aerodynamic/propulsive flowfield that are based either on potential flow methods with empirical adjustments to account for viscous effects or on purely empirical correlations. Although many of these techniques are relatively simple to implement, in general, they have only limited applicability for practical propulsion systems. Empirical predictions are unable to account for the severe nonuniformities in the velocity and temperature fields encountered by the lifting jets. Further, the empirical prediction may be limited to use with the experimental database from which it was derived. Thus, improved prediction techniques are needed to guide future STOVL development.

With the recent advances in the development of computational fluid dynamics and computing resources, a more promising approach for the powered lift problem is to use numerical algorithms based on the Navier-Stokes equations to simulate the performance of the propulsion systems. These algorithms are also subject to some limitations. First, the proper flow physics must be adequately represented by the governing equations solved by the model. In many codes, the large number of modelling assumptions removes some









of the pertinent physics. Next, the numerical accuracy of these codes cannot be neglected. Finally, the numerical code must be fully verified by comparison with experimental data.

Ultimately, the flow about an arbitrary STOVL aircraft

throughout its entire flight envelope would be analyzed; however, with current computer resources, a detailed simulation of the flow about a STOVL aircraft and its propulsion system is not possible. Instead, a simpler flowfield which isolates the physics of the jet/freestream interaction region is studied. The problem is reduced to the case of a subsonic round jet exhausting perpendicularly through a large flat plate into a subsonic crossflow. This configuration retains the essential physics of the aerodynamic/propulsive interaction problem without the added complexity of the aircraft geometry, engine inlet flows, ground effects and multiple jets.

The present study uses contemporary computational fluid

dynamics to develop and validate a computational model for a jet in a crossflow. Several investigators, for example Fearn and Weston (5-8), Schetz et al. (9) and Snyder and Orloff (10), have experimentally studied the subsonic jet in a crossflow using this simplified configuration, providing an extensive database for comparison with the computed results. Further, the experimental results reveal that the important features of the flow that must be simulated are the jet trajectory, the contrarotating vortex pair and the wake region near the plate downstream of the jet orifice.

Some of the key progress toward the understanding of the

flow phenomena associated with a jet in a crossflow is highlighted









in Chapter 2. Although only example results are emphasized, an extensive list of references (1-74) is identified. In Chapter 3, a description of the numerical approach, including the computer code, the boundary conditions and the computational grid, is given. Numerical results are presented and, when available, compared with experimental measurements in Chapter 4. The calculated results are for flows with jet to crossflow velocity ratios of 4, 6 and 8. Quantitative comparisons are made between the calculated and measured data for the jet trajectory, the pressure distribution on the plate and the properties of the contrarotating vortex pair. Flow visualization of the computed results on a graphics workstation is used to gain insight into this complex threedimensional flowfield. Finally, in Chapter 5, major conclusions and recommendations for future study are stated.














CHAPTER 2
BACKGROUND


In this chapter, the historical development of experimental, analytical and computational methods for powered lift configurations is reviewed. Emphasis is placed on the literature concerning the simplified subsonic jet/flat plate model since that is the configuration to be studied in this investigation and since the flow physics of the STOVL jets in transition are retained and isolated in this model. Further, this review focuses on investigations of subsonic round jets with velocity ratios between

2 and 10.

A description of the jet in a crossflow phenomenon, based on the experimental studies, is developed. The accumulation of data during the past several decades for the aerodynamic/propulsive interaction problem has positively influenced the design procedures for STOVL aircraft. However, Spee (11) points out that the design methods used for powered lift aerodynamics are primitive in comparison to the prediction techniques available for conventional aircraft.


Experimental Studies

The evolution of STOVL aircraft has been guided primarily through experimental investigations (see References 2,5-10,12-32). A review of the experimental work completed prior to 1979 which is presented in tabular form by Crabb, Durao and Whitelaw (12)









indicates that although a wide range of parameters and configurations have been studied, there is still insufficient information to completely describe the flowfield induced by the jet, especially in the initial mixing region. For instance, the velocity and pressure profiles at the jet orifice with the imposed crossflow remain unknown. Furthermore, these profiles are required as input boundary conditions for many calculations and the assumption of uniform exit conditions made, for example, by Patankar, Basu and Alpay (33) needs quantitative verification. In the following sections, the contributions made by the various experimental investigations are discussed. The Jet Plume

One topic of considerable interest for the jet interaction problem concerns the behavior of the jet plume itself. Several characteristics of the jet induced flowfield have already been documented either qualitatively, through flow visualization, or quantitatively, through pressure measurements. It has been established that a jet exhausting perpendicularly into a crossflow deflects, increases in lateral extent, distorts in cross-sectional shape and evolves into a flowfield that is dominated by a pair of contrarotating vortices. A schematic diagram in Figure 1 sets forth the various experimentally defined regions of the jet plume together with the associated wake region (7).

Jet trajectory. In order to address such problems as jet induced velocities and flow angularity near an aircraft in transitional flight, it is necessary to characterize the trajectory of the jet plume. The jet path for a circular jet has been









examined extensively (see References 2,13-17). Early experiments by Callaghan and Ruggeri (13) and Ruggeri, Callaghan and Bowden

(14) suggested that the velocity ratio, Uj/Um, is an important parameter in determining the penetration of the jet. Later experiments have also explored the effects of varying the jet injection angle (15) and the jet exit shape (18) on the jet penetration.

Some of the first quantitative measurements were taken by Jordinson (16) in 1956 in order to determine the path of the jet and the shape of a jet cross section. Pressure measurements were obtained in four planes aligned normal to the freestream and downstream of the jet orifice and in two planes inclined at 50 and 250 to the plate. Based on static pressure measurements, Jordinson concluded that the initial deflection of the jet is mainly a consequence of the pressure difference across the jet, but that within a few jet diameters of the orifice, the entrainment of crossflow fluid into the jet plume becomes the predominant bending mechanism. He further concluded that the mixing of jet and crossflow fluid reduces the vertical momentum at the edges of the jet causing the mixing region to deflect faster than the jet fluid that is initially unaffected by mixing and accounting for the distortion of the jet cross-sectional shape.

In 1968, Margason (15) used photographs of a water vapor marked jet to determine the path of a single jet exiting at large angles to the freestream for a range of velocity ratios. This study showed that the primary variables governing the jet path for a particular nozzle are the injection angle and the ratio of the









momentum flux across the jet orifice to the momentum flux of the freestream over an equal area. For a subsonic, unheated jet, this momentum ratio reduces to the ratio of the jet velocity to the freestream velocity. Throughout this dissertation, the term "velocity ratio" is used. Additionally, Margason's results were essentially the same for jets exhausting upward or downward and for jets near or far from adjacent surfaces. An empirical equation for the jet centerline based on the primary variables demonstrated a good fit to the photographed jet paths.

Later, Kamotani and Greber (2) studied a heated jet. They determined the jet path from maximum velocity measurements in the symmetry plane. Based on temperature measurements in the symmetry plane, they also determined that the temperature centerline does not coincide with the velocity centerline; it penetrates less and depends on the jet to crossflow density ratio, pj/p., in addition to the velocity ratio.

Fearn and Weston (5,6,8) measured the jet centerline by

placing a rake of velocity measuring probes in the symmetry plane perpendicular to an estimated jet centerline location which was deduced from the previous experiments (15). The magnitude and location of the maximum axial component of velocity at each rake location was then determined by curve fitting the measured velocity values. The locus of points of these maxima is the centerline. Experimentally, this curve was identified up to about 15 jet diameters downstream of the jet exit. The jet centerline determination was used to supplement Fearn and Weston's










investigation of the vortex pair and to provide a means of comparison with other experiments.

Contrarotating vortex pair, While early investigations of the jet plume properties focused on locating a jet trajectory, later investigations were directed toward locating and describing the vortex pair (2,4,5,17,19-21). Flow visualization of the vortex pair was used in some studies, for example the results presented in a film by the Office National D'Etudes et de Recherches Aerospatiale (19). Most of the quantitative studies inferred the strength of the vortices from velocity measurements. Margason and Fearn (4) and Antani (20) attempted to measure the vortex properties directly by using vortex meters.

The contrarotating vortex pair is the dominant and

persistent feature of a jet in a crossflow. The contrarotating vortices are formed close to the jet exit and can be detected far downstream of where the jet centerline may be measured. Specifically, these vortices were detected by Fearn and Weston (5) at 45 jet diameters downstream of the jet orifice. As the vortices propagate downstream, their strength decreases due to the diffusion of vorticity across the plane of flow symmetry.

An alternative method for describing the jet path is to

define a vortex curve to be the projection onto the symmetry plane of the location of the center of either vortex. An empirical equation for the vortex curve is presented by Fearn and Weston (5). The vortex curve does not penetrate as far into the flowfield as the jet centerline. Experimentally, the vortex trajectory can be located by examining the upwash velocity component in the region










between the two vortices. The upwash velocity is a maximum in the symmetry plane along the line joining the two vortex centers.

The vector vorticity field associated with the jet could be accurately characterized by taking finely spaced velocity measurements; however, the number of measurements that would be required is, in general, prohibitive. As a result, models have been constructed to deduce the vortex properties from the velocity field. For example, Thompson (17) calculated the variation in the vortex strength by measuring the velocity field in cross-sections perpendicular to the vortex path and then using two models to predict the vortex strength. In one model, the vortices are assumed to be discrete and quasi two-dimensional; the measured circumferential velocity distribution for a single traverse through the vortex center is used for the calculation of circulation. In the second model, the vortex strength is computed by contour integration of the tangential component of velocity around circuits of increasing size enclosing the vortex centers.

Later, two models which are capable of predicting the

location and diffusivity of the contrarotating vortices as well as their strength were developed by Fearn and Weston (15). The vortex filament model replaces the vortices with two infinite line filaments. Their strength and location are calculated from the measured upwash velocities along the axis between the vortices in the symmetry plane. Another method, the diffuse vortex model, assumes that each vortex has a Gaussian distribution of vorticity rather than being concentrated in a filament. Measured upwash velocities in the plane that is normal to the vortex path are









utilized in the model to infer the core radius of the vortices in addition to the vortex strength and spacing. Conditions on the Flat Plate

In order to understand the lift losses that are caused by

the jet interference, it is necessary to examine the conditions on the flat plate. Several experimental studies (References 6,17,19,22-29) have been directed toward this aspect of the jet flowfield. Useful information is obtained through both oil flow visualization and static pressure measurements.

Oil flow visualization on the surface of the plate indicates that the freestream flow decelerates upstream of the jet exit due to the presence of the jet and then accelerates around the jet (19,22,23). Some of these surface shear stress lines bend in toward the jet and terminate at the edge of the orifice. Other streamlines are deflected around the jet periphery and into the boundary of wake region downstream of the jet exit. Within the wake region, flow patterns indicate that some fluid is entrained forward by the jet. Directly behind this reverse flow region, wake flow moves away from the jet and into the wake boundary. The wake boundary may consist of a horseshoe vortex which is formed around the jet core. Wu, Mosher and Wright (23) noted that the photographed oil flow patterns are dependent only on the velocity ratio and that the wake region broadens with increasing velocity ratio. They also emphasized that the oil flow patterns do not necessarily describe flow above the plate boundary layer.

Static pressure measurements quantify the flow visualization results. For example, investigators (References 22-29) observe a










positive pressure region upstream of the jet exit and a large, low pressure region to the sides of the jet and behind the jet. The jet interferes with the freestream flow by blocking it, entraining it and causing a separated wake to form. Vogler (24) compared the pressure distributions for a circular jet and an equivalent diameter circular rod to find that the low pressure wake region is more extensive for the jet which indicates that entrainment as well as the blockage effect is important. Bradbury and Wood (25) suggested that the low pressure in the wake is due to boundary layer fluid being drawn up into the jet. But, the wake region extends several jet diameters above the plate, which led Wu, Mosher and Wright(23) to suggest that the freestream also separates. Relatively high pressure gradients are observed in the low pressure regions to the side and slightly downstream of the jet. These gradients may be due to the presence of the contrarotating vortices near the jet exit.

The experimental measurements, for example Fearn and Weston

(6), indicate that the velocity ratio is the dominant parameter governing characteristics of the plate pressure distribution. It was noted previously that this dimensionless parameter is the primary factor required to determine the jet trajectory. Two other dimensionless parameters, the ratio of the momentum thickness to the jet diameter and the jet Reynolds number, weakly influence the pressure distribution (25); however, their influence is detailed rather than gross.









Near-Field Measurements

In general, the measurements in the jet plume that have been discussed up to this point can be categorized as far-field measurements; that is, wind tunnel instrumentation limitations inhibited accurate measurement in the region relatively near the jet exit which is characterized by high velocity gradients and regions of backflow. A recent solution to the instrumentation problem is to use a laser Doppler anemometer (LDA) to make flow measurements. The laser Doppler system overcomes the problems of flow disturbance, probe size and directional sensitivity which are encountered by earlier instrumentation techniques. The first attempts to use a LDA to measure the velocity field associated with a jet in a crossflow (30,31) were not entirely successful; however, they provided insight into the optimization of the instrument.

Perhaps the most encouraging results for the near-field of the jet are those of Snyder and Orloff (10). A three-dimensional LDA was employed to gather data in the NASA 7- by 10-foot Wind Tunnel Facility at Ames Research Center. Their results are for a jet with a velocity ratio of 8. Further refinements of this technique should be capable of providing an adequate description of the distortion and diffusion of the shear layer in the region near the jet orifice.


Prediction Techniques

Although a large experimental database has been acquired, the number of variables and options and, thus, the number of










possible geometric configurations and design parameters is so large that an experimental database containing an adequate amount of design information is difficult, if not impossible, to accumulate. To augment the experimental database numerous theoretical prediction tools have been developed. The current state of the art prediction methods are in general based on potential flow methods which require empirical input to correct for the viscous flow characteristics. Moreover, most of these methods apply only to a restricted region of the flow and adequate verification is lacking. More recently, emphasis is being placed on the development of solutions based on the Navier-Stokes equations to simulate the entire jet/freestream interaction. Several representative models are discussed in the following sections. Analytical. Empirical and Potential Flow Models

Perhaps the earliest attempt to analyze the characteristics of a jet in a crossflow can be attributed to Chang (34) who was interested in the discharge of effluent from a pipe into a stream. She uses potential flow theory together with the concept of bound vortex filaments to calculate the shape of the separation boundary between a perpendicular cylindrical jet and a crossflowing fluid. A hand-calculated numerical solution demonstrates the horseshoe shaped cross-sectional boundary that occurs when the jet wake rolls up into a vortex pair.

A model which accounts for the interference effects of the jet by distributing sinks and doublets along the jet centerline is presented by Wooler (35). First, the jet centerline is determined through an empirical model which represents the deflection









mechanism by a pressure force and mainstream entrainment. Then, a doublet distribution is placed along the jet centerline to represent the blockage effect of the jet. The strengths of the doublets are related through experimental observations of the deformation of the initially circular jet into a jet with elliptic type cross-sections. The strengths of the sinks, which are distributed at discrete locations on axes perpendicular to the jet centerline, are proportional to the mass of air entrained by the jet. Calculations of the jet induced velocity field show good correlation with test data (36).

Alternatively, Heltsley and Parker (37) used vortex

singularities in their vortex lattice model for the jet. Once again, the initial geometry is based on experimental data. Further, a semiempirical computer program was used to simultaneously solve for all the vortex strengths as a consequence of the applied boundary conditions and the geometry. In either the sink-doublet formulation or the vortex singularity formulation, the effects of blockage and entrainment outside the jet and on the flat surface are approximated; but, the flow inside the jet is ignored.

Since the contrarotating vortex pair dominates the jet

flowfield, several models have been constructed to emulate this feature. The two empirical vortex models of Thompson (17) and Fearn and Weston (5) were already described. A two-dimensional analytical study (38) focuses on modelling the contrarotating vortex pair without incorporating empirical data. Equations governing the vortex spacing and the downstream jet velocity are formulated by approximating the forces acting on each of the









viscous vortices. Numerical solution of the governing equations yields reasonable results for the vortex trajectory, half-spacing and viscous core size. The results indicate that the vortices are formed relatively close to the jet exit and the two-dimensional analysis is inappropriate in this region.

Panel methods, which are based on potential flow theory, are commonly used to simulate the aerodynamics of conventional aircraft. Consequently, their application to powered lift configurations has been investigated (References 39-44). In panel methods, distributions of source and doublet singularities are placed on the boundary surfaces and approximated numerically by networks of quadrilateral panels which each have an arbitrary singularity strength. With the application of proper boundary conditions, a system of linear algebraic equations which can be solved for the singularity strengths is produced. Then, the entire velocity or pressure field can be computed. The assumptions that are inherent in panel methods are that the outer flow is incompressible and irrotational, boundary layer effects are neglected and separated flow is not simulated. Further, some a priori knowledge of parameters such as the jet trajectory and the distribution of entrainment is required.

Current trends are to use a panel method for the aircraft

body and to couple it with a Navier-Stokes type solver for the jet. Howell (44) used the PANAIR panel code with a Neumann boundary condition to represent the aircraft coupled with a parabolic Navier-Stokes solver with an entrainment boundary condition for the jet. The jet shape and the entrainment were calculated using the









Adler-Baron (45) method which is a semi-empirical, integral model that can be numerically solved for the internal jet flowfield. The results from this parabolic analysis are unable to account for the separated wake in a subsonic flowfield. Consequently, the computed lift losses are underestimated by about 20% of the thrust for the simple high wing aircraft with a single lift jet that was tested. Navier-Stokes Methods

A recent review by Hancock (46) stresses that the flow behavior associated with a jet in a crossflow is dominated by viscous effects. While inviscid analyses describe the general trends within the flow, they are not accurate quantitative results nor are they directly comparable to experiments. In addition, each of the techniques described above, whether viscous or inviscid, yields global rather than detailed characteristics of the flowfield. Recent solutions of the Reynolds averaged Navier-Stokes equations for a variety of propulsive flowfield applications including jet-in-ground-effects (47), internal jets (48) and upwash fountain effects (49,50) as well as a jet in a crossflow (33,51-61) offer a more promising approach.

A parabolic Navier-Stokes formulation by Baker (51-55)

utilizing a k-E turbulence model and an algebraic Reynolds stress model exhibits the expected kidney-shaped jet cross-section, a transverse vortex pair that propagates downstream and a jet trajectory that is displaced slightly upstream prior to the characteristic downstream deflection and predicts the evolution of the farfield entrainment velocity distributions. This method relies on the proper formulation of initial and farfield boundary









conditions in order to be marched in space, away from the plate. The wake induced flow separation effects are not adequately captured by the parabolic Navier-Stokes approach.

Initial elliptic, three-dimensional, Navier-Stokes solutions show improved results. Patankar, Basu and Alpay (33) employ a finite difference formulation with the three velocity components and pressure as the dependent variables coupled with a two equation turbulence model. The computations are performed in a rectangular domain about the jet with the assumption of flow symmetry imposed. Their relatively low-resolution results (a maximum grid density of 2250 points, assuming symmetry, was tested) for a three-dimensional turbulent jet demonstrate good agreement for the jet centerline and the rate of decay of the jet velocity but deviate significantly from the experimental velocity profiles along the jet axis after the jet deflects and in the reversed flow region. Other detailed flowfield results are not presented.

Other relatively coarse results are presented by Chien and Schetz (56) who formulate the steady, incompressible Navier-Stokes equations in terms of vorticity, velocity and temperature and by Demuren (57) who calculates the flow induced by a row of jets using the steady, incompressible Navier-Stokes equations with the k-e turbulence model. Notably, in the simulation by Demuren, the jet exit boundary condition is based on uniform total pressure rather than uniform vertical velocity and for a low velocity ratio, in this case R = 1.96, is similar to the jet/pipe profile measured by Andreopoulos (62). The effects of numerical diffusion are also investigated by Demuren.









Most recently, several higher-resolution computations, each employing on the order of 100,000 grid points to discretize the three-dimensional domain, have been reported. A detailed picture of the jet plume and the vorticity dynamics for a turbulent jet is given by Sykes, Lewellen and Parker (58). In their calculation, the incompressible Navier-Stokes equations are solved implicitly and the turbulence lengthscale is based on the configuration geometry, namely, the jet diameter and the distance from the center of the jet. However, since the primary motivation is the behavior of a jet remote from a wall, they do not attempt to resolve the wake region. Indeed, the lower surface is treated as a stress-free wall.

Both Reed (59) and Harloff and Lytle (61) resolve the flow near the plate. In each of these studies as well as in most of the Navier-Stokes computations mentioned previously, a stretched rectangular grid is used to discretize the physical domain. Reed employs an explicit algorithm based on the three-dimensional Navier-Stokes equations; but, after 10,000 iterations, the solution appears to be unconverged. A significant deviation exists between the jet trajectory for the computed flow and a similar experiment for flows with velocity ratios R = 4 and 8. It is concluded that replacing the explicit differencing scheme with a time accurate implicit method would allow a steady state solution to be obtained in a reasonable number of time steps.

An implicit algorithm which also incorporates all of the three-dimensional, viscous Navier-Stokes terms is applied to both subsonic and supersonic jet flows by Harloff and Lytle (61). The










subsonic computation is steady state, laminar and symmetry assumptions are applied to the computational domain. Agreement with experiment is shown for the jet centerline. The vortex curve is located by examining the upwash velocity components in the symmetry plane. This computed vortex curve is shown to be up to 2 jet diameters below the vortex locations measured by Fearn and Weston (5) for similar jet conditions. The authors speculate that turbulence, which is excluded from the simulation, is responsible for the underprediction of the vortex curve. Notably, preliminary results for a subsonic jet injected perpendicular to the freestream with a velocity ratio R = 8 indicate the presence of a secondary vortex structure in the wake region of the jet.












Cross Section to Jet Plume


JET q_


Contrarotating vortex pair


UM






z


Jet Core


Figure 1. Schematic diagram of a jet in a crossflow.














CHAPTER 3
NUMERICAL APPROACH


Thin-Layer Assumption

The flowfield induced by a jet in a crossflow is

characterized by large pressure variations, a pair of diffuse contrarotating vortices, vortex shedding on the lee side of the jet and a three-dimensional, turbulent wake containing both separated and reversed flow. A computational method based on the threedimensional, time-dependent Navier-Stokes equations would be capable of providing a total mathematical description of the flow, permitting investigation of all flow parameters. However, methods based on these equations currently have little application to practical aircraft configurations as a consequence of available computer memory which restricts gridding in the computational domain. In order to develop a numerical tool that may potentially impact STOVL design while capturing the pertinent flow features in a reasonable amount of time, several simplifying assumptions concerning the nature of the governing equations must be made.

In the numerical simulation of high Reynolds number flows, the thin-layer Navier-Stokes equations are often used. The thinlayer approximation is obtained by using order of magnitude arguments similar to those employed in boundary layer theory (75) to drop all viscous terms that contain derivatives that are parallel to the body surface. However, unlike boundary layer









theory, the thin-layer approximation retains all three momentum equations and no limitation is put on the pressure field. Thus, the thin-layer equations permit the calculation of separated and reversed flows as well as flows with large normal pressure gradients. Further, this set of equations is valid in both inviscid and viscous flow regimes.

An assessment of what is actually computed when the NavierStokes equations are applied to high Reynolds number flows provides a basis for making the thin-layer approximation (76). In order to resolve the gradients normal to the boundary layer, grid points must be highly clustered within the boundary layer. Consequently, a large amount of time is spent resolving these gradients. Then, computer resource limitations preclude the adequate resolution of gradients parallel to the surface. Since in many applications the gradients that are parallel to the surface are relatively small and are not completely resolved, the thin-layer approximation is made to exclude them from the calculation altogether.

One goal of this study is to apply existing technology, with modifications as needed, and to assess the validity of this technology for a propulsive flow problem. Several computer codes incorporating the thin-layer assumption were immediately available at the initiation of this study. The merits of this approach, in comparison with full Navier-Stokes methods, are that it reduces the effort required to calculate the viscous terms and requires a highly clustered grid in only one direction. The thin-layer approach is also superior to parabolized Navier-Stokes methods, which were also immediately available, in that the separated,









reversed wake flow may be treated and flow along the plate is calculated rather than input as a starting solution.

The application of the thin-layer assumption to the jet in a crossflow problem must be viewed with caution since the flowfield is actually viscous in all three directions. While the high gradients perpendicular to the plate within the boundary layer and in the wake may be captured, large streamwise gradients are anticipated within the wake and are excluded from the computation. Large streamwise and lateral gradients are also anticipated and neglected in the shear layer between the jet and the crossflowing fluid in the region near the jet exit (see Figure 1). The neglect of viscous terms in the near jet region may affect the initial distribution of vorticity and ultimately alter both the farfield properties of the contrarotating vortex pair and the characteristics of the horseshoe vortex. Nevertheless, the thinlayer approximation is applied and tested in this study.

The thin-layer Navier-Stokes equations can be written in conservation law form as follows:


+ F +G +H Re- S
r ++ T=Re as


where the flux vectors Q, F, G and H and the viscous fluxes, S, are given in Appendix A. The generalized coordinate variables 4, ', and T are functions of the Cartesian coordinates x, y, z and time, t. Boundary layer type coordinates must be used to resolve the viscous terms normal to the surface; in this case, the grid is









clustered in the (direction to resolve the boundary layer near the plate.


Finite Difference Equations

In any finite difference method, the numerical solution is obtained at grid points and at time levels by using a set of algebraic equations known as finite difference equations (FDE) which are derived from the governing equations. The numerical method which is used in this jet in a crossflow simulation solves the Reynolds averaged, thin-layer, Navier-Stokes equations cast in generalized coordinates which are given in equation (1). The FDEs are derived from the governing equations by using an implicit, factored method.

In contrast to explicit methods which rely only on

information that is known from previous time levels to advance the solution to the next time level, implicit algorithms utilize information from adjacent points in the flowfield at the same time level as the point under consideration. Consequently, implicit methods require the simultaneous solution of sets of linear equations and costly matrix inversions. However, in comparison to explicit methods in which numerical stability restricts the maximum allowable step size, implicit methods permit the use of a larger step size and expedite code convergence.

The finite difference method employed in this study is

contained in the F3D code (77,78). The derivation of the FDEs for the implicit, approximately factored, partially flux-split algorithm can be summarized as:









1. Transform the governing equations, in this case the thinlayer Navier-Stokes equations, from the physical,

(x,y,z,t) coordinate system to the computational space,

(t, ,,). This is achieved by by replacing the

derivatives with respect to (x,y,z,t) by derivatives with

respect to ((,r,?,t) via chain rule differentiation. The

transformed equations are stated in equation (1) with

the corresponding metric coefficients and the transformation Jacobian listed in Appendix A.

2. Replace the time derivative terms with a finite

difference formula.

3. Linearize any terms that are nonlinear with respect to

Qn+1 where Q is the flux vector given in Appendix A and

is referenced at time level n+l.

4. Employ approximate factorization followed by operator

splitting.

5. Use finite difference formulas to replace spatial

derivatives.

6. Add the numerical dissipation terms to the algorithm. More details on implementing these steps for developing the F3D algorithm are given in the following sections.

