Time-Accurate Computations of Free-Flight Aerodynamics 

of a Spinning Projectile With and Without Flow Control 

by Jubaraj Sahu 

ARL-TR-3919 September 2006 

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Army Research Laboratory 

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ARL-TR-3919 September 2006 

Time-Accurate Computations of Free-Flight Aerodynamics 

of a Spinning Projectile With and Without Flow Control 

Jubaraj Sahu 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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 Time-Accurate Computations of Free-Flight Aerodynamics of a Spinning 

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 Jubaraj Sahu (ARL) 

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U.S. Army Research Laboratory 

Weapons and Materials Research Directorate 

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14. ABSTRACT 

This report describes a new multi-disciplinary computational study undertaken to compute the flight trajectories and 

simultaneously predict the unsteady free flight aerodynamics of a spinning projectile configuration with and without 

aerodynamic flow control. Actual flight trajectories are computed with an advanced coupled computational fluid dynamics 

(CFD)-rigid body dynamics (RBD) technique. An advanced time-accurate Navier-Stokes computational technique has been 

used in CFD to compute the unsteady aerodynamics associated with the free flight of the spinning projectile at subsonic speeds. 

Computed positions and orientations of the projectile have been compared with actual data measured from free flight tests and 

are found to be generally in good agreement. The same advanced coupled procedure has been used to compute the 

aerodynamics of the spinning projectile with flow control with the use of a synthetic microjet. Unsteady numerical results 

obtained from the coupled method show the flow field, the aerodynamic forces and moments, and the flight trajectories of the 

projectile. 

15. SUBJECT TERMS

 coupled CFD-RBD; flow control; projectile aerodynamics; time accurate 

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 Jubaraj Sahu 

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iii

Contents 

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Solution Technique 3

2.1 Dual Time Stepping.........................................................................................................4 

2.2 Unsteady Jet Boundary Condition...................................................................................4 

2.3 Hybrid RANS-LES Model ..............................................................................................5 

2.4 Grid Movement ...............................................................................................................6 

2.5 Six-Degree-of-Freedom Coupling...................................................................................6 

3. Results and Discussion 7

4. Summary and Conclusions 16

5. References 18

Distribution List 20 
iv

List of Figures 

Figure 1. Projectile geometry......................................................................................................... 8 

Figure 2. Aft end geometry showing the jet location..................................................................... 8 

Figure 3. Computational grid near the projectile........................................................................... 9 

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

projectile). ............................................................................................................................... 9 

Figure 5. Computed particle traces colored by velocity, jet on, M = 0.39. ................................. 10 

Figure 6. Computed velocity magnitudes at a given instant in time............................................ 11 

Figure 7. Comparison of computed roll angle, jet off.................................................................. 11 

Figure 8. Comparison of computed Euler pitch angle, jet off for different time steps................ 12 

Figure 9. Comparison of the computed Euler yaw angle, jet off for different time steps. .......... 13 

Figure 10. Comparison of the computed z distance, jet off......................................................... 13 

Figure 11. Comparison of the computed y distance, jet off......................................................... 14 

Figure 12. Comparison of the computed Euler pitch angle, LNS, jet off and jet on (jet 

velocity = 110 m/s). ............................................................................................................. 15 

Figure 13. Comparison of the computed side force and the normal force, LNS, jet off and 

jet on (jet velocity = 110 m/s). .............................................................................................. 15 

Figure 14. Comparison of the computed y distance, LNS, jet off and jet on (jet velocity = 

110 m/s). ............................................................................................................................... 16 
v

Acknowledgments 

This work was accomplished as part of a grand challenge project jointly sponsored by the 

Department of Defense High Performance Computing Modernization program and the U.S. 

Army Research Laboratory (ARL). The author wishes to thank Dr. Sukumar Chakravarthy of 

Metacomp Technologies for his technical assistance with some of the issues on the coupling 

techniques between computational fluid dynamics and rigid body dynamics. The author also 

wishes to thank Mr. Wayne Hathaway of Arrow Tech Associates for providing expert advice and 

helping with the six-degrees-of-freedom simulations with the software, ARFDAS (Aeroballistics 

Research Facility Data Analysis System). The scientific visualization and the computational 

support of ARL’s 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, 2, 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 with 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 and drag reduction and to 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. In addition, the objective was to accomplish it through coupled 

computational fluid dynamics (CFD) and rigid body dynamics (RBD) methods. 

