Lateral Control Jet Aerodynamic Predictions for a 2.75-in 

Rocket Testbed Munition 

by Paul Weinacht 

ARL-TR-3165 April 2004 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3165 April 2004 

Lateral Control Jet Aerodynamic Predictions for a 2.75-in 

Rocket Testbed Munition 

Paul Weinacht 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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Lateral Control Jet Aerodynamic Predictions for a 2.75-in Rocket Testbed Munition 

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14. ABSTRACT 

The aerodynamic performance of lateral control jets for a 2.75-in rocket testbed munition has been investigated using a 

sophisticated numerical technique. The computational approach utilized a finite difference-based overset gridding approach to 

predict the interaction of the lateral control jet with the external flow around the munition and the resulting effect on the lateral 

control jet performance. The jet interaction effect was found to be an important contributor to the overall effectiveness of the 

lateral control jet. The circumferential location of the jet relative to the tail fins was also found to influence the jet’s 

effectiveness. A variation of 30% in the net jet moment was computed for the two fin orientations considered in the study. The 

predictions also showed that spin did not significantly influence the lateral control jet performance for the expected spin rates of 

the testbed munition; however, if the spin was increased significantly, the resulting jet interaction produced a misalignment of 

the total jet thrust relative to the nozzle. Single and tandem jet configurations were examined. For the configurations examined 

here, the tandem jet doubled the effective jet thrust, although a similar effective jet thrust could be obtained by sequential 

independent firings of both jets. Time-dependent predictions of the performance of pulsed lateral control jets indicate that 

steady-state flow is attained during the duration of the firing cycle, and the overall performance of the pulsed jets can be 

represented reasonably well by the steady-state jet-on performance. 

15. SUBJECT TERMS

aerodynamics, projectile flight mechanics, lateral control jets 

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Paul Weinacht 

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iii

Contents 

List of Figures iv

List of Tables iv

Acknowledgments v

1. Introduction 1

2. Computational Approach 4

2.1 Numerical Technique ......................................................................................................6 

2.2 Chimera Composite Overset Structured Grids................................................................7 

3. Results 11

4. Summary 25

5. References 27

List of Abbreviations and Symbols 29

Distribution List 31 
iv

List of Figures 

Figure 1. Schematic of 2.75-in rocket testbed vehicle....................................................................1 

Figure 2. Schematic of jet-interaction flow field............................................................................2

Figure 3. Overview of grid on 2.75-in testbed munition. ...............................................................8 

Figure 4. Overset grid system near jet nozzle.................................................................................8 

Figure 5. Overset grid system on surface of fin hub.......................................................................9 

Figure 6. Overset grid system near fin hub.....................................................................................9 

Figure 7. Pressure contours on 2.75-in rocket testbed, single jet. ................................................12 

Figure 8. Pressure contours on 2.75-in rocket testbed, single jet, close-up of jet region. ............12 

Figure 9. Jet interaction force distribution along length of body, single nozzle...........................13 

Figure 10. Schematic showing orientation of jet nozzle relative to the fins.................................14 

Figure 11. Comparison of jet interaction force distribution along length of body, single 

nozzle, fin orientations 1 and 2................................................................................................15 

Figure 12. Pressure contours on 2.75-in rocket testbed, dual jet. .................................................16 

Figure 13. Pressure contours on 2.75-in rocket testbed, dual jet, close-up of jet region. .............16 

Figure 14. Comparison of jet interaction force distribution along length of body, single 

and dual nozzles.......................................................................................................................17 

Figure 15. Effect of roll rate on jet interaction force distribution along length of body, 

configuration 2.........................................................................................................................18 

Figure 16. Pressure contours near jet, –20 and –215 rps roll rates, configuration 2.....................19 

Figure 17. Comparison of jet interaction normal and side force distributions along length 

of body, –20 and –215 rps roll rates, configuration 2..............................................................19 

Figure 18. Time-dependent pressure contours during jet-on transient. ........................................21 

Figure 19. Time-dependent pressure contours during jet-off transient.........................................23 

Figure 20. Variation of jet thrust, jet interaction force, and total force as a function of time, 

single nozzle.............................................................................................................................25 

List of Tables 

Table 1. Axial locations of jet nozzles..........................................................................................11 
v

Acknowledgments 

This work was supported by a grant of computer time from the Department of Defense High 

Performance Computing Major Shared Resource Center at the U.S. Army Research Laboratory.
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

The U.S. Army Research Laboratory is currently investigating smart munition technologies as a 

means of improving the lethality and accuracy of future generations of munitions. One of the 

focuses of the current effort is the development of integrated multidisciplinary design 

technologies such as structural dynamics, computational fluid dynamics (CFD), and guidance, 

navigation and control. These multidisciplinary design technologies allow complex munitions 

systems to be studied and visualized within high-performance computational environments to 

determine the nonlinear interaction of critical engineering parameters using high-fidelity physics. 

Detailed design trades can be performed on system subcomponents, enabling the optimization of 

the munitions system as a whole. Critical structural and functional shortcomings in the 

system/subsystems (such as embedded devices/electronics) can be identified and the system can 

be redesigned before any hardware is manufactured, resulting in reduced development costs and 

higher performance munitions. 

