Time-Accurate Simulations of Synthetic Jet-Based Flow 

Control for an Axisymmetric Spinning Body 

by Jubaraj Sahu 

ARL-TR-3300 September 2004 

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Army Research Laboratory 

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ARL-TR-3300 September 2004 

Time-Accurate Simulations of Synthetic Jet-Based Flow 

Control for an Axisymmetric Spinning Body 

Jubaraj Sahu 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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 Time-Accurate Simulations of Synthetic Jet-Based Flow Control for an 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

This report describes a computational study undertaken to consider the aerodynamic effect of synthetic jets as a means of 

providing the control authority needed to maneuver a projectile at a low subsonic speed. A time-accurate Navier-Stokes 

computational technique has been used to obtain numerical solutions for the unsteady jet-interaction flow field for a spinning 

projectile at a subsonic speed, Mach = 0.24, and angle of attack, 0 degree. Numerical solutions have been obtained by 

Reynolds-averaged Navier-Stokes (RANS) and hybrid RANS-large-eddy simulation turbulence models. Unsteady numerical 

results show the effect of the jet on the flow field and the aerodynamic coefficients, particularly the lift force. These numerical 

results are being used to assess if synthetic jets can be used to provide the control authority needed for maneuvering munitions 

to hit the targets with precision. 

15. SUBJECT TERMS

 flow control; projectile aerodynamics; spinning; synthetic jet; unsteady 

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 Jubaraj Sahu 

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iii

Contents 

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Solution Technique 2

2.1 CFD++ Flow Solver........................................................................................................2 

2.2 Dual Time Stepping.........................................................................................................3 

2.3 Unsteady Jet Boundary Condition...................................................................................4 

2.4 Grid Movement ...............................................................................................................4 

2.5 Hybrid RANS-LES Model ..............................................................................................5 

3. Results and Discussion 6

4. Summary and Conclusions 17

6. References 19 

Distribution List 21 
iv

List of Figures 

Figure 1. Projectile geometry..........................................................................................................7 

Figure 2. Aft-end geometry showing the jet location. ....................................................................7 

Figure 3. Computational grid near the projectile............................................................................8 

Figure 4. Schematic showing the jet actuation in one spin cycle (view from the front or the 

nose)...........................................................................................................................................9 

Figure 5. Computed particle traces colored by velocity, jet on, M = 0.24, α = 0 degree..............9 

Figure 6. Computed lift and side forces, jet on, URANS, M = 0.24, jet velocity = 31 m/s, 

α = 0 degree. ............................................................................................................................10 

Figure 7. Computed lift and side forces, jet-on, unsteady RANS, M = 0.24, jet velocity = 

69 m/s, α = 0 degree. ...............................................................................................................10 

Figure 8. Computed lift and side forces, jet-on, LNS, M = 0.24, jet velocity = 69 m/s, 

α = 0 degree. ............................................................................................................................11 

Figure 9. Comparison of computed lift force, jet-off and jet-on (jet velocity = 69 m/s), 

LNS, M = 0.24, α = 0 degree...................................................................................................12 

Figure 10. Comparison of computed lift force, URANS and LNS, M = 0.24, jet velocity = 

69 m/s, α = 0 degree. ...............................................................................................................12 

Figure 11. Computed lift force over many spin cycles, jet-off, LNS, M = 0.24, α = 0 degree, 

spin = 67 Hz.............................................................................................................................14 

Figure 12. Computed lift force over many spin cycles, LNS, jet velocity = 31 m/s, M = 0.24, 

α = 0 degree, spin = 67 Hz. .....................................................................................................14 

Figure 13. Computed lift force over many spin cycles, LNS, jet velocity = 69 m/s, M = 0.24, 

α = 0 degree, spin = 67 Hz. .....................................................................................................15 

Figure 14. Computed pitching moment over many spin cycles, LNS, jet velocity = 69 m/s, 

M = 0.24, α = 0 degree, spin = 67 Hz......................................................................................15 

Figure 15. Computed lift force over many spin cycles for different jet velocities, LNS, 

M = 0.24, α = 0 degree, spin = 67 Hz......................................................................................16 

Figure 16. Computed impulse from the lift force as a function of spin cycles for different 

jet velocities, LNS, M = 0.24, α = 0 degree, spin = 67 Hz......................................................16 

Figure 17. Computed impulse as a function of spin cycles, jet velocity = 69 m/s, LNS, 

M = 0.24, α = 0 degree, spin = 67 Hz......................................................................................17 
v

Acknowledgments 

This work was accomplished as part of a research project on micro-adaptive flow control 

sponsored and jointly funded by the Defense Advanced Research Projects Agency (DARPA) and 

the U.S. Army Research Laboratory (ARL). The author wishes to thank Sukumar Chakravarthy 

from Metacomp Technologies for his expert technical advice. The scientific visualization work 

of Richard Angelini of ARL and the computational support of the ARL Major Shared Resource 