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, CFD is 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 
2

Kral (8) have used Reynolds-averaged Navier-Stokes (RANS) 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 the amplitude of oscillation. Lee and Goldstein (9) have used 

the direct numerical simulation (DNS) on a two-dimensional synthetic jet. Although this 

simulation produced very good results, the use of DNS to model practical 3-D flows of interest is 

prohibitive because of its computing resources requirement. Even large eddy simulations (LES) 

(10) in which large eddies are computed directly and the small scales are modeled require large 

computational cost compared to RANS simulations. Although the RANS method works well for 

steady flows, the accuracy of this method for unsteady flows may be less than desired. Since the 

large energy-containing eddies are computed in 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 the 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 well the experimental data (17, 18). That research work was further extended to a 

spinning projectile at subsonic speeds (20). Results obtained for the spinning projectile 

configuration were reported at Mach 0.24, angle of attack 0 degrees and a spin rate of 67 Hz. 

The present numerical study is a big step forward and a direct extension of that research which 

now includes numerical simulation of the actual flight paths of the projectile with and without 

flow control using coupled CFD-RBD techniques. The present research allows “virtual fly-out” 

of projectiles on the supercomputers and allows numerical prediction of the actual flight paths of 

a projectile and all the associated unsteady free flight aerodynamics using coupled CFD-RBD 

techniques in an integrated manner. Sahu (21) has successfully applied such advanced coupled 

procedures to simultaneously determine the flight trajectory and the associated unsteady freeflight aerodynamics of a finned projectile at supersonic velocity. The present research is an 

extension of this work to a spinning projectile at a subsonic speed with and without flow control. 

The following sections describe the coupled numerical procedure and the computed results 

obtained for the spinning body of revolution at subsonic speeds. 
3

2. Solution Technique 

At the U.S. Army Research Laboratory (ARL), research efforts are performing real-time 

multidisciplinary-coupled CFD-RBD aerodynamic computations for the entire flight trajectory of 

a complex guided projectile system. A real-time accurate approach is used in the present work; 

however, the computations require much greater computer resources. The real-time accurate 

approach requires that the six-degrees-of-freedom (6-DOF) body dynamics be computed at each 

repetition of a flow solver. The CFD capability used here solves the Navier-Stokes equations 

and incorporates advanced boundary conditions and grid motion capabilities. The complete set 

of 3-D time-dependent Navier-Stokes equations is solved in a time-accurate manner for 

simulations of actual flights. A commercially available code, CFD++ (17, 18, 19, 20), 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. 

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

[ ] ∫ ∫ ∫ + − ⋅ =

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, the “preconditioned implicit relaxation” scheme is used to 

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

(b) relaxation, and (c) 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 topology-parameter-free formulations. 

Two-equation and higher order hybrid RANS-LES turbulence models were used to compute 

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. 
4

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 the numerical modeling of synthetic jets. Grid 

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

2.1 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. It is also applied to 

every cell and is not spatially 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 (or their discretization) are satisfied exactly; otherwise, they are satisfied 

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 good transient iteration. This may require a few internal iterations 

(between 5 and 10) to achieve, 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. 

2.2 Unsteady Jet Boundary Condition 

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

boundaries. Each BC is encoded as a basic form along with a collection of modifiers. One 

particular BC used for the simulations presented here is an “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 that varies as sin (2πft) where f is the frequency of the oscillation. 
5

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 during 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, which 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 procedure 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 

or hole. 

2.3 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, 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 if we refine the mesh, and over short time scales, they tend to be overly 

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

altogether. 

Although 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 

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. 
6

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, the rest of the flow field being 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. For computations of unsteady synthetic jet interaction flow 

fields that are of interest here, the hybrid RANS-LES or LNS technique was found to be essential 

in the accurate numerical predictions of such flows (19, 20). 

2.4 Grid Movement 

Grid velocity is 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 is attached to the 

projectile and spinning with it. Similarly, to account for RBD, the grid point velocities can be 

set as if the grid is attached to the rigid body with 6 DOF. For the RBD, the coupling refers to 

the interaction between the aerodynamic forces and moments and the dynamic response of the 

projectile or body to these forces and moments. The forces and moments are computed every 

CFD time step and transferred to a 6-DOF module that computes the body’s response to the 

forces and moments. The response is converted into translational and rotational accelerations 

that are integrated to obtain translational and rotational velocities and integrated once more to 

obtain linear position and angular orientation. The 6-DOF RBD module uses quaternions to 

define the angular orientations. However, these are easily translated into Euler angles. From 

the dynamic response, the grid point locations and grid point velocities are set. 