As part of this effort, a 2.75-in rocket has been modified as a testbed for investigating the inflight response of a flight vehicle subjected to lateral control jets (1). The testbed vehicle, shown 

in figure 1, incorporates numerous modifications including several lateral control jets/thrusters in 

the forward portion of the body, instrumentation to control, measure, and telemeter the vehicle 

flight dynamic response, and a specially designed composite body section with embedded strain 

gauges to measure the structural response. To support the experimental program, CFD has been 

performed to assess the aerodynamic performance of the lateral control jets. The results of these 

studies are detailed in the current report. 

Thruster Section

25.46

10.07

CG

Instrumented Composite

Section

Dimensions in Calibers

Wrap-around Fins

Figure 1. Schematic of 2.75-in rocket testbed vehicle. 
2

The CFD capability applied here for the lateral control jet problem has been validated in 

previous studies (2, 3). In these studies, detailed comparisons of the computed surface pressure 

distribution was made with wind tunnel measurements. The focus of the CFD effort was the 

prediction of the aerodynamic performance of lateral control jets on the testbed munition. As 

part of this effort, time-dependent aerodynamic loads produced by the lateral control jet firings 

were also determined. Lateral control jets provide a convenient means of providing control 

authority for a flight vehicle because the external aerodynamics are unaffected except during the 

controlled portion of the flight. However, the flow field that results from the interaction between 

the gaseous jet and the external flow is complicated and can modify the overall effectiveness of 

the lateral control jet. 

Figure 2 shows a schematic representation of the flow field produced by a lateral control jet in 

supersonic flight. 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 large regions of 

the flow field around the body. The jet creates a blocking effect to the oncoming flow in the 

vicinity of the nozzle, producing boundary-layer separation and recirculation upstream of the jet. 

The separation region upstream of the jet sets up a wedge-shaped area, which has the effect of 

deflecting the flow laterally around the body so that a region of high pressure is felt under the 

projectile. The details of the upstream interaction are important because it is along the upper part 

of this region that the separation shock forms. 

Figure 2. Schematic of jet-interaction flow field. 
3

Since the jet stream acts as an obstacle to the external flow, another shock wave originates 

directly upstream of the jet and above the separation region called the jet bow shock. The jetbow shock moves outward from the body and bends backward as the jet flow is deflected by the 

external flow. A strong turbulent wake results from the jet flow field interactions and extends 

over a significant portion of the projectile body. Eventually, a recompression shock forms in the 

wake downstream of the jet. The jet-bow shock also moves laterally around the jet, the effects of 

which are felt on the underbody of the projectile. If there are tail fins present, the wake behind 

the jet can also interact with these control surfaces resulting in additional changes to the jet 

effectiveness. 

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

following three components: (1) the aerodynamic force on the external body in the absence of 

the jet, (2) the force produced by the jet thrust, and (3) the aerodynamic-interaction force 

produced by the jet with the external flow field. In the current paper, all computations have been 

performed at 0° angle of attack and the transverse aerodynamic force on the external body is 0. 

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

integrated pressure at the jet exit and over the nozzle surface. The third force component 

accounts for the force produced by the interaction of the jet with the external flow field. 

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

the following equation, where Fno-jet 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: 

( ) F = Fno−jet − Fj + Fji . (1) 

The negative sign associated with the two jet forces results because the jet nozzle is located 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 Fji 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 force amplification factor, K, as shown in equation 2. 

j

j ji

F

F F

K

+ = . (2) 

An amplification factor >1 indicates that the jet interaction force amplifies or increases the total 

force produced by the jet, while an amplification factor <1 indicates that the jet interaction force 

reduces the total force produced by the jet. Similarly, a jet interaction moment amplification 
4

factor can be defined where Mj and Mji are the moment produced by the jet thrust and the jet 

interaction moment, respectively (3). 

j

j ji

m M

M M

K

+ = . (3) 

For short-duration jets which effectively produce linear and angular impulses to the body rather 

than steady-state forces or moments, a jet interaction impulse amplification factor can be defined 

in a manner analogous to the force and moment amplification factors, I and Im. 

F dt

F dt F dt

I

0

j

0

ji

0

j

∫

∫ + ∫

= ∞

∞ ∞

. (4) 

M dt

M dt M dt

I

0

j

0

ji

0

j

m

∫

∫ + ∫

= ∞

∞ ∞

. (5) 

In the current study, the CFD approach was applied to assess the aerodynamic performance of 

the lateral control jets on the testbed munition. The jet thrust, jet-interaction force and moments, 

and jet-interaction amplification factors have been determined for the testbed munition. Single 

and simultaneous dual-jet firings have been examined. The role of fin orientation relative to the 

jet for these asymmetrical fins has also been considered. Additionally, the effect of body-roll 

rate on the aerodynamic performance of the jets has been predicted. Finally, both the steadystate and short-duration time-dependent response of the lateral control jet have been assessed and 

compared. 

Comparisons of the global forces and moment produced by the jet interaction flow field were 

also made. The validation has been conducted for axisymmetric bodies and bodies with strakes 

and tail fins. A brief description of the computational approach applied this study is provided in 

the following section. 

