A Numerical Investigation of 

Supersonic Jet Interaction for 

Finned Bodies 

Mary J. Graham 

Paul Weinacht 

Julius Brandeis 

Rick Angelini 

ARL-TR-2312 DECEMBER 2000 

Approved for public release; distribution is unlimited. 

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Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-23 12 December 2000 

A Numerical Investigation of Supersonic Jet 

Interaction for Finned Bodies 

Mary J. Graham 

United States Military Academy 

Paul Weinacht 

Weapons & Materials Research Directorate, ARL 

Julius Brandeis 

Rafael, Israel 

Rick Angelini 

Computational and Information Sciences Directorate, ARL 

Approved for public release; distribution is unlimited. 
Abstract 

A detailed numerical investigation of the interaction between a lateral 

jet and the external flow has been performed for a variety of missile 

body geometries. These include non-finned axisymmetrical bodies and 

finned bodies with either SWkes or aft-mounted tail fins. The 

computations were Performed at Mach numbers 2, 4.5, and 8. To 

obtain the numerical results, both Reynolds-averaged Navier-Stokes 

and Euler techniques were aPP lied. The computational results were 

compared with results from a P reviously published wind tunnel study 

that consisted primarily of global force and moment measurements. 

The results show significant interactions of the jet-induced flow field 

with the fin surfaces, which P reduce additional effects compared 

wi& &e body alone. h agreement witi the wind tunnel study, in some 

cases the presence of lifting surfaces resulted in force and/or moment 

amplification of the jet interaction with the missile surfaces. The 

results indicate deamplification of the jet force at Mach 2 for all three 

bodies. Amplification of the jet force was also observed for high Mach 

numbers, particularly for the body with strakes. For the results 

examined here, there were only minor differences in the global force 

and moment predictions when viscous or inviscid techniques were 

used. The dependence of the interaction parameters on angle of attack 

and jet pressure was well predicted by both methods. The numerical 

techniques showed good agreement with the experiments at 

supersonic Mach nurrrbers but only fair agreement for the hypersonic, 

Mach = 8 case. 

ii 
ACKNOWLEDGMENTS 

The experimental data that formed the basis for the computational validation were 

provided, courtesy of Rafael and the Israeli Ministry of Defense Directorate of 

Defense Research and Development. Computational resources were obtained 

through a grant of computer time from the U.S. Army Research Laboratory Major 

Shared Resource Center through the U.S. Department of Defense (DOD) High 

Performance Computing (HPC) Modernization Program. Additionally, support for 

the development and parallelization of the zonal Navier-Stokes flow (ZNSFLOW) 

code has been made through the DOD HPC modernization computing software 

support initiative (CHSSI) program. 

. . . 111 
INTENTIONALLYLEFT BLANK 

iv 
Contents 

1. Introduction ...................................... 

2. Computational Technique ........................... 

2.1 Governing Equations ............................... 

2.2 Navier-Stokes Numerical Technique .................... 

2.3 Inviscid Technique ................................ 

3. Results ........................................... 

4. Scientific Visualization .............................. 

5. Summary and Conclusions ........................... 

References ............................................. 

Distribution List ........................................ 

Report Documentation Page ............................... 

Figures 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

Computational Mesh .................................. 

Chimera Gridding Near Jet Nozzle. ........................ 

Schematic of Body Geometries ........................... 

Force Amplification Factor Versus Angle of Attack, Mach 2.0, 

P, = 27 atm, Body Alone and Tail Fins ..................... 

Force Amplification Factor Versus Angle of Attack, Mach 4.5, 

P, = 38 atm, Body Alone and Tail Fins ..................... 

Force Amplification Factor Versus Angle of Attack, Mach 8, 

P,=1.9atm.BodyAloneandTailFins ..................... 

Force Amplification Factor Versus Angle of Attack, Mach 2, 

P, = 27 atm, Strakes .................................. 

Force Amplification Factor Versus Angle of Attack, Mach 4.5, 

P, = 38 atm, Strakes .................................. 

Force Amplification Factor Versus Angle of Attack, Mach 8, 

P,j=1.9atm,Strakes.. ................................ 

Force Amplification Factor as a Function of Jet Pressure, Mach 4.5, 

a=O”,StrakedBody .................................. 

Force Amplification Factor as a Function of Jet Pressure, Mach 8.0, 

a = 0”, Straked Body. ................................. 

Center of Pressure Versus Angle of Attack, Mach 2.0, P, = 27 atm, 

BodyAlone.. ...................................... 

Center of Pressure Versus Angle of Attack, Mach 4.5, P, = 38 atm, 

BodyAlone.. ...................................... 

1 

2 

2 

3 

5 

6 

14 

15 

17 

19 

23 

4 

4 

7 

10 

10 

10 

10 

11 

11 

12 

12 

12 

12 

V 
14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

Center of Pressure Versus Angle of Attack, Mach 8, l’, = 1.9 atm, 

BodyAlone........................................ 12 

Center of Pressure Versus Angle of Attack, Mach 2, P, = 27 atm, 

TailFins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 

Center of Pressure Versus Angle of Attack, Mach 4.5, P, = 38 atm, 

TailFins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 

Center of Pressure Versus Angle of Attack, Mach 8, P, = 1.9 atm, 

TailFins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 

Center of Pressure Versus Angle of Attack, Mach 2, P, = 27 atm, 

StrakedBody....................................... 13 

Center of Pressure Versus Angle of Attack, Mach 4.5, P, = 38 atm, 

StrakedBody....................................... 13 

Center of Pressure Versus Angle of Attack, Mach 8, P, = 1.9 atm, 

StrakedBody....................................... 14 

Jet Interaction Moment as a Function of Angle of Attack, Mach 4.5, 

P0j=38atm,StrakedBody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

Jet Interaction Moment as a Function of Jet Pressure, Mach 4.5, 

a = 0”; Straked Body.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

vi 
A NUMERICAL INVESTIGATION OF SUPERSONIC 

JET INTERACTION FOR FINNED BODIES 

1. Introduction 

The flow field that results from the interaction of a side (lateral) jet injection into 