Center are greatly appreciated. 
vi

INTENTIONALLY LEFT BLANK
1

1. Introduction 

The prediction of aerodynamic coefficients for projectile configurations is essential in the 

assessment of the performance of new designs. Accurate determination of aerodynamics is 

critical to the low-cost development of new, advanced guided projectiles, rockets, missiles, and 

smart munitions. Fins, canards, and jets can be used to provide control for maneuvering 

projectiles and missiles. The flow fields associated with these control mechanisms for the Army 

weapons are complex, involving three-dimensional (3-D) shock-boundary layer interactions, jet 

interaction with the free stream flow, and highly viscous dominated separated flow regions (1 

through 3). The jet interference can extend over significant portions of the projectile and must 

be modeled correctly. For missiles, jet thrusters have been studied over a number of years to 

provide high-speed aerodynamic control. These thrusters interact with the surrounding flow 

field, and the resulting jet interaction flow field again is complex. Recently, several studies have 

shown that tiny synthetic unsteady jets can significantly alter the flow field and pressure 

distributions for airfoils and cylinders (4,5,6). These synthetic jets are active control devices 

with zero net mass flux and are intended to produce the desired control of the flow field through 

momentum effects. Many parameters such as jet location, jet velocity, and actuator frequency 

can affect the flow control phenomenon. 

Smith and Glezer(4) have conducted an excellent study of the flow control by synthetic jets to 

provide increased fundamental understanding of the flow physics. Amitay et al. (5) 

experimentally investigated flow separation control on a cylinder using synthetic jet actuators. 

Their work showed that the interaction of the synthetic jet with the free stream flow resulted in a 

virtual modification of the body shape and significantly increased the lift force as a result of the 

flow reattachment. Aerodynamic flow control over an unconventional airfoil has also been 

demonstrated by Amitay et al. (6) to enhance post-stall performance via actuators operating at 

frequencies higher than the characteristic frequency of the airfoil. The synthetic jets are also 

being investigated for possible applications to improve heat transfer, reduce drag, and enhance 

mixing (7) in combustors, etc. The present analysis involves these synthetic jets for projectile 

aerodynamic control. The emphasis in the present research is to provide insight into the 

interaction of these unsteady jets with the free stream flow and to determine the feasibility of 

these jets for aerodynamic control of a subsonic spinning projectile. 

Computational and experimental data for these jet interactions are very limited. Simple theories 

cannot predict the complex flow fields associated with the jet interaction, and experimental tests 

are very expensive. To help reduce experimental costs, computational fluid dynamics (CFD) are 

being used to predict these complex flows and provide detailed pressure, force, and moment 

data. There have been several recent numerical studies (8,9) of flow separation control using 

synthetic jet actuators. He and Kral (8) have used Reynolds-averaged Navier-Stokes (RANS) 
2

models to study the effect of jet location and jet-forcing frequency on the lift and drag forces on 

an airfoil. The jet actuator was shown to increase the time-averaged lift and also to increase the 

amplitude of oscillation. Lee and Goldstein (9) have used the direct numerical simulation (DNS) 

to model a two-dimensional (2-D) synthetic jet. Although this simulation produced very good 

results, the application of DNS to practical 3-D flows of interest is prohibitive because of its 

computing resources requirement. Even large-eddy simulations (LES) (10) where large eddies 

are computed directly and the small scales are modeled require large computational cost 

compared to RANS simulations. While the RANS method works well for steady flows, the 

accuracy of this method for unsteady flows may be less than desired. Since the large energycontaining eddies are computed by the LES method, this technique is more capable of handling 

unsteady shear layers and wakes, etc. Recently, therefore, a hybrid approach (11,12) that 

combines RANS and LES has been developed to solve practical problems of interest involving 

unsteady flows at reasonable computational cost. Both RANS and hybrid RANS-LES models 

have been used in the present study. 

The advanced CFD capability used here solves the Navier-Stokes equations (13 through 16) and 

incorporates unsteady boundary conditions for simulation of the synthetic jets (17,18). Also, a 

hybrid RANS-LES turbulence model (12) was used for accurate numerical prediction of 

unsteady jet flows. Sahu (19) used these advanced techniques and performed numerical flow 

field computations for steady and unsteady jets for a non-spinning projectile configuration at a 

low subsonic speed. Computed lift forces attributable to the unsteady synthetic jets were found 

to match the experimental data well (17,18). The present numerical study is a direct extension of 

that research to a spinning projectile at subsonic speeds and atmospheric flight conditions. The 

following sections describe the numerical procedure and the computed results obtained. Results 

obtained for the spinning projectile configuration are presented at Mach 0.24, angle of attack 

0 degree, and a spin rate of 67 Hz. 

2. Solution Technique 

2.1 CFD++ Flow Solver 

The complete set of 3-D time-dependent Navier-Stokes equations is solved in a time-accurate 

manner for simulations of unsteady jets. A commercially available code, CFD++ (14,15,16), is 

used for the time-accurate unsteady CFD simulations. The basic numerical framework in the 

code contains unified grid, unified physics, and unified computing features. The user is referred 

to these references for details of the basic numerical framework. Here, only a brief synopsis of 

this framework and methodology is given. 