For differencing the thin-layer Navier-Stokes equations, it is often advantageous to difference around a known base solution so that the resulting differences have a smaller, smoother variation and possibly, less differencing error. Thus,









8,(Q-Qo)+86(F-Fo)+8,(G-Go) +(H-Ho) -Re- 6 (S-So) =

- Qo-a Fo- Go- H +Re-I So (2) in which the subscript o signifies the base solution, 8 is some general difference operator and a is the differential operator. For this application, the freestream solution is used as the base solution so that equation (2) reduces to


6 (Q-Q.)+8 (F-F F)+ 61 (G- G )+68 (H- H )-Re 1 (S-S )=0 (3) Time Differencing

The time derivative in equation (3) is replaced by using a general, linear, two step method for integrating a first order ordinary differential equation that may be expressed as


eun- 1-(l + 2e)un+( + e)un + 1 =At + l-fn- 1+(1+o -8)fn+ fn +11 (4) for some constants E, * and 0 and some function f = f(u) such that a u = f(u). This general formula produces several two or three time level, explicit or implicit schemes, such as Leapfrog or CrankNicolson, depending on the value of the constants. In the F3D algorithm c = #= 0 so that equation (4) becomes (Un+ 1 -un)=Atn+ l[fn+ 1+(_1-)fn] (5) For application to the thin-layer Navier-Stokes equations, equation

(5) takes the form
(Qn+l _ Qn) = Atn+l[fn+l + (1-0)fn] (6) in which f = -[a (F-F ) + q (G-G ) + a (H-H ) - Re-l (S-S )]. The value of the constant Ois 1 or 0.5 for first or second order accuracy, respectively. For steady state computations, the final









solution is independent of the time step making first order accuracy sufficient; then, 0= 1. Linearization of Nonlinear Terms

Next, all terms that are nonlinear with respect to Qn+l are linearized. As in the Beam and Warming algorithm (79) time linearization is used to approximate the fluxes, F, G and H, at the (n + l)st time level. Since F, G and H are homogeneous functions of Q of degree one, let A = aQF, B = G and C = aQH. Then the linearization is accomplished by using a truncated Taylor series of the form

Fn+l = Fn + AnAQn + O(At)2

Gn+l = Gn + BnAQn + O(At)2 (7) Hn+l = Hn + CnAQn + O(At)2

in which A, B and C are the flux Jacobians and AQn = Qn+1 - Qn. Thus, the error introduced by the linearization is on the order of

(At)2. The flux Jacobians contain only information that is known from the previous time level. The viscous flux term, Sn+l, is linearized by the method set forth in Pulliam and Steger (80). When these linearized terms are substituted into equation (6) and rearranged, it becomes


+eAtn+1 (A n +Bn1 +cn Re J-1 n J)]n I+At A P) + IC ( --Re- J M J Q =
-Atn+[D Fn-F.)+3 (Gn-G )+3 Hn-H )-Re-I8 S-S )] 8


where I is the identity matrix.









Approximate Factorization

In deriving FDEs with the implicit, factored method, the next step is to approximately factor the left hand side (LHS) of equation (8). By expressing the differential operator, L = (Ana + BnB + Cna - Re-lj-1Mnj), as the sum of three differential operators such that L = L + n + L , the unfactored form of equation (8) can be rewritten as

[I + eAt(L + L + LC)]AQn = RHS (9) in which the right hand side is denoted by RHS. Since fluxsplitting in the freestream direction, t, is employed within F3D, the operator L is split so that L = L+ + L-; details of the fluxsplitting are given in the next section. With flux-splitting included, the final unfactored form of equation (8) is summarized

as
+ n
[I+At (L+L +L +L AQ =RHS
(10)

Beginning with the unfactored form of equation (10), the LHS is replaced with two implicit factors giving
+ n
[I+At (L + L; ][I+At(L +L)] AQ =RHS with splitting error
2 4 _
E=(0At) L + +L+L


Thus, the splitting error due to the two factor approximation is order (At)2 and is comparable to the error due to the time linearization. Provided that the chosen finite difference scheme is no more than second order accurate in time, the cross terms contained in E can be disregarded. This implicit, two factored









scheme is shown by Ying (81) to be unconditionally stable for a model wave equation when upwind differencing is applied in the freestream direction, t, and central differences are employed in the Sand C directions.

One advantage of the approximate factorization is that instead of solving a block matrix on the order of dimension ((x TJ x C), two, two-dimensional problems are factored out and individually inverted resulting in a significant computational savings. The algorithm is split into a sequence of problems such that
[I+0At (L +L)]Q = RHS

I+6At L +L )]AQn =AQ (12) where AQ* represents an intermediate variable. Notably, the first factor results in a lower triangular matrix while the second factor is an upper triangular matrix.

A comparison of the F3D algorithm with other approximately factored methods for the thin-layer Navier-Stokes equations shows that the F3D numerical scheme is a reasonable approach for this jet in a crossflow simulation. First, in comparison to the standard, three-factor Beam and Warming algorithm for the thin-layer NavierStokes equations (79), the two-factor method outlined above and employed in F3D provides some computational savings. In addition, a conceptual advantage of F3D is that the three-factor Beam and Warming method is unconditionally unstable without added dissipation terms. It is suspected that this instability will have an increasing impact on fine grid computations as the grid density









is increased in that more and more dissipation would need to be added to maintain stability with the Beam and Warming algorithm. Another more cost effective alternative to F3D might be the diagonalized version of the Beam and Warming scheme (82); but, the time accuracy is only approximately O(Atl/2) for this scheme. Flux Splitting

Prior to replacing the spatial derivatives with finite

differences, flux vector splitting is applied in the freestream, , direction and then utilized in forming L+ and L-. Since the flux vector splitting is confined to the t direction, consider F = AQ where A = aQF

since F is a homogeneous function of Q of degree one. In generalized coordinates, the eigenvalues of A are given by


X =X2 =X3 =U
X 2 +t J
2 22

5 =U + + t +x y zC
2 2 2
Us=U- x+y+zC

in which c = (p/p)0.5 is the local speed of sound and the contravarient velocity, U, and metric terms are given in the Appendix. Some similarity transformation, T, exists which reduces A to a diagonal matrix A containing the eigenvalues. Then, T-1AT = A or A = TAT-1

The matrix of eigenvalues, A, is split into two matrices, one that contains only positive elements and one that contains only negative elements, which permits the following equality to be stated









F = AQ = (TAT-1)Q = (TA+T-1)Q + (TA-T-1)Q = A+Q + A-Q = F+ + F(13)

The flux vector F is split.

It is noted that the splitting of A is not unique. In F3D, the implicit LHS terms are differenced using simple flux vector splitting as described by Steger and Warming (83). Flux vector splitting can introduce oscillations into the solution at locations where the eigenvalues change sign and the corresponding fluxes are discontinuous. To alleviate this problem, transition operators are employed in F3D (81,84). The transitional scheme for the explicit RHS, 4, derivatives, which is identical to flux splitting except at the point where an eigenvalue changes sign, is summarized in Reference 77.

Since A+ has only positive eigenvalues, information is only propagated in the positive t direction (downstream) and backward differencing is used. Similarly, A- has negative eigenvalues, information is only propagated in the negative 4 direction (upstream) and forward differencing is appropriate. The flux vector F has both positive and negative eigenvalues for subsonic flows so that without the use of flux vector splitting, central differencing would have to be used for the t direction. Thus, one advantage of flux vector splitting is that it permits the use of one sided difference formulas which possess superior dispersive and dissipative characteristics than central differencing schemes. Spatial Differencing

All time and spatial derivatives are replaced with finite difference formulas. For the spatial differences, in general,








second order upwind differences, denoted by t, are used in the t direction and second order central differences, denoted by In and

6 , are used in the T and C directions in this algorithm. A midpoint operator, 6, is used with the viscous terms. Examples of second order, three-point forward and backward differences, second order central differences and the midpoint operator are

f (-Q�++ 4Q - 3Q
f +2, 1 k,n + 4Qi+1, 1 k, n- 3i, j, k, n)
0 2At
(3 4 Q)
(3Q, I k,n -4Q 1, 1 kn i-2,j,k, n
2,g
(Qi, j+ 1, k,n - i, j- 1, k, n)
7Q 2Ail
(Q +Q
i,J, k+- ,n i, j, k- i,n
Q = 2 2(14)



where T = A = A = 1 in the computational domain.

Then, with the flux splitting and the difference operator notations defined, the two implicit factors (see equation 12) can be written as
I+OAt (L +L)]=6 A + C-Re- i( J MJ)
f
I+eAt L +L )]=8 A-+B (15)
( 4 4 71(15) Also, the explicit RHS is

RHS=-At 8 1 ( --F. + 8 (F -F- + ( G - G) +

Hn - H) - Re (Sn - S) (16)











The numerical solution of both the Euler and the NavierStokes equations requires the addition of numerical dissipation, especially for the Euler equations since they exclude natural dissipation. For the Navier-Stokes equations, artificial dissipation is needed to damp high frequency, nonlinear instabilities in the flow, such as shocks, and to control the odd/even decoupling of grid points which is associated with central dfference schemes. The artificial dissipation must be added judiciously in order to control numerical instabilities without smearing the flow physics. In general, there are two methods for introducing artificial dissipation. First, it is introduced by upwind difference schemes, in this case through the flux splitting, and there is little user control over the amount of dissipation that is added. Next, with central differencing schemes, the numerical dissipation is intentionally added.
The implicit and explicit smoothing terms, denoted by D. and
1
De, are added to the centrally differenced, 1 and C directions to control nonlinear instabilities. These numerical dissipation operators are written as a combination of second and fourth order differences and have the form

Dei = (At) J 2 4

Di, =(At).j 1(28IBIP8+2. 5C4B IB BJ where

P= 2
K 1+8 lp(17)









and where IBI is either the absolute value of the matrix B or an approximation of it, p is the nondimensional fluid pressure, �2 is O(1) and C4 is 0(0.1). The coefficient of the second difference increases and the coefficient of the fourth difference decreases as P gets large; thus, in the vicinity of a shock, the effect of the fourth difference is reduced. Actually, the fourth order difference terms can introduce oscillations near shock waves and may be dropped from the computation near them. Code Implementation

The final form of the implicit, approximately factored

FDE that is derived using the steps described above and implemented in the F3D code is



I+ h68 A )n + h6 Cn - h Re-' J- MnJ- Di,
-1n
) _An n n
x I+h6(A) +h B -Di, AQ =


-At e( F n- F++6 f[(F-)n-Fj +6 (Gn-G + S(Hn- H.)- Re Sn-S

-De(Q Q )

(18)


where h = At or At/2 for first or second order time accuracy, respectively. In this equation, 8 is typically a three-point, second order accurate, central difference operator, 9 is a midpoint operator that is used for the viscous terms and the operators Sf and 8b are forward and backward, three-point difference operators. The flux F is split into F+ and F- according to its eigenvalues, the matrices A+, A-, B, C and M result from the local linearization of









the fluxes about the previous time-level and J is the Jacobian of the coordinate transformation. De and Di are the dissipation
91
operators for the central space differenced directions.

The computed flows may be steady or unsteady, viscous or inviscid, laminar or turbulent. In the resulting code, the convergence rate of the algorithm may be accelerated for steady state applications by applying a space varying time step such that At
At=
1+%/J

Since the Jacobian, J, is the inverse of the cell volume for a three-dimensional grid, the time step is dependent on the local grid spacing, rendering the calculation nontime-accurate. Srinivasan, Chyu and Steger (85) compared the convergence rates for a time-accurate computation in which the same At is used everywhere and a nontime-accurate solution to find that the spatial varying time step improves the convergence rate by a factor of 2. For laminar calculations, the coefficient of viscosity, R, is obtained from Sutherland's law. Turbulence modelling is not used in this study. The code, which is fully vectorized for supercomputers, requires 10-4 CPU seconds/iteration/grid point on the CRAY-2 computer.


Computational Grid

In order to implement the numerical algorithm described above, the continuous spatial domain must be replaced by a set of discrete grid points. Consequently, a three-dimensional computational domain, shown in Figure 2(a), is set up for the









calculations. A right handed Cartesian coordinate system is used with x increasing in the freestream direction and z pointing away from the plate. The rectangular grid is exponentially clustered within the boundary layer of the flat plate and around the jet exit which is centered at the origin of the Cartesian coordinate system. Each grid point is indexed with the labels i, j and-k. The index i ranges from 1 to imax for which i = 1 corresponds to the upstream boundary x = Xmin and i = imax corresponds to the outflow boundary x = xmax. Similarly, the index j spans the domain laterally and k increases in the z direction.

Since the experimental measurements that are used for

comparison with these numerical calculations are taken for jets exhausting through a circular orifice, the jet exit is approximated by a right-angled polygon of nearly equal area for this Cartesian grid system. A partial sketch of the grid on the flat plate in Figure 2(b) illustrates the representation of the jet exit. Although the relatively sparse experimental measurements indicate that the global features of the flow induced by a jet in a crossflow are steady and symmetric, the secondary flow features of the flowfield are assumed to be unsteady and assymmetric. As sketched in Figure 2(b), no symmetry assumptions are made for the computational domain; this permits the investigation of flowfield symmetry about the y = 0.0 plane. Nevertheless, in accord with previous experimental studies, the y = 0.0 plane is referred to as the symmetry plane.

In Table 1, the number of grid points in each direction and the limits of the physical domain are specified for two









representative grids. Initial calculations are made on a relatively coarse grid with 39 (freestream) x 35 (normal to the y =

0.0 plane) x 32 (normal to the flat plate) points. The coarse grid computations are made on a CRAY-XMP computer. In order to investigate the effects of the grid on the computation and to provide good correlation with the experimental data, the grid density is increased from 39 x 35 x 32 (43,680 points) to 55 x 55 x 50 (151,250 points). Calculations on the refined grids are performed on the CRAY-2 computer.
The quantities Xmin, xmax' Ymax and zmax are

nondimensionalized by the jet diameter, D, and define the minimum and maximum boundary locations. Since the domain is symmetric about the y-axis, ymin = -ymax. Further, Zmin corresponds to the flat plate, z = 0.0. The method for determining the values of the boundary locations will be examined further in Chapter 4.



Table 1. Details of the Physical Domain for Two Grids.



GRID MAXIMUM INDEX VALUE DOMAIN SIZE

i j k xmin/D xmax/D ymax/D zmax/D


1 39 35 32 -5.0 15.0 9.5 18.0 2 55 55 50 -6.0 15.0 9.5 18.0




For Grid 1, the points are distributed throughout the domain such that









imax = 39: (8)upstream + (7) jet + (24)downstream

jmax = 35: (14)left + (7) jet + (14)right

kmax = 32: boundary layer + (28)

where the subscripts upstream, downstream, left and right denote the position with respect to the jet orifice and the subscripts jet exit and boundary layer denote points that are contained in the jet exit and the boundary layer respectively. Similarly, for Grid 2,
imax = 55: (12)upstream + (9)jet + (34)downstream

imax = 55: (23)left + (9)jet + (23)right
kmax = 50: (10)boundary layer + (40)

The details of the minimum and maximum grid spacing are presented for the representative grids in Table 2. For both cases, the minimum spacing occurs within the jet exit and at the first grid point within the boundary layer while the maximum spacing occurs at the edges of the domain. In Table 2, Ax1/D refers to the spacing upstream of the jet and Ax2/D refers to the spacing downstream.


Table 2. Details of the Grid Structure.









In order to perform the calculations, the physical domain is transformed to a computational space with uniform grid spacing as shown in Figure 2(c). In the transformed grid system, 4 corresponds to the freestream direction such that t = 1 coincides with the upstream boundary where x = Xmin, 1 spans the domain laterally and ( is the direction normal to the body surface such that C= 1 maps to the solid surface. The i, j and k indexing is retained in the computational space.


Boundary Conditions

A unique numerical solution of the thin-layer Navier-Stokes equations set forth in equation (1) is obtainable only if the proper physical and numerical boundary conditions are imposed; thus, the correct specification of boundary conditions is critical to the success of the jet in a crossflow simulation. Moreover, a good initial choice of boundary conditions may speed up the convergence of the algorithm. These boundary conditions are not only dependent on the problem physics but also on the grid topology.

The boundary conditions that are required for the jet in a crossflow problem with the previously described domain consist of

1) freestream conditions on the upstream boundary, t = 1,

2) extrapolation on the downstream boundary, t= max' 3) solid surface conditions on the flat plate, ( = 1,

4) jet exit profile for Ix2 +y2I /D 0.25 and 1 = 1,

5) freestream conditions for the top surface, = max' and

6) extrapolation on the lateral surfaces, 4= 1 and nmax*









In general, these conditions are implemented explicitly in modular form within the code.

A laminar boundary layer profile is input on the upstream boundary. The variables on the downstream boundary are obtained through an explicit, linear space and time extrapolation by


t+1 t t
i, jmax,k =2Qi, jmax -1,k i, jmax -2,k

where Q is a vector in the numerical code containing the five flow variables defined in Appendix A, jmax corresponds to the max plane and t refers to the time level. Similarly, a zero normal gradient boundary condition is applied to the lateral surfaces of the computational domain.

The boundary conditions on the flat plate are dependent on whether or not the flow is viscous. For viscous calculations, the no-slip velocity condition is imposed at the plate; specifically, all velocity components are set to zero at the wall. The density is then calculated by zeroeth order extrapolation from the adjacent points interior to the flowfield. On the other hand, for inviscid computations, a flow tangency condition is applied; fluid cannot penetrate the solid surface. The pressure on the solid surface is obtained as the solution of the normal momentum equation.

The precise physical boundary condition at the jet exit is unknown. Therefore, some assumptions must be made about the character of the flow at the orifice which may lead to discrepancies between the computed and measured results in the near jet region. Previous authors have imposed the jet exit condition by specifying uniform vertical velocity (33) or uniform total









pressure (57) or by initiating the calculation inside the jet nozzle. In this simulation, a uniform velocity, constant pressure profile jet is injected from the lower boundary into the flowfield. Since the circular orifice is represented by a rectangular grid, care is taken to match the input mass flow and momentum of the jet with the experiment. The second type of boundary condition, uniform total pressure, may ultimately prove to be more effective but guidance, in the form of experimental measurements for jets with high velocity ratios, is needed to insure its validity. Similarly, calculation of the flow inside the pipe may provide a more realistic jet profile. But, the cost of the calculation would be greatly increased with the present finite difference formulation since a two part (flow internal to the nozzle and the external flowfield) iterative solution would be required to accurately predict the pressure profile at the jet exit with the freestream influence.










































Figure 2.


I


Grid structure. a) Physical space; b) Computational space; c) Enlargement of the jet exit. The shaded region, which is 2% larger than the circular jet exit, denotes the area included in the jet exit for the fine grid calculation.














CHAPTER 4
RESULTS AND DISCUSSION


The purpose of this study is twofold, first, to adapt the F3D code for use in propulsive flowfield simulations and to validate it both qualitatively and quantitatively through experimental comparisons and second, to provide insight into the physics of the three-dimensional lifting jet aerodynamic/propulsive interaction. In this chapter, several applications of the numerical algorithm are described. The test cases were chosen to correspond closely with the wind tunnel tests of Fearn and Weston (5-7). First, the computed results are quantitatively compared with the measured data by examining the jet trajectory, the properties of the contrarotating vortex pair and the plate pressure distribution. Then, through flow visualization, the flow physics are qualitatively investigated.

Calculations are presented for the jet/flat plate

configuration with the jet injected perpendicular to the cross stream for the velocity ratios, R = M /M 0, of 4, 6 and 8. The computed flows have freestream Mach numbers of 0.19, 0.13 and 0.12 and Reynolds numbers based on the freestream velocity and the jet diameter of 500,000, 350,000 and 350,000, respectively. In most of the numerical calculations, the flow is assumed to be steady and viscous. A summary of the viscous test cases is given in Table 3.











Table 3. Summary of Viscous Test Cases.


CASE R M. RE GRID SIZE DOMAIN SIZE Xmini Xmax Ymax Zmax


1 4 0.20 500,000 39x35x32 -5.0 15.0 9.5 18.0 2 4 0.19 500,000 55x55x50 -6.0 15.0 9.5 18.0 3 6 0.13 350,000 39x35x32 -5.0 15.0 9.5 18.0 4 6 0.13 350,000 55x55x50 -6.0 15.0 9.5 18.0 5 8 0.12 350,000 39x35x32 -5.0 15.0 10.0 20.0




In addition, inviscid calculations were made corresponding to the viscous cases 1 and 2. Obviously, the inviscid calculations cannot predict the separated flow region downstream of the jet. Furthermore, the inviscid solution contains numerical viscosity which tends to introduce diffusion throughout the entire physical domain while the actual viscous effects are concentrated in areas such as the boundary layer and the shear layer between the jet and the freestream. The purpose of the inviscid computations is to isolate features of the flow which are dominated by effects such as pressure rather than viscosity and to identify regions of the flow, if any, that might be adequately modelled with a reduced set of governing equations.

The numerical results are computed on a three-dimensional computational grid which has the features described in Chapter 3









and which has the grid density and the domain limits, nondimensionalized by the jet diameter, that are given in Table 3 for each of the cases. Initially, the dimensions of the computational domain were chosen by examining the empirical jet paths and by estimating the amount of "blockage" by the jet. In order to ensure that these boundary locations are not adversely influencing the numerical predictions, computations are made in which the boundaries are moved progressively farther away from(or closer to) the jet until the flow is relatively unaffected by the boundary locations. Two criteria for determining a sufficient domain size are enforced. First, the jet centerline is required to exhibit good agreement with the experimental jet centerline and with the centerline location calculated on the next larger domain. Second, the top and lateral boundaries are positioned so that the flow direction, at these locations on the next larger domain tested, is nearly aligned with the freestream flow.

As discussed previously, the domain sizes are minimized; however, for the convenience of point by point comparison of the flow variables, some cases, such as Case 1 and 3, are computed on the larger domain. It should be noted that the boundary location optimization has not been completed for the R - 8 jet, Case 5. Both the lack of grid refinement and the minimal domain sizing may significantly influence the detailed results for the R = 8 case in which flow variables will exhibit higher gradients than for the lower velocity ratio jets. But, the results for selected global properties such as the jet centerline are examined when reasonable agreement is demonstrated. It is anticipated that positioning the









upper boundary so that zmax is between 20D and 22D would improve the accuracy of future solutions.

In each of the cases studied, the flow is assumed to be

laminar; no turbulence model is applied. In experimental studies of flows with relatively low velocity ratios, both Crabb, Durao and Whitelaw (12) and Andreopoulos (62) found regions in which the turbulence field is anisotropic which suggests that the commonly used turbulence models, such as eddy viscosity models and k-e models, will not be able to account for all the features of the jet flowfield. Thus, the exclusion of turbulence from the calculation permits the evaluation of the numerical model without having to estimate the errors associated with an inadequate turbulence model.

In this laminar simulation, a Blasius boundary layer profile is computed upstream of and beside the jet. Since boundary layer velocity profiles and separation point locations are dependent on the characteristics of the boundary layer, complete quantitative agreement between the measured turbulent flow and the calculated flow is not anticipated. On the other hand, the experiments of Bradbury and Wood (25) revealed that the momentum thickness and the jet Reynolds number only weakly influenced the pressure distribution on the plate. Thus, the global features of the jet in a crossflow such as the jet centerline, the contrarotating vortex pair and the pressure distribution on the plate that are depicted in Figure 1 may be relatively insensitive to secondary parameters and might be adequately captured in a laminar simulation.

At each grid point, the density, p, the three components of momentum, pu, pv, and pw, and the total energy, e, are computed.









The solutions are considered to be converged when the L2 average norm of the residual is O(10-7). A typical convergence history for the steady state solution for Case 2 is shown in Figure 3. Since a space varying time step is used for the steady state calculations, a comparison between the convergence rates for all of the test cases would not be instructive.


Grid Dependence

In order to examine the grid dependence of the solution, profiles of the streamwise velocity component are constructed at several locations downstream of the jet orifice. For presentation, the vertical profiles of streamwise velocity in the symmetry plane at the downstream locations x/D = 1.0, 2.0, 3.0, 4.0 and 8.0 are shown in Figure 4(a)-(e) for the R = 4 jet and in Figure 5(a)-(e) for the R = 6 jet. In both cases, the coarse and fine grid results are significantly different, particularly in the boundary layer region.

The velocity vector field within the symmetry plane for the R = 4 jet is presented as Figure 6 in order to aid in the interpretation of the scalar velocity profiles. The vector field indicates the onset flow, the jet plume which is deflected by the freestream, a reversed flow region behind the jet, an upwash velocity component which lies in the symmetry plane between the contrarotating vortices shown in Figure 1 and finally, the predominantly streamwise flow. The vector field that is shown is confined to the symmetry plane since there are no lateral velocity components in this plane.









In the region of maximum streamwise velocity, which closely corresponds to the location of the jet trajectory, the velocity peaks are more pronounced and tend to be greater in magnitude on the fine grid than on the coarse grid. Although these peaks do not reflect the same magnitude, in general, their vertical location appears to agree well. At the most downstream location, Figures 4(e) and 5(e), this difference is minimized due to the decreasing impact of the jet on the surrounding flow and to the increased similarity between the two clustered grids far downstream. A similar grid dependence was seen in the calculations by Demuren

(57) who found that only the predictions on the finest grid he tested (approximately 50,000 grid points for the three-dimensional domain) showed significant velocity peaks in the central region of the deflected jet while this peak is smeared out on the less refined grids. He also noted that the direction of the grid refinements, for example increasing the grid in the lateral direction without increasing the number of points in the other two directions, altered the results.

Next, large discrepancies arise in the streamwise velocity profiles within 3.0 jet diameters of the flat plate. Actually, the distinct flow features that are captured on the fine grid cannot be resolved on the coarse grid. At the most upstream location, Figures 4(a) and 5(a), both the coarse and fine grid solutions show a region of reversed flow. However, with the grid refinement, there is a decrease in magnitude of the reversed flow velocity component near the edge of the boundary layer and the minimum velocity peak occurs nearly a jet diameter farther away from the









plate. On the fine grid, the minimum velocity peak that is closest to the plate is due to flow reversal within the jet wake while the second, or more negative, minimum velocity peak corresponds to the upwash between the contrarotating vortices. Since the boundary layer is not resolved and the numerical dissipation is increased on the coarse grid, the coarse grid solutions show only one minimum velocity peak with no distinction between the wake flow and the upwash.

In addition, the fine grid computation predicts a smaller

streamwise region of reversed flow. For example for R - 4, between x/D = 2.0 and 3.0 shown in Figures 4(b) and (c), the fine grid solution exhibits full forward recovery of the velocity field near the plate while the coarse solution still indicates a large region of reversed flow. At x/D = 4.0 which is shown in Figure 4(d), some distinction is noticed between the reversed wake flow and the entrained flow in the coarse grid solution. But, the experimental measurements (7) which are plotted for the planes x/D = 4.0 and 8.0 on Figures 4(d) and (e) show that the fine grid solution more accurately characterizes the flow. Similar behavior is observed for R = 6 and is illustrated in Figures 5(b)-(e). Once the flow reversal within the wake ceases, the point of minimum velocity represents upwash.

At the most downstream location shown, x/D = 8.0, the fine grid results for the R = 4 jet which are displayed in Figure 4(e) overshoot the experimental data in the region near the flat plate. This may be caused by the application of freestream boundary conditions on the top surface of the computational domain since it









is not uncommon for numerical codes to predict higher velocities near the surface in order to conserve mass and momentum in the calculation. The application of characteristic boundary conditions could be investigated to alleviate the overshoot. On the other hand, the overshoot in streamwise velocity may be caused by the rapid convergence of streamlines toward the symmetry plane which tends to accelerate the flow in the symmetry plane. In this latter case, the governing equations, grid sizing and turbulence are all suspect.


Flow in the Symmetry Plane

The computed velocity profile in the symmetry plane was described in the previous section. This particular view of the flowfield provides some information on the trajectory of the jet plume and on the penetration of the contrarotating vortex pair. In addition, a large amount of experimental and computational data is available for comparison in this plane. The plots in Figures 6-9 show the velocity vector field and the Mach number contours for the cases R = 4 and 6. These plots, which show that the jet extends a short distance into the flow before being deflected downstream by the crossflow, are similar to the qualitative data presented as plots of the scalar field and the velocity vector field by Sykes, Lewellen and Parker (58) based on their Navier-Stokes computations.