a supersonic external flow, called the jet interaction flow field, has been the 

subject of several experimental [l-5] and numerical [6-111 investigations. The 

typical jet interaction flow field is complicated because the jet interrupts the 

oncoming external flow. The qualitative features of the jet interaction flow field 

include regions of shock-boundary layer interaction and flow separation that 

have an effect on the overall flow around the body. In our previous work 1111, a 

detailed numerical study was performed for non-finned axisymrnetrical bodies. 

In this paper, results are presented for missiles with several body geometries. 

The finned missile configurations are body strakes and aft-mounted fins. It was 

shown previously [ll] that for a finless body, deamplification occurred partially 

because the jet bow shock wrapped around and interacted with the flow 

underneath the body. The presence of strakes has the effect of blocking the wraparound phenomenon and channeling the high pressure flow down the body, 

thereby allowing amplification. The purpose of the current research is to develop 

a reliable computational capability to assess the performance of control jets and 

to obtain a quantitative understanding of the flow phenomenon produced by 

control jets in the presence of strakes and/or fins and to demonstrate that 

computational fluid dynamics (CFD) simulation is capable of predicting the 

important features of jet interaction phenomenon. 

This report primarily addresses viscous techniques, but we have also examined 

the ability to use and the feasibility of using inviscid techniques to predict the 

same forces and moments. Numerical predictions of the supersonic viscous flow 

have been obtained via an existing Reynolds-averaged Navier-Stokes solver, and 

the inviscid flow simulations were performed with a three-dimensional (3-D) 

multi-zone Euler technique. This work demonstrates that numerical techniques 

are sufficiently mature to be a useful predictive tool in designing jet-control 

systems and for flow diagnostics that cannot be made in the experiments. 

Comparisons between viscous and inviscid results can shed new light on the 

significance of viscous effects (i.e., separation of flow because of shock 

interaction) in overall vehicle forces and moments. 

In addition to the use of several different geometries, the parameters varied in 

this study are Mach number, angle of attack, and jet stagnation pressure. The 

Mach numbers are 2.0, 4.5, and 8. The angles of attack range from -10 to 10 

degrees. The jet stagnation pressure varies from 3.6 psi to 72 psi. This report 

1 
shows that the numerical results corroborate the experimental findings, in which 

the presence of strakes caused large control force amplification. This permits the 

use of smaller control jets and therefore results in propellant volume and weight 

savings. 

In the present study, numerical approaches have been applied to investigate the 

jet interaction phenomena for flight bodies that have lifting surfaces with a single 

lateral jet in supersonic flight and to demonstrate the advantages to force 

amplification factor in the presence of these surfaces. An overset grid approach 

has been applied to more easily resolve the geometry and flow physics 

associated with the jet interaction problem. All the numerical results have been 

validated via global force and moment data from a recently published 

experimental investigation 151. 

2. Computational Technique 

2.1 Governing Equations 

The nonreacting compressible viscous flow, which obeys the Newtonian law of 

friction about a flight vehicle, is governed by the equations of mass, momentum, 

and energy conservation, i.e., the Navier-Stokes equations. For these 

computations, the complete set of 3-D, time-dependent, generalized geometry, 

Reynolds-averaged, thin layer, Navier-Stokes equations for generalized 

coordinates 5, q, and c are used and can be written as follows 1121: 

5 ={(~,y,z,t), ?J =q(x,y,z,t), and 6 ={(x,y,z,t) are the longitudinal 

coordinate (direction along the body), the circumferential coordinate (direction 

around the body), and the nearly normal coordinate (outward direction from the 

body surface), respectively. 

The inviscid flux vectors i, $, and &, and the viscous term j are functions of 

the dependent variable qr = (p, pu, pv, pw, e). The inviscid flux vectors and the 

source term are shown next. Details of the thin layer viscous term are available in 

the literature. 

The local pressure is related to the dependent variables by applying the ideal gas 

law: 
p= (a-1&-0.5&+V~+w2)1 (2) 

1 in which y is the ratio of specific heats. Density, p , is referenced to po; and the 

total energy, e , to p-u: . 

The form of the mass-averaged Navier-Stokes equations requires a model for the 

turbulent eddy viscosity. There are numerous approaches for determining the 

turbulent viscosity. The turbulent contributions are supplied through the 

algebraic two-layer eddy viscosity model developed by Baldwin and Lomax [131, 

which is patterned after the model of Cebeci [141. 

2.2 Navier-Stokes Numerical Technique 

The time-dependent Navier-Stokes equations are solved via a time-iterative 

solution technique to obtain the final steady state converged solution. The 

particular time-marching technique applied here is the implicit, partially fluxsplit, upwind numerical scheme developed by Steger et al. [15,161 and is based 

on the flux-split approach of Steger and Warming 1171. This scheme employs 

central differencing in the normal and circumferential directions, q and c, 

respectively, and flux splitting in the stream-wise direction, < . 