The 3-D, time-dependent RANS equations are solved by the following finite volume method: 
3

[ ] ∫ ∫ ∫ + − ⋅ =

V V

dV dA dV

t

W F G H

∂

∂ (1) 

in which W is the vector of conservative variables, F and G are the inviscid and viscous flux 

vectors, respectively, H is the vector of source terms, V is the cell volume, and A is the surface 

area of the cell face. 

For low speed flows considered here, “preconditioned implicit relaxation” scheme is used to 

achieve faster convergence. It combines three basic ideas: (1) implicit local time stepping, (2) 

relaxation, and (3) preconditioning. Preconditioning the equations ideally equalizes the 

eigenvalues of the inviscid flux Jacobians and removes the stiffness arising from large 

discrepancies between the flow and sound velocities at low speeds. The use of an implicit 

scheme circumvents the stringent stability limits suffered by their explicit counterparts, and 

successive relaxation allows cells to be revised as information becomes available and thus aids 

convergence. These features of the code have been extremely useful in the present numerical 

simulations at very low subsonic speeds. Second order discretization was used for the flow 

variables and the turbulent viscosity equation. The turbulence closure is based on topologyparameter-free formulations. Two-equation and higher order hybrid RANS-LES turbulence 

models were used for the computation of turbulent flows. These models are ideally suited to 

unstructured bookkeeping and massively parallel processing because of their independence from 

constraints related to the placement of boundaries and/or zonal interfaces. 

For computations of unsteady jet interaction flow fields that are of interest here, dual time 

stepping (as described next) was used to achieve the desired time accuracy. In addition, special 

jet boundary conditions were developed and used for numerical modeling of synthetic jets. Grid 

was actually moved to take into account the spinning motion of the projectile. 

2.2 Dual Time Stepping 

The “dual time-stepping mode” of the code was used to perform the transient flow simulations. 

The term “dual time step” implies the use of two time steps. The first is an “outer” or global 

(and physical) time step that corresponds to the time discretization of the physical time variation 

term. This time step can be chosen directly by the user and is typically set to a value to represent 

1/100 of the period of oscillation expected or forced in the transient flow. This time step is 

applied to every cell (not separately varying). 

An artificial or “inner” or “local” time variation term is added to the basic physical equations. 

This time step and corresponding “inner iteration” strategy is chosen to help satisfy the physical 

transient equations to the desired degree. If the inner iterations converge, then the outer physical 

transient equations (their discretization) are satisfied exactly; otherwise, approximately. For the 

inner iterations, the time step is allowed to vary spatially. Also, relaxation with multi-grid 

(algebraic) acceleration is employed to reduce the residues of the physical transient equations. It 

is found that an order of magnitude reduction in the residues is usually sufficient to produce a 
4

good transient iteration. This may require a few internal iterations to achieve (between 3 and 10, 

depending on the magnitude of the outer time step), the nature of the problem, the nature of the 

boundary conditions, and the consistency of the mesh with respect to the physics at hand. 

2.3 Unsteady Jet Boundary Condition 

A large number of boundary conditions (BC) are available and can be specified at the 

appropriate boundaries. Each boundary condition is encoded as a basic form along with a 

collection of modifiers. One particular BC used for the simulations presented here is a 

“oscillating jet” BC. In its basic form, it is a steady inflow/outflow BC wherein the user supplies 

the velocity normal to the boundary along with static temperature and any turbulence quantities. 

When the velocity provided is negative, it is considered to be an inflow and when it is positive, it 

is treated as an outflow. In the case of inflow, the static temperature and turbulence quantities 

are used along with the inflow velocity. In the case of outflow, only the velocity is used. At 

inflow, the tangential component of velocity is set to zero, and at outflow, the tangential 

component is extrapolated from the interior. At outflow, all primitive variables except normal 

velocity are extrapolated from the interior. At inflow, the static pressure is taken from the 

interior. 

The first modifier available for this BC allows the velocity to oscillate. The base velocity is 

multiplied by an amplitude which varies as sin (2πft) where f is the frequency of the oscillation. 

Thus, the oscillating velocity can cycle from being positive to being negative and back within 

each period (or from being negative to positive and back, based on the sign of the input for the 

basic BC formulation). 

A second modifier permits the steady or oscillating inflow/outflow to be “on” over certain time 

intervals and “off” during other intervals. During “on” periods, the basic or the basic multiplied 

by the oscillating amplitude multiplier (first modifier) is used. The user provides the ranges of 

time during which the jet is on. The user also provides a repetition time period (e.g., the time 

period corresponding to one spin rotation of the projectile). Within each time period, therefore, 

there are sets of starting and ending times that define when the jet is on. During “off” periods, 

the amplitude is set to zero. In parts of the cycle when the jet is off, the boundary condition thus 

reverts to the condition of inviscid surface tangency. This allows the flow to slip past the 

boundary, as would exist (in the form of a shear layer) if the jet were emanating from a 

cavity/hole. 