2. Computational Approach

The nonreacting compressible Newtonian viscous flow about a flight vehicle is governed by the 

equations of mass, momentum, and energy conservation—the Navier-Stokes equations. For 

these computations, the complete set of three-dimensional, time-dependent, generalizedgeometry, Reynolds-averaged, thin-layer, Navier-Stokes equations for generalized coordinates 

ξ , η, and ζ are used and can be written as follows (4): 
5

ξ η ζ ∂ζ

∂

+ = ∂

∂

+

∂

∂

+

∂

∂

+

∂

∂ S

ˆ

Re

1 Hˆ Gˆ Eˆ Fˆ

t

qˆ , (6) 

where ξ = ξ (x, y, z,t) , η =η(x, y, z,t), and ζ = ζ (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, Eˆ , Fˆ , and Gˆ , and the viscous term S

ˆ are functions of the dependent 

variable, J qˆ ( , u, v, w,e)/ T = ρ ρ ρ ρ . The source term, Hˆ , incorporates the effects of the 

centrifugal and Coriolis forces from the rotating-coordinate frame produced by rolling motion. 

The inviscid-flux vectors and the source term are shown below. Details of the thin-layer viscous 

term are available in the literature. 

( ) 































+

+

+

+

=

e p U

wU p

vU p

uU p

U

J

1 Eˆ

z

y

x

ρ ξ

ρ ξ

ρ ξ

ρ

 

( ) 































+

+

+

+

=

e p V

wV p

vV p

uV p

V

J

1 Fˆ

z

y

x

ρ η

ρ η

ρ η

ρ

( ) 































+

+

+

+

=

e p W

wW p

vW p

uW p

W

J

1 Gˆ

z

y

x

ρ ζ

ρ ζ

ρ ζ

ρ

 

































+ +

=

x y z

z

y

x

u f v f w f

f

f

f

0

J

1 Hˆ

ρ ρ ρ

ρ

ρ

ρ

, (7) 

where 

x y z

x y z

x y z

W u v w

V u v w

U u v w

ζ ζ ζ

η η η

ξ ξ ξ

= + +

= + +

= + +

 (8)

J(y x x y )

J(z y y z ) J(x z z x )

J(z x x z ) J(x y y x )

J(x y y x ) J(y z z y )

J(y z z y ) J(z x x z )

z

x y

y z

z x

x y

η ξ η ξ

η ξ η ξ η ξ η ξ

ζ ξ ζ ξ ζ ξ ζ ξ

η ζ η ζ ζ ξ ζ ξ

η ζ η ζ η ζ η ζ

ζ

ζ ζ

η η

ξ η

ξ ξ

= −

= − = −

= − = −

= − = −

= − = −

 )] J 1/[x (y z y z ) x (y z y z ) x (y z y z = ξ η ζ − ζ η + η ξ ζ − ζ ξ + ζ ξ η − η ξ . (9) 

The pressure, p, can be related to the dependent variables by applying the ideal gas law. 
6

( )





 



 = − − + + 2 2 2

u v w

2 p ( 1) e ρ γ . (10) 

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 (5), which is patterned after the model of Cebeci (6). 

The Coriolis and centrifugal acceleration terms due to the rotating coordinate system which are 

contained in the source term, Hˆ , are shown in the following: 

f 2 u ( R) r r r r r r = Ω× + Ω× Ω× . (11) 

The Coriolis acceleration is a function of the angular velocity of the coordinate frame with 

respect to the inertial frame, Ω

r

, and the fluid velocity vector, u

r

, which can be represented by 

the velocity components, u, v, and w. The centripetal acceleration is a function of the angular 

velocity of the rotating frame, Ω

r

, and the displacement vector, R

r

, between the axis of rotation 

and the particular location in the flow field. The acceleration vector, f

r

, can be written in terms 

of its components along the x, y, and z axes, fx, fy, and fz. The angular velocity of the rotating 

coordinate frame due to pure rolling motion is written as follows: 

xi r r Ω = Ω . (12) 

For the case of rolling motion, the source term takes the following form: 

[ ] 2 3 4 5

T 0,H ,H ,H ,H

J

1 Hˆ = , (13) 

where 

H 0 2 =

H 2 w y

2

3 = − Ωρ − ρΩ

 z H 2 v

2

4 = Ωρ − ρΩ

H (y v z w) 2

5 = −Ω ρ + ρ . (14) 

2.1 Numerical Technique 

The time-dependent Navier-Stokes equations are solved using a time-iterative solution technique 

to obtain the final steady-state converged solution. The particular time-marching technique 

applied here is the implicit, partially flux-split, upwind numerical scheme developed by Ying et 

al. (7) and Sahu and Steger (8), and is based on the flux-splitting approach of Steger and 

Warming (9). In its original form, the technique was referred to as the F3D technique. This 

scheme utilizes central differencing in the normal and circumferential directions, η and ζ , 

respectively, and flux splitting in the streamwise direction, ξ . Rather than directly invert the 
7

implicit equation, a two-factor implicit technique similar to that of Steger and Buning (10) is 

utilized. The resulting factored implicit equation is shown as follows: 

( ) Mˆ J i D | q RHS

J

1

Re

1 D i h ˆ C i h ˆ A i h ˆ I i h *

b i

n

b

n

b

n

b

n b

b = 



 



 + + + − − + δ ξ δ ζ δ ζ ζ , (15) 

( ) n *

b i

n

b

n f

b B i D | qˆ q ˆ A i h ˆ I i h ∆ = 

  + + − − δ ξ δ η η , (16) 

and 

( ) ( ) 



 



 = − ∆ + + + + − − + − n

e

n n n n n f n b

b S D qˆ ˆ

Re

1 Eˆ Eˆ Fˆ Gˆ Hˆ RHS i t δ ξ δ ξ δ η δ ζ δ , (17) 

where h = ∆t . Here, δ is typically a three-point, second-order accurate central difference 

operator, δ is a midpoint operator used with the viscous terms, and the operators f δ ζ and b δ ζ are 

the forward and backward, respectively, three-point difference operators. The Jacobian matrices 

Aˆ , Bˆ , Cˆ , Dˆ , and Mˆ result from local linearization of the fluxes about the previous time level 

(11) (e.g., ( ) ( ) n 1 n n n 1 n 2 A qˆ qˆ t Eˆ = Eˆ + ˆ − + Ο ∆ + + , where 

q

Eˆ Aˆ

∂

∂ = ). 