2.2.1 Chimera Composite Overset Structured Grids 

To more easily model the geometry and resolve the flow physics associated with 

the lateral jet problem, the Chimera composite overset grid technique has been 

applied. The Chimera technique is a domain decomposition approach that allows 

the entire flow field to be meshed into a collection of independent grids, in which 

each piece is gridded separately and overset into a main grid. In current 

computations, the flight body with lateral jet was subdivided into three distinct 

grids: one for the body, one adjacent to the jet, and one for the jet nozzle. Overset 

grids are not required to join in any special way. Usually, a major grid covers the 

main domain (the external flow field about the projectile), and minor grids are 

generated to resolve the rest of the body or sections of the body (the jet and the 

nozzle regions). 

. 

Figure 1 displays the computational mesh, showing the main grid for the 

projectile body along with an overset grid to better capture the physics of the jet 

interaction with the external flow. The overset jet grid is seen here residing on 

top of the jet exit as a cylinder with a radius larger than the jet nozzle opening 

itself. A third grid, used to model the jet nozzle, resides underneath the jet grid. 

The communication between the nozzle grid and the jet grid is point-to-point 

zonal, however. Figure 2 is a “close-up” of the grid near the jet port, which is 

covered by the nozzle grid and jet grid. It also shows the Chimera grid for the jet 

and the projectile body. 
Figure 1. Computational Mesh. 

..................... ‘ ... _ ................. 

....... ...... .. .^. . , .. .: ... 

Figure 2. Chimera Gridding Near Jet Nozzle. 

4 
2.2.2 Grid Refinement and Computational Time 

The baseline computational grid used to generate the published results consisted 

of approximately 1.6 million points: 153 by 93 by 70 for the projectile 

(background) grid in the longitudinal, circumferential, and normal directions, 

respectively; 65 by 21 by 50 for the cylindrically shaped jet grid in the radial, 

circumferential, and normal directions, respectively; and 25 by 21 by 21 for the 

nozzle grid in the radial, circumferential, and normal directions. The 

computational mesh employed a mirror plane of symmetry in the circumferential 

direction. A grid refinement study was performed to assess grid convergence for 

the baseline computational grid. Coarse grid solutions were obtained with 115 by 

107 by 58, 49 by 16 by 38, and 19 by 16 by 16 grids for the projectile, jet, and 

nozzle meshes, respectively. Fine grid solutions were obtained with 189 by 167 

by 84, 78 by 25 by 60, and 30 by 25 by 25 grids for the projectile, jet, and nozzle 

meshes, respectively. 

Comparisons were made of the forces, moments, amplification factor and center 

of pressure shift. For the finer grid, there is a 1.1% difference in the jet interaction 

force and a 1% difference in the moments, amplification factor, and the center of 

pressure shift. For the coarser grid, there is a 1.2% difference in the jet interaction 

force and a 1.1% to 2.0% difference in the moments, amplification factor, and the 

center of pressure shift. 

A complete simulation on the baseline grid took approximately 250 central 

processing unit (CPU) hours on a Silicon Graphics, Inc. (SGI) Origin 2000. The 

solutions were run in parallel with eight processors, and a parallel acceleration of 

5.4 was obtained. 

x 2.3 Inviscid Technique 

The flow field solution is obtained here by running the CFD code named INCA@ 

1221. The code is presently run in the Euler (inviscid) mode, but INCA@ is a 

multi-block, Navier-Stokes solver with broad capabilities. The field equations are 

solved via an implicit finite volume method. The evaluation of the inviscid terms 

is based on flux splitting in combination with upwind biasing. 

The numerical solution was run until convergence was reached. The degree of 

convergence of the solution was judged from the maximum field residue 

occurring at each time step for each block. In general, convergence level is 

different in each block, especially for cases with jet activation. 

Convergence level is indicated by the logarithm of the largest residual (root 

mean square, averaged for the field variables). At Mach = 4.5, for the 

configuration without the jet, convergence of the maximum residual to lo-l3 was 

obtained in fewer than 1000 iterations for each of the four blocks. For a typical 

“jet-on” case, convergence to levels better than 10-l’ is achieved within 1000 

5 
cycles. The convergence properties of the Mach = 8 solutions were very similar to 

those described for the Mach = 4.5 solutions. From the point of view of integral 

parameters, such as forces and moments obtained from pressure integration, 

there is no variation after the residual decreased three or four orders of 

magnitude. 

2.3.1 Computational Grid Generation 

The grid is divided into four blocks. This is the minimum number of blocks 

needed to cover the given topology. The final grid topology used in the inviscid 

computation of the flow field over the configuration without the jet has about 

70,000 grid points. The grid available at this stage is of sufficient resolution for 

numerical computation of inviscid (Euler) flow about the described missile 

configuration when the jet is not activated. The jet nozzle has a diameter of 0.5 

cm, which is comparable in size to one finite difference cell. To obtain better 

resolution, local grid refinement in the vicinity of the jet nozzle was performed 

off line, and the number of grid points defining the nozzle has been quadrupled. 

Away from the body surface, the grid becomes coarser. Although this will affect 

the definition of shocks (the shocks are formed of several grid points, which 

when far apart will give smeared and wavy appearance), the solution close to the 

body surface will not be affected in the supersonic calculations. The refined grid, 

giving good definition of the nozzle size and location, consists of about 120,000 

points. The block structure described earlier is retained. 

2.3.2 Grid Refinement Study (grid independence) 

For the configuration presented here, the “jet-off” case was computed with and 

without the local grid refinement performed to better resolve the area containing 

the nozzle and the wings’ leading edges (120,000 points versus 70,000 points). No 

difference in solution was observed within plotting accuracy. Grid sensitivity 

analyses were performed for a prototype configuration, different from the 

present configuration only in some dimensions. For that configuration (not 

presented), several grids were used, the finest being coarser than the unrefined 

grid used in the present report. The graphical results for two successive grids 

were indistinguishable. 