2.4 Grid Movement 

CFD++ code has many features related to grid movement. Grid velocity can be assigned to each 

mesh point. This general capability can be tailored for many specific situations. For example, 

the grid point velocities can be specified to correspond to a spinning projectile. In this case, the 

grid speeds are assigned as if the grid were attached to the projectile and spinning with it. 
5

A proper treatment of grid motion requires careful attention to the details of the implementation 

of the algorithm applied to every mesh point and mesh cell so that no spurious numerical effects 

are created. For example, a required consistency condition is that free stream uniform flow be 

preserved for arbitrary meshes and arbitrary mesh velocities. This condition is satisfied in the 

code. 

Another important aspect concerns the way that boundary conditions are affected by grid 

velocities. Two significant classes of boundary conditions (slip or no slip at a wall, and far field 

boundary) are considered here. Both are treated in a manner that works seamlessly with or 

without mesh velocities. In both cases, consider the contra-variant velocity, which includes the 

effect of grid motion. Effectively, it is the normal component of the velocity relative to the 

mesh. At the body surface, it is set to zero. For a no-slip wall, one adds the requirement that the 

tangential component of the velocity is equal to the mesh velocity by default. Effectively, this 

assures that the velocity of the mesh is equal to the velocity of the flow at the body. At a farfield boundary, the sign of the contra-variant velocity determines inflow (negative sign) or 

outflow (positive sign). The magnitude of the contra-variant velocity is compared with the local 

speed of sound, helping to define one of four possibilities: supersonic inflow, supersonic 

outflow, subsonic inflow, subsonic outflow. The characteristics theory is used to determine what 

and how much information is applied as the boundary condition for each type. A careful 

treatment of the grid motion results in its use in a transparent manner. 

2.5 Hybrid RANS-LES Model 

Currently, the two most popular forms of turbulence closure, namely, ensemble-averaged models 

(typically based on the RANS equations) and LES with a sub-grid-scale model, both face a 

number of unresolved difficulties. Specifically, existing LES models have met with problems 

related to the accurate resolution of the near-wall turbulent stresses. In the near-wall region, the 

foundations of large-eddy simulation are less secure, since the sizes of the (anisotropic) near-wall 

eddies approach those of the Kolmogorov scale, requiring a mesh resolution approaching that of 

a direct numerical simulation. On the other hand, existing ensemble-averaged turbulence models 

are limited by their empirical calibration. Their representation of small-scale flow physics 

cannot be improved by mesh refinement, and over short time scales, they tend to be overly 

dissipative with respect to perturbations around the mean, often suppressing unsteady motion 

altogether. 

While LES is an increasingly powerful tool for unsteady turbulent flow prediction, it is still 

prohibitively expensive. To bring LES closer to becoming a design tool, a hybrid RANS-LES 

approach based on limited numerical scales (LNS) has been recently developed by Metacomp 

Technologies (12). This approach combines the best features of RANS and LES in a single 

modeling framework. The LNS model is formulated from an algebraic or differential Reynoldsstress model, in which the sub-grid stresses are limited by the numerically computed local length 

scale and velocity scale products. LNS thus behaves like its parent RANS model on RANS-type 
6

grids but reverts to an anisotropic LES sub-grid model as the mesh is refined locally, thereby 

reaching the correct (DNS) fine grid limit. Locally embedded regions of LES may be achieved 

automatically through local grid refinement, while the superior near-wall stress predictions of the 

RANS model are preserved, removing the need of ad hoc, topography-parameter-based wall 

damping. 

The LNS hybrid formulation is well suited to the simulation of unsteady flows, including mixing 

flows, and contains no additional empirical constants beyond those appearing in the original 

RANS and LES sub-grid models. With this method, a regular RANS-type grid is used except in 

isolated flow regions where denser, LES-type mesh is used to resolve critical unsteady flow 

features. The hybrid model transitions smoothly between an LES calculation and a cubic k-ε

model, depending on grid fineness. A somewhat finer grid was placed around the body and near 

the jet, but the rest of the flow field was occupied by a coarser, RANS-like mesh. 

To date, the LNS technique has been used successfully on a number of unsteady flows. 

Examples include flows over cavities, flows around blunt bodies, flows around airfoils and 

wings at high angle of attack, separation suppression using synthetic jets, forced and natural 

convection flows in a room, and mixing flows in nozzles. 

3. Results and Discussion 

Time-accurate unsteady numerical computations using viscous Navier-Stokes methods were 

performed to predict the flow field and aerodynamic coefficients on the subsonic projectile for 

jet-on conditions. Three-dimensional numerical computations have been performed for the 

projectile configuration with jet interaction with CFD++ code at a subsonic Mach number of 

0.24, angle of attack 0 degree, and a spin rate of 67 Hz. The preconditioned version of the 

CFD++ code was used to obtain an efficient numerical solution at low speed. For modeling of 

the unsteady synthetic jets, both unsteady RANS and a hybrid RANS-LES approach (12) were 

used. For computations of these unsteady jets, full 3-D computations are performed and no 

symmetry was used. 