The two-factor implicit algorithm involves two sweeps through the grid at each time step. The 

first sweep involves inverting the block tridiagonal system of equations shown in equation 15 

along constant η grid lines to determine the intermediate solution variable * qˆ . During the 

second sweep, a second block tridiagonal system of equations (equation 16) is inverted along 

grid lines of constant ζ to determine the dependent variable n ∆qˆ . The two-factor implicit 

algorithm reduces the computational requirements of the approach compared with the threefactor, central-difference, implicit algorithm of Beam and Warming (11). The algorithm 

contains additional numerical smoothing terms, i η

D , i ζ D , and De , to suppress numerical 

oscillations associated with the odd-even decoupling produced by the central differencing in the

η and ζ directions (7). 

2.2 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, where each piece is gridded separately and overset into a 

background grid. Overset grids are not required to join in any special way. Usually there is a 

major grid that 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. In current computations, 

a mixture of point-to-point zonal overlaps and Chimera overset gridding was utilized. As shown 

in figure 3, five large grid blocks with point-to-point zonal overlaps between blocks were used 
8

to form the background grid around the munition body. Near the jet nozzle (figure 4) and on the 

fins (figures 5 and 6), Chimera overset grids were used to discretize the flow field near these 

geometric features. 

Figure 3. Overview of grid on 2.75-in testbed munition. 

Figure 4. Overset grid system near jet nozzle. 
9

Figure 5. Overset grid system on surface of fin hub. 

Figure 6. Overset grid system near fin hub. 
10

Figure 4 displays the computational mesh near the nozzle region, showing the background 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 residing on top of the jet exit as a 

cylinder with a radius larger than the jet hole itself. A third grid, used to model the jet nozzle, 

resides underneath the jet grid. Point-to-point zonal overlaps are used for communication 

between the nozzle grid and the jet grid. A hole has been cut into the background grid by the 

scheme, and background grid points within the hole region are excluded from the solution 

process. The overset grid system used here for the jet region is very similar to that used in prior 

validation studies for the lateral control jet problem (2, 3). 

The computational model included the jet nozzle region downstream of the throat with the inflow 

plane located at the nozzle throat. Sonic flow was across at the throat of the nozzle, with a total 

temperature of 2500 K and a total pressure of 950 atmospheres. Subsequent calculations of the 

thrust produced at the nozzle exit (including contributions from the pressure at the nozzle throat, 

the momentum flux through the nozzle throat, and the pressure on the interior surface of the 

nozzle) indicated a jet-thrust impulse of 4.1 N-s over a duration of 6 ms. This correlated well 

with the nominal design impulse of 4 N-s. Because of the high jet total pressure relative to the 

external freestream pressure, the flow in the nozzle throat and the resulting jet thrust were 

essentially independent of the external flow conditions examined here. 

Figures 5 and 6 show portions of the computational mesh used on the fins. Each fin was 

embedded in a box grid, with two additional nearbody grids used to discretize the flow field on 

the main fin surface and on the fin tip. For the complete grid system, 11.5-million grid points 

were used with 7.1-million points on the fins. In some cases, the mirror-plane symmetry of the 

flow field on the axisymmetric forebody was exploited to reduce the number of grid points by 

~20% for the supersonic flow results described. 

The thin-layer Navier-Stokes equations, shown previously in equations 15–17, have been 

modified for the Chimera scheme (12). The single and overset grid versions of the algorithm are 

identical except for the variable ib, which accommodates the possibility of having arbitrary holes 

in the grid. The array ib has the values of either 0 (for hole points) or 1 (for conventional field 

points—a mesh point within the computational domain updated by the flow solver or boundary 

conditions). Thus, points inside a hole are not updated (i.e., 0 qˆ n ∆ = ), and the solution values 

on intergrid boundary points are supplied via interpolation from corresponding solutions in the 

overlap region of neighboring grid systems (13). 

A complete simulation of a single steady-state lateral jet firing took ~750 CPU hours on an SGI 

Origin 3000. The solutions were run in parallel using eight processors, and a parallel speed-up to 

5.4 was obtained. 
11

3. Results 

The computational approach has been applied to investigate the performance of the lateral 

control jet on the testbed munition shown in figure 1 at a flight speed of Mach 2. All predictions 

were made at 0° angle of attack for ambient sea-level flight conditions. During the course of the 

testbed munition design process, various axial locations for the jet thrusters were proposed, 

including in-line tandem jets. Table 1 presents the jet axial locations for the two configurations 

examined here. Configuration 1 is a tandem jet configuration, with the jets located in the same 

circumferential plane as shown in figure 1. Configuration 2 is a single jet configuration. 

Analyses have been performed to assess both the steady-state performance of the lateral jet and 

the time-dependent evolution of the pulsed jet. Steady-state jet-on simulations are discussed first 

and provide a basis for examining the time-dependent pulsed jet results presented later. 