3. Results 

Validation of the computational approach for the jet interaction problem was 

accomplished by a comparison of the predictions with data from a previously 

published wind tunnel investigation [5]. Supplemental experimental results for 

the validation were provided courtesy of Rafael and the Ministry of Defense, 

Directorate of Defense Research & Development. The experiments were 

conducted at the Israeli Aircraft Industries (IAI) trisonic wind tunnel facility at 

6 
~ 

. 

i . 

Mach numbers of 2 and 4.5 and in the IA1 hypersonic wind tunnel facility at 

Mach 8. While the experimental investigation was quite extensive in scope, the 

current computational study focused on normal jet injection from a single nozzle 

geometry at Mach 2, 4.5, and 8. Five different configurations were examined in 

the experimental study; this report examines three. Global force and moment 

comparisons were made to validate the computational approach. 

Figure 3 shows the three body geometries addressed in this study. Each of the 

models used has a sharp, ogive-shaped nose section of 2 calibers and a 

cylindrical afterbody of 3.8 calibers mounted on the midsection for a total length 

of 5.8 calibers. The reference diameter of the models was 50 mm. For all three 

geometries, the jet nozzle was 2.5 calibers down stream from the nose tip. A 

single 5-nun circular nozzle that was designed to achieve sonic flow at the exit 

was examined here, although additional geometries were considered in the 

experiment. The strake and aft-mounted fins have the same exposed semi-span 

of 25 mm and leading edge sweep angle of 45 degrees. Configuration 1 is an 

axisymmetrical body-alone configuration used as a reference configuration for 

comparison. Configuration 2 has an aft-mounted tail fin. The root leading edge of 

each tail surface is 220 mm from the nose tip. Configuration 3 contains strakes 

spanning 65% of the body’s length. The root leading edge is 100 mm from the 

nose tip. Global force and moment wind tunnel experiments were performed on 

these bodies. 

CONRUJRATON1 

_._._...--...........-........_.._._._...-....._...__._._...--.-...--.-. -. 

-25 33- . 

CONFlGURATlON2 i 

l.0 

..-....-...-.....--...............-..-......-..-.-...--.-......-........ . 

I 4.4 . 

{I 

CONFWRAllON3/ [1 

05 -...______._-_---...-..-.-....-.........-.-...-....-.--------.-----.--.... . 

-20 - I 

Figure 3. Schematic of Body Geometries. 

For the jet interaction problem, the total force acting on the body can be 

decomposed into three components: the aerodynamic force on the external body 

in the absence of the jet, the force produced at the jet exit, and the aerodynamic 

interaction force produced by the jet with the external flow field. In this work, 

the aerodynamic force on the external body is typically produced when the body 

is at an angle of attack with the free stream flow. The force produced at the jet 

exit results from a combination of the momentum flux through the jet nozzle and 

the integrated pressure at the jet exit. Given that the exit conditions for the jet are 

7 
fixed as a boundary condition for the computations, this force component can be 

explicitly calculated before the flow field computation. The third force 

component’accounts for the force produced by the interaction of the jet with 

external flow field. 

The relationship of these three force components to the total force, F , can be 

described by the following equation, in which F,,- jrr is the force in the absence 

of the jet, Fj is the force produced at the jet exit, and Fji is the jet interaction 

force: 

(3) 

The negative sign associated with the two jet forces results because the jet exit 

hole is situated on the upper surface of the body in the current study and 

produces a downward force when activated. The jet-off force component 

typically produces an upward force for positive angles of attack. Note that a 

positive value of Fji indicates that the interaction force produces an effect that 

augments the jet force Fj , while a negative value of Fjj indicates a reduction in 

the total force produced by the jet. The jet interaction force accounts for the 

complete interaction produced by the jet with the external flow field and may 

vary with angle of attack and jet mass flow rate. 

The relative magnitudes of the jet force and the jet interaction force can be 

compared through a jet interaction amplification factor, K, as shown in 

Equation 4: 

K= 

Fj + Fji 

Fi 

An amplification factor greater than 1 indicates that the jet interaction force 

amplifies or increases the total force produced by the jet, while an amplification 

factor small than 1 indicates that the jet interaction force reduces the total force 

produced by the jet. 

In addition to the jet interaction forces, the interaction moments were calculated. 

In the present case, the center of moments is taken to be the center of the jet 

nozzle, the assumed center of gravity. The moments considered are pure 

interaction moment effects which are calculated as follows: 

in which c,,,,,eu” is the moment coefficient for the configuration with the jet and 

C n,,,,-,l/ is the moment coefficient for the configuration without the jet. 
. 

The moment interaction can also be assessed by the examination of the center of 

pressure shift produced by the interaction. The center of pressure shift in calibers 

is defined as follows with a positive center of pressure shift indicating a 

rearward movement or a corresponding nose-down pitching moment. 

AXCP = 

- AC,,, 

Cf.lj + c,. ,I 

Figures 4 through 6 display the variation of the force amplification factor with 

angle of attack at Mach 24.5, and 8 for the body alone and body with tail fins. 

This comparison is meaningful because ahead of the tail fins (for these 

supersonic flow cases), the force amplification is essentially identical for both 

bodies. The differences in the force amplification factor occur only over the aft 

1.4 calibers of the body that contain the fins. The predicted results at Mach 2 and 

4.5 show excellent agreement with experiment between -10 and 10 degrees angle 

of attack. At Mach 8, the results show an under-prediction of the jet amplification 

factor, although the trend with angle of attack is consistent with the experimental 

data. 