The subsonic projectile is a 1.8-caliber ogive-cylinder configuration (see figure 1). Here, the 

primary interest is in the development and application of CFD techniques for accurate simulation 

of projectile flow field in the presence of unsteady jets. The first step here was to obtain the 

steady state results for the same projectile without the jet. The converged jet-off steady state 

solution was then used as the starting condition for the computation of time-accurate unsteady 

flow field for the projectile with synthetic jets. Computations were also performed for the steady 

jet cases. The jet locations on the projectile are shown in figure 2. The jet conditions were 

specified at the exit of the jet for both steady (fixed jet velocity) and unsteady (sinusoidal 

variation in jet velocity) jets. The jet conditions specified include the jet pressure, density, and 
7

velocity components. The flow field inside the tiny jet cavity is not computed. For the unsteady 

jet case, time-dependent jet boundary conditions are applied at the jet exit. Numerical 

computations have been made for these jet cases at a Mach number, M = 0.24, an angle of attack, 

α = 0 degree, and a spin rate of 67 Hz. The jet width was 0.32 mm, the jet slot half-angle was 

18 degrees, and the peak jet velocities used were 31 and 69 m/s operating at a frequency of 

1000 Hz. 

Figure 1. Projectile geometry. 

 

Figure 2. Aft-end geometry showing the jet location. 

A computational grid expanded near the vicinity of the projectile is shown in figure 3. Grid 

points are clustered near the jet as well as the boundary layer regions to capture the high gradient 

flow regions. The computational grid has 211 points in the streamwise direction, 241 in the 

circumferential direction, and 80 in the normal direction. The unsteady simulation took 

thousands of hours of central processing unit time on a Silicon Graphics (SGI) Origin computer 

running with 16 to 32 processors. 
8

Figure 3. Computational grid near the projectile. 

Unsteady time-accurate CFD computations require huge computer resources. All the results 

presented here were obtained from unsteady numerical computations with a single synthetic jet 

on a 40–mm subsonic grenade (figure 1). Looking from the front (see figure 4), the projectile 

spins clockwise at a rate of 67 Hz. The jet actuation corresponds to one-fourth of the spin cycle 

from -45 to +45 degrees with 0 degree being the positive y-axis. The jet is off during the 

remaining three-fourths of the spin cycle. Time-accurate unsteady simulation even with a single 

synthetic jet requires a large amount of resources. The unsteady CFD modeling technique 

required about 600 time steps to resolve a full spin cycle. The unsteady synthetic jet operates at 

a high frequency of 1000 Hz. For the part of the spin cycle when the jet is on, the jet operated 

for approximately for four cycles. Time-accurate CFD modeling of each jet cycle required more 

than 40 time steps. The actual computing time for one full spin cycle of the projectile was about 

50 hours using 16 processors (i.e., 800 processor-hours) on an SGI Origin 3000 or an IBM SP3 

system for a mesh size about 4,000,000 grid points. Multiple spin cycles and thus, a large 

number of synthetic jet operations were required to reach the desired periodic time-accurate 

unsteady result. As will be seen later, some cases were run for as many as 60 spin cycles, 

requiring more than 48,000 processor hours of computer time. Many flow field solutions were 

saved at regular intermittent time intervals to produce “movies” to gain insight into the physical 

phenomenon resulting from the synthetic jet interactions. Computed particle traces emanating 

from the jet into the wake are shown in figure 5 at a given instant in time for M = 0.24 and 

α = 0 degree. These traces are colored by the velocity magnitude. The particle traces emanating 

from the jet interact with the wake flow, making it highly unsteady. It shows the flow in the base 

region to be asymmetric because of the interaction of the unsteady jet. The shear layer from the 

free stream flow resulting from the step corner up stream from the base interacts with the 

unsteady jet and deteriorates just a short distance down stream from the jet. The unsteady jet 

substantially alters the flow field near the jet and the base region, which in turn affected the 

forces and moments even at 0 degree angle of attack. 
9

Z

Y

Jet-on

t=0 

t=3.73 ms

Figure 4. Schematic showing the jet actuation in one spin cycle (view 

from the front or nose). 

Figure 5. Computed particle traces colored by velocity, jet on, M = 0.24, 

α = 0 degree. 

The computed surface pressures from the unsteady flow fields are integrated to obtain the 

aerodynamic forces and moments. Computed results have been obtained with unsteady RANS 

(URANS) as well as the hybrid RANS-LES approach referred here as the LNS for the jet-on 

conditions. The unsteady jet is applied while the projectile is spinning. For comparison, the jetoff computations have been performed in a time-accurate mode with the LNS model. Also, the 

jet-off URANS calculations were first obtained and the jets were activated, beginning at time, 

t = 28 ms. The computed normal or lift force (FY) and side force (FZ) are shown in figures 6 and 

7 as a function of time for two different jet velocities, 31 and 69 m/s, respectively. These 

computed results clearly indicate the unsteady nature of the flow field. These forces are found to 
10

change as a function of time. Changes in the side force are slightly smaller than changes in the 

lift force. As shown in figure 6, URANS calculations clearly show the effect of the jet on the lift 

and side forces. When the jet is turned off, the levels of these forces drop to the same levels 

before the jet activation corresponding to the jet-off wake flow. Similar behavior can be seen in 

figure 7 for the bigger jet case. The effect of the jet is stronger, as evidenced by the larger peak 

magnitudes seen in the forces. 