2.5 Six-Degree-of-Freedom Coupling 

In CFD++, two modes are available to help simulate RBD: an uncoupled mode and a coupled 

mode. The coupling refers to the interaction between the aerodynamic forces and moments and 

the dynamic response of the projectile or body to these forces and moments. In both modes, the 

forces and moments are computed every time step and reported to the user. In the coupled mode, 

the forces and moments are passed to a 6-DOF module which computes the body’s response to 

the forces and moments. The response is converted into translational and rotational accelerations 

which are integrated to result in translational and rotational velocities and integrated once more 

to result in linear position and angular orientation. The 6-DOF RBD module uses quaternions to 

define the angular orientations. However, these are easily translated into Euler angles. From the 
7

dynamic response, the grid point locations and grid point velocities are set. In the uncoupled 

mode, the forces and moments are not coupled with the RBD module. The motion of the 

projectile is kinematics only and depends on the initial linear and angular velocities prescribed. 

Typically, we begin with a computation performed in “steady state mode” with the grid 

velocities prescribed to account only for the translational motion component of the complete set 

of initial conditions to be prescribed. At this stage, we also impose the angular orientations from 

the initial conditions. The complete set of initial conditions includes translational and rotational 

velocity components, along with initial position and angular orientation. With a fixed translational velocity, we obtain the steady state solution. This becomes the initial condition for the 

next step which involves adding just the spin component of the projectile. With the addition of 

spin, time-accurate calculations are performed for a few cycles of spin until converged periodic 

forces and moments are obtained. A sufficient number of time steps are performed so that the 

angular orientation for the spin axis corresponds to the prescribed initial conditions. All this is 

performed in an uncoupled mode. The angular velocity initial conditions associated with the 

non-spin rotational modes are then added. The mesh is translated back to the desired initial 

position, the non-spin rotational velocity initial conditions are turned on, and computations are 

performed in the coupled mode. 

3. Results and Discussion 

Time-accurate unsteady numerical computations were performed with Navier-Stokes and 

coupled 6-DOF methods to predict the flow field and aerodynamic coefficients and the flight 

paths of a subsonic spin-stabilized projectile for both jet-off and jet-on conditions at a subsonic 

speed, M = 0.39. The preconditioned version of the CFD++ code was used to obtain 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. In all cases, full 3-D computations 

were 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-RBD techniques for accurate 

simulation of projectile flow field with and without flow control using unsteady microjets. 

The first step here was to obtain the steady state results for the same projectile without the jet 

with the grid velocities prescribed. Also imposed were the angular orientations at this stage. 

Corresponding converged jet-off steady state solution was then used as the starting condition, 

along with the other initial conditions for the computation of coupled CFD-RBD runs. Synthetic 

jets were activated at a specified point in the trajectory. The jet locations on the projectile are 

shown in figure 2. The jet conditions were specified at the exit of the jet (sinusoidal variation in 

jet velocity). The jet conditions specified include the jet pressure, density, and velocity components. 
8

The flow field inside the tiny jet cavity is not computed. For the unsteady jets, time-dependent 

jet boundary conditions are applied at the jet exit. Numerical computations have been made for 

these jet cases at an initial Mach number, M = 0.39, initial angle of attack, α = 2 degrees, and an 

initial spin rate of 434 Hz. The jet width was 0.32 mm, the jet slot half-angle was 18 degrees, and 

the peak jet velocity used was 110 m/s operating at a frequency of 1000 Hz. The jet-off coupled 

calculations are started from the same initial conditions. 

Figure 1. Projectile geometry. 

Jet

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 gradients 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 coupled numerical simulations took 

thousands of hours of central processing unit time on an IBM SP1 P4 computer running with 32 to 

64 processors. 

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 (see figure 1). Looking from the back of the projectile (see figure 4), 

 1Not an acronym 
9

we see that the projectile spins clockwise. 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. The unsteady CFD modeling technique required about 

180 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 

four cycles. 

Figure 3. Computational grid near the projectile. 

Z

Y

Jet-on

Figure 4. Schematic showing the jet actuation in one spin 

cycle (view from the back of the projectile). 
10

Computed particle traces emanating from the jet into the wake are shown in figure 5 at a given 

instant in time for Mach = 0.39. 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 breaks down just a short distance down stream from the jet. The unsteady 

jet substantially alters the flow field near the jet and the base region that in turn affected the forces 

and moments even at 0 degree angle of attack. 

Figure 6 shows the velocity magnitude contours at a given time or at a given location in the 

trajectory. It clearly shows the orientation of the body at that instant in time and the resulting 

asymmetric flow field in the wake because of the body at angle of attack. The orientation of the 

projectile of course changes from one instant in time to another as the projectile flies down range. 