The results for the body with tail fins show an increasing trend with angle of 

attack. For positive angles of attack, the force amplification factor for the tail fins 

is similar in magnitude to the body-alone results. However, at negative angles of 

attack, the body with tail fins shows a much stronger deamplification than for 

the body alone. The differences in the behavior at positive and negative angles of 

attack are attributable to an interaction of the jet wake on the tail fins. At positive 

angles of attack, the jet wake is directed away from the tail fins, while at negative 

angles of attack, the jet wake is convected downward onto the tail fins, thus 

producing a stronger interaction and deamplification. Ahead of the tail fins, the 

distribution of the jet interaction force is identical to the body-alone results. Very 

little additional interaction over the tail fins is seen at a = 10”. However, at 

a= -lo”, there is a significant interaction over the tail fins; which results in 

deamplification. 

An increasing trend in force amplification factor is noted with increasing Mach 

number in Figures 4 through 6. However, the force amplification factor also 

varies with jet pressure. No attempt to’ scale the jet pressure has been made for 

the results presented in Figures 4 through 6. 

. 

l 

Figures 7 through 9 display the variation of the force amplification factor with 

angle of attack at Mach 2,4.5, and 8 for the straked body. At Mach 2 and 4.5, the 

results are in good agreement with the experimental data across the range of 

angles of attack. At Mach 8, the computed results do not compare as well with 

the experimental data. At Mach 4.5 and Mach 8, results were obtained with an 

inviscid code, as well as with the Navier-Stokes code. At Mach 4.5, there is very 

9 
little difference in the predicted results with either code. Similar results are found 

at Mach 8, where both codes show a trend that is different than the experimental 

data. The similarity between the predicted results is significant, considering the 

results were obtained with different codes and computational grids. 

5 

12 

i= 

i 

l n NAVIER-STOKES 

a to 0 0 EXPERIMENT 

0 

’ 0.6- 

8 

; 0.4- 

TAIL FINS 

y 02-rno .. 

M - Degrees 

Figure 4. Force Amplification Factor 

Versus Angle of Attack, Mach 2.0, 

PL,i = 27atm’, Body Alone and Tail Fins. 

. 2.0- 

d . l NAMER-STOKE 

5 !a- 0 0 EXPERIMENT 

o l.6- 

z 

i 2 l.4- 

Q L2- 

g IO2 0.8 - TAIL FINS 

y 0.6 I, 

-a0 -8.0 -40 0.0 40 8.0 a0 

a - Degrees 

Figure 6. Force Amplification Factor 

Versus Angle of Attack, Mach 8, 

P,,, = 1.9 atm, Body Alone and Tail Fins. 

id 0.6 

l l NAVIER-STOKES 

o 0 EXPERIMENT 

i TAIL FINS 

1 

1 I I 1 I I 1 

-a0 -a0 -40 0.0 40 a0 a0 

a - Degrees 

Figure 5. Force Amplification Factor Versus 

Angle of Attack, Mach 4.5, P, = 38 atm, Body 

Alone and Tail Fins. 

p 

L2- 

n m NAVIER-STOKES 

Dfl EXPERIMENT 

a l.o0 

: 021 

-no 4.0 -40 0.0 4.0 80 l2.0 

a - Degrees 

Figure 7. Force Amplification Factor Versus 

Angle of Attack, Mach 2, P, = 27 atm, 

Strakes. 

A comparison of the jet interaction force distribution was performed for the body 

alone and the body with strakes at 0 degrees angle of attack at Mach 2,4.5, and 8. 

For all three Mach numbers, the presence of the strakes amplifies the jet 

interaction effect. Near the jet exit, the high pressure behind the jet bow shock 

acts not only on the body but on the adjacent fins as well. This produces an 

additional force augmentation relative to the body alone. At Mach 2, the effect of 

the low pressure region behind the jet is also increased for the &raked body, thus 

producing a force component that results in a deamplification of the jet force. 

‘atm = atmospheres 

10 
l 

n n NAVIER-STOKES 

LO . . EULER 

:: Dfl EXPERIMENT 

; 0.8 

42.0 -8.0 -4.0 0.0 40 8.0 a0 

a - Degrees 

Figure 8. Force Amplification Factor 

Versus Angle of Attack, Mach 4.5, 

P,i = 38 aim, Strakes. 

The force amplification factor for the straked body also shows an apparent 

increasing trend with Mach number as did the results for the body alone and 

body with tail fins shown previously. Again, it may be difficult to draw a general 

conclusion from these results alone because of the dependence of the 

amplification factor on jet pressure. Figures 10 and 11 show the variation of the 

force amplification factor with jet pressure at zero degrees angle of attack for 

Mach 4.5 and 8, respectively. The force amplification factor for both Mach 

numbers decreases with increasing jet pressure because the force attributable to 

the jet increases faster than the jet interaction force. The sharp decreasing trend in 

force amplification as jet pressure increases is captured by both the inviscid and 

viscous simulations. At Mach 4.5, it is seen that the predicted force amplification 

factor as ,a function of jet pressure exhibits the same quantitative variation as the 

experiment, slightly under-predicting the magnitude. The computed results at 

Mach 8 have the same qualities as the experiment but under-predict the 

magnitude. It was anticipated that a viscous code would close the gap between 

the Euler code predictions and experimental results. However, the results appear 

to suggest that perhaps the differences are not attributable to the viscous effects. 