-0.02

0

0.02

0.04

0.06

20 30 40 50 60

Time (ms)

Forces

Fy

Fz

Figure 6. Computed lift and side forces, jet on, URANS, M = 0.24, jet velocity 

= 31 m/s, α = 0 degree. 

-0.1

-0.06

-0.02

0.02

0.06

0.1

20 30 40 50 60 70 80

Time (ms)

Forces

Fy Fz

Figure 7. Computed lift and side forces, jet on, URANS, M = 0.24, jet velocity 

= 69 m/s, α = 0 degree. 
11

Computed results were also obtained with the hybrid RANS-LES or LNS approach for the jet-on 

conditions. Figure 8 shows the lift and side forces obtained with LNS as a function of time for 

the bigger jet case. Here again, the jet was turned on at time, t = 28 ms, starting with an URANS 

solution. The mean values of both FY and FZ forces are non-zero because of the jet interaction. 

The LNS model predicts large amplitudes of oscillations in the forces. For comparison purposes, 

jet-off calculations were obtained with the LNS model, starting with the same initial URANS 

solution. A comparison of the computed lift force with LNS for jet-off and jet-on conditions (see 

figure 9) clearly indicates that the wake is unsteady. Also, the amplitudes of oscillations are 

large for jet-off and jet-on cases. Figure 10 shows the comparison of the predicted lift force with 

URANS and LNS models. As indicated earlier, the URANS approach clearly shows when the 

jet is on and when it is off during the spin cycle. The effect caused by the jet for the LNS case is 

not as easily seen. It is hidden in these oscillations. However, the mean value of the lift force 

seems to be close to zero when the jet is off during the spin cycle. In general, the levels of the 

lift force oscillations predicted by the LNS models are larger than those predicted by the 

URANS. 

-0.4

-0.2

0

0.2

0.4

0.6

0.8

20 30 40 50 60 70

Time (ms)

Forces

Fy Fz

Figure 8. Computed lift and side forces, jet-on, LNS, M = 0.24, jet velocity = 

69 m/s, α = 0 degree. 
12

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

20 30 40 50

Time (ms)

Fy

Jet-on

Jet-off

Figure 9. Comparison of computed lift force, jet-off and jet-on (jet velocity = 

69 m/s), LNS, M = 0.24, α = 0 degree. 

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

20 30 40 50

Time (ms)

Fy

LNS

URANS

Figure 10. Comparison of computed lift force, URANS and LNS, M = 0.24, jet 

velocity = 69 m/s, α = 0 degree. 
13

Based on the comparisons for the non-spinning cases, the LNS method predicted the level of 

forces that closely matched the data (19). In the current study, the addition of spin as well as the 

jet actuation for part of the spin cycle further complicates the analysis of the CFD results with 

LNS. Whether the jet is on or off, the level of oscillations seen is quite large and the effect of the 

jet cannot be easily seen in the instantaneous time histories of the unsteady forces and moments. 

To get the net effect of the jet, unsteady computations were run for many spin cycles of the 

projectile both with and without the synthetic jets. For the synthetic jet cases, two jet velocities 

of 31 and 69 m/s (peak values) were used. The CFD results are plotted over only one spin cycle; 

each subsequent spin cycle was superimposed and a time-averaged result is then obtained over 

one spin cycle. In all these cases, the jet is on for one-fourth of the spin cycle (time, t = 0 to 

3.73 ms) and is off for the remainder (three-fourths) of the spin cycle. Figures 11 through 15 

show the time-averaged results over a spin cycle. Figure 11 shows the averaged computed lift 

force for the jet-off case after 5, 10, 15, and 20 spin cycles, and as expected, it is close to zero, 

especially after 20 spin cycles. 

Figure 12 shows the computed lift force again averaged over many spin cycles for the peak jet 

velocity of 31 m/s. The jet effect can clearly be seen when the jet is on (t = 0 to 3.73 ms) even 

after 5 or 10 spin cycles. The net lift is ~0.07 newton because of the jet actuation and seems to 

have converged after 20 spin cycles. For the remainder of the spin cycle, the jet is off; however, 

the effect of the jet on the wake still persists, and this figure shows that lift force is still available 

because of the jet interaction. The magnitude of the lift force is, of course, reduced when the jet 

is off. The force and moment results for a bigger jet (peak jet velocity of 69 m/s) are shown in 

figures 13 and 14, respectively. As shown in figure 13, the lift force requires at least five spin 