The small, inner, nearly triangular contours which are

located just above the jet exit in Figures 7 and 9 denote a region of high velocity. This region, which reaches several jet diameters into the flow, is part of the jet core that is identified in









Figure 1. The velocity field exhibits relatively slow changes within the jet core in comparison to the high velocity gradients found in the shear layer that exists at the interference between the jet and the crossflowing fluid. Distortion, caused by the crossflow, and rapid diffusion of the shear layer eventually erode the jet core.

In recent unpublished research, Krothapalli, Lourenco and

Buchlin (73) are conducting an experimental study of the separated flow region upstream of the jet near the nozzle exit. Their study has uncovered an unsteady separated flow region which is dominated by a horse-shoe vortex system near the plane of the jet exit. The downward component of the computed velocity vectors that immediately precede the jet exit and are close to the plate is attributed to the presence of such a vortex system (see Figures 6 and 8). However, the true nature of this horse-shoe vortex system is masked by the assumption of steady flow in the computation and is further tainted by the assumptions made in the thin-layer Navier-Stokes equations and the grid resolution.

The features of the computed velocity field farther from the plate agree well with the detailed symmetry plane velocity surveys by Fearn and Weston (7,8) and Fearn and Benson (32). Two lines of computed symmetry plane velocity vectors taken near the jet exit are presented in Figures 10 for the case R = 4. One rake of data is centered at (x/D, z/D) = ( 0.66, 0.96) and rotated an angle

9 = 880 from the Z-axis and the other is centered at (x/D, z/D) = (1.25, 1.67) and 9= 36'. The vector field exhibits the same general trends as the experimental measurements of Fearn









and Benson (32) which are plotted in Figure 11 although a greater upstream component of velocity is predicted for the region downstream of the jet in the computation. Jet Centerline

The determination of the trajectory of the jet plume was a primary objective for many of the experimental and theoretical studies. In this study, the determination of the jet trajectory provides a means of comparison between the calculated results and the experimental data. The jet path, called the jet centerline, is defined to be the locus of points of maximum velocity in the direction of the jet within the plane of flow symmetry, y = 0.0. Fearn and Weston (5-7) located the jet centerline in a similar manner experimentally by measuring the maximum velocity component in the jet direction. Strict attention must be paid to the centerline definition since alternative definitions may result in different jet paths. For example, in a given vertical plane, the point of maximum velocity magnitude and the point of maximum velocity in the direction of the jet do not necessarily coincide; thus, taking the jet centerline as the locus of points of maximum velocity magnitude in vertical planes is inconsistent with the present definition.

In all of the cases examined, the computed centerline

compares favorably with Fearn and Weston's experimental data and with the empirically described jet centerline (69) as demonstrated in Figures 12(a)-(c) using the data for the most refined grids for R = 4, 6 and 8 where the empirical curve is given by








bc
z =aR x
c c (19) with the empirical parameters a = 0.9772, b = 0.9113 and c = 0.3346. As the velocity ratio is increased, the jet extends further into the flow. Examination of the computed jet centerlines for R = 4, 6 and 8 that are plotted in Figure 12 illustrates the variation of the jet penetration with the velocity ratio.

Only small deviations occur between the jet paths calculated on the coarse grid (43,680 points) and the paths calculated on the fine grid (151,250 points), indicating that the jet trajectory may be captured with a relatively small number of grid points. Furthermore, for the R = 4, inviscid cases, the only detectable differences in jet trajectory from the viscous calculations are a tendency for the centerline to deflect on the order of 0.1D upstream in the near jet region prior to the characteristic downstream deflection and to penetrate less than 0.5D farther into the freestream far downstream of the jet exit, x/D > 10.0. This agrees with the experimental conclusion that pressure, rather than viscosity, is the primary mechanism governing the initial deflection of the jet (16).

The plot of the velocity vectors in the symmetry plane for the R = 4 inviscid computation in Figure 13 shows the similarity between the viscous and inviscid results despite the slip condition on the plate. For the inviscid computation, the jet tends to expand in the streamwise direction as it leaves the jet exit which explains the tendency for the jet centerline to bend forward initially. Vertical profiles of the velocity component in the









freestream direction taken in the symmetry plane for various downstream locations are plotted in Figures 14(a)-(e) for a viscous and an inviscid computation of a jet with a velocity ratio R = 4. Both cases are computed on the most refined grid. And, the figures correspond to the results presented in Figure 4. Despite the differences in the two flows close to the flat plate, the overall agreement of the velocity profiles is quite good. At each downstream location, the locations of maximum u/u are nearly identical although the magnitude of u/u is generally lower for the inviscid computation.

Decay of the Jet Velocity

As indicated by the streamwise velocity plots, the magnitude of velocity along the jet centerline decreases with increasing downstream distance. Figures 15(a) and (b) show the variation of u /u with s/D for the velocity ratios R = 4 and 6, respectively. Here, u is the velocity at the jet exit, uc denotes the velocity magnitude along the jet centerline and s is the arclength along the jet centerline which is nondimensionalized by the diameter of the jet exit, D. Originally, the rate of decay of the velocity along the jet centerline was plotted in order to provide a comparison between the present computation and the Navier-Stokes computations made by Patankar, Basu and Alpay (33). Thus, the nondimensional quantities that are plotted are chosen to correspond to the format of the computational results of Patankar, Basu and Alpay (33) who provide a comparison with the experimental data from Chassaing et al. (74).









In both cases, R = 4 and 6, there is a significant

difference between the experimental results of Chassaing et al.

(74) and Fearn and Weston (7); moreover, the results of the present calculation predict a more rapid decay of the jet centerline velocity, u c, than either of the two experiments. One difference between the two experiments is the jet exit geometry. In the experiment by Chassaing et al. (74), the jet exhausts into the freestream through a pipe while the flow measured by Fearn and Weston (7) discharges into the freestream through a hole in a flat plate. As discussed by Moussa, Trischka and Eskinazi (1), the effect of the geometrical configuration may account for the slight difference in the jet centerlines as well as the different velocity decay rates for the two jets.

In both numerical experiments, the jet is assumed to
discharge from a hole in a flat plate with a constant velocity, uj, leading to speculation that the numerical results would compare more favorably with the jet centerline velocity rates for the Fearn and Weston (7) experiment. On the contrary, the coarse grid computations (only 2,250 grid points are used in comparison to 151,250 grid points in a similar domain for the present study) of Reference 33 demonstrate excellent agreement with the measured decay rate by Chassaing et al. (74). In the present calculation, the coarse grid results (43,680 grid points) indicate a more rapid decay of the centerline velocity than the fine grid solution (see Figures 4 and 5) even though the jet trajectory is the same for both predictions. So, grid effects are expected to alter the centerline velocity decay rate.









The rate of velocity decay for the R = 4 inviscid case is also shown in Figure 15(a). Although the computed jet centerline locations for the viscous and inviscid computations are similar, the centerline velocity, Uc, in the inviscid case is shown to decrease more rapidly. As expected in the jet core (at small values of s/D), the velocity ratio remains close to unity in all cases and is not plotted.

In light of the previous observations, it may be speculated that the jet's development and final steady state properties are strongly dependent on the initial flow conditions. Despite the apparent insensitivity of the jet path to grid density and viscous effects, the centerline velocity decay rate is sensitive to both the grid density and viscous effects. Furthermore, since the vorticity distribution at the jet exit is as important as the momentum distribution, the jet exit boundary condition is also expected to alter the properties of the contrarotating vortex pair far downstream of the jet exit.


Contrarotating Vortex Pair

A pair of contrarotating vortices is identified both

experimentally and computationally as the dominant and persistent feature of the flow far downstream of the jet exit. The most natural way to examine the contrarotating vortex pair is in a cross section that is oriented perpendicular to the jet plume. Since the computed results should closely approximate the measured jet characteristics, the vortices are expected to lie relatively close









to the empirically predicted vortex curve of Fearn and Weston (5) which is located in the XZ-plane by
bc
zv =aR xv (20) where the empirical parameters are a = 0.3515, b = 1.1220 and c = 0.4293. A planar cut is made through the jet plume perpendicular to this curve as shown in the illustration in Figure 16. Then, the computed values are interpolated onto this plane, which is referenced as a vortex cross section, and resolved into the components of the local two-dimensional coordinate system.

The velocity field in a vortex cross section is displayed in Figure 17 for the R = 4 viscous computation with the location of the cross section specified by the (x,z) location of the (x vz v) origin and the rotation angle of the plane, p. The velocity field shows the freestream fluid approaching the jet plume, the fluid sheared around the jet by the crossflow and the characteristic vortex pair.

Inviscid computations for R = 4 reveal a flow that contains a vortical structure but not necessarily the characteristic contrarotating vortex pair. A velocity vector plot of the flow in a vortex cross section centered at x/D - 3.0, z/D = 2.67 and 4 = 210 is shown in Figure 18. The vectors in this cross section are located, scaled and viewed identically to the viscous results that are shown in Figure 17. The computed velocity vector profile exhibits assymmetry. The development of assymmetry in an otherwise symmetric F3D calculation has also been observed by Ying (81); and, Buning (84) notes that the upwind differencing scheme in the









freestream direction, t, can produce assymmetry in a symmetric problem.

Vortex Models

Filament Model. As discussed briefly in Chapter 2 and

reiterated here, Fearn and Weston (5) present models which use the measured velocity field taken in vortex cross sections in order to infer the properties of the contrarotating vortex pair. In the filament vortex model, the measured velocities taken in vortex cross sections but restricted to the plane of symmetry, y = 0.0, are used to calculate the strength and location of two infinite straight vortex filaments. This model is two-dimensional and gives the properties of the vortex pair at the vortex cross section. No description of the distribution of vorticity within the cross section is given. This model requires relatively few velocity measurements as inputs.

Diffuse Vortex Model. In the second model, the diffuse vortex model, the vorticity is no longer concentrated within a filament; each vortex is assumed to have a Gaussian distribution of vorticity. The measured velocities within a vortex cross section are required as inputs in order to compute the vortex location, strength and diffusivity. It is assumed that the velocity in the vortex cross section can be represented as the vector sum of the component of the freestream velocity in the plane of the cross section and the velocity induced by the contrarotating vortex pair so that









p -02r22 P22
-o 1-e 1- e U
V=2 r I r2 0g2 - Uz sin ,v(2
V -(21)


where To is the strength of a single isolated diffuse vortex, P is a diffusion constant, r1 and r2 are the distances to an arbitrary point from the two vortex centers and 6 denotes unit vectors in the directions defined in the model geometry that is sketched in Figure 19. A least squares best fit between the velocity data and the model is used to obtain the parameters that describe the vortex geometry and strength in equation (21). The model results are presented in terms of the effective vortex strength,


f= 2 0o(r 0) r dr dO
2 0
2


the effective vortex spacing,


h f f Yv (r18) rdrdo
hr _R 0Y
2


(23)


and the vortex core size, rc, which is defined to be the distance from the axis of a single Gaussian distribution of vorticity to the location of maximum tangential speed. In these definitions, 0(rl) is the distribution of vorticity for a pair of Gaussian vortices. The effective vortex strength and spacing and the vortex core size are related to the vortex model parameters by








F=erf(pho))
h0
h=o
erf(Pho)
-1
r = 1.121

where

erf (pho) =2 x- 0.5 o h e-t 2 dt (24)
(24)

is the error function. The diffuse vortex model is also twodimensional and only describes the component of vorticity that is perpendicular to the vortex cross-section. Comparison with Experiment

The diffuse vortex model is implemented for use with the velocity data from the computations in order to infer the vortex penetration, spacing, strength and core radius and to make direct comparisons with the vortex properties which were inferred for the experimental data (5). This is the first known simulation of a jet in a crossflow using the Navier-Stokes equations to provide such an extensive comparison of the vortex properties. Overall, fair agreement between the computations and the experiment is demonstrated.

The curves, drawn with a solid line, on Figures 20 and 21 are computed using the two-dimensional analytical model of Karagozian and Greber (38). The vortex penetration and separation curves are determined by fitting the model results to a power curve which has the form of equation (20). The parameters for the penetration and separation curves are found to be a - 0.480, b = 1.06, c = 0.385 and a = 0.4261, b = 0.365, c = 0.367, respectively. Although these model results for the vortex location









show reasonable agreement with the experiment and the computations, the model is unable to accurately predict the vortex strength in the region of interest, x/D < 15, since the two-dimensional assumptions are violated in the initial jet mixing region. Thus, in the Navier-Stokes simulations, effort must be placed on accurately characterizing the vortex properties in the near jet region while the farfield properties could be modelled using a simpler method.

The vortex curves for the R = 4 and 6 jets, which are

computed on the refined grid and shown in Figures 20(a) and 21(a), are within 0.3 to 0.4 jet diameters of the experimental curve. In each case, the curve for the computed data is closer to the plate than the experimental curve and shows less curvature. Further, the vortices have less separation for the computed data as indicated by the lower, flatter curves in Figures 20(b) and 21(b).

In addition to being closer together and closer to the

plate, the vortex strength for the computed flows shown in Figures 20(c) and 21(c) is nearly 0.1 r/2DU. below the vortex strength that was inferred for the experiment at x/D = 3.0 and is approximately 0.4 r/2DU greater than the experiment by x/D = 10.0. The slower rate of diffusion of vorticity for the computation is also emphasized by the slightly slower rate of growth for the core radius which is plotted in Figures 20(d) and 21(d).

For the velocity ratio R = 4, the results computed on the coarse grid as well as the fine grid are plotted on Figures 20(a)-(d). The lack of grid resolution clearly impacts the solution. For instance, the computed vortices exhibit even less









penetration into the flow; but, they are spaced farther apart and are generally weaker than predicted in the fine grid calculation and weaker than in the experiment. In short, the coarse grid solution is more diffuse than the actual flow and the amount of diffusion is reduced in the fine grid computation. Thus, the grid dependence of the solution clearly affects the properties of the contrarotating vortex pair. Further, the diffuse vortex model did not provide a good fit for the input velocity data from the R = 6 and 8 coarse grid computations which suggests that the grid effects are intensified at the higher velocity ratios.

Only speculative remarks can be made concerning the factors that cause the rate of diffusion of the contrarotating vortices to deviate from the experiment. First, the grid dependence of the solution must be a contributing factor. Next, in some instances such as the vortex spacing at downstream locations x/D > 5, the coarse grid solution appears to agree better with the experiment than with the fine grid solution. Thus, while the numerical diffusion is reduced as the mesh is refined, either turbulence is needed to adjust the diffusion rate or some of the viscous terms that are neglected in the thin-layer assumption must be included in the calculation.


Surface Pressure Distribution

The surface pressure distribution can be examined through measurements at discrete points as well as through plots of constant pressure contours. A direct comparison between the calculated and experimental pressure coefficients, Cp, on the flat









plate along the rays that are located by the angle 8,which is measured from the negative x-axis and is defined in Figure 22, is made in Figures 23 , 24 and 25 for R = 4, 6 and 8 using the solution computed on the most refined grid for each case. Three ray plots are presented in each of the figures: (a) the Cp along the upstream ray, 8 = 00, (b) the C along the ray that is perpendicular to the freestream, 0 = 900 and (c) the Cp along the downstream ray, 0 = 1800. The results along the ray 8 = -90o are not included because they are nearly identical to the 0 = 900 values. The data used in these comparisons are from Fearn and Weston (6).

Upstream of the jet exit, there is a positive pressure gradient. From the experiment, it is noted that the maximum pressure attained in front of the jet decreases as the effective velocity ratio increases. However, in the numerical solution, severe fluctuations in the pressure field exist at the interface between the jet exit and the onset flow so that the correct flow behavior cannot be identified. These spikes are magnified as the velocity ratio is increased and influence an even greater area of the flow in the coarse grid computations. Despite the reduced oscillations in the pressure field as the mesh is refined, this numerical anomaly is probably a consequence of the boundary condition discontinuity which occurs at the jet edge with the specification of a uniform velocity/constant pressure jet. Thus, to provide better agreement between the numerical results and the experiment in the region extending approximately one jet diameter










away from the jet exit, a more realistic and continuous jet profile should be investigated.

Along the ray that is perpendicular to the freestream, the largest pressure gradient occurs close to the side of the jet. Once again, the computed C near the jet edge is more negative than the experimental C . In addition, the extreme low pressure at the jet edge is balanced by an increased pressure farther away from the jet before the two C profiles exhibit good agreement. The low pressures to the side of the jet seem to correspond to the presence of the contrarotating vortices near the base of the jet.

Similar numerically induced errors occur near the jet exit along the 1800 ray. In the wake region, the initial pressures are relatively low although they are significantly higher than the pressures near the sides of the jet. The experimental data show that the minimum C increases as the velocity ratio increases indicating that the pressure recovery in the wake is more rapid at the higher velocity ratios. It is interesting that in both the computed and the experimental results, there appears to be a "glitch" or a small, slow changing pressure gradient in the profile. The calculated pressures in the wake are, in general, higher than the measured pressures. This may be the result of comparing the laminar, calculated flow with the turbulent measured flow. Since the pressure on the plate is related to the properties of the vortex pair, the wake pressure distribution is also subject to the liabilities of the computation that were identified for the prediction of the vortex properties.









Although the region near the jet exit exhibits spatial fluctuations in pressure and the wake pressures are high, the overall pressure field on the plate corresponds to the pressure field expected for a jet in a crossflow. Constant pressure contours for the computed dataset are plotted alongside the identical experimentally determined contours (6) in Figures 26(a) and 27 for the high resolution cases R = 4 and 6. Most of the physical effects that were described by the ray plots are also evident in the contour plots. In each case, there is a higher pressure region upstream of the jet exit and a lower pressure region to the sides and downstream of the jet exit. The contours show that the region of lowest pressure, that was detected along the ray at 0 = 90*, is actually centered slightly aft of 0 - 900. In each case, the contours indicate that a more rapid pressure recovery in the wake occurs in the computations. As expected, the pressure recovery in the wake region is even higher for the inviscid computation than for the viscous one while the overall plate pressure disribution is the same for both calculations (see Figure 26(b)). Finally, better agreement between the location of the computed contours and the experimental contours is demonstrated for the lower velocity ratio, R = 4 which suggests that the grid resolution and/or the turbulence may have an increased effect at the higher velocity ratio.








Flow Visualization

Streamlines

Several of the experimental studies relied heavily on flow visualization in order to provide a physical description of the flowfield. For example, Margason (15) studied a water vapor marked jet to define a jet trajectory, McMahon and Mosher (22) examined oil flow traces on the flat plate to gain insight into the complex interaction between blockage, entrainment and wake phenomena near the jet exit and the Office National D'Itudes et de Recherches Aerospatiale (O.N.E.R.A.) (19) used water tunnel flow visualization to look at flow behavior in the boundary layer and the wake region as well as in the contrarotating vortex pair. Similar flow visualization of the computational results is accomplished by constructing particle traces on a graphics workstation. For steady flow, particle traces or equivalently, three-dimensional streamlines, are computed by releasing a particle that moves with the local fluid velocity originating from selected spatial locations. It should be noted that the figures presented in this dissertation are black and white reprints of the original color flow visualization photographs. Although the following discussion describes all of the flow characteristics observed by the author, some of the interpretation of the flow features is not as clearly visible without the use of color.

Computed particle traces for the cases R = 4, 6 and 8 are

shown in Figures 28, 29(a)-(e) and 30. In each case, particles are released both upstream of the jet exit within the boundary layer at a height of 0.68* for R = 4 and 6 and 1.18* for R = 8 and from the









jet exit itself. Fluid from the jet exit, which will be referred to as jet fluid, exhausts into the flowfield and is deflected by the freestream. In studying the jet centerline, it was shown that the jet penetrates further into the flowfield before being deflected as the effective velocity ratio is increased.

In Figure 29(b), the trajectories of particles released from the jet exit in the symmetry plane are extracted from the detailed flow visualization shown in Figure 29(a) for the R = 6 jet. Although the separation of the composite particle trace facilitates visualization and discussion, similar partitioning for the other two cases provides the same qualitative information. Examination of the location of the fore and aft faces of the jet fluid shows that the jet area contracts in this plane. The fluid on the front face interacts with the freestream in a very thin interaction region or shear layer. Thus, it undergoes rapid, threedimensional, viscous erosion. In contrast, the fluid on the aft face is protected from interaction with the freestream so that fluid on the aft face follows a more vertical trajectory before being deflected. Moreover, the pressure imbalance resulting from the relatively high pressure upstream of the jet and the low pressure behind the jet contributes to the curvature of the streamlines. It is worth noting that the instantaneous streamline emanating from the origin coincides with the computed jet centerline using the present centerline definition; thus, this flow visualization suggests that the jet centerline consists of jet fluid.









Near the plate, the freestream fluid is deflected around the sides of the jet into the low pressure aft region. However, as the curvature of the jet streamlines compensates for the pressure gradient and the front face of the jet becomes more blunt from the shear, the freestream fluid begins to flow over the top of the jet rather than around it. Eventually, the jet fluid follows a path closely aligned with the freestream. Viewing the particle traces that originate in the jet exit and that are perpendicular to the freestream on the line x = 0.0, Figures 29(c) and (d), indicates the increased spread of the jet in the transverse plane. Actually, the fluid originating in the circular jet orifice is distorted by the crossflow into a deformed elliptical cross-sectional shape with the major axis of the ellipse aligned perpendicular to the freestream.

The flow patterns highlighted in Figure 29(e) are composed strictly of fluid which originated in the flat plate boundary layer. They indicate that the boundary layer fluid approaches the jet, moves around it staying relatively close to the plate and some of the fluid is swept up into the aft side of the jet plume which consists of the original jet fluid together with the entrained fluid. The distribution of this fluid in the region aft of the jet is strongly related to the velocity ratio. For the R = 4 jet (Figure 28), two loosely structured vortex cores become evident as the boundary layer fluid spirals around toward the base of the jet and into the jet plume. At the increased velocity ratios, R = 6 and 8, the structure of the vortex cores is more cohesive; the fluid particles follow a clearly defined path toward and upward









along the aft side of the jet plume. Hence, the flow patterns indicate that the vortex cores contain some entrained boundary layer fluid. The boundary layer fluid which moves through the reversed flow region and between the two vortex cores is drawn into the low pressure region in the jet plume close to the aft face of the jet. Finally, some of the deflected boundary layer fluid continues moving downstream before being caught up in the jet plume. This flow feature is emphasized in the O.N.E.R.A. (19) water tunnel flow visualization film. Entrainment

A unique feature of the computational experiment, in contrast to the physical experiment, is the ability to trace a particle path backwards in time to find where in the flowfield that particular particle originated. In this manner, the entrainment of fluid into the jet plume, and specifically into the contrarotating vortex pair, can be documented in a qualitative sense through flow visualization. Figures 31(a) and (b) show two views of the negative time history of particles released within a vortex cross section for the R = 6 jet. In Figure 31(a), the flow is viewed at an angle with the freestream flowing approximately from the left while Figure 31(b) shows the same flow viewed at an angle from nearly upstream of the jet. The square patch is centered at the jet exit, has the dimensions of one jet diameter and is aligned with the freestream. Only one of the contrarotating vortices is shown. This particular vortex section is centered at x/D = 5.0, z/D = 5.2 and rotated an angle p = 660. All of the particles shown are equally distributed within a core radius of the location of the









vortex center that is predicted by the diffuse vortex model in this plane; these parameters are xv = 5.1, yv - 1.4, zv = 4.9 and rc = 1.4. Clearly, the vortex core contains jet fluid, entrained boundary layer fluid and entrained freestream fluid. Topological Considerations

The interpretation of the flow patterns created by

continuous vector fields such as velocity, vorticity and skin friction from both experiments and computations can be aided by topological concepts. The purpose of this dissertation is not to fully investigate the topology of a jet in a crossflow; that has already been described in detail by Hunt et al. (86) and Foss (87). Rather, some basic topological classifications are introduced to characterize the visualized flow features.

A general autonomous system of ordinary differential equations is given by


dx
d- F(x,y)
dt
dy
d =G(x,y) (25)
dt (25) where F and G are functions of the variables x and y. A solution of this system, x(t), y(t), is a curve, C, or trajectory in the XY-plane, also called the phase plane, whose orientation is defined as positive in the direction of increasing t. The slope of the path C is
dy G(x,y)
dx F(x,y) (26) If G(x,y) and F(x,y) are both zero at some point, say Po' then the slope is undefined and P0 is called a critical point of the flow.









Generally, topological discussions of fluid flows focus on simple critical points. A simple critical point is a critical point that is isolated (one that lies in an arbitrarily small disk that contains no other critical point) and that has nonzero, linearly independent terms in the first order Taylor series expansions of F and G about P0 . By locating the critical point at the origin of the local coordinate system, the first order Taylor series expansions give


F(x,y) = ax+ by
G(x, y) = cx + dy (27) Then, the coupled differential equations expressed in (4) become


dx
- ax +by
dt
dy
- = cx + dy
dt (28) These equations can be uncoupled and solved as two linear equations by assuming solutions of the form (x,y) = (A,B)ekt. The two unique solutions are
(x11 Y) =(bA 1 -a)Ale 1 It
(x2. 2)=Ih, _)Ae 2 (x 2, 2 2 - a) A2e 2 (29) in which
2
a+d (a+d)
1 = -+ -(ad-bc)
1 2 4
2
1 a+ d _-ad) (ad - bc)
2 2 4 Several types of simple critical points exist, and the
values of X1 and 12 determine the flow behavior at each critical point. If X1 and 12 are both real and are both positive or both









negative, then the critical point is a node. A proper node has every path approach Po in a definite direction as t approaches either - or -0 and, given any direction, there is a path approaching Po. When X1 and 12 are negative, the direction of the vector field is towards P0 as t increases. This is an attractive node which is also called a node of separation. A node of attachment occurs when the k1 and X2 are both positive and the vector field is directed away from P0 . A further classification of nodes as well as other critical points is based on stability. If all curves that are close to Po at some instant remain close to P0 for all time, then the critical point is stable.

A critical point that is not stable is defined to be

unstable. An example of a critical point that is unstable is a saddle point. The critical point is a saddle point when 1 and X2 are both real but have opposite signs. Two lines, called the separatrixes, pass through P . The path orientation is positive along one separatrix and negative along the other. All other paths avoid the critical point.
A third type of behavior occurs when l1 and X2 are complex conjugates. Then, the critical point is a spiral node. An infinite number of paths form spirals about P0 with P0 as an asymptotic point. The trajectory spirals away from the critical point if the real part of the complex conjugates is positive or flows into the critical point if it is negative. Several examples of simple, two-dimensional critical points are sketched in Figure 32.









A primary motivation for applying topological concepts to the observed flow patterns is the relation between the number of nodes and saddle points that can exist in the flow. For the jet in a crossflow, which is topologically equivalent to the flow at the intersection of two pipes, Hunt et al. (86) show that on the surface the number of saddle points must exceed the number of nodes by one or

IN - JS = 1 (30) where N denotes a node and S denotes a saddle point. The flow in a cross-sectional plane such as the symmetry plane satisfies a different summation rule. Specifically, there must be an equal number of nodes and saddle points so that N - S = 0 (31) The application of these kinematical principles, in particular the summation rule, simplifies the interpretation of complicated flow patterns by categorizing the flow structures and removing the ambiguity that can occur when inferring flow patterns from the visualized results.

Observed Flow Patterns

Surface Shear Stress. Surface particle traces may be

constructed by restricting particles to the plane just above the flat plate; that is, the normal velocity component is set to zero during the time integration. These surface particle lines are analogous to the surface shear stress lines that are seen in oil flow surface visualization experiments. The surface particle lines represent a continuous vector field that has continuous derivatives. Further, the vector field is zero at only a finite









number of points and is located in a plane that is a small distance above the surface. Thus, the rules of topology apply.