Table 1. Axial locations of jet nozzles. 

Axial location of Jet Nozzle From Nose 

(Calibers) 

Configuration Forward Jet Rear Jet 

1 5.817 6.461 

2 6.066 NA 

Note: NA = not applicable. 

As a baseline for comparisons, computations were first performed for a single jet firing of the 

forward-most jet of configuration 1. Fluid dynamic modeling of the cavity flow associated with 

the inactive rear jet is incorporated in the simulation. Figure 7 shows the pressure contours on 

the body surface and in a vertical plane containing the jet. The most apparent effect of the jet on 

the pressure field is in the vicinity of the jet itself. A close-up of the jet region is shown in 

figure 8. The high-pressure region in front of the jet and the low-pressure region behind the jet 

are readily apparent. Jet effects further downstream of the jet are less apparent in the pressure 

field, but the small asymmetries produced by the jet on the flow field downstream of the jet 

produce important aerodynamic effects that are apparent in the force’s distribution. 

Figure 9 shows the jet interaction force distribution along the body. In the immediate vicinity of 

the jet (X/D = 5.8), there is a significant increase in the force distribution due to the high pressure 

near the jet nozzle produced by the flow field immediately behind the jet bow shock. Behind the 

jet, the force drops rapidly due to the combined effect of the low-pressure region behind the jet 

and the high-pressure region produced by the jet bow shock as it wraps around the body. Some 

recovery of the force is produced by the recompression downstream of the low-pressure region 

behind the jet. From the cylindrical portion of the body to the aft of the recompression, the jet 

interaction force shows relatively little variation. The interaction of the jet wake with the tail 

fins produces an additional reduction in the jet interaction force that is approximately the same 
12

Figure 7. Pressure contours on 2.75-in rocket testbed, single jet. 

Figure 8. Pressure contours on 2.75-in rocket testbed, single jet, close-up of jet region. 

magnitude as the jet interaction force on the cylindrical portion of the body. The net jet 

interaction force for the complete body is negative, indicating that the jet interaction opposes the 

thrust produced by the jet.
13

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25 30

X/D

Nji C

Figure 9. Jet interaction force distribution along length of body, single nozzle. 

The jet interaction force generated by the tailfins appears, from flow-visualization studies, to be 

due to a small local angle of attack produced by the jet wake. This local angle of attack is 

oriented such that a lift force is produced on the fins that oppose the jet thrust. The local angle of 

attack varies as a result of a velocity gradient across the wake. 

As discussed previously, the jet interaction amplification factor provides a means of assessing 

the size of the jet interaction force relative to the force produced by the jet thrust itself. For the 

single nozzle case, a jet amplification factor of K = 0.79 was computed, indicating that the 

effective thrust produced by the jet is reduced by 21% due to the jet interaction effect. 

Although these lateral control jets produce control forces, the moment can be an important 

contributor for altering the trajectory due to the aerodynamic jump produced by the resulting 

angular rate (14). As previously shown, the jet interaction force has two important 

contributors—on the body near the jet and on the fin hub. Each of these reduce the effective jet 

force. However, when the moment is considered, both the direction of the applied force and the 

location of the applied force relative to the center of gravity are important. In this case, the 

contribution to the jet interaction moment on the body near the jet opposes the moment produced 

by the jet while the contribution to the jet interaction moment from the fin hub provides a 

moment component that augments to the moment produced by the jet thrust. The jet interaction 
14

moment produced by the fin hub is more significant than that from the body because of the large 

moment arm between the center of gravity and the fin hub. The amplification factor for the 

moment of Km = 1.18 was computed for this case, indicating that the moment is increased by 

18% compared with the moment produced by the jet thrust itself. 

The predicted results demonstrate that the interaction of the jet wake with the tailfins plays an 

important role in the overall effectiveness of the jet, despite the large distance between the fins 

and the jet nozzle. As discussed previously, the jet wake produces an asymmetrical velocity 

gradient across the flow field near the fin hub. It is therefore likely that the orientation of the fins 

relative to the jet may be an important variable in the overall performance of the jet. 

In the current study, two tailfin orientations have been considered, as shown in figure 10. In 

orientation 1, the center of the jet wake passes between two adjacent fins. In orientation 2, the 

fins are rotated 60° with respect to orientation 1 so that one of the fins is close to the center of the 

jet wake. The results shown previously in figures 7–9 correspond to fin orientation 1. 

Orientation 1 Orientation 2

Figure 10. Schematic showing orientation of jet nozzle relative to the fins. 

Figure 11 shows a comparison of the jet interaction force distribution along the body for both fin 

orientations. In front of the fins, the jet interaction force is unaffected by the fin orientation. 

However, on the fin hub, the jet interaction force is significantly different for the two fin 

orientations, with a significant reduction in the jet interaction force for the second fin orientation. 

Jet interaction force and moment amplification factors of K = 0.87 and Km = 0.91 were computed 

for fin orientation 2. While the differences in the net jet forces between the two fin orientations 
15

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25 30

X/D

Nji C

Orientation 1

Orientation 2

Figure 11. Comparison of jet interaction force distribution along length of body, single nozzle, fin orientations 1 

and 2. 

is 10%, there is nearly a 30% difference in the net moment with amplification of the moment for 

orientation 1 and deamplification for orientation 2. 