cycles before reaching a more consistent level. The lift force is on the order of 0.17 when the jet 

is on (t = 0 to about 4 ms), and then decreases gradually. The mean level of the lift force when 

the jet is off during the spin cycle is on the order of 0.07 newton. The variation in the pitching 

moment (see figure 14) is very similar and seems to settle after 20 spin cycles. Figure 15 shows 

the computed averaged lift force after 50 and 60 spin cycles for jet velocities 31 and 69 m/s, 

respectively. It clearly shows that the larger jet produces larger lift force than the smaller jet 

when the jet is activated. Again, one can see a contribution of the jet to the lift force for both 

cases, even when the jet is off (time from 4 to 15 ms). The lift force can be integrated over time 

to obtain the impulse. Figure 16 shows the impulse obtained from the lift force as a function of 

the spin cycles for both jets. As seen here, in both cases, it takes about 30 to 40 spin cycles 

before the impulse achieves an asymptote at a fixed value. The impulse is larger for the larger 

jet. Figure 17 shows the impulse obtained from both the lift force and the side force for the 

larger jet case. The impulse from the side force drops close to zero at about 30 spin cycles, then 

rises somewhat, and reaches an asymptote at a value slightly higher than zero. 
14

-0.2

-0.1

0

0.1

0.2

0 5 10 15

Time (ms)

Forces (Newton)

Fy (5)

Fy (10)

Fy (15)

Fy (20)

Figure 11. Computed lift force over many spin cycles, jet-off, LNS, M = 0.24, 

α = 0 degree, spin = 67 Hz. 

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15

Time (ms)

Forces (Newton)

Fy (5)

Fy (10)

Fy (20)

Fy (30)

Fy (40)

Fy (50)

Figure 12. Computed lift force over many spin cycles, LNS, jet velocity = 31 m/s, 

M = 0.24, α = 0 degree, spin = 67 Hz. 
15

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15

Time (ms)

Forces (Newton)

Fy (2)

Fy (5)

Fy (10)

Fy (20)

Fy (30)

Fy (40)

Figure 13. Computed lift force over many spin cycles, LNS, jet velocity = 69 m/s, 

M = 0.24, α = 0 degree, spin = 67 Hz. 

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0 5 10 15

Time (ms)

Pitching Moment (N-m)

Mz (2)

Mz (5)

Mz (10)

Mz (20)

Mz (30)

Mz (40)

Figure 14. Computed pitching moment over many spin cycles, LNS, jet 

velocity =69 m/s, M = 0.24, α = 0 degree, spin = 67 Hz. 
16

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15

Time (ms)

Forces (Newton)

Fy (50), Vj= 31 m/s

Fy (60), Vj = 69 m/s 

Figure 15. Computed lift force over many spin cycles for different jet velocities, 

LNS, M = 0.24, α = 0 degree, spin = 67 Hz. 

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Number of Spin Cycles

Impulse (N-s) * 10**3

Vj = 69 m/s

Vj = 31 m/s

Figure 16. Computed impulse from the lift force as a function of spin cycles for 

different jet velocities, LNS, M = 0.24, α = 0 degree, spin = 67 Hz. 
17

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Number of Spin Cycles

Impulse (N-s) * 10**3

From Fy

From Fz

Figure 17. Computed impulse as a function of spin cycles, jet velocity = 

69 m/s, LNS, M = 0.24, α = 0 degree, spin = 67 Hz. 

The present computed results are currently being used in six-degree-freedom (6-DOF) analyses 

to determine the feasibility of the synthetic jets to provide control authority for the projectile; the 

early indications are that they can. Also, a firing test was just recently completed to show that 

the synthetic jets can provide the control authority needed for maneuvering a subsonic projectile 

to its target. 

4. Summary and Conclusions 

This report describes a computational study undertaken to determine the aerodynamic effect of 

tiny synthetic jets as a means to provide the control authority needed to maneuver a spinning 

projectile at low subsonic speeds. Computed results have been obtained at a low subsonic speed, 

M = 0.24 and 0 degree angle of attack, for a subsonic spinning projectile via time-accurate 

Navier-Stokes computational technique and advanced turbulence models. The unsteady jet is 

shown in this and earlier studies to substantially alter the flow field near the jet and the base 

region of the projectile, which in turn affects the forces and moments even at 0 degree angle of 

attack. It is easier to see the effect of the jet in the RANS predictions for both jets (jet velocities 

of 31 and 69 m/s). Also, RANS predictions clearly show that the jet effect disappears for the 

part of the spin cycle when the jet is off. In other words, it reverts to the usual wake without the 

jet. The level of the lift force predicted by the RANS-LES model is larger than that predicted by 

the URANS technique. This was also observed earlier in the unsteady calculations for the nonspinning projectile case (19). 
18

In general, it is more difficult to easily see the effect of the jet in the unsteady RANS-LES 

results. Time averaging was required with these unsteady results over many spin cycles. The net 

time-averaged results obtained over the time period corresponding to one spin cycle clearly show 

the effect of the synthetic jets on the lift as well as the side forces. It was found that it takes 10 to 

20 full spin cycles before the results converge to a periodic state. The effect of the jet in this 

case is clearly seen when the jet is on during the spin cycle. However, these results show that 

there is an effect on the lift force (although reduced) for the remainder of the spin cycle when the 

jet is off. This is a result of the wake effects that persist from one spin cycle to another. The 

large jet is found to provide larger lift force when the jet is on. The level of lift force for the jetoff period is almost similar. The impulse obtained from the predicted forces for both jets seems 

to reach an asymptote after 30 spin cycles (the impulse for the larger jet being larger). 