The surface particle paths taken at a height of roughly 0.68* above the plate for R = 4 and 6 are shown in Figures 33 and 34 respectively. Results for the R = 8 jet are excluded because the first grid point in the coarse grid is located too far above the surface for the restricted particle traces to mirror the surface shear stress patterns. In both of the cases shown, fluid from upstream of the jet exit approaches the jet, deflects around the jet periphery and moves into the boundary of the wake. In the wake region, some flow is entrained forward by the jet. Behind this reversed flow, fluid moves downstream within the wake. The oil flow patterns observed by McMahon and Mosher (22), described in Chapter 2 and presented as Figure 35 for the case R - 8, exhibit similar behavior. In both the computed and experimental results, the wake broadens as the velocity ratio is increased.

The summation rule of topology for a jet in a crossflow can be satisfied by the existence of a single saddle point in the flowfield. However, for the range of effective velocity ratios studied, several critical points are apparent. A sketch of the flow structures, called the phase portrait of the flow, is shown in Figure 36. If the jet is assumed to behave like a solid body near the flat plate, then the number of nodes must equal the number of saddle points at the jet exit. Saddle points located at the solid surface are called half saddles and denoted by S'. In the phase portrait constructed from the computed surface particle traces,









there are four half-saddles and two nodes around the jet with a net attachment saddle located downstream of the jet exit so that N - S - I S'= -1 (32) which indicates that the observed flow is kinematically possible. In this flow, it is the aft saddle point that supplies forward flow to the aft half-saddle and entrainment into the jet. The two spiral nodes correspond to the low pressure region at the sides of the jet and feed flow into the contrarotating vortices. A larger system of self-cancelling nodes and saddles may exist than the one depicted by the phase portrait constructed from these computations. Although a more detailed analysis of the flow near the plate with a greater grid density is needed to unveil any secondary topological flow features, the current analysis verifies that the surface flow patterns are compatible with a continuous velocity field for a jet in a crossflow.

Symmetry Plane Flow. The flow topology in the symmetry

plane can be investigated by restricting the particle traces to the plane y = 0.0. By assuming that the jet can be adequately represented as a point jet, that is, by collapsing the jet to a single point, a very simple portrait of the flow pattern that is shown in Figure 37 for the R = 6 jet can be constructed. The phase portrait that is sketched in Figure 38 contains two half-saddles on the plate and a spiral node of separation that is located slightly above and forward of the aft half-saddle. The aft half-saddle corresponds to the attachment saddle that was observed in the computed surface shear stress pattern. Flow from the spiral node fills the region below the jet plume. Since the number of nodes





77



equals the number of saddles, the flow in the symmetry plane satisfies the summation rule in equation (31). Thus, the flow visualization is validated for at least two planes in the flowfield.






















-3.



-4.



-5.



-6.



-7.


Figure 3.


0. 500. 1000. 1500. 2000. 2500. 3000.
ITERATION



Convergence history for a nontime-accurate computation. This history is for the case R = 4 with 151,250 grid points.






















16.

14. 12. 10.


-1.0 -.5 .0 .5 1.0
u/Uinf


Figure 4.


Vertical profiles of the velocity component in the freestream direction taken in the symmetry plane at several locations downstream of the jet exit using two
grid structures. R = 4. a) x/D = 1.0.


1.5




























-.5 .0 .5 1.0
u/Uinf


-.5 .0 .5 1.0
u/Uinf


Continued; b) x/D = 2.0; c) x/D = 3.0.


16. 14. 12. 10.


-1.0


-1.0


1.5


1.5


Figure 4.




























-1.0 -.5 .0 .5 1.0
u/Uinf


-.5 .0 .5 1.0
u/Uinf


Continued; d) x/D = 4.0; e) x/D = 8.0.


1.5


1.5


-1.0


Figure 4.











































-.5 .0 .5 1.0
u/Uinf


Figure 5.


Vertical profiles of the velocity component in the freestream direction taken in the symmetry plane at several locations downstream of the jet exit using two grid structures. R = 6. a) x/D =1.0.


16.

14.

12. 10.


-1.0


1.5





























-1.0 -.5 .0 .5 1.0
u/Uinf


-1.0


-.5 .0 .5 1.0
u/Uinf


Continued; b) x/D = 2.0; c) x/D = 3.0.


16.

14. 12. 10.


1.5


1.5


Figure 5.





84


16. 14. 39 x 35 x 32 Grid
----- 55 x 55 x 50 Grid
12. 10. 8.
NJ
6.

4.

2.

0.
-1.0 -.5 .0 .5 1.0 1.5
u/Uinf

(d)




16.

14.
14. . 39 x 35 x 32Grid 12. ------ 55 x 55 x 50 Grid 10.

8.

6.




2.

0. --- 1- 1.5-------1.0 -.5 .0 .5 1.0 1.5
u/Uinf


(e)

Figure 5. Continued; d) x/D = 4.0; e) x/D = 8.0.

















































-6 -4 -2 0 2 4 6 8 10


Figure 6.


Velocity vector field in the symmetry plane for R = 4. In the region shown, x/D = [-6, 10] and z/D = [0, 16].


















































-4 -2 0 2 4 6 8 10


Figure 7.


Mach number contours in the symmetry plane for R = 4. In the region shown, x/D = [-6, 10] and z/D = [0, 16]. The contours range from 0.05 to 0.95 with an increment of 0.05. High and low Mach number regions are denoted by H and L, respectively.


OL
-6









































-6 -4


Figure 8.


87























____ _-4 - - --2 0 2 46 810 Velocity vector field in the symetry plane for R =6. In the region shown, x/D = [-6, 10] and z/D = [0, 16) .
___ ___ _-..---/ -- - 7 7 /

___.____.,,7,, V7/ / / /

- _- _ -/'i// l l I / / .../.I\\\III// /

,--,///







22 04 6 8 10

Velcit vetorfield in the symmetry plane for R = 6. In the region shown, x/D = [-6, 10] and z/D = [0, 16].

















































-6 -4 -2 0 2 4 6


8 10


Mach number contours in the symmetry plane for R = 6. In the region shown, x/D = [-6, 10] and z/D = [0, 161. The contours range from 0.05 to 0.95 with an increment of 0.05. High and low Mach number regions are denoted by H and L, respectively.


Figure 9.



















































x/D


Figure 10.


Symmetry plane velocities for R = 4. The two lines of vectors are centered at x/D = 0.67, z/D = 0.96 and S= 880 and at x/D = 1.25, z/D = 1.67 and p = 360.



















































x/D


Figure 11.


Measured symmetry plane velocities for R = 4 from Fearn and Benson (32). The two rakes of vectors are centered at x/D = 0.67, z/D = 0.96 and p = 880 and at x/D = 1.25, z/D = 1.67 and P = 360.




Full Text

PAGE 1

APPLICATION OF A THREE-DIMENSIONAL NAVIER-STOKES MODEL FOR A SUBSONIC JET IN A CROSSFLOW BY KARLIN REN6e ROTH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL ' THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 irXV F LfBRARlES

PAGE 2

ACKNOWLEDGMENTS Without the continued guidance, encouragement and support of my advisor. Dr. Richard L. Fearn, this work would not have been possible. His suggestions gave me new ideas and new approaches which enabled me to understand the physical problem and to relate the computed results to the experimental data. I am especially grateful for the advantages that he gave me by arranging for me to complete my research at the NASA Ames Research Center. Special thanks are extended to Mr. Richard J. Margason and to all the members of the Fixed Wing Aerodynamics Branch at the Ames Research Center for the opportunity to work with them and for their assistance during the past three years. The computer facilities that they made available to me were essential for calculating the flow and for visualizing the computed flowfield. Appreciation is extended to Dr. James Ross and to Dr. William Van Dalsem for their many helpful discussions on both aerodynamics and conputational fluid dynamics. I also want to recognize the efforts of Mr. Siddharth Thakur who performed some of the data analysis required for making comparisons between the vortex properties for the computation and the experiment. This project was supported by NASA Grant NCC 2-403. ii

PAGE 3

TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF SYMBOLS v ABSTRACT ix CHAPTERS 1 INTRODUCTION 1 2 BACKGROUND 5 Experimental Studies 5 The Jet Plioine 6 Conditions on the Flat Plate 11 Near-Field Measurements 13 Prediction Techniques 13 Analytical, Empirical and Potential Flow Models 14 Navier-Stokes Methods 17 3 NUMERICAL APPROACH 22 Thin-Layer Assumption 22 Finite Difference Equations 23 Time Differencing 27 Linearization of Nonlinear Terms 28 Approximate Factorization 29 Flux Splitting 31 Spatial Differencing 32 Numerical Dissipation 34 Code Implementation 35 Computational Grid 36 Boundary Conditions 40 4 RESULTS AND DISCUSSION 44 Grid Dependence 48 Flow in the Symmetry Plane 51 Jet Centerline 53 Decay of the Jet Velocity 55 Contrarotating Vortex Pair 57 Vortex Models 59 Comparison with Experiment 61 Surface Pressure Distribution 63 iii

PAGE 4

Flow Visualization 67 Streamlines 67 Entrainment "^0 Topological Considerations 71 Observed Flow Patterns "74 5 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY 129 APPENDIX 136 REFERENCES 138 BIOGRAPHICAL SKETCH 145 iv

PAGE 5

LIST OF SYMBOLS A,B Arbitrary constants A,B,C,M Flux Jacobians resulting from the local linearization of the fluxes about the previous time level a,b, c Constants in the equation for the empirical vortex curve a,b,c,d Coefficients for first order Taylor series expansion of F and G about c Local speed of sound, (yp/p)^-^ C Curve or path in the phase plane C Pressure coefficient, {p~Poo)/*3oo P D Jet diameter D^,D^ Niomerical dissipation operators for the central space differenced directions e Total energy # Unit vector E Splitting error for the approximate factorization f Arbitrary function F,G Arbitrary functions F,G,H,Q Flux vectors for the thin-layer Navier-Stokes equations h Constant which has the value At or At/2 in the F3D difference algorithm h Effective vortex spacing I Identity matrix J Jacobian of the coordinate transformation L Differential operator M Mach number N Node

PAGE 6

0 Order operator p Fluid pressure p Critical point o q Fluid dynamic pressure Q Vector containing the fluid density, three components of momentum and total energy in F3D Pr Prandtl number r Radius of jet exit r Distance to an arbitrary point from a vortex center R Velocity ratio, M^/M^ r Vortex core radius c Re Reynolds n\amber based on freestream velocity and jet diameter, U^D/v s Arclength S Viscous flux vector for the thin-layer Navier-Stokes equations S Saddle point Half-saddle t Time T Similarity transformation natrix U Velocity U,V,W Contravarient velocity components u,v,w Cartesian velocity components V Velocity vector x,y,z Cartesian coordinates P Parameter in the numerical dissipation based on the fluid pressure p Diffusion constant 3 Differential vi

PAGE 7

5 Finite difference operator §* Boundary layer displacement thickness A Denotes an incremental change in a variable, often a first order forward difference At Time step £^,6^ Constants in the numerical dissipation operator e,(j),9 Arbtrary constants in the time differencing formula Y Ratio of specific heat r Vortex strength Strength of an isolated diffuse vortex (p Rotation angle for a cross-section perpendicular to the vortex curve K Coefficient of thermal conductivity X Eigenvalues of the flux Jacobian, A X Parameter in phase plane solution A Diagonal matrix containing the eigenvalues of the flux Jacobian, A \l Fluid viscosity 6 Angle defined for the diffuse vortex model 9 Angle measured in degrees from the negative x-axis V Fluid kinematic viscosity ^,ri,^ Generalized coordinates Jt Pi p Fluid density T Time variable in generalized coordinate system vii

PAGE 8

± Vorticity distribution for a pair of Gaussian oo mnni uy OUDSCj-ipX-o C ^ 1^ 1 1 ^ JJ1^ -i -i V Qr^or'i'Pw a 1 Hpal" ion in tllG ^ 1— ^ ^ Jy ct ^ w wo ^ A A computational plane j rnax min o V 1,2,3 Denote unique values of a variable oo Freestream condition Superscripts b Backward difference f Forward difference n Time level +,Denote variables associated with positive or negative eigenvalues after flux splitting viii

PAGE 9

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 APPLICATION OF A THREE-DIMENSIONAL NAVIER-STOKES MODEL FOR A SUBSONIC JET IN A CROSSFLOW By Karlin Ren6e Roth December, 1988 Chairman: Richard L. Fearn Major Department: Aerospace Engineering, Mechanics and Engineering Science The aerodynamic/propulsive interaction between a subsonic jet exhausting perpendicularly through a flat plate into a crossflow is investigated numerically. An approximately factored, partially flux split, implicit solver for the three-dimensional thin-layer Navier-Stokes equations is used for the computations. This algorithm is applied to flows with a range of jet to crossflow velocity ratios between 4 and 8 . Both qualitative and quantitative agreement between the computed results and the existing experimental database are demonstrated. Qualitatively, all of the pertinent flow features, including the jet trajectory, the contrarotating vortex pair and the wake region near the flat plate, are captured numerically. Flow visualization of the computed results on a graphics workstation is instructive for understanding the coitplex ix

PAGE 10

1 interaction between the jet and the crossflowing fluid in the region near the jet exit. The computed jet centerlines deviate less than one jet diameter from the empirically predicted and experimentally located centerlines for all cases tested, including an inviscid computation. This indicates that the location of the jet centerline is relatively insensitive to grid structure and is determined primarily by pressure effects rather than viscous effects . The computed velocity field is analyzed so that a direct comparison can be made with the experimentally determined properties of the contrarotating vortex pair. The upwash velocities in vortex cross sections are used in a vortex model to infer the vortex strength, location and core radius. Good agreement between the vortex properties calculated for the simulation and the experiment is demonstrated for the most refined computations . X

PAGE 11

CHAPTER 1 INTRODUCTION During the past 50 years, a considerable number of theoretical, experimental and computational investigations have been directed toward the flowfield induced by the injection of a turbulent jet of fluid into a crossflow. Perhaps the simplest illustration of this flow phenomenon is the emission from a chimney on a windy day (1) . In gas turbine combustors, the mixing of jets in a crossflow plays a dominant role in determining the temperature profile of gas leaving the combustor, and hence, the performance and durability of the turbine (2) . Yet another example of a jet in a crossflow is given by the discharge of effluent into a waterway (3) . The present study is motivated by the aerodynamic/propulsive interaction problem associated with Short Take-Off and Vertical Landing (STOVL) aircraft. In the transition between hover and wingborne flight, STOVL aircraft rely on the direct thrust of lift jets to supplement the aerodynamic lift. As a result, a relatively high velocity jet is injected into a crosswind created by the forward motion of the aircraft during transition. The interaction of the lifting jets with the flow over the aerodynamic lifting surface produces a complex flowfield around the aircraft. This interaction usually results in a loss of lift and an increase in nose-up pitching moment with increasing forward flight velocity when compared with 1

PAGE 12

2 the results of a linear model for the lift generated by the forward flight of the aircraft added to the direct lift of the jets without interference (4) . Thus, the aerodynamic/propulsive interference effect poses a significant problem for STOVL aerodynamics. Traditionally, the development of STOVL aircraft has been guided primarily by experimental investigations. In order to reduce the time and expense of the design and testing of these vehicles, researchers have developed prediction techniques for the aerodynamic/propulsive flowfield that are based either on potential flow methods with empirical adjustments to account for viscous effects or on purely empirical correlations. Although many of these techniques are relatively simple to implement, in general, they have only limited applicability for practical propulsion systems. Empirical predictions are unable to account for the severe nonunif ormities in the velocity and temperature fields encountered by the lifting jets. Further, the empirical prediction may be limited to use with the experimental database from which it was derived. Thus, improved prediction techniques are needed to guide future STOVL development . With the recent advances in the development of computational fluid dynamics and computing resources, a more promising approach for the powered lift problem is to use numerical algorithms based on the Navier-Stokes equations to simulate the performance of the propulsion systems . These algorithms are also subject to some limitations. First, the proper flow physics must be adequately represented by the governing equations solved by the model. In many codes, the large number of modelling assijmptions removes some

PAGE 13

of the pertinent physics. Next, the numerical accuracy of these codes cannot be neglected. Finally, the numerical code must be fully verified by comparison with experimental data. Ultimately, the flow about an arbitrary STOVL aircraft throughout its entire flight envelope would be analyzed; however, with current computer resources, a detailed simulation of the flow about a STOVL aircraft and its propulsion system is not possible. Instead, a simpler flowfield which isolates the physics of the jet/freestream interaction region is studied. The problem is reduced to the case of a subsonic round jet exhausting perpendicularly through a large flat plate into a subsonic crossflow. This configuration retains the essential physics of the aerodynamic/propulsive interaction problem without the added complexity of the aircraft geometry, engine inlet flows, ground effects and multiple jets. The present study uses contenporary conputational fluid dynamics to develop and validate a computational model for a jet in a crossflow. Several investigators, for example Fearn and Weston (5-8), Schetz et al. (9) and Snyder and Orloff (10), have experimentally studied the subsonic jet in a crossflow using this simplified configuration, providing an extensive database for comparison with the computed results. Further, the experimental results reveal that the important features of the flow that must be simulated are the jet trajectory, the contrarotating vortex pair and the wake region near the plate downstream of the jet orifice. Some of the key progress toward the understanding of the flow phenomena associated with a jet in a crossflow is highlighted

PAGE 14

4 in Chapter 2. Although only example results are emphasized, an extensive list of references (1-74) is identified. In Chapter 3, a description of the numerical approach, including the computer code, the boundary conditions and the computational grid, is given. Numerical results are presented and, when available, compared with experimental measurements in Chapter 4. The calculated results are for flows with jet to crossflow velocity ratios of 4, 6 and 8. Quantitative comparisons are made between the calculated and measured data for the jet trajectory, the pressure distribution on the plate and the properties of the contrarotating vortex pair. Flow visualization of the computed results on a graphics workstation is used to gain insight into this complex threedimensional flowfield. Finally, in Chapter 5, major conclusions and recommendations for future study are stated.

PAGE 15

CHAPTER 2 BACKGROUND In this chapter, the historical development of experimental, analytical and computational methods for powered lift configurations is reviewed. Emphasis is placed on the literature concerning the simplified subsonic jet/flat plate model since that is the configuration to be studied in this investigation and since the flow physics of the STOVL jets in transition are retained and isolated in this model. Further, this review focuses on investigations of subsonic round jets with velocity ratios between 2 and 10. A description of the jet in a crossflow phenomenon, based on the experimental studies, is developed. The acciamulation of data during the past several decades for the aerodynamic /propulsive interaction problem has positively influenced the design procedures for STOVL aircraft. However, Spee (11) points out that the design methods used for powered lift aerodynamics are primitive in comparison to the prediction techniques available for conventional aircraft . Ex perimental Studies The evolution of STOVL aircraft has been guided primarily through experimental investigations (see References 2,5-10,12-32). A review of the experimental work completed prior to 1979 which is presented in tabular form by Crabb, Durao and Whitelaw (12) 5

PAGE 16

6 indicates that although a wide range of parameters and configurations have been studied, there is still insufficient information to completely describe the flowfield induced by the jet, especially in the initial mixing region. For instance, the velocity and pressure profiles at the jet orifice with the imposed crossflow remain unknown. Furthermore, these profiles are required as input boundary conditions for many calculations and the assiamption of uniform exit conditions made, for example, by Patankar, Basu and Alpay (33) needs quantitative verification. In the following sections, the contributions made by the various experimental investigations are discussed. The Jet Plume One topic of considerable interest for the jet interaction problem concerns the behavior of the jet plume itself. Several characteristics of the jet induced flowfield have already been documented either qualitatively, through flow visualization, or quantitatively, through pressure measurements. It has been established that a jet exhausting perpendicularly into a crossflow deflects, increases in lateral extent, distorts in cross-sectional shape and evolves into a flowfield that is dominated by a pair of contrarotating vortices. A schematic diagram in Figure 1 sets forth the various experimentally defined regions of the jet plume together with the associated wake region (7) . Jet tra jectory. In order to address such problems as jet induced velocities and flow angularity near an aircraft in transitional flight, it is necessary to characterize the trajectory of the jet plume. The jet path for a circular jet has been

PAGE 17

7 examined extensively (see References 2,13-17). Early experiments by Callaghan and Rugger i (13) and Ruggeri, Callaghan and Bowden (14) suggested that the velocity ratio, Uj/Uoo, is an inportant parameter in determining the penetration of the jet. Later experiments have also explored the effects of varying the jet injection angle (15) and the jet exit shape (18) on the jet penetration. Some of the first quantitative measurements were taken by Jordinson (16) in 1956 in order to determine the path of the jet and the shape of a jet cross section. Pressure measurements were obtained in four planes aligned normal to the freestream and downstream of the jet orifice and in two planes inclined at 5° and 25° to the plate. Based on static pressure measurements, Jordinson concluded that the initial deflection of the jet is mainly a consequence of the pressure difference across the jet, but that within a few jet diameters of the orifice, the entrainment of crossflow fluid into the jet pl\m\e becomes the predominant bending mechanism. He further concluded that the mixing of jet and crossflow fluid reduces the vertical momentum at the edges of the jet causing the mixing region to deflect faster than the jet fluid that is initially unaffected by mixing and accounting for the distortion of the jet cross-sectional shape. In 1968, Margason (15) used photographs of a water vapor marked jet to determine the path of a single jet exiting at large angles to the freestream for a range of velocity ratios. This study showed that the primary variables governing the jet path for a particular nozzle are the injection angle and the ratio of the

PAGE 18

8 momentum flux across the jet orifice to the momentum flux of the freestream over an equal area. For a subsonic, unheated jet, this momentiam ratio reduces to the ratio of the jet velocity to the freestream velocity. Throughout this dissertation, the term "velocity ratio" is used. Additionally, Margason's results were essentially the same for jets exhausting upward or downward and for jets near or far from adjacent surfaces. An empirical equation for the jet centerline based on the primary variables demonstrated a good fit to the photographed jet paths. Later, Kamotani and Greber (2) studied a heated jet. They determined the jet path from maximum velocity measurements in the symmetry plane. Based on temperature measurements in the symmetry plane, they also determined that the temperature centerline does not coincide with the velocity centerline; it penetrates less and depends on the jet to crossflow density ratio, Pj/p^, in addition to the velocity ratio. Fearn and Weston (5,6,8) measured the jet centerline by placing a rake of velocity measuring probes in the symmetry plane perpendicular to an estimated jet centerline location which was deduced from the previous experiments (15) . The magnitude and location of the maximum axial component of velocity at each rake location was then determined by curve fitting the measured velocity values. The locus of points of these maxima is the centerline. Experimentally, this curve was identified up to about 15 jet diameters downstream of the jet exit. The jet centerline determination was used to supplement Fearn and Weston's

PAGE 19

9 investigation of the vortex pair and to provide a means of comparison with other experiments. Contrarotating vortex pair. While early investigations of the jet plume properties focused on locating a jet trajectory, later investigations were directed toward locating and describing the vortex pair (2,4,5,17,19-21). Flow visualization of the vortex pair was used in some studies, for example the results presented in a film by the Office National D' Etudes et de Recherches Aerospatiale (19) . Most of the quantitative studies inferred the strength of the vortices from velocity measurements. Margason and Fearn (4) and Antani (20) attempted to measure the vortex properties directly by using vortex meters . The contrarotating vortex pair is the dominant and persistent feature of a jet in a crossflow. The contrarotating vortices are formed close to the jet exit and can be detected far downstream of where the jet centerline may be measured. Specifically, these vortices were detected by Fearn and Weston (5) at 45 jet diameters downstream of the jet orifice. As the vortices propagate downstream, their strength decreases due to the diffusion of vorticity across the plane of flow symmetry. An alternative method for describing the jet path is to define a vortex curve to be the projection onto the symmetry plane of the location of the center of either vortex. An empirical equation for the vortex curve is presented by Fearn and Weston (5) . The vortex curve does not penetrate as far into the flowfield as the jet centerline. Experimentally, the vortex trajectory can be located by examining the upwash velocity component in the region 1 i

PAGE 20

10 between the two vortices. The upwash velocity is a maximum in the symmetry plane along the line joining the two vortex centers. The vector vorticity field associated with the jet could be accurately characterized by taking finely spaced velocity measurements; however, the number of measurements that would be required is, in general, prohibitive. As a result, models have been constructed to deduce the vortex properties from the velocity field. For example, Thompson (17) calculated the variation in the vortex strength by measuring the velocity field in cross-sections perpendicular to the vortex path and then using two models to predict the vortex strength. In one model, the vortices are assumed to be discrete and quasi two-dimensional; the measured circumferential velocity distribution for a single traverse through the vortex center is used for the calculation of circulation. In the second model, the vortex strength is computed by contour integration of the tangential component of velocity around circuits of increasing size enclosing the vortex centers. Later, two models which are capable of predicting the location and diffusivity of the contrarotating vortices as well as their strength were developed by Fearn and Weston (15) . The vortex filament model replaces the vortices with two infinite line filaments. Their strength and location are calculated from the measured upwash velocities along the axis between the vortices in the symmetry plane. Another method, the diffuse vortex model, assumes that each vortex has a Gaussian distribution of vorticity rather than being concentrated in a filament. Measured upwash velocities in the plane that is normal to the vortex path are

PAGE 21

11 utilized in the model to infer the core radius of the vortices in addition to the vortex strength and spacing. Conditions on the Flat Plate In order to understand the lift losses that are caused by the jet interference, it is necessary to examine the conditions on the flat plate. Several experimental studies (References 6,17,19,22-29) have been directed toward this aspect of the jet flowfield. Useful information is obtained through both oil flow visualization and static pressure measurements. Oil flow visualization on the surface of the plate indicates that the freestream flow decelerates upstream of the jet exit due to the presence of the jet and then accelerates around the jet (19,22,23). Some of these surface shear stress lines bend in toward the jet and terminate at the edge of the orifice. Other streamlines are deflected around the jet periphery and into the boundary of wake region downstream of the jet exit . Within the wake region, flow patterns indicate that some fluid is entrained forward by the jet . Directly behind this reverse flow region, wake flow moves away from the jet and into the wake boundary. The wake boundary may consist of a horseshoe vortex which is formed around the jet core. Wu, Mosher and Wright (23) noted that the photographed oil flow patterns are dependent only on the velocity ratio and that the wake region broadens with increasing velocity ratio. They also emphasized that the oil flow patterns do not necessarily describe flow above the plate boundary layer. Static pressure measurements quantify the flow visualization results. For example, investigators (References 22-29) observe a

PAGE 22

12 positive pressure region upstream of the jet exit and a large, low pressure region to the sides of the jet and behind the jet . The jet interferes with the freestream flow by blocking it, entraining it and causing a separated wake to form. Vogler (24) compared the pressure distributions for a circular jet and an equivalent diameter circular rod to find that the low pressure wake region is more extensive for the jet which indicates that entrainment as well as the blockage effect is important. Bradbury and Wood (25) suggested that the low pressure in the wake is due to boundary layer fluid being drawn up into the jet. But, the wake region extends several jet diameters above the plate, which led Wu, Mosher and Wright (23) to suggest that the freestream also separates. Relatively high pressure gradients are observed in the low pressure regions to the side and slightly downstream of the jet. These gradients may be due to the presence of the contrarotating vortices near the jet exit . The experimental measurements, for example Fearn and Weston (6), indicate that the velocity ratio is the dominant parameter governing characteristics of the plate pressure distribution. It was noted previously that this dimensionless parameter is the primary factor required to determine the jet trajectory. Two other dimensionless parameters, the ratio of the momentiom thickness to the jet diameter and the jet Reynolds number, weakly influence the pressure distribution (25); however, their influence is detailed rather than gross .