Potentially, the discrepancy may be related to the difference between the 

measured and calculated values of the net jet thrust, even though the former was 

corrected for various factors. These values are used, respectively, in calculating 

the experimental and computed amplification factor, K . 

yJ +y&& i , 

42.0 -8.0 -4.0 0.0 4.0 a0 a0 

a - Degrees 

Figure 9. Force Amplification Factor Versus 

Angle of Attack, Mach 8, P, = 1.9 aim, 

Strakes. 

Figures 12 through 20 show the center of pressure shift as a function of angle of 

attack at Mach 2,4.5, and 8 for each of the three bodies. In general, the predicted 

variation of AXCP with angle of attack is in good agreement with the 

experimental data. The largest variation in AXCP occurs at Mach 2. For all three 

bodies at Mach 2 and 4.5, AXCP is positive, indicating a rearward shift in center 

of pressure. At Mach 8, AXCP takes positive and negative values for negative 

and positive angles of attack, respectively. 

11 
y 10 -EXPERIMENT 

0.0 200.0 I 400.0 I 800.0 I 800.0 I 4 I Y LO looo.0 l200.0 I , I I I I I 

0.0 lo.0 20.0 30.0 40.0 50.0 60.0 70.0 

P 01 + - Psi P . - Psi 01 

Figure 10. Force Amplification Factor Figure 11. Force Amplification Factor 

as a Function of Jet Pressure, Mach 4.5, as a Function of Jet Pressure, Mach 8.0, 

a = 0”, Straked Body. ct = 0”, Straked Body. 

20 0.5 

m n NAVIER-STOKES 

D---El EXPERIMENT 

n N NAVIER-STOKES 

0.4 D--o EXPERIMENT 

l.5 

g 10 a 53 0.3 

a a 02 

0.5 

0.1 

0.0 0.0 

40 -8.0 -4.0 0.0 40 a0 eo -120 -8.0 -4.0 0.0 4.0 8.0 QO 

a - Degrees a - Degrees 

Figure 12. Center of Pressure Versus 

Angle of Attack, Mach 2, P, = 27 atm, 

Body Alone. 

Figure 13. Center of Pressure Versus 

Angle of Attack, Mach 4.5, P, = 38 al-m, 

Body Alone. 

too - 4.S0.75 - 

4.0 - 

35- 

0.50 - NAVIER-STOKES 3.0- 

$ 025- 6 

x =- 

a a 2.0- 0.00 - l5- n = NAVIER-ST0 

-025- 10- 

D--E) EXPERIMENT 

-0.50 , I I I I I 0.5 I I , I I I 

-120 -8.0 -4.0 0.0 4.0 a0 a0 -QO -8.0 -4.0 0.0 40 a0 QO 

a - Degrees a - Degrees 

Figure 14. Center of Pressure Versus 

Angle of Attack, Mach 8, Poi = 1.9 aim, 

Body Alone. 

Figure 15. Center of Pressure Versus 

Angle of Attack, Mach 2, P,, = 27 atm, 

Tail Fins. 

12 

. 
l.5 n n NAVIER-STOKES 

20 

, ~fl EXPERIMENT m 

15 

LO 

10 n n NAVIER-STOKES 

-0.5 I I I I I I I -10 I I I I I I I 

-l20 -8.0 -4.0 0.0 4.0 8.0 a0 -120 -80 -4.0 0.0 40 8.0 a0 

a - Degrees 

Figure 16. Center of Pressure Versus 

Angle of Attack, Mach 4.5, P, = 38 atm, 

Tail Fins. 

a - Degrees 

Figure 17. Center of Pressure Versus 

Angle of Attack, Mach 8, Poj = 1.9 atm, 

Tail Fins. 

45 

4.0 

3.5 

15 1 n n NAVIER-STOKES 

~fl EXPERIMENT 

10 

6 x= =O 

a 

1 . 4 

20 

2 

0.5 

l.5 0.0 

10 

+=-4 t2.0 

a - Degrees 

0.5 I I I 1 I I I 

420 -0.0 -4.0 0.0 40 a0 TLO 

a - Degrees 

Figure 18. Center of Pressure Versus Figure 19. Center of Pressure Versus 

Angle of Attack, Mach 2, P, = 27 atm, Angle of Attack, Mach 4.5, P, = 38 atm, 

Straked Body. Straked Body. 

Figures 21 and 22 show the corresponding jet interaction moment as a function of 

angle of attack at Mach 4.5 and 8, respectively, for the &raked body. At Mach 4.5, 

the Navier-Stokes and inviscid predictions are essentially equivalent and agree 

well with the experimental data. Here, there is an increasing trend with angle of 

attack for the interaction moment that is predicted by CFD and observed in the 

experiment. Although the increase in the moment is relatively small, it is 

attributable to the jet wake effects. At lower angles of attack, the jet wake effects 

interact with the rear of the vehicle much more strongly than they do at higher 

angles of attack. At Mach 8, the Navier-Stokes results appear to be in better 

agreement with the experimental data than do the inviscid results, although the 

moment is relatively small ( AXCP at a = 0” is about a quarter caliber). 

13 
0.5 1 n n NAVIER-STOKES 

I)--o EXPERIMENT 

--40 4.0 4.0 o!o 410 8lO $0 

a - Degrees 

02- 

--- NAVIER-STOKES 

0.0 - 

- EULER 

- EXPERIMENT 

-0.6 - 

-0.8 1 I I I 

-x).0 -5.0 0.0 5.0 lo.0 

a - Degrees 

Figure 20. Center of Pressure Versus Figure 21. Jet Interaction Moment as a 

Angle of Attack, Mach 8, P, = 1.9 atm, Function of Angle of Attack, P, = 38 atm, 

Straked Body. Straked Body. 