This work demonstrates a method to accurately predict the time-accurate unsteady aerodynamics 

of synthetic jets on an M203 grenade-launched projectile. This capability has provided 

fundamental understanding of fluid dynamic mechanisms associated with the interaction of the 

unsteady synthetic jets and the projectile flow fields. The predicted forces and moments are now 

being used in the 6-DOF simulations to assess if these jets can provide adequate control authority 

to steer a projectile to its target. 
19

6. References 

1. Sahu, J.; Heavey, K. R.; Ferry, E. N. Computational Fluid Dynamics for Multiple Projectile 

Configurations. Proceedings of the 3rd Overset Composite Grid and Solution Technology 

Symposium, Los Alamos, NM, October 1996. 

2. Sahu, J.; Heavey, K. R.; Nietubicz, C. J. Time-Dependent Navier-Stokes Computations for 

Submunitions in Relative Motion. 6th International Symposium on Computational Fluid 

Dynamics, Lake Tahoe, NV, September 1995. 

3. Meakin, R. L. Computations of the Unsteady Flow About a Generic Wing/Pylon/FinnedStore Configuration; AIAA 92-4568-CP, August 1992. 

4. Smith B. L.; Glezer, A. The Formation and Evolution of Synthetic Jets. Journal of Physics 

of Fluids September 1998, 10 (9). 

5. Amitay, M.; Kibens, V.; Parekh, D.; Glezer, A. The Dynamics of Flow Reattachment over a 

Thick Airfoil Controlled by Synthetic Jet Actuators; AIAA Paper No. 99-1001, January 1999. 

6. Amitay, M.; Smith, D. R.; Kibens, V.; Parekh, D.; Glezer, A. Aerodynamic Flow Control 

over an Unconventional Airfoil using Synthetic Jet Actuators. AIAA Journal 2001, 39, 361-

370. 

7. Davis, S. A.; Glezer, A. The Manipulation of Large- and Small-Scales in Coaxial Jets using 

Synthetic Jet Actuators; AIAA Paper No. 2000-0403, January 2000. 

8. He, Y.; Kral, L. Post-Stall Control on an Airfoil using Localized Jet Actuators; AIAA Paper 

No. 2000-0408, January 2000. 

9. Lee, C. Y.; Goldstein, D. B. Two-Dimensional Synthetic Jet Simulation; AIAA Paper No. 

2000-0406, January 2000. 

10. Avancha, R.; Pletcher, R. H. Large Eddy Simulation of the Turbulent Flow Past a Backward 

Facing Step; AIAA paper No. 2000-0542, January 2000. 

11. Arunajatesan, S.; Sinha, N. Towards Hybrid LES-RANS Computations of Cavity Flowfields; 

AIAA Paper No. 2000-0401, January 2000. 

12. Batten, P.; Goldberg U.; Chakravarthy, S. Sub-grid Turbulence Modeling for Unsteady Flow 

with Acoustic Resonance; AIAA Paper 00-0473, 38th AIAA Aerospace Sciences Meeting, 

Reno, NV, January 2000. 

13. Pulliam, T. H.; Steger, J. L. On Implicit Finite-Difference Simulations of ThreeDimensional Flow. AIAA Journal February 1982, 18 (2), 159–167. 
20

14. Peroomian, O.; Chakravarthy, S.; Goldberg, U. A ‘Grid-Transparent’ Methodology for 

CFD; AIAA Paper 97-07245, 1997. 

15. Peroomian, O.; Chakravarthy, S.; Palaniswamy, S.; Goldberg, U. Convergence Acceleration 

for Unified-Grid Formulation Using Preconditioned Implicit Relaxation; AIAA Paper 98-

0116, 1998. 

16. Goldberg, U. C.; Peroomian, O.; Chakravarthy, S. A Wall-Distance-Free K-E Model With 

Enhanced Near-Wall Treatment. ASME Journal of Fluids Engineering 1998, 120, 457-462. 

17. Rinehart, C.; McMichael, J. M.; Glezer, A. Synthetic Jet-Based Lift Generation and 

Circulation Control on Axisymmetric Bodies; AIAA Paper No. 2002-3168. 

18. McMichael, J. GTRI, Private Communications 

19. Sahu, J. Unsteady Numerical Simulations of Subsonic Flow over a Projectile with Jet 

Interaction; AIAA Paper 2003-1352, Reno, NV, 6-9 January 2003. 
21

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