PAGE 23

13 Near-Field Measurements In general, the measurements in the jet plume that have been discussed up to this point can be categorized as far-field measurements; that is, wind tunnel instrumentation limitations inhibited accurate measurement in the region relatively near the jet exit which is characterized by high velocity gradients and regions of backflow. A recent solution to the instrumentation problem is to use a laser Doppler anemometer (LDA) to make flow measurements. The laser Doppler system overcomes the problems of flow disturbance, probe size and directional sensitivity which are encountered by earlier instrxmientation techniques. The first attempts to use a LDA to measure the velocity field associated with a jet in a crossflow (30,31) were not entirely successful; however, they provided insight into the optimization of the instrument . Perhaps the most encouraging results for the near-field of the jet are those of Snyder and Orloff (10) . A three-dimensional LDA was employed to gather data in the NASA 7by 10-foot Wind Tunnel Facility at Ames Research Center. Their results are for a jet with a velocity ratio of 8. Further refinements of this technique should be capable of providing an adequate description of the distortion and diffusion of the shear layer in the region near the jet orifice. Prediction Techniques Although a large experimental database has been acquired, the number of variables and options and, thus, the number of

PAGE 24

14 possible geometric configurations and design parameters is so large that an experimental database containing an adequate amount of design information is difficult, if not impossible, to accumulate. To augment the experimental database numerous theoretical prediction tools have been developed. The current state of the art prediction methods are in general based on potential flow methods which require empirical input to correct for the viscous flow characteristics. Moreover, most of these methods apply only to a restricted region of the flow and adequate verification is lacking. More recently, emphasis is being placed on the development of solutions based on the Navier-Stokes equations to simulate the entire jet/f reestream interaction. Several representative models are discussed in the following sections. Analytical. Empirical and Potential Flow Models Perhaps the earliest attempt to analyze the characteristics of a jet in a crossflow can be attributed to Chang (34) who was interested in the discharge of effluent from a pipe into a stream. She uses potential flow theory together with the concept of bound vortex filaments to calculate the shape of the separation boundary between a perpendicular cylindrical jet and a crossflowing fluid. A hand-calculated numerical solution demonstrates the horseshoe shaped cross-sectional boundary that occurs when the jet wake rolls up into a vortex pair. A model which accounts for the interference effects of the jet by distributing sinks and doublets along the jet centerline is presented by Wooler (35) . First, the jet centerline is determined through an empirical model which represents the deflection

PAGE 25

15 mechanism by a pressure force and mainstream entrainment. Then, a doublet distribution is placed along the jet centerline to represent the blockage effect of the jet. The strengths of the doublets are related through experimental observations of the deformation of the initially circular jet into a jet with elliptic type cross-sections. The strengths of the sinks, which are distributed at discrete locations on axes perpendicular to the jet centerline, are proportional to the mass of air entrained by the jet . Calculations of the jet induced velocity field show good correlation with test data (36) . Alternatively, Heltsley and Parker (37) used vortex singularities in their vortex lattice model for the jet. Once again, the initial geometry is based on experimental data. Further, a semiempirical computer program was used to simultaneously solve for all the vortex strengths as a consequence of the applied boundary conditions and the geometry. In either the sink-doublet formulation or the vortex singularity formulation, the effects of blockage and entrainment outside the jet and on the flat surface are approximated; but, the flow inside the jet is ignored. Since the contrarotating vortex pair dominates the jet flowfield, several models have been constructed to emulate this feature. The two empirical vortex models of Thompson (17) and Fearn and Weston (5) were already described. A two-dimensional analytical study (38) focuses on modelling the contrarotating vortex pair without incorporating empirical data. Equations governing the vortex spacing and the downstream jet velocity are formulated by approximating the forces acting on each of the

PAGE 26

16 viscous vortices . Numerical solution of the governing equations yields reasonable results for the vortex trajectory, half-spacing and viscous core size. The results indicate that the vortices are formed relatively close to the jet exit and the two-dimensional analysis is inappropriate in this region. Panel methods, which are based on potential flow theory, are commonly used to simulate the aerodynamics of conventional aircraft. Consequently, their application to powered lift configurations has been investigated (References 39-44) . In panel methods, distributions of source and doiiblet singularities are placed on the boundary surfaces and approximated numerically by networks of quadrilateral panels which each have an arbitrary singularity strength. With the application of proper boundary conditions, a system of linear algebraic equations which can be solved for the singularity strengths is produced. Then, the entire velocity or pressure field can be computed. The assumptions that are inherent in panel methods are that the outer flow is incompressible and irrotational, boundary layer effects are neglected and separated flow is not simulated. Further, some a priori knowledge of parameters such as the jet trajectory and the distribution of entrainment is required. Current trends are to use a panel method for the aircraft body and to couple it with a Navier-Stokes type solver for the jet. Howell (44) used the PANAIR panel code with a Neumann boundary condition to represent the aircraft coupled with a parabolic Navier-Stokes solver with an entrainment boundary condition for the jet. The jet shape and the entrainment were calculated using the J

PAGE 27

17 Adler-Baron (45) method which is a semi-empirical, integral model that can be numerically solved for the internal jet flowfield. The results from this parabolic analysis are unable to account for the separated wake in a siobsonic flowfield. Consequently, the computed lift losses are underestimated by about 20% of the thrust for the simple high wing aircraft with a single lift jet that was tested. Navier-Stokes Methods A recent review by Hancock (46) stresses that the flow behavior associated with a jet in a crossflow is dominated by viscous effects. While inviscid analyses describe the general trends within the flow, they are not accurate quantitative results nor are they directly comparable to experiments. In addition, each of the techniques described above, whether viscous or inviscid, yields global rather than detailed characteristics of the flowfield. Recent solutions of the Reynolds averaged Navier-Stokes equations for a variety of propulsive flowfield applications including jet-in-ground-ef fects (47), internal jets (48) and upwash fountain effects (49,50) as well as a jet in a crossflow (33,51-61) offer a more promising approach. A parabolic Navier-Stokes formulation by Baker (51-55) utilizing a k-e turbulence model and an algebraic Reynolds stress model exhibits the expected kidney-shaped jet cross-section, a transverse vortex pair that propagates downstream and a jet trajectory that is displaced slightly upstream prior to the characteristic downstream deflection and predicts the evolution of the farfield entrainment velocity distributions. This method relies on the proper formulation of initial and farfield boundary

PAGE 28

18 conditions in order to be marched in space, away from the plate. The wake induced flow separation effects are not adequately captured by the parabolic Navier-Stokes approach. Initial elliptic, three-dimensional, Navier-Stokes solutions show improved results. Patankar, Basu and Alpay (33) employ a finite difference formulation with the three velocity components and pressure as the dependent variables coupled with a two equation turbulence model . The computations are performed in a rectangular domain about the jet with the assumption of flow symmetry imposed. Their relatively low-resolution results (a maximum grid density of 2250 points, assuming symmetry, was tested) for a three-dimensional turbulent jet demonstrate good agreement for the jet centerline and the rate of decay of the jet velocity but deviate significantly from the experimental velocity profiles along the jet axis after the jet deflects and in the reversed flow region. Other detailed flowfield results are not presented. Other relatively coarse results are presented by Chien and Schetz (56) who formulate the steady, inconpressible Navier-Stokes equations in terms of vorticity, velocity and tenperature and by Demuren (57) who calculates the flow induced by a row of jets using the steady, incompressible Navier-Stokes equations with the k-e turbulence model. Notably, in the simulation by Demuren, the jet exit boundary condition is based on uniform total pressure rather than uniform vertical velocity and for a low velocity ratio, in this case R = 1.96, is similar to the jet/pipe profile measured by Andreopoulos (62) . The effects of niamerical diffusion are also investigated by Demuren.

PAGE 29

19 Most recently, several higher-resolution computations, each employing on the order of 100,000 grid points to discretize the three-dimensional domain, have been reported. A detailed picture of the jet plume and the vorticity dynamics for a turbulent jet is given by Sykes, Lewellen and Parker (58) . In their calculation, the incompressible Navier-Stokes equations are solved implicitly and the turbulence lengthscale is based on the configuration geometry, namely, the jet diameter and the distance from the center of the jet. However, since the primary motivation is the behavior of a jet remote from a wall, they do not attempt to resolve the wake region. Indeed, the lower surface is treated as a stress-free wall. Both Reed (59) and Harloff and Lytle (61) resolve the flow near the plate. In each of these studies as well as in most of the Navier-Stokes computations mentioned previously, a stretched rectangular grid is used to discretize the physical domain. Reed err^loys an explicit algorithm based on the three-dimensional Navier-Stokes equations; but, after 10,000 iterations, the solution appears to be unconverged. A significant deviation exists • between the jet trajectory for the computed flow and a similar experiment for flows with velocity ratios R = 4 and 8. It is concluded that replacing the explicit differencing scheme with a time accurate implicit method would allow a steady state solution to be obtained in a reasonable number of time steps. An implicit algorithm which also incorporates all of the three-dimensional, viscous Navier-Stokes terms is applied to both subsonic and supersonic jet flows by Harloff and Lytle (61) . The

PAGE 30

20 s\absonic computation is steady state, laminar and symmetry assumptions are applied to the computational domain. Agreement with experiment is shown for the jet centerline. The vortex curve is located by examining the upwash velocity components in the symmetry plane. This computed vortex curve is shown to be up to 2 jet diameters below the vortex locations measured by Fearn and Weston (5) for similar jet conditions. The authors speculate that turbulence, which is excluded from the simulation, is responsible for the underprediction of the vortex curve. Notably, preliminary results for a subsonic jet injected perpendicular to the freestream with a velocity ratio R = 8 indicate the presence of a secondary vortex structure in the wake region of the jet .

PAGE 31

21 e 3

PAGE 32

CHAPTER 3 NUMERICAL APPROACH Thin-Layer Assumption The flowfield induced by a jet in a crossflow is characterized by large pressure variations, a pair of diffuse contrarotating vortices, vortex shedding on the lee side of the jet and a three-dimensional, turbulent wake containing both separated and reversed flow. A corrputational method based on the threedimensional, time-dependent Navier-Stokes equations would be capable of providing a total mathematical description of the flow, permitting investigation of all flow parameters. However, methods based on these equations currently have little application to practical aircraft configurations as a consequence of available computer memory which restricts gridding in the computational domain. In order to develop a numerical tool that may potentially impact STOVL design while capturing the pertinent flow features in a reasonable amount of time, several simplifying assumptions concerning the nature of the governing equations must be made. In the numerical simulation of high Reynolds niomber flows, the thin-layer Navier-Stokes equations are often used. The thinlayer approximation is obtained by using order of magnitude arguments similar to those employed in boundary layer theory (75) to drop all viscous terms that contain derivatives that are parallel to the body surface. However, unlike boundary layer 22

PAGE 33

23 theory, the thin-layer approximation retains all three momentum equations and no limitation is put on the pressure field. Thus, the thin-layer equations permit the calculation of separated and reversed flows as well as flows with large normal pressure gradients. Further, this set of equations is valid in both inviscid and viscous flow regimes. An assessment of what is actually computed when the NavierStokes equations are applied to high Reynolds number flows provides a basis for making the thin-layer approximation (76) . In order to resolve the gradients normal to the boundary layer, grid points must be highly clustered within the boundary layer. Consequently, a large amount of time is spent resolving these gradients . Then, computer resource limitations preclude the adequate resolution of gradients parallel to the surface. Since in many applications the gradients that are parallel to the surface are relatively small and are not completely resolved, the thin-layer approximation is made to exclude them from the calculation altogether. One goal of this study is to apply existing technology, with modifications as needed, and to assess the validity of this technology for a propulsive flow problem. Several computer codes incorporating the thin-layer ass\imption were immediately available at the initiation of this study. The merits of this approach, in comparison with full Navier-Stokes methods, are that it reduces the effort required to calculate the viscous terms and requires a highly clustered grid in only one direction. The thin-layer approach is also superior to parabolized Navier-Stokes methods, which were also immediately available, in that the separated.

PAGE 34

24 reversed wake flow may be treated and flow along the plate is calculated rather than input as a starting solution. The application of the thin-layer assumption to the jet in a crossflow problem must be viewed with caution since the flowfield is actually viscous in all three directions. While the high gradients perpendicular to the plate within the boundary layer and in the wake may be captured, large streamwise gradients are anticipated within the wake and are excluded from the computation. Large streamwise and lateral gradients are also anticipated and neglected in the shear layer between the jet and the crossflowing fluid in the region near the jet exit (see Figure 1) . The neglect of viscous terms in the near jet region may affect the initial distribution of vorticity and ultimately alter both the farfield properties of the contrarotating vortex pair and the characteristics of the horseshoe vortex. Nevertheless, the thinlayer approximation is applied and tested in this study. The thin-layer Navier-Stokes equations can be written in conservation law form as follows: ax an a; " ac where the flux vectors Q, F, G and H and the viscous fluxes, S, are given in Appendix A. The generalized coordinate variables ^, Tj, ^ and T are functions of the Cartesian coordinates x, y, z and time, t. Boundary layer type coordinates must be used to resolve the viscous terms normal to the surface; in this case, the grid is

PAGE 35

25 clustered in the ^direction to resolve the boundary layer near the plate. Finite Difference Equations In any finite difference method, the numerical solution is obtained at grid points and at time levels by using a set of algebraic equations known as finite difference equations (FDE) which are derived from the governing equations. The numerical method which is used in this jet in a crossflow simulation solves the Reynolds averaged, thin-layer, Navier-Stokes equations cast in generalized coordinates which are given in equation (1) . The FDEs are derived from the governing equations by using an inplicit, factored method. In contrast to explicit methods which rely only on information that is known from previous time levels to advance the solution to the next time level, implicit algorithms utilize information from adjacent points in the flowfield at the same time level as the point under consideration. Consequently, implicit methods require the simultaneous solution of sets of linear equations and costly matrix inversions. However, in comparison to explicit methods in which niimerical stability restricts the maximum allowable step size, implicit methods permit the use of a larger step size and expedite code convergence. The finite difference method employed in this study is contained in the F3D code (77,78). The derivation of the FDEs for the implicit, approximately factored, partially flux-split algorithm can be summarized as:

PAGE 36

26 1. Transform the governing equations, in this case the thinlayer Navier-Stokes equations, from the physical, (x,y,2,t) coordinate system to the computational space, (^'HfC/'t) • This is achieved by by replacing the derivatives with respect to (x,y,z,t) by derivatives with respect to (4,T1,^,T) via chain rule differentiation. The transformed equations are stated in equation (1) with the corresponding metric coefficients and the transformation Jacobian listed in Appendix A. 2. Replace the time derivative terms with a finite difference formula. 3. Linearize any terms that are nonlinear with respect to Q^"*"^ where Q is the flux vector given in Appendix A and is referenced at time level n+1. 4. Employ approximate factorization followed by operator splitting. 5. Use finite difference formulas to replace spatial derivatives . 6. Add the numerical dissipation terms to the algorithm. More details on implementing these steps for developing the F3D algorithm are given in the following sections . For differencing the thin-layer Navier-Stokes equations, it is often advantageous to difference around a known base solution so that the resulting differences have a smaller, smoother variation and possibly, less differencing error. Thus,

PAGE 37

27 5^(Q-Q,) + 5^(F-Fj + 5^(G-Gj+5^(H-Hj-Re-^5^(S-Sj = (2) in which the subscript o signifies the base solution, 5 is some general difference operator and d is the differential operator. For this application, the freestream solution is used as the base solution so that equation (2) reduces to 5^(Q-Qj+5 (F-Fj+5 (G-Gj+5 (H-Hj-Re-^5 (S-SJ = 0 ^ ^ ^ ; ; (3) Time Differencing The time derivative in equation (3) is replaced by using a general, linear, two step method for integrating a first order ordinary differential equation that may be expressed as eu"'-(1 + 2e)u" + (i +e)u"^ ' = At" '[- and 0 and some function f = f (u) such that d^u = f (u) . This general formula produces several two or three time level, explicit or implicit schemes, such as Leapfrog or CrankNicolson, depending on the value of the constants. In the F3D algorithm e = = 0 so that equation (4) becomes (u"^^-u") = At""^0f""^(l-e)f"] (5) For application to the thin-layer Navier-Stokes equations, equation (5) takes the form (Qn+1 _ qti) = ^tn+l[ef"+l + (1-9) f"] (6) in which f = -[a^(F-F^) + a^(G-G^) + a^(H-H^) Re'^a^ (S-S^) ] . The value of the constant 6 is 1 or 0.5 for first or second order accuracy, respectively. For steady state computations, the final

PAGE 38

28 solution is independent of the time step making first order accuracy sufficient; then, 9= 1. Linearization of Nonlinear Terms Next, all terms that are nonlinear with respect to Q are linearized. As in the Beam and Warming algorithm (79) time linearization is used to approximate the fluxes, F, G and H, at the (n + l)st time level. Since F, G and H are homogeneous functions of Q of degree one, let A = 9qF, B = 3qG and C = 3qH. Then the linearization is accomplished by using a truncated Taylor series of the form in which A, B and C are the flux Jacobians and AQ" = Q^i+l Q^. Thus, the error introduced by the linearization is on the order of (At) 2. The flux Jacobians contain only information that is known from the previous time level. The viscous flux term, S^"*"^, is linearized by the method set forth in Pulliam and Steger (80) . When these linearized terms are substituted into equation (6) and rearranged, it becomes pn+l = + A^AQ^ + 0(At)2 Qn+l = G" + B"AQn + 0(At)2 (7) Hn+1 = + c"AQ" + 0(At)2 1+ GAt -At n + 11 [a^(F" F J +a^(G" g J + a^(H" hJ -Re" ^ a^(s" S J] (8) where I is the identity matrix.

PAGE 39

29 Approximate Factorization In deriving FDEs with the implicit, factored method, the next step is to approximately factor the left hand side (LHS) of equation (8) . By expressing the differential operator, L = (A"3^ + B^d^ + C"3^ Re-l3^J-lM"J) , as the sum of three differential operators such that L = + + L^, the unfactored form of equation (8) can be rewritten as [I + eAt(L^ ^1 L^)]AQri = RHS (9) in which the right hand side is denoted by RHS. Since fluxsplitting in the freestream direction, ^, is employed within F3D, the operator is split so that = L"*" + L~; details of the fluxsplitting are given in the next section. With flux-splitting included, the final unfactored form of equation (8) is siimmarized as [l-HeAt(L; + L-+L^ + Lj]AQ"=RHS ^^^^ Beginning with the unfactored form of equation (10) , the LHS is replaced with two implicit factors giving [l + eAt(L^^-H Lj]|-I + eAt(L+ L^)jAQ"=RHS ^^^^ with splitting error E = (eAt) (l^L-+l;^L^+L-L^+L^lJ Thus, the splitting error due to the two factor approximation is order (At)^ and is comparable to the error due to the time linearization. Provided that the chosen finite difference scheme is no more than second order accurate in time, the cross terms contained in E can be disregarded. This implicit, two factored

PAGE 40

30 scheme is shown by Ying (81) to be unconditionally stable for a model wave equation when upwind differencing is applied in the freestream direction, ^, and central differences are employed in the Tl and ^ directions . instead of solving a block matrix on the order of dimension {^x T| X ^) , two, two-dimensional problems are factored out and individually inverted resulting in a significant conputational savings. The algorithm is split into a sequence of problems such factor results in a lower triangular matrix while the second factor is an upper triangular matrix. A comparison of the F3D algorithm with other approximately factored methods for the thin-layer Navier-Stokes equations shows that the F3D numerical scheme is a reasonable approach for this jet in a crossflow simulation. First, in comparison to the standard, three-factor Beam and Warming algorithm for the thin-layer NavierStokes equations (79), the two-factor method outlined above and employed in F3D provides some computational savings. In addition, a conceptual advantage of F3D is that the three-factor Beam and Warming method is unconditionally unstable without added dissipation terms. It is suspected that this instability will have an increasing impact on fine grid computations as the grid density One advantage of the approximate factorization is that that (12) where AQ* represents an intermediate variable. Notably, the first

PAGE 41

31 is increased in that more and more dissipation would need to be added to maintain stability with the Beam and Warming algorithm. Another more cost effective alternative to F3D might be the diagonalized version of the Beam and Warming scheme (82); but, the time accuracy is only approximately 0{At^^^) for this scheme. Flux Splitting Prior to replacing the spatial derivatives with finite differences, flux vector splitting is applied in the freestream, ^, direction and then utilized in forming L"*" and L~. Since the flux vector splitting is confined to the ^ direction, consider F = AQ where A = 9qF since F is a homogeneous function of Q of degree one. In generalized coordinates, the eigenvalues of A are given by 7 2 2 2 I +1 +t C 7 2 2 2 in which c = Cyp/p)*^-^ is the local speed of sound and the contravarient velocity, U, and metric terms are given in the Appendix. Some similarity transformation, T, exists which reduces A to a diagonal matrix A containing the eigenvalues. Then, T"1aT = a or a = TAT"1 The matrix of eigenvalues. A, is split into two matrices, one that contains only positive elements and one that contains only negative elements, which permits the following equality to be stated

PAGE 42

32 F = AQ = (TAT"1)Q = (TA"'"T"1)Q + (TA-T"1)Q = A+Q + A'Q = F+ + F" (13) The flux vector F is split. It is noted that the splitting of A is not unique. In F3D, the implicit LHS terms are differenced using simple flux vector splitting as described by Steger and Warming (83) . Flux vector splitting can introduce oscillations into the solution at locations where the eigenvalues change sign and the corresponding fluxes are discontinuous . To alleviate this problem, transition operators are en^jloyed in F3D (81,84). The transitional scheme for the explicit RHS, %, derivatives, which is identical to flux splitting except at the point where an eigenvalue changes sign, is summarized in Reference 77. Since A"*" has only positive eigenvalues, information is only propagated in the positive ^ direction (downstream) and bacJcward differencing is used. Similarly, A~ has negative eigenvalues, information is only propagated in the negative ^ direction (upstream) and forward differencing is appropriate. The flux vector F has both positive and negative eigenvalues for subsonic flows so that without the use of flux vector splitting, central differencing would have to be used for the ^ direction. Thus, one advantage of flux vector splitting is that it permits the use of one sided difference formulas which possess superior dispersive and dissipative characteristics than central differencing schemes. Spatial Differencing All time and spatial derivatives are replaced with finite difference formulas. For the spatial differences, in general.

PAGE 43

33 second order upwind differences, denoted by 5^, are used in the ^ direction and second order central differences, denoted by 5^ and 5^, are used in the Tj and ^ directions in this algorithm. A midpoint operator, 6, is used with the viscous terms. Examples of second order, three-point forward and backward differences, second order central differences and the midpoint operator are -f ( ~ + 2, j, k, n ^^i + 1, i k, n ~ ^^i, j, k, n) V= i^^ (^Q. . 4Q. +Q. \ V 1, 1 k, n 1 1, > k, n i2, ], k, O O = V 1. j + 1, k, n 1, : 1, k, = 2^ _ I 1, 3, k + n 1, j, k -5-, n ) 5 0 = -^: r 2 (14) where = Atj = = 1 in the computational domain. Then, with the flux splitting and the difference operator notations defined, the two implicit factors (see equation 12) can be written as [l + e At (^L^ + L J j = 5^ + 6 Re" ^ B ^( J~ ^ M j) L U 4 ^ (15) Also, the explicit RHS is RHS=At{5j(F^)" f:]+ ^l^Y-f f:]+ 5^(g" G J + 5_(H"-Hj-Re-^Bfs"-sJ-l ^ C J (16)

PAGE 44

34 Numerical Dissipation The numerical solution of both the Euler and the NavierStokes equations requires the addition of niomerical dissipation, especially for the Euler equations since they exclude natural dissipation. For the Navier-Stokes equations, artificial dissipation is needed to damp high frequency, nonlinear instabilities in the flow, such as shocks, and to control the odd/even decoupling of grid points which is associated with central dfference schemes. The artificial dissipation must be added judiciously in order to control numerical instabilities without smearing the flow physics. In general, there are two methods for introducing artificial dissipation. First, it is introduced by upwind difference schemes, in this case through the flux splitting, and there is little user control over the amount of dissipation that is added. Next, with central differencing schemes, the nxamerical dissipation is intentionally added. The implicit and explicit smoothing terms, denoted by and are added to the centrally differenced, T| and ^ directions to control nonlinear instabilities. These numerical dissipation operators are written as a combination of second and fourth order differences and have the form n n where

PAGE 45

35 and where |B| is either the absolute value of the matrix B or an approximation of it, p is the nondimensional fluid pressure, £2 is 0(1) and is O{0.1). The coefficient of the second difference increases and the coefficient of the fourth difference decreases as P gets large; thus, in the vicinity of a shock, the effect of the fourth difference is reduced. Actually, the fourth order difference terms can introduce oscillations near shock waves and may be dropped from the computation near them. Code Im plementation The final form of the implicit, approximately factored FDE that is derived using the steps described above and implemented in the F3D code is I+ha^CA"") +h5c"-hRe"^B j"^m"j-D 1 x|^i + h6^^(A")" + hS^e" D . Jaq" = At|6'|(F")" F:]+5^[(f-)" -f:] + 6^(g" G J +5^(h" hJ Re-^ S^(s" S j} (18) where h = At or At/2 for first or second order time accuracy, respectively. In this equation, 6 is typically a three-point, second order accurate, central difference operator, 5 is a midpoint operator that is used for the viscous terms and the operators 5^ and 5^ are forward and backward, three-point difference operators. The flux F is split into F"*" and F~ according to its eigenvalues, the matrices A+, A", B, C and M result from the local linearization of

PAGE 46

36 the fluxes about the previous time-level and J is the Jacobian of the coordinate transformation. and are the dissipation operators for the central space differenced directions. The computed flows may be steady or unsteady, viscous or inviscid, laminar or turbulent. In the resulting code, the convergence rate of the algorithm may be accelerated for steady state applications by applying a space varying time step such that At= 1 + Vj Since the Jacobian, J, is the inverse of the cell volume for a three-dimensional grid, the time step is dependent on the local grid spacing, rendering the calculation nontime-accurate . Srinivasan, Chyu and Steger (85) compared the convergence rates for a time-accurate computation in which the same At is used everywhere and a nontime-accurate solution to find that the spatial varying time step improves the convergence rate by a factor of 2 , For laminar calculations, the coefficient of viscosity, \l, is obtained from Sutherland's law. Turbulence modelling is not used in this study. The code, which is fully vectorized for supercomputers, requires lO""^ CPU seconds/iteration/grid point on the CRAY-2 computer . Computa tional Grid In order to implement the numerical algorithm described above, the continuous spatial domain must be replaced by a set of discrete grid points. Consequently, a three-dimensional computational domain, shown in Figure 2(a), is set up for the

PAGE 47

37 calculations. A right handed Cartesian coordinate system is used with X increasing in the freestream direction and z pointing away from the plate . The rectangular grid is exponentially clustered within the boundary layer of the flat plate and around the jet exit which is centered at the origin of the Cartesian coordinate system. Each grid point is indexed with the labels i, j and k. The index i ranges from 1 to ^^^^ for which i = 1 corresponds to the upstream boundary x = x . and i = i__^ corresponds to the outflow boundary X = Xj^gjjSimilarly, the index j spans the domain laterally and k increases in the z direction. Since the experimental measurements that are used for comparison with these numerical calculations are taken for jets exhausting through a circular orifice, the jet exit is approximated by a right-angled polygon of nearly equal area for this Cartesian grid system. A partial sketch of the grid on the flat plate in Figure 2 (b) illustrates the representation of the jet exit . Although the relatively sparse experimental measurements indicate that the global features of the flow induced by a jet in a crossflow are steady and symmetric, the secondary flow features of the flowfield are assumed to be unsteady and assymmetric. As sketched in Figure 2 (b) , no symmetry assumptions are made for the computational domain; this permits the investigation of flowfield symmetry about the y = 0.0 plane. Nevertheless, in accord with previous experimental studies, the y = 0.0 plane is referred to as the symmetry plane. In Table 1, the number of grid points in each direction and the limits of the physical domain are specified for two