E 

2 -0.3 - 

- EXPERIMENT 

-0.6 I I I 1 I I 

0.0 zoo.0 4co.o 600.0 800.0 looo.0 mo.0 

P - - Psi 01 

Figure 22. Jet Interaction Moment as a Function of Jet Pressure, Mach 4.5, a = 0” , 

Straked Body. 

The Navier-Stokes computations were performed under the viscous assumption, 

which may account for the closer comparison than the Euler results. As indicated 

in Brandeis’ results [5], these graphs show that at small angles of attack, the jet 

interaction’s effects produce a nose-up pitching moment. 

4. Scientific Visualization 

Scientific visualization provides a methodology to explore, define, and present 

the results of computations generated on high performance, parallel computers. 

Through the use of real-time 3-D interactive exploration and animated depictions 

of their data, researchers are able to extrude and present previously 

unobservable information. The results of a visual analysis can be used to 

compare math contours resulting from a computation with schlieren 

photographs from live experiments. Scientific visualization provides technical 

14 
. 

I t 

innovations that present data in unique and nontraditional ways. Through these 

new techniques, more visual information is made available to the researcher, 

improving comprehension and providing new insight to the problem. 

The computational results for this problem, which required 250 hours of CPU 

time to process, were analyzed and reported by a variety of 2-D and 3-D 

graphical techniques. The use of scientific visualization techniques was essential 

for analyzing a data set of this magnitude; for this particular problem, 

approximately 7.5 gigabytes of data, encompassing 236 time steps, were 

generated. The resulting visualization required the interactive display and 

manipulation of more than 1.5 million cells of data per time step. The ability to 

visualize and verify these results and share those findings with colleagues and 

customers was an invaluable part of the engineering process. 

A commercially available scientific visualization package “EnSightTM” was used 

to post-process the results of this computation. This package was able to visually 

display the time steps (236) of data that were generated as the computation 

converged to a steady state solution. EnsightTM uses many of the same parallel 

programming techniques that were used to generate the data, therefore taking 

full advantage of the multi-processing architecture of the visualization server to 

improve the efficiency and interactive response during the visual analysis 

process. Reducing 7.5 gigabytes of computed data into a single animation 

allowed for visual inspection of the evolution of the computation over time and 

provided real-time analysis of the jet-thruster phenomena. 

Scientific visualization techniques were used to demonstrate the physical forces 

and interaction in and around the lateral jet thruster. This visual analysis clearly 

indicated the jet-on/jet-off state of the lateral jet thruster during the course of the 

computation. Also, the effects of the jet wake as it interacted with the aftmounted fins of the vehicle could be visually analyzed. The use of pressure 

values to color-code the results of the 3-D analysis provided additional insight 

into the forces acting on the vehicle body, and the results of the jet thruster could 

be indicated. 

5. Summary and Conclusions 

A computational approach has been validated with experimental data for bodies 

with lateral control jets in supersonic flight at varying Mach numbers and angles 

of attack and for different jet stagnation pressures. The bodies include bodyalone configurations, a body with aft-mounted tail fins, and a body with strakes. 

The results show significant interactions of the jet-induced flow field with the fin 

surfaces, which produce additional effects compared with the body alone. In 

15 
agreement with the experiment in some cases, the presence of lifting surfaces 

resulted in force and/or moment amplification of the jet interaction with the 

missile surfaces. The results indicate deamplification of the jet force at Mach 2 for 

all three bodies. Amplification of the jet force was also observed for high Mach 

numbers, particularly for the body with strakes. For the results examined here, 

there were only minor differences in the global force and moment predictions 

with viscous or inviscid techniques. This similarity indicates that the viscous 

effects are small for these configurations, especially in view of the large lifting 

surfaces that receive the pressure forces. Both techniques correctly predicted the 

dependence of the interaction parameters on angle of attack and jet pressure. The 

results indicate that for the purpose of overall design of configurations with jet 

force control, the inviscid methods may be both sufficient and expedient. Viscous 

computations are, however, imperative when the near field close to the jet is 

considered. 

16 
References 

1. Brandeis, J., and J. Gill, “Experimental Investigation of Side-Jet Steering for 

Supersonic and Hypersonic Missiles,” Journal of Spacecraft and Rockets, Vol. 33, 

No. 3, pp. 346-352, May-June 1996. 

2. Spaid, F.W., and E.E. Zukoski, “A Study of the Interaction of Gaseous Jets 

from Transverse Slots with Supersonic External Flows,” AMA Journal, Vol. 6, 

No. 2, pp. 205-212, February 1968. 

3. Gilman, R.G, “Control Jet Interaction Investigation,” Journal of Spncecraft and 

Rockets, Vol. 8, No. 4, pp. 334-339, April 1971. 

4. Spaid, F.W., “Two-Dimensional Jet Interaction Studies at Large Values of 

Reynolds and Mach Numbers,” AMA Journal, Vol. 13, No. 11, pp. 1430-1434, 

November 1975. 

5. Brandeis, J., and J. Gill, “Experimental Investigation of Super- and 

Hypersonic Jet Interaction on Missile Configurations,” Journal of Spacecraft 

and Rockets, Vol. 35, No. 3, May-June 1998. 

6. McMaster, D.L., J.S. Shang, and W.C Golbitz, “Supersonic, Transverse Jet 

from a Rotating Ogive Cylinder in a Hypersonic Flow,” Journal of Spacecraft 

and Rockets, Vol. 26, No. 1, pp. 24-30, January-February 1989. 