For some of the firings, it was envisioned that two lateral control jets would be firing in tandem 

to increase the net control force. The predictive technique was applied to determine the 

effectiveness of this configuration. Figures 12 and 13 show the predicted pressure contours 

produced by the simultaneous firing of the two lateral control jets. The fins are oriented in 

orientation 1 for these computations. Although the presence of the second jet is apparent in the 

plots of the pressure contours, the forward-located jet appears to produce the strongest effect in 

interacting and redirecting the oncoming flow. 

Figure 14 shows a comparison of the distribution of the jet interaction force along the body for 

both the single and dual jet configurations. The rearward-located jet appears to have little 

influence on the force distribution in the vicinity of the forward-most jet. However, the effect of 

the second jet is much stronger in the low-pressure region behind the jets producing a larger jet 

interaction force that opposes the jet thrust. The effect of the wake on the tail fins is also 

stronger for the dual jet configuration producing an additional force deamplification. The net 

effect of dual jet firings is to produce approximately twice the net force of the single jet. Jet 
16

Figure 12. Pressure contours on 2.75-in rocket testbed, dual jet. 

Figure 13. Pressure contours on 2.75-in rocket testbed, dual jet, close-up of jet region. 

interaction force and moment amplification factors of K = 0.82 and Km = 1.17 were computed 

for this configuration, which are very similar to the results obtained for the single nozzle results 

for fin orientation 1. Although the dual jet firings produce approximately twice the net force and 

moment of a single jet, these results indicate that there is no significant benefit or disadvantage 
17

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25 30

X/D

Nji C

Single Nozzle

Dual Nozzle

Figure 14. Comparison of jet interaction force distribution along length of body, single and dual nozzles. 

of simultaneous firings of the dual lateral control jet compared with sequential firings of the 

same jets as far as the resulting control force and moment. 

Typically, the 2.75-in rocket testbed flies with a small roll rate due to the geometric and 

aerodynamic asymmetries of the fins. Trajectory simulations indicate that the expected roll rate 

is ~ –20 rev/s (counterclockwise rotation of 20 rev/s when viewing the flight body looking 

downrange) for the testbed munition at Mach 2. Predictions of the effect of roll rate on the 

lateral control jet performance were made using the computational procedure. Figure 15 shows a 

comparison of the distribution of the jet interaction normal force along the body for 0, –20, and 

–215 rev/s. Compared to the zero-roll rate results, there is almost no effect of roll rate on the 

lateral jet performance at –20 rev/s. This is not too surprising because the fins rotate only about 

15° in the time it takes for a fluid particle to travel the distance from the jet to the fin hub. 

However, roll rate effects can become more significant if the roll rate is increased. Figure 15 

also shows the jet interaction normal force distribution for a roll rate of –215 rev/s, which is an 

order of magnitude higher than the expected roll rate. Most of the difference occurs several 

calibers downstream of the jet and on the fins. In particular, the jet interaction force near the 

recompression region behind the jet shows a somewhat-larger deamplification compared with the 

lower spin rates. In part, this is due to the fact that the recompression region has rotated around 
18

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25 30

X/D

Nji C

-20 RPS

-215 RPS

0 RPS

Figure 15. Effect of roll rate on jet interaction force distribution along length of body, configuration 2. 

the body and acts on the side of the body rather than on the top. This rotation of the flow field as 

it interacts on the fins also produces an amplification of the jet interaction normal force on the 

fins instead of the deamplification observed for the 0 and –20 rev/s spin rates. 

The rotation of the jet interaction flow field due to the spin can be seen in the surface pressure 

profiles shown in figure 16. At the –20 rev/s spin rate, the flow field is nearly symmetric. As 

the spin rate increases to –215 rev/s, the asymmetries behind the jet produced by the spin rate 

become more apparent. 

The asymmetries in the surface pressure on the axisymmetric part of the body produce a side 

force as shown in figure 17. Due to symmetry, the side force on axisymmetric part of the body 

at zero spin rate is nonexistent, although a small side force is produced on the fins due to the 

asymmetric arrangement of the wraparound fins. A small side force is produced as the spin rate 

is increased to –20 rev/s. However, this side force is quite small compared with the zero-spin jet 

interaction normal force also displayed on this figure. For the larger spin rate of –215 rev/s, the 

side force produced by the jet interaction becomes much larger and has nearly the same 

magnitude as the jet interaction normal force for this spin rate. The additional side force 

produced by the jet interaction results in a misalignment of the total force produced by the jet. 
19

Roll Rate = –215 RPS 

Roll Rate = –20 RPS 

Figure 16. Pressure contours near jet, –20 and –215 rps roll rates, configuration 2. 

Figure 17. Comparison of jet interaction normal and side force distributions along length of body, –20 and –215 rps 

roll rates, configuration 2.
20

Based on the computed jet interaction side and normal forces and the jet thrust, this 

misalignment is ~5° for the –215 rev/s spin rate. 

As noted previously, for the current application, the lateral control jet is pulsed over a short 

duration (6 ms). To properly assess the performance of the pulsed jet, time-dependent 

computations were executed to quantify the time-dependent response of the jet using 

configuration 2. Steady-state jet-on computations were also performed for configuration 2 as a 

baseline for comparison, although only slight differences where found between configuration 2 

and the single-jet configuration 1 results presented previously. Figures 18 and 19 show the 

pressure contours during the jet-on and jet-off transients. During the jet-on transients, the 

disturbance produced by the jet effects the flow field nearest the jet first and gradually sweeps 

down the body to the tailfins. During the jet-off transients, again the flow field nearest the jet is 

effected first and the disturbances are gradually convected downstream to the tailfins as the flow 

field returns to the jet-off steady state. 