PAGE 48

38 representative grids . Initial calculations are made on a relatively coarse grid with 39 (freestream) x 35 (normal to the y = 0.0 plane) x 32 (normal to the flat plate) points. The coarse grid computations are made on a CRAY-XMP computer. In order to investigate the effects of the grid on the computation and to provide good correlation with the experimental data, the grid density is increased from 39 x 35 x 32 (43,680 points) to 55 x 55 x 50 (151,250 points). Calculations on the refined grids are performed on the CRAY-2 computer . The quantities x„. , x^,^, y^,„ and z , are ^ mxn' max' -'max max nondimensionalized by the jet diameter, D, and define the minimum and maximum boundary locations. Since the domain is symmetric about the y-axis, y^^^^^ = -y^^.Further, z^^^^ corresponds to the flat plate, z = 0.0. The method for determining the values of the boundary locations will be examined further in Chapter 4 . Table 1. Details of the Physical Domain for Two Grids. GRID MAXIMl i M INDEX j VALUE k 'Wn/D DOMAI^ '^max/^ SIZE y /D ^max' z /D max 1 39 35 32 -5.0 15.0 9.5 18.0 2 55 55 50 -6.0 15.0 9.5 18.0 For Grid 1, the points are distributed throughout the domain such that

PAGE 49

39 ^itiax 39 : upstream jet ^ ^^'^^ downstream ^max ~ 35: <^^)left ^ ("^^jet ^ right V = max 32: ^ ^ boundary layer ^ ^^^^ where the subscripts upstream, downstream, left and right denote the position with respect to the jet orifice and the subscripts jet exit and boundary layer denote points that are contained in the jet exit and the boundary layer respectively. Similarly, for Grid 2, i_-.. = 55: (12)^p3t^33^ + (9) + <34)^^^^3^^^^ <23)ieft ^ jet ^ ^^^^ right max j =55 -"max ^max = 50 (10) boundary layer ^ ^^0) The details of the minimiam and maximum grid spacing are presented for the representative grids in Table 2. For both cases, the minimum spacing occurs within the jet exit and at the first grid point within the boundary layer while the maximxam spacing occurs at the edges of the domain. In Table 2, Ax^/D refers to the spacing upstream of the jet and AX2/D refers to the spacing downstream. Table 2. Details of the Grid Structure. GRID MINIMI Ax/D M GRID S Ay/D PACING Az/D MJ Ax^/D \XIMUM GR AX2/D ID SPACIl Ay/D Az/D 1 0.167 0.167 0.100 1.250 1.454 1.563 1.719 2 0.125 0.125 0.005 0.808 1.005 0.877 1.200

PAGE 50

In order to perform the calculations, the physical domain is transformed to a computational space with uniform grid spacing as shown in Figure 2(c) . In the transformed grid system, ^ corresponds to the freestream direction such that ^ = 1 coincides with the upstream boundary where x = Xj^^j^^^f 'H spans the domain laterally and ^ is the direction normal to the body surface such that C= 1 maps to the solid surface. The i, j and k indexing is retained in the computational space . Boundary Conditions A unique numerical solution of the thin-layer Navier-Stokes equations set forth in equation (1) is obtainable only if the proper physical and niamerical boundary conditions are imposed; thus, the correct specification of boundary conditions is critical to the success of the jet in a crossflow simulation. Moreover, a good initial choice of boundary conditions may speed up the convergence of the algorithm. These boundary conditions are not only dependent on the problem physics but also on the grid topology. The boundary conditions that are required for the jet in a crossflow problem with the previously described domain consist of 1) freestream conditions on the upstream boundary, ^ = 1, 2) extrapolation on the downstream boundary, ^= K^^r 3) solid surface conditions on the flat plate, C, = 1, 4) jet exit profile for | x^ +y2|/D < 0.25 and C = 1/ 5) freestream conditions for the top surface, ^= and 6) extrapolation on the lateral surfaces, Tl = 1 and n ' 'max

PAGE 51

41 In general, these conditions are iitplemented explicitly in modular form within the code . A laminar boundary layer profile is input on the upstream boundary. The variables on the downstream boundary are obtained through an explicit, linear space and time extrapolation by t + 1 t t Q. . = 20, . Q 1, J max , k 1, ] max 1, k i, 3 max 2, k where Q is a vector in the numerical code containing the five flow variables defined in Appendix A, jmax corresponds to the ^^j^ plane and t refers to the time level. Similarly, a zero normal gradient boundary condition is applied to the lateral surfaces of the computational domain. The boundary conditions on the flat plate are dependent on whether or not the flow is viscous. For viscous calculations, the no-slip velocity condition is imposed at the plate; specifically, all velocity components are set to zero at the wall. The density is then calculated by zeroeth order extrapolation from the adjacent points interior to the flowfield. On the other hand, for inviscid computations, a flow tangency condition is applied; fluid cannot penetrate the solid surface. The pressure on the solid surface is obtained as the solution of the normal momentum equation. The precise physical boundary condition at the jet exit is unknown. Therefore, some ass\amptions must be made about the character of the flow at the orifice which may lead to discrepancies between the computed and measured results in the near jet region. Previous authors have imposed the jet exit condition by specifying uniform vertical velocity (33) or uniform total j

PAGE 52

42 pressure (57) or by initiating the calculation inside the jet nozzle. In this simulation, a uniform velocity, constant pressure profile jet is injected from the lower boundary into the flowfield. Since the circular orifice is represented by a rectangular grid, care is taken to match the input mass flow and momentum of the jet with the experiment. The second type of boundary condition, uniform total pressure, may ultimately prove to be more effective but guidance, in the form of experimental measurements for jets with high velocity ratios, is needed to insure its validity. Similarly, calculation of the flow inside the pipe may provide a more realistic jet profile. But, the cost of the calculation would be greatly increased with the present finite difference formulation since a two part (flow internal to the nozzle and the external flowfield) iterative solution would be required to accurately predict the pressure profile at the jet exit with the freestream influence .

PAGE 53

43 7 c o •H +-> (0 4J +J C -H « X I* M i OJ 0) XIT) TD Om (tJ 3 x: CO u c (0
PAGE 54

CHAPTER 4 RESULTS AND DISCUSSION The purpose of this study is twofold, first, to adapt the F3D code for use in propulsive flowfield simulations and to validate it both qualitatively and quantitatively through experimental comparisons and second, to provide insight into the physics of the three-dimensional lifting jet aerodynamic/propulsive interaction. In this chapter, several applications of the numerical algorithm are described. The test cases were chosen to correspond closely with the wind tunnel tests of Fearn and Weston (5-7) . First, the computed results are quantitatively compared with the measured data by examining the jet trajectory, the properties of the contrarotating vortex pair and the plate pressure distribution. Then, through flow visualization, the flow physics are qualitatively investigated. Calculations are presented for the jet/flat plate configuration with the jet injected perpendicular to the cross stream for the velocity ratios, R = M^/M^, of 4, 6 and 8. The computed flows have freestream Mach numbers of 0.19, 0.13 and 0.12 and Reynolds numbers based on the freestream velocity and the jet diameter of 500,000, 350,000 and 350,000, respectively. In most of the numerical calculations, the flow is assumed to be steady and viscous. A summary of the viscous test cases is given in Table 3. 44

PAGE 55

45 Table 3. Summary of Viscous Test Cases, CASE R Moo RE GRID SIZE DC ^min )MAIN ^max SIZE Vmax ^max 1 4 0.20 500, 000 39x35x32 -5.0 15.0 9.5 18.0 2 4 0.19 500,000 55x55x50 -6.0 15.0 9.5 18.0 3 6 0.13 350,000 39x35x32 -5.0 15.0 9.5 18.0 4 6 0.13 350,000 55x55x50 -6.0 15.0 9.5 18.0 5 8 0.12 350,000 39x35x32 -5.0 15.0 10.0 20.0 In addition, inviscid calculations were made corresponding to the viscous cases 1 and 2 . Obviously, the inviscid calculations cannot predict the separated flow region downstream of the jet . Furthermore, the inviscid solution contains numerical viscosity which tends to introduce diffusion throughout the entire physical domain while the actual viscous effects are concentrated in areas such as the boundary layer and the shear layer between the jet and the freestream. The purpose of the inviscid computations is to isolate features of the flow which are dominated by effects such as pressure rather than viscosity and to identify regions of the flow, if any, that might be adequately modelled with a reduced set of governing equations. The numerical results are computed on a three-dimensional computational grid which has the features described in Chapter 3

PAGE 56

46 and which has the grid density and the domain limits, nondimensionalized by the jet diameter, that are given in Table 3 for each of the cases. Initially, the dimensions of the computational domain were chosen by examining the empirical jet paths and by estimating the amount of "blockage" by the jet. In order to ensure that these boundary locations are not adversely influencing the numerical predictions, computations are made in which the boundaries are moved progressively farther away from (or closer to) the jet until the flow is relatively unaffected by the boundary locations . Two criteria for determining a sufficient domain size are enforced. First, the jet centerline is required to exhibit good agreement with the experimental jet centerline and with the centerline location calculated on the next larger domain. Second, the top and lateral boundaries are positioned so that the flow direction, at these locations on the next larger domain tested, is nearly aligned with the freestream flow. As discussed previously, the domain sizes are minimized; however, for the convenience of point by point comparison of the flow variables, some cases, such as Case 1 and 3, are computed on the larger domain. It should be noted that the boundary location optimization has not been completed for the R = 8 jet. Case 5. Both the lack of grid refinement and the minimal domain sizing may significantly influence the detailed results for the R = 8 case in which flow variables will exhibit higher gradients than for the lower velocity ratio jets. But, the results for selected global properties such as the jet centerline are examined when reasonable agreement is demonstrated. It is anticipated that positioning the

PAGE 57

47 upper boundary so that Zj^^, is between 20D and 22D would improve the accuracy of future solutions. In each of the cases studied, the flow is assumed to be laminar; no turbulence model is applied. In experimental studies of flows with relatively low velocity ratios, both Crabb, Durao and Whitelaw (12) and Andreopoulos (62) found regions in which the turbulence field is anisotropic which suggests that the commonly used turbulence models, such as eddy viscosity models and k-e models, will not be able to account for all the features of the jet flowfield. Thus, the exclusion of turbulence from the calculation permits the evaluation of the numerical model without having to estimate the errors associated with an inadequate turbulence model. In this laminar simulation, a Blasius boundary layer profile is computed upstream of and beside the jet. Since boundary layer velocity profiles and separation point locations are dependent on the characteristics of the boundary layer, complete quantitative agreement between the measured turbulent flow and the calculated flow is not anticipated. On the other hand, the experiments of Bradbury and Wood (25) revealed that the momentum thickness and the jet Reynolds number only weakly influenced the pressure distribution on the plate. Thus, the global features of the jet in a crossflow such as the jet centerline, the contrarotating vortex pair and the pressure distribution on the plate that are depicted in Figure 1 may be relatively insensitive to secondary parameters and might be adequately captured in a laminar simulation. At each grid point, the density, p, the three components of momentum, pu, pv, and pw, and the total energy, e, are computed.

PAGE 58

48 The solutions are considered to be converged when the L2 average norm of the residual is 0(10""^). A typical convergence history for the steady state solution for Case 2 is shown in Figure 3. Since a space varying time step is used for the steady state calculations, a comparison between the convergence rates for all of the test cases would not be instructive. Grid Dependence In order to examine the grid dependence of the solution, profiles of the streamwise velocity component are constructed at several locations downstream of the jet orifice. For presentation, the vertical profiles of streamwise velocity in the symmetry plane at the downstream locations x/D = 1.0, 2.0, 3.0, 4.0 and 8.0 are shown in Figure 4 (a) -(e) for the R = 4 jet and in Figure 5 (a) -(e) for the R = 6 jet. In both cases, the coarse and fine grid results are significantly different, particularly in the boundary layer region. The velocity vector field within the symmetry plane for the R = 4 jet is presented as Figure 6 in order to aid in the interpretation of the scalar velocity profiles. The vector field indicates the onset flow, the jet plume which is deflected by the freestream, a reversed flow region behind the jet, an upwash velocity component which lies in the symmetry plane between the contrarotating vortices shown in Figure 1 and finally, the predominantly streamwise flow. The vector field that is shown is confined to the symmetry plane since there are no lateral velocity components in this plane.

PAGE 59

49 In the region of maximum streamwise velocity, which closely corresponds to the location of the jet trajectory, the velocity peaks are more pronounced and tend to be greater in magnitude on the fine grid than on the coarse grid. Although these peaks do not reflect the same magnitude, in general, their vertical location appears to agree well. At the most downstream location, Figures 4(e) and 5(e), this difference is minimized due to the decreasing impact of the jet on the surrounding flow and to the increased similarity between the two clustered grids far downstream. A similar grid dependence was seen in the calculations by Demuren (57) who found that only the predictions on the finest grid he tested (approximately 50,000 grid points for the three-dimensional domain) showed significant velocity peaks in the central region of the deflected jet while this peak is smeared out on the less refined grids. He also noted that the direction of the grid refinements, for example increasing the grid in the lateral direction without increasing the number of points in the other two directions, altered the results. Next, large discrepancies arise in the streamwise velocity profiles within 3.0 jet diameters of the flat plate. Actually, the distinct flow features that are captured on the fine grid cannot be resolved on the coarse grid. At the most upstream location. Figures 4(a) and 5(a), both the coarse and fine grid solutions show a region of reversed flow. However, with the grid refinement, there is a decrease in magnitude of the reversed flow velocity component near the edge of the boundary layer and the minimiom velocity peak occurs nearly a jet diameter farther away from the

PAGE 60

50 plate. On the fine grid, the minimum velocity peak that is closest to the plate is due to flow reversal within the jet wake while the second, or more negative, minimum velocity peak corresponds to the upwash between the contrarotating vortices. Since the boundary layer is not resolved and the numerical dissipation is increased on the coarse grid, the coarse grid solutions show only one minimum velocity peak with no distinction between the wake flow and the upwash. In addition, the fine grid computation predicts a smaller streamwise region of reversed flow. For exanple for R = 4, between x/D = 2.0 and 3.0 shown in Figures 4(b) and (c) , the fine grid solution exhibits full forward recovery of the velocity field near the plate while the coarse solution still indicates a large region of reversed flow. At x/D = 4.0 which is shown in Figure 4(d), some distinction is noticed between the reversed wake flow and the entrained flow in the coarse grid solution. But, the experimental measurements (7) which are plotted for the planes x/D = 4.0 and 8.0 on Figures 4(d) and (e) show that the fine grid solution more accurately characterizes the flow. Similar behavior is observed for R = 6 and is illustrated in Figures 5(b) -(e) , Once the flow reversal within the wake ceases, the point of minimum velocity represents upwash. At the most downstream location shown, x/D = 8.0, the fine grid results for the R = 4 jet which are displayed in Figure 4(e) overshoot the experimental data in the region near the flat plate. This may be caused by the application of freestream boundary conditions on the top surface of the computational domain since it

PAGE 61

51 is not uncommon for niomerical codes to predict higher velocities near the surface in order to conserve mass and momentum in the calculation. The application of characteristic boundary conditions could be investigated to alleviate the overshoot. On the other hand, the overshoot in streamwise velocity may be caused by the rapid convergence of streamlines toward the symmetry plane which tends to accelerate the flow in the symmetry plane. In this latter case, the governing equations, grid sizing and turbulence are all suspect , Flow in the Symmetry Plane The computed velocity profile in the symmetry plane was described in the previous section. This particular view of the flowfield provides some information on the trajectory of the jet plume and on the penetration of the contrarotating vortex pair. In addition, a large amount of experimental and coirputational data is available for comparison in this plane. The plots in Figures 6-9 show the velocity vector field and the Mach number contours for the cases R = 4 and 6. These plots, which show that the jet extends a short distance into the flow before being deflected downstream by the crossflow, are similar to the qualitative data presented as plots of the scalar field and the velocity vector field by Sykes, Lewellen and Parker (58) based on their Navier-Stokes computations. The small, inner, nearly triangular contours which are located just above the jet exit in Figures 7 and 9 denote a region of high velocity. This region, which reaches several jet diameters into the flow, is part of the jet core that is identified in

PAGE 62

52 Figure 1. The velocity field exhibits relatively slow changes within the jet core in comparison to the high velocity gradients found in the shear layer that exists at the interference between the jet and the crossflowing fluid. Distortion, caused by the crossflow, and rapid diffusion of the shear layer eventually erode the jet core. In recent unpublished research, Krothapalli, Lourenco and Buchlin (73) are conducting an experimental study of the separated flow region upstream of the jet near the nozzle exit. Their study has uncovered an unsteady separated flow region which is dominated by a horse-shoe vortex system near the plane of the jet exit . The downward component of the confuted velocity vectors that immediately precede the jet exit and are close to the plate is attributed to the presence of such a vortex system (see Figures 6 and 8) . However, the true nature of this horse-shoe vortex system is masked by the assunption of steady flow in the computation and is further tainted by the assumptions made in the thin-layer Navier-Stokes equations and the grid resolution. The features of the computed velocity field farther from the plate agree well with the detailed symmetry plane velocity surveys by Fearn and Weston (7,8) and Fearn and Benson (32). Two lines of computed symmetry plane velocity vectors taken near the jet exit are presented in Figures 10 for the case R = 4. One rake of data is centered at (x/D, z/D) = ( 0.66, 0.96) and rotated an angle


PAGE 63

53 and Benson (32) which are plotted in Figure 11 although a greater upstream component of velocity is predicted for the region downstream of the jet in the computation. Jet Cen terline The determination of the trajectory of the jet plume was a primary objective for many of the experimental and theoretical studies. In this study, the determination of the jet trajectory provides a means of comparison between the calculated results and the experimental data. The jet path, called the jet centerline, is defined to be the locus of points of maximum velocity in the direction of the jet within the plane of flow symmetry, y = 0.0. Fearn and Weston (5-7) located the jet centerline in a similar manner experimentally by measuring the maximum velocity component in the jet direction. Strict attention must be paid to the centerline definition since alternative definitions may result in different jet paths. For example, in a given vertical plane, the point of maximum velocity magnitude and the point of maximum velocity in the direction of the jet do not necessarily coincide; thus, taking the jet centerline as the locus of points of maximiam velocity magnitude in vertical planes is inconsistent with the present definition. In all of the cases examined, the confuted centerline compares favorably with Fearn and Weston's experimental data and with the empirically described jet centerline (69) as demonstrated in Figures 12 (a) -(c) using the data for the most refined grids for R = 4, 6 and 8 where the empirical curve is given by

PAGE 64

54 with the empirical parameters a = 0.9772, b = 0.9113 and c = 0.3346. As the velocity ratio is increased, the jet extends further into the flow. Examination of the computed jet centerlines for R = 4, 6 and 8 that are plotted in Figure 12 illustrates the variation of the jet penetration with the velocity ratio. Only small deviations occur between the jet paths calculated on the coarse grid (43,680 points) and the paths calculated on the fine grid (151,250 points), indicating that the jet trajectory may be captured with a relatively small number of grid points. Furthermore, for the R = 4, inviscid cases, the only detectable differences in jet trajectory from the viscous calculations are a tendency for the centerline to deflect on the order of O.ID upstream in the near jet region prior to the characteristic downstream deflection and to penetrate less than 0.5D farther into the freestream far downstream of the jet exit, x/D > 10.0. This agrees with the experimental conclusion that pressure, rather than viscosity, is the primary mechanism governing the initial deflection of the jet (16) . The plot of the velocity vectors in the symmetry plane for the R = 4 inviscid computation in Figure 13 shows the similarity between the viscous and inviscid results despite the slip condition on the plate. For the inviscid computation, the jet tends to expand in the streamwise direction as it leaves the jet exit which explains the tendency for the jet centerline to bend forward initially. Vertical profiles of the velocity component in the

PAGE 65

55 freestream direction taken in the symmetry plane for various downstream locations are plotted in Figures 14 (a) -(e) for a viscous and an inviscid computation of a jet with a velocity ratio R = 4. Both cases are computed on the most refined grid. And, the figures correspond to the results presented in Figure 4 . Despite the differences in the two flows close to the flat plate, the overall agreement of the velocity profiles is quite good. At each downstream location, the locations of maximiam u/u are nearly identical although the magnitude of u/u^ is generally lower for the inviscid computation. Decay o f the Jet Velocity As indicated by the streamwise velocity plots, the magnitude of velocity along the jet centerline decreases with increasing downstream distance. Figures 15(a) and (b) show the variation of Uj/u^ with s/D for the velocity ratios R = 4 and 6, respectively. Here, u^ is the velocity at the jet exit, u^ denotes the velocity magnitude along the jet centerline and s is the arclength along the jet centerline which is nondimensionalized by the diameter of the jet exit, D. Originally, the rate of decay of the velocity along the jet centerline was plotted in order to provide a conparison between the present computation and the Navier-Stokes computations made by Patankar, Basu and Alpay (33) . Thus, the nondimensional quantities that are plotted are chosen to correspond to the format of the computational results of Patankar, Basu and Alpay (33) who provide a comparison with the experimental data from Chassaing et al. (74) .

PAGE 66

56 In both cases, R = 4 and 6, there is a significant difference between the experimental results of Chassaing et al, (74) and Fearn and Weston (7); moreover, the results of the present calculation predict a more rapid decay of the jet centerline velocity, u^, than either of the two experiments. One difference between the two experiments is the jet exit geometry. In the experiment by Chassaing et al. (74), the jet exhausts into the freestream through a pipe while the flow measured by Fearn and Weston (7) discharges into the freestream through a hole in a flat plate. As discussed by Moussa, Trischka and Eskinazi (1), the effect of the geometrical conf icfuration may account for the slight difference in the jet centerlines as well as the different velocity decay rates for the two jets. In both numerical experiments, the jet is assimed to discharge from a hole in a flat plate with a constant velocity, Uj, leading to speculation that the numerical results would compare more favorably with the jet centerline velocity rates for the Fearn and Weston (7) experiment. On the contrary, the coarse grid computations (only 2,250 grid points are used in comparison to 151,250 grid points in a similar domain for the present study) of Reference 33 demonstrate excellent agreement with the measured decay rate by Chassaing et al. (74), In the present calculation, the coarse grid results (43,680 grid points) indicate a more rapid decay of the centerline velocity than the fine grid solution (see Figures 4 and 5) even though the jet trajectory is the same for both predictions. So, grid effects are expected to alter the centerline velocity decay rate.

PAGE 67

57 The rate of velocity decay for the R = 4 inviscid case is also shown in Figure 15 (a) . Although the computed jet centerline locations for the viscous and inviscid computations are similar, the centerline velocity, u^, in the inviscid case is shown to decrease more rapidly. As expected in the jet core (at small values of s/D) , the velocity ratio remains close to unity in all cases and is not plotted. In light of the previous observations, it may be speculated that the jet's development and final steady state properties are strongly dependent on the initial flow conditions. Despite the apparent insensitivity of the jet path to grid density and viscous effects, the centerline velocity decay rate is sensitive to both the grid density and viscous effects. Furthermore, since the vorticity distribution at the jet exit is as important as the momentum distribution, the jet exit boundary condition is also expected to alter the properties of the contrarotating vortex pair far downstream of the jet exit. Contrarotati ng Vortex Pair A pair of contrarotating vortices is identified both experimentally and computationally as the dominant and persistent feature of the flow far downstream of the jet exit. The most natural way to examine the contrarotating vortex pair is in a cross section that is oriented perpendicular to the jet plxome. Since the computed results should closely approximate the measured jet characteristics, the vortices are expected to lie relatively close

PAGE 68

58 1 to the empirically predicted vortex curve of Fearn and Weston (5) which is located in the XZ-plane by Zv aR ^20) where the empirical parameters are a = 0.3515, b = 1.1220 and c = 0.4293. A planar cut is made through the jet plume perpendicular to this curve as shown in the illustration in Figure 16. Then, the computed values are interpolated onto this plane, which is referenced as a vortex cross section, and resolved into the components of the local two-dimensional coordinate system. The velocity field in a vortex cross section is displayed in Figure 17 for the R = 4 viscous computation with the location of the cross section specified by the (x,z) location of the (x^,z^) origin and the rotation angle of the plane, (p. The velocity field shows the freestream fluid approaching the jet plume, the fluid sheared around the jet by the crossflow and the characteristic vortex pair. Inviscid computations for R = 4 reveal a flow that contains a vortical structure but not necessarily the characteristic contrarotating vortex pair. A velocity vector plot of the flow in a vortex cross section centered at x/D = 3.0, z/D =2,67 and


PAGE 69

59 I freestream direction, ^, can produce assymmetry in a symmetric problem. Vortex Models Filament Model. As discussed briefly in Chapter 2 and reiterated here, Fearn and Weston (5) present models which use the measured velocity field taken in vortex cross sections in order to infer the properties of the contrarotating vortex pair. In the filament vortex model, the measured velocities taken in vortex cross sections but restricted to the plane of symmetry, y = 0.0, are used to calculate the strength and location of two infinite straight vortex filaments. This model is two-dimensional and gives the properties of the vortex pair at the vortex cross section. No description of the distribution of vorticity within the cross section is given. This model requires relatively few velocity measurements as inputs. Diffuse Vortex Model . In the second model, the diffuse vortex model, the vorticity is no longer concentrated within a filament; each vortex is assumed to have a Gaussian distribution of vorticity. The measured velocities within a vortex cross section are required as inputs in order to compute the vortex location, strength and diffusivity. It is assiamed that the velocity in the vortex cross section can be represented as the vector sum of the component of the freestream velocity in the plane of the cross section and the velocity induced by the contrarotating vortex pair so that

PAGE 70

60 V = 1 e R 2^ -P r 2k 1 e -P 2. (21) where is the strength of a single isolated diffuse vortex, (3 is a diffusion constant, r^ and r2 are the distances to an arbitrary point from the two vortex centers and e denotes unit vectors in the directions defined in the model geometry that is sketched in Figure 19. A least squares best fit between the velocity data and the model is used to obtain the parameters that describe the vortex geometry and strength in equation (21) . The model results are presented in terms of the effective vortex strength, % — oo r = J J '^(^i^) ^ ~ ^ ° (22) the effective vortex spacing, -— oo ^=rl_n I y,to(r^e) rdrde 2 ° (23) and the vortex core size, r^, which is defined to be the distance from the axis of a single Gaussian distribution of vorticity to the location of maximum tangential speed. In these definitions, (0(r^6) is the distribution of vorticity for a pair of Gaussian vortices. The effective vortex strength and spacing and the vortex core size are related to the vortex model parameters by

PAGE 71

61 r = erf(ph„) erf(phj r^ = 1.121p~^ where erffph ^ =2 Jc"°-M^^° e"^' dt ^ °^ ^0 (24) is the error function. The diffuse vortex model is also twodimensional and only describes the component of vorticity that is perpendicular to the vortex cross-section. Comparison with Ex periment The diffuse vortex model is implemented for use with the velocity data from the computations in order to infer the vortex penetration, spacing, strength and core radius and to make direct comparisons with the vortex properties which were inferred for the experimental data (5) . This is the first known simulation of a jet in a crossflow using the Navier-Stokes equations to provide such an extensive comparison of the vortex properties. Overall, fair agreement between the computations and the experiment is demonstrated. The curves, drawn with a solid line, on Figures 20 and 21 are computed using the two-dimensional analytical model of Karagozian and Greber (38) . The vortex penetration and separation curves are determined by fitting the model results to a power curve which has the form of equation (20) . The parameters for the penetration and separation curves are found to be a = 0.480, b = 1.06, c = 0.385 and a = 0.4261, b = 0.365, c = 0.367, respectively. Although these model results for the vortex location