7. Chamberlain, R., “Calculation of Three-Dimensional Jet Interaction Flow 

Fields,” AIAA Paper 90-2099, July 1990. 

8. Hsieh, T., and A.B. Wardlaw, “Numerical Simulation of Cross Jets in 

Hypersonic Flow Over a Biconic Body,” AIAA Paper 94-0165, January 1994. 

9. Sahu, J., “Numerical Computations of Three Dimensional Jet Interaction Flow 

Fields,” AIAA Paper 94-3521, August 1994. 

10. Srivastava, B., “Aerodynamic Performance of Supersonic Missile Body- and 

Wing Tip-Mounted Lateral Jets,” Journal of Spacecraft and Rockets, Vol. 35, 

No. 3, pp. 278-286, May-June 1998. 

11. Graham, M.J., and P. Weinacht, “A Numerical Investigation of Supersonic Jet 

Interaction for Axisymmetric Bodies,” Journal of Spacecraft and Rockets, Vol. 37, 

No. 5, pp. 675-683, September-October 2000. 

17 
12. Pulliam, T.H., and J.L. Steger, “On Implicit Finite-Difference Simulations of 

Three-Dimensional Flow,” AIAA Journal, Vol. 18, No. 2, pp. 159-167, February 

1982. 

13. Baldwin, B.S., and H. Lomax, “Thin Layer Approximation and Algebraic 

Model for Separated Turbulent Flows,” AIAA Paper 78-257, January 1978. 

14. Cebeci, T., “Calculation of Compressible Turbulent Boundary Layers with 

Heat and Mass Transfer,” AIAA Paper 70-741, June-July 1970. 

15. Ying, S.X., J.L. Steger, L.B. Schiff, and D. Baganoff, “Numerical Simulation of 

Unsteady, Viscous, High-Angle-of-Attack Flows Using a Partially Flux-Split 

Algorithm,” AIAA Paper 86-2179, August 1986. 

16. Sahu, J., and J.L. Steger, “Numerical Simulation of Three-Dimensional 

Transonic Flows,” International Journal for Numerical Methods in Fluids, Vol. 10, 

1990, pp. 855-873. 

17. Steger, J.L., and R.F. Warming, “Flux Vector Splitting of the Inviscid 

Gasdynamic Equations with Application to Finite-Difference Methods,” 

Journal of Computational Physics, Vol. 40,198l pp. 263-293. 

18. Steger, J.L., and P. Buning, “Developments in the Simulation of Compressible 

Inviscid and Viscous Flow on Supercomputers,” NASA TM 86674,1985. 

19. Beam, R., and R.F. Warming, “An Implicit Factored Scheme for the 

Compressible Navier-Stokes Equations,” AIAA Journal, Vol. 16, No. 4, 1978, 

pp.393-402. 

20. Benek, J.A., T.L. Donegan, and N.E. Suhs, “Extended Chimera Grid 

embedding Scheme with Application to Viscous Flows,” AIAA Paper No. 87- 

1126-Cl?, 1987. 

21. Meakin, R.L., “Composite Overset Structured Grids,” CRC Handbook of Grid 

Generation, CRC Press, Inc., Boca Raton, FL, 1997. 

22. Anonymous, “INCA: 3D Multi-Zone Navier-Stokes Flow Analysis with 

Finite Rate Chemistry,” User’s Manual, Amtec Engineering, Inc., Bellevue, 

Washington, January 1992. 

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Program Development Corporation, 300 Hamilton Avenue, White Plains, 

New York, August 1994. 

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. . TITLE AND SUBTITLE 

A Numerical Investigation of Supersonic Jet Interaction for Finned Bodies 

;. AUTHOR(S) 

3. REPORT TYPE AND DATES COVERED 

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PR: lL162618AHSO 

Graham, M.J. (U.S. Military Academy); Weinacht, P. (ARL); Brandeis, J. (Rafael); 

Angelini, R. (ARL) 

‘. PERFORMING ORGANIZATION NAME(S) AND ADDRESS 

U.S. Army Research Laboratory 

Weapons & Materials Research Directorate 

Aberdeen Proving Ground, MD 21005-5066 

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3. ABSTRACT (Maximum 200 words) 

A detailed numerical investigation of the interaction between a lateral jet and the external flow has been performed for a variety of 

missile body geometries. These include non-finned axisymmetrical bodies and finned bodies with either snakes or aft-mounted 

tail fins. The computations were performed at Mach numbers 2, 4.5, and 8. To obtain the numerical results, both Reynoldsaveraged Navier-Stokes and Euler techniques were applied. The computational results were compared with results from a 

previously published wind tunnel study that consisted primarily of global force and moment measurements. The results show 

significant interactions of the jet-induced flow field with the fin surfaces, which produce additional effects compared with the 

body alone. In agreement with the wind tunnel study, in some cases the presence of lifting surfaces resulted in force and/or 

moment amplification of the jet interaction with the missile surfaces. The results indicate deamplification of the jet force at Mach 

2 for all three bodies. Amplification of the jet force was also observed for high Mach numbers, particularly for the body with 

strakes. For the results examined here, there were only minor differences in the global force and moment predictions when 

viscous or inviscid techniques were used. The dependence of the interaction parameters on angle of attack and jet pressure was 

well predicted by both methods. The numerical techniques showed good agreement with the experiments at supersonic Mach 

numbers but only fair agreement for the hypersonic, Mach = 8 case. 

4. SUBJECT TERMS 

computational fluid dynamics 

lateral jet 

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