Figure 20 shows the jet thrust, the jet interaction force, and total force coefficients as a function 

of time. The results show that after ~2.5 ms, a steady-state jet-on flow field was obtained. This 

was ascertained by monitoring the jet interaction force distribution along the body until steadystate was reached. The results were also compared with the steady-state results discussed 

previously. It should be noted that 2.5 ms is approximately the same time required for a fluid 

particle in the undisturbed freestream to travel the length of the body. After 6 ms, the jet is 

turned off and an additional 2.5 ms are required for the flow field to return to the steady-state jetoff flow. For ~1.5 ms after the jet is turned on or turned off, the variation in the jet interaction 

force is predominately on the axisymmetric part of the body and the forces on the fins are 

relatively unaffected because the disturbances from the jet have not reached the fins. The 

transients 1.5 ms after the jet is turned on or off are for the most part due to transients on the fins, 

although some transients on the body are still apparent as steady flow is established on the entire 

body. 

Jet interaction linear and angular impulse amplification factors of I = 0.72 and Im = 1.21 were 

computed for the time-dependent pulsed jets. These compare favorably with the jet interaction 

force and moment amplification factors for the steady-state jet for the same configuration. 

Essentially, the linear and angular impulses produced by the time-dependent pulsed jet are 

similar to the impulse that would be produced if the steady-state jet force and moment were to be 

applied over the 6-ms duration of the pulsed jet. Because the jet interaction force takes some 

time to develop when the jet is turned on and off, the linear impulse due to the pulsed jet occurs 

slightly earlier than if the jet force acted as a square-wave pulse over the 6-ms duration of the jet 

pulse.
21

(a) 0.1 ms 

(b) 0.25 ms 

(c) 0.5 ms 

Figure 18. Time-dependent pressure contours during jet-on transient. 
22

(d) 1.0 ms 

(e) 1.5 ms 

(f) 2.5 ms 

Figure 18. Time-dependent pressure contours during jet-on transient (continued). 
23

(a) 6.0 ms 

(b) 6.1 ms 

(c) 6.25 ms 

Figure 19. Time-dependent pressure contours during jet-off transient. 
24

(d) 6.5 ms 

(e) 7.0 ms 

(f) 10.0 ms 

Figure 19. Time-dependent pressure contours during jet-off transient (continued).
25

-0.20

0.00

0.20

0.40

0.60

0.80

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Time - Seconds

Force Coefficient

Total Force

Jet Thrust

Jet Interaction

Figure 20. Variation of jet thrust, jet interaction force, and total force as a function of time, single nozzle. 

4. Summary

The aerodynamic performance of lateral control jets for a testbed munition has been investigated 

using a sophisticated numerical technique. Using the numerical approach, the interaction of the 

jet with the external flow around the munition has been captured and its effect on the lateral 

control jet performance has been assessed. Important design variables such as single and tandem 

jet geometries, tail fin orientation relative to the jet, and munition spin rate have been 

investigated. Both steady-state and time-dependent pulsed jet performances have been predicted 

using the computational approach. 

The forces and moments produced by the interaction of the jet with the external flow over the 

body were significant contributors to the total force and moment produced by the lateral control 

jets. Thus, benchtop characterization of the jet thrust yielded only a portion of the total force 

produced by the jet. The most significant jet interaction forces and moments were created on 

two distinct parts of the body. Near the jet, the high pressure around the jet orifice and the low 

pressure in the wake jet behind the jet interacted to produce competing forces that for the flow 
26

conditions presented here produced a net deamplification or reduction of the thrust produced by 

the jet. Significant forces and moments were also produced by the interaction of the jet wake 

with the tail fins of the testbed munition. The relative magnitude of the forces and moments on 

the tail fins was found to be dependent on factors such on the geometric orientation of the fins 

relative to the jet and projectile spin rate. In some cases, amplification of the jet moment was 

observed even when force deamplification occurred, due to the interaction of the wake with the 

rear mounted tail fins. 

The fin orientation relative to the jet was also found to be an important parameter in the 

performance of the lateral control jets. When one of the fins was aligned in the same 

circumferential plane as the jet, deamplification of jet force and moment was observed. 

However, when the fins were oriented such that the circumferential plane containing the jet was 

located between adjacent fins, a net amplification of the jet moment was observed, increasing 

nearly 30% compared with the other fin orientation. 

The effect of spin on the performance of the lateral control jets was also investigated. For the 

expected range of spin rates for the testbed munition, the predictions showed that spin rate had 

very little effect on the performance of the lateral control jets. However, if the spin rate was 

increased to an order of magnitude, the performance of the lateral control jet showed a much 

larger dependence on the spin rate including a misalignment of the net thrust relative to the jet 

nozzle. 

A tandem jet arrangement was also investigated as a means of increasing the control authority by 

firing multiple jets simultaneously. The results showed a doubling of net force produced by 

tandem jet firings. While doubling of the jet thrust was expected, the results showed that the jet 

interaction force also doubled, resulting in nearly identical jet force amplification factors for the 

tandem jet arrangement compared with the single jet configuration. The implication here was 

that no synergistic effect was observed for the tandem jet firings, and virtually the same net jet 

force could be obtained by sequential firings of the individual jets. 