PAGE 72

62 show reasonable agreement with the experiment and the computations, the model is unable to accurately predict the vortex strength in the region of interest, x/D < 15, since the two-dimensional assumptions are violated in the initial jet mixing region. Thus, in the Navier-Stokes simulations, effort must be placed on accurately characterizing the vortex properties in the near jet region while the farfield properties could be modelled using a simpler method. The vortex curves for the R = 4 and 6 jets, which are computed on the refined grid and shown in Figures 20(a) and 21(a), are within 0.3 to 0.4 jet diameters of the experimental curve. In each case, the curve for the computed data is closer to the plate than the experimental curve and shows less curvature. Further, the vortices have less separation for the computed data as indicated by the lower, flatter curves in Figures 20(b) and 21(b), In addition to being closer together and closer to the plate, the vortex strength for the computed flows shown in Figures 20(c) and 21(c) is nearly 0.1 r/2DU^ below the vortex strength that was inferred for the experiment at x/D =3.0 and is approximately 0,4 r/2DU^ greater than the experiment by x/D = 10.0. The slower rate of diffusion of vorticity for the computation is also enphasized by the slightly slower rate of growth for the core radius which is plotted in Figures 20(d) and 21(d). For the velocity ratio R = 4, the results computed on the coarse grid as well as the fine grid are plotted on Figures 20 (a) -(d). The lack of grid resolution clearly impacts the solution. For instance, the computed vortices exhibit even less

PAGE 73

63 penetration into the flow; but, they are spaced farther apart and are generally weaker than predicted in the fine grid calculation and weaker than in the experiment. In short, the coarse grid solution is more diffuse than the actual flow and the amount of diffusion is reduced in the fine grid computation. Thus, the grid dependence of the solution clearly affects the properties of the contrarotating vortex pair. Further, the diffuse vortex model did not provide a good fit for the input velocity data from the R = 6 and 8 coarse grid computations which suggests that the grid effects are intensified at the higher velocity ratios. Only speculative remarks can be made concerning the factors that cause the rate of diffusion of the contrarotating vortices to deviate from the experiment. First, the grid dependence of the solution must be a contributing factor. Next, in some instances such as the vortex spacing at downstream locations x/D > 5, the coarse grid solution appears to agree better with the experiment than with the fine grid solution. Thus, while the numerical diffusion is reduced as the mesh is refined, either turbulence is needed to adjust the diffusion rate or some of the viscous terms that are neglected in the thin-layer assiomption must be included in the calculation. Surface Pressure D istribution The surface pressure distribution can be examined through measurements at discrete points as well as through plots of constant pressure contours. A direct comparison between the calculated and experimental pressure coefficients, C , on the flat

PAGE 74

64 plate along the rays that are located by the angle 9 , which is measured from the negative x-axis and is defined in Figure 22, is made in Figures 23 , 24 and 25 for R = 4, 6 and 8 using the solution computed on the most refined grid for each case. Three ray plots are presented in each of the figures: (a) the Cp along the upstream ray, 9 = 0°, (b) the Cp along the ray that is perpendicular to the freestream, 9 = 90° and (c) the Cp along the downstream ray, 9 = 180°. The results along the ray 9 = -90° are not included because they are nearly identical to the 9 = 90° values . The data used in these comparisons are from Fearn and Weston (6) . Upstream of the jet exit, there is a positive pressure gradient. From the experiment, it is noted that the maximum pressure attained in front of the jet decreases as the effective velocity ratio increases. However, in the numerical solution, severe fluctuations in the pressure field exist at the interface between the jet exit and the onset flow so that the correct flow behavior cannot be identified. These spikes are magnified as the velocity ratio is increased and influence an even greater area of the flow in the coarse grid computations. Despite the reduced oscillations in the pressure field as the mesh is refined, this numerical anomaly is probably a consequence of the boundary condition discontinuity which occurs at the jet edge with the specification of a uniform velocity/constant pressure jet. Thus, to provide better agreement between the numerical results and the experiment in the region extending approximately one jet diameter

PAGE 75

65 away from the jet exit, a more realistic and continuous jet profile should be investigated. Along the ray that is perpendicular to the freestream, the largest pressure gradient occurs close to the side of the jet. Once again, the computed Cp near the jet edge is more negative than the experimental Cp. In addition, the extreme low pressure at the jet edge is balanced by an increased pressure farther away from the jet before the two Cp profiles exhibit good agreement. The low pressures to the side of the jet seem to correspond to the presence of the contrarotating vortices near the base of the jet. Similar numerically induced errors occur near the jet exit along the 180° ray. In the wake region, the initial pressures are relatively low although they are significantly higher than the pressures near the sides of the jet . The experimental data show that the minimum Cp increases as the velocity ratio increases indicating that the pressure recovery in the wake is more rapid at the higher velocity ratios. It is interesting that in both the computed and the experimental results, there appears to be a "glitch" or a small, slow changing pressure gradient in the profile. The calculated pressures in the wake are, in general, higher than the measured pressures . This may be the result of comparing the laminar, calculated flow with the turbulent measured flow. Since the pressure on the plate is related to the properties of the vortex pair, the wake pressure distribution is also subject to the liabilities of the computation that were identified for the prediction of the vortex properties.

PAGE 76

66 Although the region near the jet exit exhibits spatial fluctuations in pressure and the wake pressures are high, the overall pressure field on the plate corresponds to the pressure field expected for a jet in a crossflow. Constant pressure contours for the computed dataset are plotted alongside the identical experimentally determined contours (6) in Figures 26(a) and 27 for the high resolution cases R = 4 and 6. Most of the physical effects that were described by the ray plots are also evident in the contour plots. In each case, there is a higher pressure region upstream of the jet exit and a lower pressure region to the sides and downstream of the jet exit . The contours show that the region of lowest pressure, that was detected along the ray at 9 = 90°, is actually centered slightly aft of 9 = 90°. In each case, the contours indicate that a more rapid pressure recovery in the wake occurs in the computations . As expected, the pressure recovery in the wake region is even higher for the inviscid computation than for the viscous one while the overall plate pressure disribution is the same for both calculations (see Figure 26(b)). Finally, better agreement between the location of the computed contours and the experimental contours is demonstrated for the lower velocity ratio, R = 4 which suggests that the grid resolution and/or the turbulence may have an increased effect at the higher velocity ratio.

PAGE 77

67 Flow Visu alization Streamlines Several of the experimental studies relied heavily on flow visualization in order to provide a physical description of the flowfield. For example, Margason (15) studied a water vapor marked jet to define a jet trajectory, McMahon and Mosher (22) examined oil flow traces on the flat plate to gain insight into the complex interaction between blockage, entrainment and wake phenomena near the jet exit and the Office National D'fetudes et de Recherches Aerospatiale (O.N.E.R.A.) (19) used water tunnel flow visualization to look at flow behavior in the boundary layer and the wake region as well as in the contrarotating vortex pair. Similar flow visualization of the computational results is accomplished by constructing particle traces on a graphics workstation. For steady flow, particle traces or equivalently, three-dimensional streamlines, are computed by releasing a particle that moves with the local fluid velocity originating from selected spatial locations. It should be noted that the figures presented in this dissertation are black and white reprints of the original color flow visualization photographs. Although the following discussion describes all of the flow characteristics observed by the author, some of the interpretation of the flow features is not as clearly visible without the use of color. Computed particle traces for the cases R = 4, 6 and 8 are shown in Figures 28, 29 (a) -(e) and 30. In each case, particles are released both upstream of the jet exit within the boundary layer at a height of 0.66* for R = 4 and 6 and l.l5* for R = 8 and from the

PAGE 78

68 jet exit itself. Fluid from the jet exit, which will be referred to as jet fluid, exhausts into the flowfield and is deflected by the freestream. In studying the jet centerline, it was shown that the jet penetrates further into the flowfield before being deflected as the effective velocity ratio is increased. In Figure 29(b), the trajectories of particles released from the jet exit in the symmetry plane are extracted from the detailed flow visualization shown in Figure 29(a) for the R = 6 jet. Although the separation of the composite particle trace facilitates visualization and discussion, similar partitioning for the other two cases provides the same qualitative information. Examination of the location of the fore and aft faces of the jet fluid shows that the jet area contracts in this plane. The fluid on the front face interacts with the freestream in a very thin interaction region or shear layer. Thus, it undergoes rapid, threedimensional, viscous erosion. In contrast, the fluid on the aft face is protected from interaction with the freestream so that fluid on the aft face follows a more vertical trajectory before being deflected. Moreover, the pressure imbalance resulting from the relatively high pressure upstream of the jet and the low pressure behind the jet contributes to the curvature of the streamlines . It is worth noting that the instantaneous streamline emanating from the origin coincides with the computed jet centerline using the present centerline definition; thus, this flow visualization suggests that the jet centerline consists of jet fluid.

PAGE 79

69 Near the plate, the freest ream fluid is deflected around the sides of the jet into the low pressure aft region. However, as the curvature of the jet streamlines compensates for the pressure gradient and the front face of the jet becomes more blunt from the shear, the freestream fluid begins to flow over the top of the jet rather than around it. Eventually, the jet fluid follows a path closely aligned with the freestream. Viewing the particle traces that originate in the jet exit and that are perpendicular to the freestream on the line x = 0.0, Figures 29(c) and (d) , indicates the increased spread of the jet in the transverse plane. Actually, the fluid originating in the circular jet orifice is distorted by the crossflow into a deformed elliptical cross-sectional shape with the major axis of the ellipse aligned perpendicular to the freestream. The flow patterns highlighted in Figure 29(e) are composed strictly of fluid which originated in the flat plate boundary layer. They indicate that the boundary layer fluid approaches the jet, moves around it staying relatively close to the plate and some of the fluid is swept up into the aft side of the jet plume which consists of the original jet fluid together with the entrained fluid. The distribution of this fluid in the region aft of the jet is strongly related to the velocity ratio. For the R = 4 jet (Figure 28) , two loosely structured vortex cores become evident as the boundary layer fluid spirals around toward the base of the jet and into the jet plume. At the increased velocity ratios, R = 6 and 8, the structure of the vortex cores is more cohesive; the fluid particles follow a clearly defined path toward and upward

PAGE 80

70 along the aft side of the jet plume. Hence, the flow patterns indicate that the vortex cores contain some entrained boundary layer fluid. The boundary layer fluid which moves through the reversed flow region and between the two vortex cores is drawn into the low pressure region in the jet plume close to the aft face of the jet. Finally, some of the deflected boundary layer fluid continues moving downstream before being caught up in the jet plume. This flow feature is emphasized in the O.N.E.R.A. (19) water tunnel flow visualization film. Entrainment A unique feature of the computational experiment, in contrast to the physical experiment, is the ability to trace a particle path backwards in time to find where in the flowfield that particular particle originated. In this manner, the entrainment of fluid into the jet plume, and specifically into the contrarotating vortex pair, can be documented in a qualitative sense through flow visualization. Figures 31(a) and (b) show two views of the negative time history of particles released within a vortex cross section for the R = 6 jet. In Figure 31(a), the flow is viewed at an angle with the freestream flowing approximately from the left while Figure 31 (b) shows the same flow viewed at an angle from nearly upstream of the jet. The square patch is centered at the jet exit, has the dimensions of one jet diameter and is aligned with the freestream. Only one of the contrarotating vortices is shown. This particular vortex section is centered at x/D = 5.0, z/D = 5.2 and rotated an angle


PAGE 81

71 vortex center that is predicted by the diffuse vortex model in this plane; these parameters are x^=5.1, y^=1.4, z^=4.9 and r^ = 1.4. Clearly, the vortex core contains jet fluid, entrained boundary layer fluid and entrained freestream fluid. Topological Cons iderations The interpretation of the flow patterns created by continuous vector fields such as velocity, vorticity and skin friction from both experiments and computations can be aided by topological concepts. The purpose of this dissertation is not to fully investigate the topology of a jet in a crossflow; that has already been described in detail by Hunt et al. (86) and Foss (87). Rather, some basic topological classifications are introduced to characterize the visualized flow features. A general autonomous system of ordinary differential equations is given by f = F(x,y) dy dt=^(^'>') (25) where F and G are functions of the variables x and y. A solution of this system, x(t), y(t), is a curve, C, or trajectory in the XY-plane, also called the phase plane, whose orientation is defined as positive in the direction of increasing t. The slope of the path C is dy ^ G(x, y) dx " F(x, y) ^26) If G(x,y) and F(x,y) are both zero at some point, say P^, then the slope is undefined and P^ is called a critical point of the fl ow . i

PAGE 82

72 Generally, topological discussions of fluid flows focus on simple critical points. A simple critical point is a critical point that is isolated (one that lies in an arbitrarily small disk that contains no other critical point) and that has nonzero, linearly independent terms in the first order Taylor series expansions of F and G about P^. By locating the critical point at the origin of the local coordinate system, the first order Taylor series expansions give F(x, y) = ax + by G(x, y) = cx+ dy ^27) Then, the coupled differential equations expressed in (4) become dx , , — = ax+by — = cx + dy (28) These equations can be uncoupled and solved as two linear equations by assuming solutions of the form (x,y) = (A,B)e^''^. The two unique solutions are X t (x^, y^) = (b,X^ a)A^e ^ X t y,) = (bA, a) A,e ^ 2 2 2 (29) in which 1 _ a + d -J (a + d) -(ad be) '(a + d)^ -(ad be) Several types of simple critical points exist, and the values of and X2 determine the flow behavior at each critical point. If A,^ and ^2 are both real and are both positive or both

PAGE 83

73 .1 1 negative, then the critical point is a node. A proper node has every path approach in a definite direction as t approaches either °° or -«> and, given any direction, there is a path approaching P^. When X,^ and X2 are negative, the direction of the vector field is towards P^ as t increases. This is an attractive node which is also called a node of separation. A node of attachment occurs when the X,^ and ^2 are both positive and the vector field is directed away from P^. A further classification of nodes as well as other critical points is based on stability. If all curves that are close to P^ at some instant remain close to P^ for all time, then the critical point is stable, A critical point that is not stable is defined to be unstable. An example of a critical point that is unstable is a saddle point . The critical point is a saddle point when and ^2 are both real but have opposite signs. Two lines, called the separatrixes, pass through P^ . The path orientation is positive along one separatrix and negative along the other. All other paths avoid the critical point. A third type of behavior occurs when X,^ and ^2 are complex conjugates. Then, the critical point is a spiral node. An infinite number of paths form spirals about P with P as an ^ ^ 00 asymptotic point. The trajectory spirals away from the critical point if the real part of the coitplex conjugates is positive or flows into the critical point if it is negative. Several examples of simple, two-dimensional critical points are sketched in Figure 32. i

PAGE 84

74 A primary motivation for applying topological concepts to the observed flow patterns is the relation between the niomber of nodes and saddle points that can exist in the flow. For the jet in a crossflow, which is topologically equivalent to the flow at the intersection of two pipes. Hunt et al. (86) show that on the surface the number of saddle points must exceed the n\amber of nodes by one or 1^-1^ = -^ (30) where N denotes a node and S denotes a saddle point. The flow in a cross-sectional plane such as the symmetry plane satisfies a different s\immation rule. Specifically, there must be an equal number of nodes and saddle points so that 1^-1^ = ° (31) The application of these kinematical principles, in particular the summation rule, simplifies the interpretation of complicated flow patterns by categorizing the flow structures and removing the ambiguity that can occur when inferring flow patterns from the visualized results. Observed Flow Patterns Surface Shear Stress. Surface particle traces may be constructed by restricting particles to the plane just above the flat plate; that is, the normal velocity component is set to zero during the time integration. These surface particle lines are analogous to the surface shear stress lines that are seen in oil flow surface visualization experiments. The surface particle lines represent a continuous vector field that has continuous derivatives. Further, the vector field is zero at only a finite

PAGE 85

75 nimiber of points and is located in a plane that is a small distance above the surface. Thus, the rules of topology apply. The surface particle paths taken at a height of roughly 0 . 65* above the plate for R = 4 and 6 are shown in Figures 33 and 34 respectively. Results for the R = 8 jet are excluded because the first grid point in the coarse grid is located too far above the surface for the restricted particle traces to mirror the surface shear stress patterns. In both of the cases shown, fluid from upstream of the jet exit approaches the jet, deflects around the jet periphery and moves into the boundary of the wake. In the wake region, some flow is entrained forward by the jet. Behind this reversed flow, fluid moves downstream within the wake. The oil flow patterns observed by McMahon and Mosher (22), described in Chapter 2 and presented as Figure 35 for the case R = 8, exhibit similar behavior. In both the conputed and experimental results, the wake broadens as the velocity ratio is increased. The summation rule of topology for a jet in a crossflow can be satisfied by the existence of a single saddle point in the flowfield. However, for the range of effective velocity ratios studied, several critical points are apparent. A sketch of the flow structures, called the phase portrait of the flow, is shown in Figure 36. If the jet is assumed to behave like a solid body near the flat plate, then the number of nodes must equal the number of saddle points at the jet exit. Saddle points located at the solid surface are called half saddles and denoted by S'. In the phase portrait constructed from the coitputed surface particle traces.

PAGE 86

76 there are four half-saddles and two nodes around the jet with a net attachment saddle located downstream of the jet exit so that which indicates that the observed flow is kinematically possible. In this flow, it is the aft saddle point that supplies forward flow to the aft half-saddle and entrainment into the jet. The two spiral nodes correspond to the low pressure region at the sides of the jet and feed flow into the contrarotating vortices. A larger system of self-cancelling nodes and saddles may exist than the one depicted by the phase portrait constructed from these computations. Although a more detailed analysis of the flow near the plate with a greater grid density is needed to unveil any secondary topological flow features, the current analysis verifies that the surface flow patterns are conpatible with a continuous velocity field for a jet in a crossflow. Symmetry Plane Flow. The flow topology in the symmetry plane can be investigated by restricting the particle traces to the plane y = 0.0. By assuming that the jet can be adequately represented as a point jet, that is, by collapsing the jet to a single point, a very simple portrait of the flow pattern that is shown in Figure 37 for the R = 6 jet can be constructed. The phase portrait that is sketched in Figure 38 contains two half-saddles on the plate and a spiral node of separation that is located slightly above and forward of the aft half -saddle. The aft half-saddle corresponds to the attachment saddle that was observed in the computed surface shear stress pattern. Flow from the spiral node fills the region below the jet plume. Since the nujnber of nodes

PAGE 87

77 equals the number of saddles, the flow in the symmetry plane satisfies the summation rule in equation (31) . Thus, the flow visualization is validated for at least two planes in the f lowf ield.

PAGE 88

78 500. 1000. 1500. 2000. 2500. ITERATION Figure 3 . Convergence history for a nontime-accurate computation. This history is for the case R with 151,250 grid points.

PAGE 89

79 (a) Figure 4 . Vertical profiles of the velocity component in the freestream direction taken in the symmetry plane at several locations downstream of the jet exit using two grid structures. R = 4. a) x/D = 1.0.

PAGE 90

80 Figure 4 . Continued; (c) b) x/D = 2.0; c) x/D =3.0.

PAGE 91

81 Q N Q u/Uinf (d) 16. 14 . 12. 10. 8. 6. 4 . 2. 0. 39 X 35 X 32 Grid 55 X 55 X 50 Grid Fearn and Weston (1978) -1.0 -.5 0 .5 1.0 1.5 u/Uinf (e) Figure 4. Continued; d) x/D = 4.0; e) x/D = 8.0,

PAGE 92

82 (a) Figure 5. Vertical profiles of the velocity component in the freestream direction taken in the symmetry plane at several locations downstream of the jet exit using two grid structures. R = 6. a) x/D =1.0.

PAGE 93

83 D 1 1 1 1 39 X 35 X 32 Grid 55 X 55 X 50 Grid . ... — 1 ...... .1 1 ~ * \ > " " Vs '.^^^ — — > -1.0 -.5 .0 .5 1.0 1.5 u/Uinf (c) Figure 5. Continued; b) x/D = 2.0; c) x/D =3.0.

PAGE 94

84 Figure 5. Continued; (e) d) x/D = 4.0; e) x/D =8.0.

PAGE 95

85 1 Figure 6. Velocity vector field in the symmetry plane for R = 4. In the region shown, x/D = [-6, 10] and z/D = [0, 16] ."

PAGE 96

86 Figure 7. Mach number contours in the syrnmetry plane for R = 4. In the region shown, x/D = [-6, 10] and z/D = [0, 16] . The contours range from 0.05 to 0.95 with an increment of 0.05. High and low Mach number regions are denoted by H and L, respectively.

PAGE 97

87 16 14 12 10 8 / / / / / 1 1 / / 1 1 / / 1 f / / 1 / / / / / / / / -6 -4 -2 8 10 Figure 8. Velocity vector field in the syinmetry plane for R = 6. In the region shown, x/D = [-6, 10] and z/D = [0, 16] ,

PAGE 98

88 Figure 9. Mach number contours in the symmetry plane for R = 6. In the region shown, x/D = [-6, 10] and z/D = [0, 16]. The contours range from 0.05 to 0.95 with an increment of 0.05. High and low Mach nximber regions are denoted by H and L, respectively.

PAGE 99

89

PAGE 100

90 1 ° 2 0 -1 0 x/D Figure 11. Measured symmetry plane velocities for R = 4 from Fearn and Benson (32) . The two rakes of vectors are centered at x/D = 0.67, z/D =0.96 and 9 = 88° and at x/D = 1.25, z/D = 1,67 and


PAGE 101

91 Figure 12. Empirical, experimental and computed jet centerlines a) R = 4; b) R = 6.

PAGE 102

92

PAGE 103

93 16 14 12 10 / /, / / / / / / / -6 -4 8 10 Figure 13. Velocity vector field in the symmetry plane for the R = 4, inviscid calculation. In the region shown x/D = [-6,10] and z/D = [0,16]

PAGE 104

94 Q N 16. 14 . 12. 10. Viscous Inviscid
PAGE 105

95

PAGE 106

96 Figure 14. Continued; (e) d) x/D = 4.0; e) x/D =8.0.

PAGE 107

97 CD 2.5 — R = 4 -R = 4, Inviscid O R = 3.95, Chassaing et al. (1 974) R » 4. Fearn and Weston (1 978) 5. 7.5 10 s/D 12.5 15. 17.5 (a) u D 7 . 6. 5. 4 . 3. 2. 1. 0. o R = 6 R = 6.35, Chassaing et al. (1974) R = 6, Fearn and Weston (1 978) 2.5 5. 7.5 10 s/D 12.5 15, 17,5 (b) Figure 15. Velocity decay along the jet centerline, b) R = 6. a) R = 4; r

PAGE 108

98

PAGE 109

99 i I I ////// /////// J I i /////// J i V //////" " " ' / / / / / ^ / / / / / ^ / / / / / ^ III//' I I I i J ^ I I I I r \ \ ^ t / \ t / \ t / t I I / 1 I I ! \ \ \ \-^// y \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I \ \ \ \ I I Figure 17. Velocity vectors in a vortex cross section for R = 4 This cross section is centered at x/D = 3.0, z/D = 2.67 and (p= 21°.

PAGE 110

100 1 1 1 ' T 1— , t 1 ^ t f \ / / / III'' ^ ^ \ \ \ \ / / ill'' f s s . . ' ^ \ \ \ \ / / /III' \ \ / / / / / / ' ^ ^ \ \ \ \ \ / / / / / / / ^ " " N \ \ \ \ \ / / / / / / / ^ \ \ \ \ \ \ / / / / / / / / 1 \ ^ \ \ \ \ \ \ / / / / / / ' \ t / ^ \ \ \ \ \ \ / / Jin' 1 1 1 ' M \ \ \ \ / / / / 1 \ / f \ 'MM \ / 1 / t \ ^ M M I \ i ^M / / 1 \ 1 / / 1 1 l \ \ 1 1 \ \ \ \ \ 1 / \ 1 I "M / / , / / / / 1 \ Figure 18. Velocity vectors in a vortex cross section for the R = 4, inviscid case. This cross section is centered at x/D = 3.0, z/D = 2.67 and
PAGE 111

101 Figure 19. Geometry for the diffuse vortex model.

PAGE 112

102 12 10. N 10. x/D 15, 20 (a) 12 10 Q ^ 6. 4 . O 39 X 35 X 32 Grid 55 X 55 X 50 Grid A Fearn and Weston (1 974) Karagozian and Greber (1986) A, 10. x/D 15, 20 (b) Figure 20. Properties of the contrarotating vortex pair inferred using the diffuse vortex model. R = 4. a) Vortex penetration; b) Vortex spacing.

PAGE 113

103 4. 3. O 39 X 35 X 32 Grid 55 X 55 X 50 Grid A Fearn and Weston (1974) O CN 9 a A 1. ° o o o A, 0. 5. 10 . x/D 15, 20 (c) 12 . 10 8. O 39 X 35 X 32 Grid 55 X 55 X 50 Grid A Fearn and Weston (1974) o u 6. 8 £ 8 8 0° S ^ Ai 0. 10. x/D 15. 20 (d) Figure 20. Continued; c) Vortex strength; d) Vortex core size.

PAGE 114

104 12 10, A 55 X 55 X 50 Grid Fearn and Weston (1 974) Karagozian and Greber (1986) 10. x/D 15, 20, (b) Figure 21. Properties of the contrarotating vortex pair inferred using the diffuse vortex model. R = 6. a) Vortex penetration; b) Vortex spacing.

PAGE 115

105 55 X 55 X 50 Grid A Fearn and Weston (1974) 0. 10 . x/D 15, 20, (c) 55 X 55 X 50 Grid A Fearn and Weston (1974) p 6 D 10. x/D 15, 20, (d) Figure 21. Continued; c) Vortex strength; d) Vortex core size.

PAGE 116

106 y A 180° X -90° f Figure 22. Definition sketch for pressure coefficient locations. o e — O dOOQ^ 0 Fearn and Weston (1975) — B — Computed -8. -7. -6. -5. -4. -3. -2. -1. 0. x/D (a) Figure 23. Computed and experimental pressure coefficients on the plate for R = 4. a) 9 = 0°.

PAGE 118

Figure 24. Computed and experimental pressure coefficients on the plate for R = 6. a) 9 = 0°.

PAGE 119

109 . J3 n n n n — a — n. n — _ ._. -a cj — a H H J^OOOOo c ) O " U c [ O Fearn and Weston (1975) — B — Ckjmputed 0. 1. 2. 3. 4. 5. 6. 7. 8. x/D (c) Figure 24. Continued; b) 9 = 90°; c) 9 = 180°.

PAGE 120

110 Figure 25. Coitputed and experimental pressure coefficients on the plate for R = 8. a) 9 = 0°.

PAGE 121

Ill

PAGE 122

112 p x/D (a) Figure 26. Plate pressure distribution for R = 4. a) The pressure contours for the measurements by Fearn and Weston (6) are plotted in the half-plane y > 0 and the computational results are plotted in the half-plane y < 0.

PAGE 123

113 Figure 26. Continued; b) The pressure contours for the inviscid calculation are plotted in the half-plane y > 0 and the viscous results are plotted in the half -plane y < 0.

PAGE 124

114 x/D Figure 27. Plate pressure distribution for R = 6. The pressure contours for the measurements by Fearn and Weston (6) are plotted in the half-plane y > 0 and the computational results are plotted in the half-plane y < 0.

PAGE 125

115 Figure 28. Particle traces for R = 4. Particles are released within the boundary layer upstream of the jet exit and from the jet exit itself.

PAGE 126

116 (a) Figure 29. Particle traces for R = 6. a) Particles are released within the boundary layer upstream of the jet exit and from the jet exit itself.

PAGE 127

117 (c) Figure 29. Continued; b) Particles are released at the jet exit in the symmetry plane; c) Particles are released at the jet exit along the line x = 0.0 and viewed from upstream of the jet . r'

PAGE 128

118 (e) Figure 29. Continued; d) Particles are released at the jet exit along the line x = 0.0 and viewed from the side. e) Particles are released within the boundary layer < upstream of the jet.

PAGE 129

Figure 30. Particle traces for R = 8. Particles are released within the boundary layer upstream of the jet exit and from the jet exit itself .

PAGE 130

120 (a) Figure 31. Negative time particle traces for a group of uniforrtay spaced particles released from the vortex cross section centered at x/D = 5.0, z/D = 5.2 and


PAGE 131

121

PAGE 132

122 STABLE NODE OF SEPARATION UNSTABLE NODE OF ATTACHMENT UNSTABLE SADDLE POINT STABLE SPIRAL NODE UNSTABLE SPIRAL NODE Figure 32. Two-dimensional simple critical points.

PAGE 133

123 Figure 33. Surface particle traces for R = 4.

PAGE 134

124 Figure 34. Surface particle traces for R = 6.

PAGE 135

125 Figure 35. Oil flow visualization of a jet in a crossflow by McMahon and Mosher (Reference 22) for R = 8.

PAGE 136

126 Figure 36. Flat plate phase portrait. In this pattern, there are four half-saddles and two nodes around the jet exit and one saddle point downstream of the jet exit.

PAGE 137

127

PAGE 138

128 Figure 38. Symmetry plane phase portrait. This pattern contains two half -saddles and one node.

PAGE 139

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY A three-dimensional, thin-layer Navier-Stokes algorithm was used to calculate the aerodynamic/propulsive interaction between a siabsonic jet exhausting perpendicularly through a flat plate into a crossflow. Computed results were presented and analyzed for flows with velocity ratios of R = 4, 6 and 8. All of the pertinent flow physics, including the jet trajectory, the contrarotating vortex pair and the wake region near the flat plate downstream of the jet exit, were captured numerically. Several major conclusions are drawn from this research. In addition, future research directions of this work are projected in the areas of both niomerics and physics. The following observations are made: 1. Calculations are presented for computational grids with two different grid densities. Examination of the flow in the symmetry plane indicated that more of the actual flow physics are captured on the higher resolution grid. Further, the increased numerical diffusion on the coarse mesh significantly altered the vortex properties. Thus, using the current clustered rectangular grid structure, a grid density of at least 150,000 points is required to provide good correlation with the experimental data in the high 129

PAGE 140

130 gradient regions of the flow such as the shear layer between the jet and the freestream and the wake. 2. Three-dimensional flow visualization on a color graphics workstation is required in order to understand the complex interaction between the jet and the crossflow, especially in the near jet region. The off-surface particle traces present a comprehensive picture of the flow that cannot be constructed from the relatively sparse experimental measurements. Notably, the visualization shows that the jet centerline is comprised primarily of original jet fluid while the cores of the contrarotating vortices contain fluid that originated within the flat plate boundary layer. In addition, the surface particle traces are topologically correct and agree with the experimental oil flow patterns . 3. The jet centerline can be predicted with a small niamber of grid points and is relatively insensitive to viscous effects. This indicates that the jet trajectory is dominated primarily by the pressure effects. Therefore, although a niomerical simulation should accurately capture the jet centerline, the accuracy of the simulation cannot be assessed on the prediction of the centerline alone . 4. To the author's knowledge, this is the first numerical solution of the Navier-Stokes equations for a jet in a crossflow in which the computed vortex properties are directly conpared with the vortex properties determined for the measured flow. Overall, reasonable agreement between the computation and the experiment for the vortex strength, location and diffusivity calculated with the

PAGE 141

131 diffuse vortex model is demonstrated. The predicted contrarotating vortices are slightly closer together and closer to the plate and the vortex strength is greater than in the experiment. These deviations are more pronounced on the coarse mesh. 5. Although the wake region is characteristic of a jet wake rather than a wake behind a solid blockage, the predicted pressure distribution on the plate downstream of the jet for these laminar calculations is higher than in the experimental measurements. With the exception of the rapid pressure recovery in the wake, the general trends of the pressure contours on the plate match the experiment . 6. A full three-dimensional computational space, with no symmetry imposed, is used in these simulations so that symmetry in the n\americal solution can be investigated. In the present steady state, laminar computations, none of the global jet features such as the contrarotating vortices or the wake exhibit assymmetry. Thus, future calculations could be made using only half of the computational domain and applying a symmetry boundary condition on the y = 0 plane. On the other hand, some of the secondary features of a jet in a crossflow may not be symmetric. For example, if the computations are extended to include unsteadiness and to capture the horse-shoe vortex or the shed vortex behind the jet, the flow near the plate is expected to be assymmetric. In short, care should be taken, especially for investigations directed toward fluid mechanics evaluation, when determining the appropriateness of the symmetry assiimption.

PAGE 142

132 7. The uniform velocity/constant pressure jet exit boundary condition is inadequate. A detailed experimental survey of the jet exit profile is needed to guide the selection of the jet boundary conditions for future simulations. Also, the representation of the circular jet orifice on a rectangular grid did not significantly influence the computations provided that the mass flow and momentum matched the experiment. 8. The viscous flow problem that is studied in this research is confined to laminar flow assumptions. Some of the discrepancies between the n\americal solution and the experimental data may be due to the presence of turbulent fluid in the physical flow. For example, in the experiment, the flat plate boundary layer is turbulent. The flow visualization indicates that some of this fluid is entrained into the vortex cores; hence, the characteristics of the boundary layer affect the vortex properties. Also, the wake region contains turbulent fluid. The rapid pressure recovery exhibited in the wake in this simulation may be due to neglecting turbulence. In future applications, the implementation of a simple and robust turbulence model may be needed to capture the turbulent flow behavior. 9. The results of the present investigation indicate that while the thin-layer equations are necessary for the computation, they may not be adequate. Even though the solution of the thin-layer equations yields reasonable results for the global features of the jet in a crossflow, for completeness, the effects of adding both the streamwise and lateral viscous terms should be investigated. Since the physical flow is fully three-dimensional, these

PAGE 143

133 additional viscous terms may be needed to capture secondary flow features such as the horse-shoe vortex. Additionally, the inclusion of viscous terms in the streamwise and lateral directions may improve the spreading rate for the contrarotating vortex pair. It is unclear whether the extended equations must be fully threedimensional or if adding the thin-layer terms in the remaining two directions would be sufficient . 10. The jet in a crossflow is computed by assuming that the flow is steady in order to decrease the amount of time needed for code convergence. To fully resolve the detailed, unsteady features of the flowfield, time accurate computations need to be made. Other researchers are currently working to develop the hardware and software that will be necessary for analyzing large, threedimensional, unsteady datasets. The computational results presented in this study seem to indicate several areas of concern rather than providing a definitive statement on the mechanisms governing the flowfield induced by a subsonic jet in a crossflow. Thus, the major inpact of this study is the knowledge base that can be applied in future attempts to analyze propulsive flowfields using the Navier-Stokes equations. With the advantage of hindsight and with the developments in conputational fluid dynamics since the initiation of this particular study, pertinent suggestions covering four major areas: grid structure, application of boundary conditions, selection of governing equations and turbulence modelling, can be made for initiating a follow on study. i

PAGE 144

134 In future studies, probably the most important first step toward realizing improved computational results for the jet in a crossflow will be refining the computational grid. Increasing the grid density will be both imperative and interesting, not only to eliminate the grid dependence but also to uncover more detailed flow features. A refined grid will be needed even before the influence of the governing equations and turbulence modelling can be analyzed. In addition, new gridding schemes which could place a higher density of points within the jet plume while maintaining a relatively sparse grid structure in the inviscid, outer flow region should be explored. Ideally, a solution adaptive grid should be used in order to optimize the grid point location for resolving the high gradient flow regions and to minimize the required number of grid points without increasing the numerical diffusion. In conjunction with the improved grid resolution near the jet orifice, an improved jet exit boundary condition will need to be applied since the niamerical diffusion caused by the grid spacing will no longer compensate for the constant velocity/constant pressure condition. It is suggested that the boundary condition be specified in terms of constant total pressure. Next, provided that the computational grid is sufficiently dense, the effect of adding additional viscous terms to the governing equations can be tested. A first approach might be to add the streamwise thin-layer terms, then the lateral thin-layer terms and finally, if needed, all of the viscous terms. While the flow certainly has turbulent features, it is believed that turbulence is a secondary effect. The effects of a

PAGE 145

135 turbulence model should be evaluated after the refinements to the grid, boundary conditions and equations are made. Undoubtably, future investigations will be directed toward the aerodynamic /propulsive interaction problem associated with realistic STOVL aircraft configurations . As complex geometries are studied and new technologies such as moving grids are introduced for use in simulating the motion of the aircraft with respect to the ground, the risk of missing the actual physics in the numerical solution is greatly increased. Since the present results provide an excellent baseline case for comparison with future jet in a crossflow calculations, code changes can be tested and verified on the simpler jet flat plate configuration prior to costly simulations of the entire aircraft.

PAGE 146

APPENDIX THIN-LAYER NAVIER-STOKES EQUATIONS The flux terms for the thin-layer Navier-Stokes equations written in the generalized coordinates (x,^,Tl,C) in Equation (1) are given by pU puU + 4^p pVU+^yP pwU + ^ (e+p)U-^^p Q = J 1 G = J 1 p pu pv F = pw .e . f-pv puV+Tl^p pvV + TlyP pwV + n^P _(e + p)V-Tl^V 1 and H = J 1 pw puW+ ;^p PVW+ CyP pwW+ C^p (e + p)W -C^P_ and the viscous flux in the ^ direction is given by S = J 0. 0 ^{Cx + Cy + C'J + l(^x + Cy + C,w^)C^ (si + ^ y + Oh 5h( + v^ + w^) ^ + K Pr^(Y 1)" ' (a^) J 136

PAGE 147

137 The pressure p is defined as P = (Y1) [e-0. 5p(u^ +w^)] in which e denotes the total energy. The Jacobian of the coordinate transformation, J, is a scalar variable which is calculated using the metric terms, Tl^, and so on; expressions for the Jacobian and the metrics are J' = 1 / J = x^p^z ^ y Y^z ^ y + x^p^z ^ y ^2 The contravariant velocity components U, V and W are related to the Cartesian velocity components u, v and w by U = ^^u + ^yV + ^^w + ^^ V=Tl^U+TlyV+Tl^W + Tl^ w=Cx" + CyV + C^w+C^ The viscosity is expressed by \l, Pr is the Prandtl number, K is the coefficient of thermal conductivity and y is the ratio of specific heats .

PAGE 148

REFERENCES 1. Moussa, Z., Trischka, J. W. and Eskinazi, S.: The Near Field in the Mixing of a Round Jet with a Cross Stream. J. Fluid Mech., Vol. 80, Part 1, pp. 49-80, 1977. 2. Kamotani, Y. and Greber, I.: Experiments on a Turbulent Jet in a Cross-Flow. AIAA J., Vol. 10, No. 11, pp. 1425-1429, 1972. 3. McGuirk, J. J. and Rodi, W. : Depth Averaged Model for the Near Field of Side Discharges into Open Channel Flow. J. Fluid Mech., Vol. 86, pp. 761-781, 1978. 4. Margason, R. J. and Fearn, R. L.: Jet-Wake Characteristics and Their Induced Aerodynamic Effects on V/STOL Aircraft. NASA SP218, 1969. 5. Fearn, R. L. and Weston, R. P.: Vorticity Associated with a Jet in a Crossflow. AIAA J., Vol. 12, No. 12, pp. 1666-1671, 1974. 6. Fearn, R. L. and Weston, R. P.: Induced Pressure Distribution of a Jet in a Crossflow. NASA TN D-7616, 1975. 7. Fearn, R. L. and Weston, R. P.: Induced Velocity Field of a Jet in a Crossflow. NASA TP-1087, 1978. 8. Fearn, R. L. and Weston, R. P.: Velocity Field of a Round Jet in a Crossflow for Various Jet Injection Angles and Velocity Ratios. NASA TP-1506, 1979. 9. Schetz, J. A., Jakubowski, A. K. and Aoyagi, K. : Jet Trajectories and Surface Pressures Induced on a Body of Revolution with Various Dual Jet Configurations. AIAA Paper 83-0080, 1983. 10. Snyder, P. and Orloff, K. L.: Three-Dimensional Laser Doppler Anemometer Measurements of a Jet in a Crossflow. NASA TM 85997, 1984. 11. Spee, B. M. : Technical Evaluation Report on the AGARD Fluid Dynamics Panel Symposium on Fluid Dynamics of Jets with Applications to V/STOL. AGARD Advisory Report No. 187, 1982. 12. Crabb, D., Durao, D. F. G. and Whitelaw, J. H.: A Jet Normal to a Crossflow. Imperial College, London, FS/78/35, 1979. 13. Callaghan, E. E. and Ruggeri, R. S.: Investigation of the Penetration of an Air Jet Directed Perpendicularly to an Air Stream. NACA TN 1615, 1948. 138

PAGE 149

139 14. Ruggeri, R. S., Callaghan, E. E. and Bowden, D. T.: Penetration of Air Jets Issuing from Circular, Square and Elliptical Orifices Directed Perpendicularly to an Air Stream. NACA TN 2019, 1950. 15. Margason, R. J.: The Path of a Jet Directed at Large Angles to a Subsonic Free Stream. NASA TN D-4919, 1968. 16. Jordinson, R. : Flow in a Jet Directed Normal to the Wind. R. and M. No. 3074, British A.R.C. Technical Report, 1958. 17. Thompson, A. M. : The Flow Induced by Jets Exhausting Normally from a Plane Wall into an Airstream. Ph.D. Thesis, University of London, 1971. 18. Mosher, D. K.: An Experimental Investigation of a Turbulent Jet in a Cross Flow. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1970. 19. O.N.E.R.A. Motion-Picture Film No. 575: Flows with Large Velocity Fluctuations. O.N.E.R.A., Chatillon, France, 1968. 20. Antani, D.: An Experimental Investigation of the Vortices and the Wake Region Associated with a Jet in Crossflow. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1977. 21. Harms, L. : Experimental Investigation of the Flow Field of a Hot Turbulent Jet with Lateral Flow Part II. NASA TT F15706, 1974. 22. McMahon, H. M. and Mosher, D. K. : Experimental Investigation of Pressures Induced on a Flat Plate by a Jet Issuing into a Subsonic Crosswind. NASA SP-218, 1969. 23. Wu, J. C, Mosher, D. K. and Wright, M. A.: Experimental and Analytical Investigations of Jets Exhausting into a Deflecting Stream. AIAA Paper 69-223, 1969. 24. Vogler, R. D.: Surface Pressure Distributions Induced on a Flat Plate by a Cold Air Jet Issuing Perpendicularly From the Plate and Normal to a Low-Speed Free-Stream Flow, NASA TN1629, 1963. 25. Bradbury, L. J. S. and Wood, M. N. : The Static Pressure Distribution Around a Circular Jet Exhausting Normally from a Plane Wall into an Airstream. CP. No. 822, Brit. A.R.C, 1965. 26. Ousterhout, D. S.: An Experimental Investigation of a Cold Jet Emitting from a Body of Revolution into a Subsonic Free Stream. NASA CR-2089, 1972. 27. Soullier, A.: Testing at SI. MA for Basic Investigation on Jet Interactions — Distribution of Pressures and Velocities in the Jet Using the Ideal Standard Nozzle (In Unheated State) . NASA TT F-14702, 1972.

PAGE 150

140 28. Soullier, A.: Testing at SI. MA for Basic Investigation on Jet Interactions — Distribution of Pressures Around the Jet Orifice. NASA TT F-14066, 1972. 29. Kuhlman, J. M. , Ousterhout, D. S. and Warcup, R. W. : Experimental Investigation of Effect of Jet Decay Rate on JetInduced Pressures on a Flat Plate. NASA CR-2979, 1978. 30. Fearn, R., Doddington, H. and Westphal, R. : LDV Studies of a Jet in a Crossflow. NADC Technical Report 80238-60, 1981. 31. Aoyagi, K. and Snyder, P. K. : Experimental Investigation of a Jet Inclined to a Subsonic Crossflow. AIAA Paper 81-2610, 1981. 32. Fearn, R. L. and Benson, J. P.: Velocity Field Near the Jet Orifice of a Round Jet in A Crossflow. NASA CR 152293, 1979. 33. Patankar, S. V., Basu, D. K. and Alpay, S. A.: Prediction of the Three-Dimensional Velocity Field of a Deflected Turbulent Jet. Trans. ASME J. Fluids Engng., Vol. 99, No. 4, pp. 758762, 1977. 34. Chang, H.: Aufrollung lines zylindrischen Strables durch Querwind. Ph.D. Thesis, Gottingen, 1942. 35. Wooler, P. T. : Development of an Analytical Model for the Flow of a Jet into a Subsonic Crosswind. NASA SP-218, 1969. 36. Wooler, P. T., Kao, H. C, Schwendemann, M. F., Wasson, H. R. and Ziegler, H.: V/STOL Aircraft Aerodynamic Prediction Methods Investigation. Technical Report AFFDL-TR-72-26, Vol. I IV, 1972. 37. Heltsley, F. L., and Parker, R. L. : Application of the VortexLattice Method to Represent a Jet Exhausting from a Flat Plate into a Crossf lowing Stream. AEDC-TR-73-57, 1973. 38. Karagozian, A. and Greber, I.: An Analytical Model for the Vorticity Associated with a Transverse Jet. AIAA Paper 841662, 1984. 39. Rubbert, P. E. : Calculation of Jet Interference Effects on V/STOL Aircraft by a Nonplanar Potential Flow Method. NASA SP218, 1969. 40. Furlong, K. L. and Fearn, R. L. : A Lifting Surface Computer Code with Jet-in-Crossf low Interference Effects. NASA CR 166524, Vol. I II, 1983. 41. Maskew, B., Strash, D., Nathman, J. and Dvorak, F. A.: Investigation to Advance Prediction Techniques of the Low-Speed Aerodynamics of V/STOL Aircraft. NASA CR-166479, 1983. 42. Perkins, S. C. and Mendenhall, M. R. : Computer Programs to Predict Induced Effects of Jets Exhausting Into a Crossflow. NASA CR 166591, 1984.

PAGE 151

141 43. Kotansky, D. R. and Glaze, L. W. : STOL Landing Thrust-Reverser Jet Flowfields. NASA CP-24 62, 1985. 44. Howell, G. A.: Automated Surface and Plume Simulation Procedure for Use with Aerodynamic Panel Codes . NASA CR 177420, 1986. 45. Adler, D. and Baron, A.: Prediction of a Three-Dimensional Circular Turbulent Jet in a Crossflow. AIAA J., Vol. 17, No. 2, pp. 168-174, 1978. 46. Hancock, G. J. : A Review of the Aerodynamics of a Jet in a Cross Flow. Aeronautical J., Vol. 91, No. 905, pp. 201-213, 1987. 47. Van Dalsem, W. R. : Study of Jet in Ground Effect with Crossflow Using the Fortified Navier-Stokes Scheme. AIAA Paper 87-2279, 1987. 48. Claus, R. W, : Numerical Calculation of Subsonic Jets in Crossflow with Reduced Numerical Diffusion. AIAA Paper 851441, 1985. 49. Childs, R. E. and Nixon, D.: Unsteady Three-Dimensional Simulations of VTOL Upwash Fountain Turbulence. NASA CP-2462, 1985. 50. Rizk, M. H. and Menon, S.: Numerical Investigation of V/STOL Jet Induced Interactions. NASA CP-2462, 1985. 51. Baker, A. J. and Manhardt, P. D.: A Numerical Viscous-Inviscid Interaction Algorithm for Prediction of Near-Field V/STOL JetInduced Flowfields. NADC Report 77167-30, 1979. 52. Baker, A. J. and Orzechowski, J. A.: Prediction of Turbulent Near-Field Evolution of a Jet in a Crossflow Using a PNS Solver. NADC Report 85177, 1986. 53. Baker, A. J., Orzechowski, J. A. and Manhardt, P. D.: A Nvmierical Three-Dimensional Turbulent Simulation of a Subsonic V/STOL Jet in Cross-Flow Using a Finite Element Algorithm. NADC Report 79021-60, 1981. 54. Baker, A. J., Orzechowski, J. A., Manhardt, P. D. and Yen, K. T. : A Three-Dimensional Finite Element Algorithm for Prediction of V/STOL Jet-Induced Flowfields. AGARD-CP-308, 1982. 55. Baker, A. J., Snyder, P. K. and Orzechowski, J. A.: Three Dimensional Nearfield Characterization of a VSTOL Jet in Turbulent Crossflow. AIAA Paper 87-0051, 1987. 56. Chien, C. J. and Schetz, J. A.: Numerical Solution of the Three-Dimensional Navier-Stokes Equations with Application to Channel Flows and a Bouyant Jet in a Cross-Flow. Trans. ASME E: J. Appl. Mech., Vol. 42, pp. 575-579, 1975.

PAGE 152

142 57. Demuren, A. 0.: Numerical Calculations of Steady ThreeDimensional Turbulent Jets in Cross Flow. Comp. Meth. Appl. Mech. and Engng., Vol. 37, pp. 309-328, 1983. 58. Sykes, R. I., Lewellen, W. S. and Parker, S. F.: On the Vorticity Dynamics of a Turbulent Jet in a Crossflow. J, Fluid Mech., Vol. 168, pp. 393-413, 1986. 59. Reed, C. L. : PropulsiveJet Flow Field Analysis Using the Three-Dimensional Navier-Stokes Equations . NASA Contract Report, 1987. 60. Roth, K. R. : Numerical Simulation of a Subsonic Jet in a Crossflow. SAE Paper 872343, 1987. 61. Harloff, G. J. and Lytle, J. K. : Three-Dimensional Viscous Flow Computations of a Circular Jet in a Subsonic and Supersonic Cross Flow. AIAA Paper 88-3703, 1988. 62. Andreopoulos, J.: Measurements in a Jet-Pipe Flow Issuing Perpendicularly into a Cross Stream. Trans. ASME J, Fluids Engng., Vol. 104, pp. 493-499, 1982. 63. Abramovich, G. N. : The Theory of Turbulent Jets. M.I.T. Press, Cambridge, MA, 1963. 64. Keffer, J. F. and Baines, W. D.: The Round Turbulent Jet in a Cross-wind. J. Fluid Mech., Vol. 15, Part 4, 1963. 65. Analysis of a Jet in a Subsonic Crosswind. NASA SP-218, 1969. 66. Thompson, J. F,: Two Approaches to the Three-Dimensional Jetin-Cross Wind Problem. A Vortex Lattice Model and a Numerical Solution of the Navier-Stokes Equations. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1971. 67. Eskinazi, S.: Fluid Mechanics and Thermodynamics of Our Environment. Academic Press, New York, 1975. 68. Bradbury, L. J. S.: Some Aspects of Jet Dynamics and Their Implications for VTOL Research. AGARD-CP-308, 1981. 69. Fearn, R. L. : Progress Towards a Model to Describe Jet/Aerodynamic-Surface Interference Effects. AIAA J., Vol. 22, No. 6, pp. 752-753, 1984. 70. Nunn, R. H.: Vorticity Growth and Decay in the Jet in Cross Flow. AIAA J., Vol. 23, No. 3, pp. 473-475, 1985. 71. Margason, R. : Propulsion-Induced Effects Caused by Out-ofGround Effects. NASA TM 100032, 1987. 72. Margason, R. and Kuhn, R. : Application of Empirical and Linear Methods to VSTOL Powered-Lift Aerodynamics. SAE Paper 872341, 1987.

PAGE 153

143 11 73. Krothapalli, A., Lourenco, L. and Buchlin, J. M. : The Structure of the Separated Flow Region Upstream of a Jet in a Cross Flow. Unpublished abstract. 74. Chassaing, P., George, J., Claria, A. and Sananes, F. : Physical Characteristics of Subsonic Jets in a Cross-Stream. J. Fluid Mech., Vol. 62, Part 1, pp. 41-64, 1974. 75. Prandtl, L.: Ueber die ausgebildete Turbulenz . Proceedings of the 2nd International Congress for Applied Mechanics, Zurich, pp. 62-74, 1926. 76. Baldwin, B. S. and Lomax, H,: Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows. AIAA Paper 78257, 1978. 77. Steger, J. L., Ying, S. X. and Schiff, L. B. : A Partially Flux-Split Algorithm for Numerical Simulation of Compressible Inviscid and Viscous Flow. Proceedings of the Workshop on Computational Fluid Dynamics, Institute of Nonlinear Sciences, University of California, Davis, CA, 1986. 78. Ying, S. X., Steger, J. L., Schiff, L. B. and Baganoff, D.: Numerical Simulation of Unsteady, Viscous, High Angle of Attack Flows Using a Partially Flux-Split Algorithm. AIAA Paper 862179, 1986. 79. Beam, R. and Warming, R. F.: An Iitplicit Finite-Difference Algorithm for Hyperbolic Systems in Conservation Law Form. J. Con^ut. Phys., Vol. 22, pp. 87-110, 1976. 80. Pulliam, T. H. and Steger, J. L. : On Implicit FiniteDifference Simulations of Three-Dimensional Flow. AIAA Paper 78-10, 1978. 81. Ying, S. X.: Three-Dimensional Implicit Approximately Factored Schemes for the Equations of Gasdynamics. Ph.D. Thesis, Stanford University, Stanford, CA, 1986. 82. Pulliam, T. H. and Chaussee, D. S.: A Diagonal Form of an Implicit Approximate-Factorization Algorithm. J. Comp. Physics, Vol. 39, No. 2, pp. 347 363, 1981, 83. Steger, J. L. and Warming, R. F.: Flux-Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite Difference Methods. J. Comput . Phys., Vol. 40, No. 2, pp. 263-293, 1981. 84. Buning, P. G. : Computation of Inviscid Transonic Flow Using Flux Vector Splitting in Generalized Coordinates. Ph.D. Thesis, Stanford University, Stanford, CA, 1983. 85. Srinivasan, G. R., Chyu, W. J. and Steger, J. L. : Conputation of Simple Three-Dimensional Wing-Vortex Interaction in Transonic Flow. AIAA Paper 81-1206, 1981.

PAGE 154

144 86. Hunt, J. C. R., Abell, C. J., Peterka, J. A. and Woo, H. : Kinematical Studies of the Flows Around Free or Surface-Mounted Obstacles: Applying Topology to Flow Visualization. J. Fluid Mech., Vol. 86, Part 1, pp. 179-200, 1978. 87. Foss, J. F. : Interaction Region Phenomena for the Jet in a Cross-Flow Problem. SFB 80/E/161, 1980.

PAGE 155

I BIOGRAPHICAL SKETCH Karlin Ren6e Roth, the oldest of three children in the family of Mr. and Mrs. Richard M. Roth, was born on October 18, 1961, in Jeannette, Pennsylvania. She received her diploma from Hempfield Area High School in Greensburg, Pennsylvania, in 1979. Following graduation she attended the Indiana University of Pennsylvania, where she received a Bachelor of Science degree, with honors, in applied mathematics in 1983. Karlin continued her education by enrolling in the graduate program in the Engineering Sciences Department at the University of Florida. She received a Master of Science degree from the University of Florida in 1986. Her thesis project, titled "Stability of Closed Loop Filaments as Computational Elements for a Three-Dimensional Vortex Filament Algorithm," was sponsored by NASA Grant NSG-2288. Her dissertation research was supported by NASA Grant NCC 2-403. 145

PAGE 156

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard L. Fearn, Chair Professor of Aerospace Engineering, Mechanics and Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ^ Chen-Chi Hsu Professor of Aerospace Engineering, Mechanics and Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. L. )fessor Mechanics Hairtffiacic of Aerospace Engineering, and Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy David W. Mikolaitis Assistant Professor of Aerospace Engineering, Mechanics and Engineering Sciences

PAGE 157

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. TomTT-PT^hih Associat/5 Professor of Mechanical Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1988 Dean, Graduate School