Time-dependent computations were also performed to assess the performance of the pulsed jets. 

The predictions showed that steady-state flow fields were obtained for about half of the 6 ms 

duration of the jet pulse. During the transient phases of start up and shut down of the jets, the 

transient effects were observed first near the nozzle and lastly on the tail fins on the aft of the 

body. The jet interaction linear and angular impulse factors for the pulsed jets were very similar 

to the jet interaction amplification factors for the steady-state jets. Thus, the steady-state jet 

results could be used to provide meaningful assessments of the performance of the pulsed jets. 
27

5. References 

1. Brown, T. G.; Muller, P.; Brosseau, T.; Weinacht, P.; Childers, M.; Newill, J. Accuracy 

Control of a Rocket. Proceedings of the 20th International Symposium on Ballistics, 

Orlando, FL, 23–27 September 2002. 

2. Graham, M. J.; Weinacht, P. Numerical Investigation of Supersonic Jet Interaction for 

Axisymmetric Bodies. Journal of Spacecraft and Rockets 2000, 37 (5), 675–683. 

3. Graham, M. J.; Weinacht, P.; Brandeis, J. Numerical Investigation of Supersonic Jet 

Interaction for Finned Bodies. Journal of Spacecraft and Rockets 2002, 39 (3), 376–383. 

4. Pulliam, T. H.; Steger, J. L. On Implicit Finite-Difference Simulations of ThreeDimensional Flow. AIAA Journal 1982, 18 (2), 159–167. 

5. Baldwin, B. S.; Lomax, H. Thin Layer Approximation and Algebraic Model for Separated 

Turbulent Flows. AIAA Paper 78-257; 16th AIAA Aerospace Sciences Meeting, Huntsville, 

AL, January 1978. 

6. Cebeci, T. Calculation of Compressible Turbulent Boundary Layers With Heat and Mass 

Transfer. AIAA Paper 70-741; June–July 1970. 

7. Ying, S. X.; Steger, J. L.; Schiff, L. B.; Baganoff, D. Numerical Simulation of Unsteady, 

Viscous, High-Angle-of-Attack Flows Using a Partially Flux-Split Algorithm. AIAA Paper 

86-2179; AIAA 13th Atmospheric Flight Mechanics Conference, Williamsburg, VA, August 

1986. 

8. Sahu, J.; Steger, J. L. Numerical Simulation of Three-Dimensional Transonic Flows. 

International Journal for Numerical Methods in Fluids 1990, 10, 55–873. 

9. Steger, J. L.; Warming, R. F. Flux Vector Splitting of the Inviscid Gasdynamic Equations 

With Application to Finite-Difference Methods. Journal of Computational Physics 1981, 

40, 263–293. 

10. Steger, J. L.; Buning, P. Developments in the Simulation of Compressible Inviscid and 

Viscous Flow on Supercomputers. NASA TM 86674, Moffett Field, CA, 1985. 

11. Beam, R.; Warming, R. F. An Implicit Factored Scheme for the Compressible NavierStokes Equations. AIAA Journal 1978, 16 (4), 85–129. 

12. Benek, J. A.; Donegan, T. L.; Suhs, N. E. Extended Chimera Grid Embedding Scheme With 

Application to Viscous Flows. AIAA Paper No. 87-1126-CP, 1987. 
28

13. Meakin, R. L. Composite Overset Structured Grids. In CRC Handbook of Grid Generation; 

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

14. Guidos, B. J.; Cooper, G. R. The Effect of a Simple Lateral Impulse on Kinetic Energy 

Projectiles in Flight. ARL-TR-2076; U.S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, December 1999. 
29

List of Abbreviations and Symbols 

C Normal force coefficient, N

ref

2 V S

2

1

F

ρ

N ji C Jet interaction normal force coefficient 

Yji C Jet interaction side force coefficient 

D Reference diameter 

e Total energy per unit volume 

Gˆ F, ˆ E, ˆ Flux vectors in transformed coordinates 

f

r

 Acceleration vector due to rotating frame 

x y z f ,f ,f Component of acceleration vector f

r

 in the x , y, and z directions 

F Force 

F Jet thrust j

F Jet interaction force ji

Fno− jet Jet-off force 

Hˆ Source term in Navier-Stokes equations due to rotating coordinate frame 

x i Unit vector along body longitudinal axis 

I Jet interaction linear impulse amplification factor 

mI Jet interaction angular impulse amplification factor 

J Jacobian 

K Jet interaction force amplification factor 

K m Jet interaction moment amplification factor 

M Moment due to jet thrust j

M Jet interaction moment ji
30

p Pressure, as used in Navier-Stokes equations 

qˆ Vector of dependent variables scaled by Jacobian 

R

r

 Displacement vector between the axis of rotation and location in flow field 

Re Reynolds number, ∞ ρ ∞ μ ∞ a D /

S

ˆ Viscous flux vector 

S Reference area, ref 4

D S

2

ref

π

=

t Time 

u, v,w Velocity components in the three coordinate directions 

U Contravariant velocity components ,V,W

V Reference velocity 

x Axial, horizontal, a , y, z nd vertical coordinates 

X Dimensional axial coordinate 

Greek Symbols 

γ Ratio of specific heats 

μ Viscosity 

ρ Density 

Ω Roll rate of body/coordinate system 

ξ,η,ζ Transformed coordinates in Navier-Stokes equations 
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