This includes the Euler pitch and yaw angles as well. The computed surface pressures from the 

unsteady dynamic 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 RANSLES approach referred here as the LNS for the jet-on conditions. The unsteady jet is applied while 

the projectile is spinning (see figure 4). As pointed out earlier, the jet-off computa-tions have been 

made in a time-accurate dynamic coupled mode. Numerical calculations with the microjet are also 

computed with the coupled approach. Computational results for the jet-off case obtained with the 

coupled procedure are shown next. For these coupled simulations, the aerody-namic forces and 

moments were completely obtained through CFD as the projectile flew. The simulation started 

from the first station away from the muzzle where the actual data were measured. The first station 

was situated about 4.9 m from the muzzle. 

Figure 5. Computed particle traces colored by velocity, jet on, M = 0.39. 
11

Figure 6. Computed velocity magnitudes at a given instant in time. 

Figure 7 shows the computed roll angles as a function of the x-distance or the range. The roll 

angles shown here are the accumulated values to include 360 degrees for every spin cycle of the 

projectile. Computed roll angles are compared with the experimentally observed values as well 

as data obtained from the 6-DOF analysis of the flight results from ARFDAS (22) and are found 

to be in good agreement. 

-2000.00

2000.00

6000.00

10000.00

14000.00

18000.00

22000.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Range, m

Phi, deg

ARFDAS Fit

Experimental Data

CFD

Figure 7. Comparison of computed roll angle, jet off. 
12

Figure 8 shows the variation of the Euler pitch angle with distance traveled. As seen in this 

figure, both the amplitude and frequency in the Euler pitch angle variation are predicted very 

well by the computed results and match well with the data from the flight tests for a distance as 

far as 20 meters. For the rest of the trajectory, the frequency of the motion is predicted very 

well, as seen by the comparison with the data, but the amplitude does not compare as well with 

the data. Two sets of CFD results are shown in this figure, one set obtained with the original 

time step and another one with a smaller time step reduced by half from the original time step. 

As seen in this figure, reducing the time step by half has a negligible effect on the accuracy of 

the computed results. Figure 9 shows similar behavior with Euler yaw angle with x distance. 

The agreement between the computed results and the experimental data again is generally good 

except for the amplitude comparison at greater ranges. In figures 8 and 9, we have slow and fast 

mode frequencies and both compare well with the data and the ARFDAS fits. 

-8.00

-7.00

-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Range, m

Theta-FP, deg

ARFDAS Fit

Experimental Data

CFD

CFD(SM-DT)

Figure 8. Comparison of computed Euler pitch angle, jet off for different time steps. 

Figures 10 and 11 show the computed z and y distances, respectively, as a function of the 

x distance or range. Computed y and z distances were obtained with two different time steps 

(DT) of 0.00007875 and 0.00002625 and are compared with the actual measured data and the fits 

obtained through ARFDAS fits. The effect of time step on the computed y and z distances are 

very small, indicating that the smaller time step is good enough to provide the desired time 

accuracy. As shown in figure 10, the computed z distance compares very well with the data 

obtained from the flight test. The computed y distance is shown in figure 11. As can be seen, the 

computed y distance matches very well with the data in the early part of the trajectory, i.e., until 

about 30 m. The comparison of the computed result with the data is not so good for the rest of the 

trajectory. This small discrepancy can be attributed to errors in the computational geometric 
13

modeling of the actual flight body, which can result in small differences in the prediction of the 

side force and Magnus moment for a spinning body, especially at a low subsonic speed. 

Additional research in these areas can possibly further improve the accuracy of the computed 

results. Although not shown here, the effect of inner time step iterarions was also studied and a 

calculation with 10 inner iterations yielded essentially the same results as were obtained with five 

sub-iterations. 

-8.00

-7.00

-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Range, m

Psi-FP, deg

ARFDAS Fit

Experimental Data

CFD (SM_DT)

CFD

Figure 9. Comparison of the computed Euler yaw angle, jet off for different time steps. 

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Range, m

Z-Coord, m

ARFDAS Fit

Experimental Data

CFD

CFD (SM_DT)

Figure 10. Comparison of the computed z distance, jet off. 
14

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Range, m

Y-C oord, m

ARFDAS Fit

Experimental Data

CFD

CFD (SM_DT)

Figure 11. Comparison of the computed y distance, jet off. 

A few results obtained for the spinning projectile with flow control are presented next. The 

unsteady synthetic microjet is applied while the projectile is spinning (see figure 4) so as to get a 

side force in this case. The synthetic jet was activated at approximately 41 meters into the trajectory. Figure 12 shows the variation of the Euler pitch angle with distance traveled. As seen in this 

figure, the effect of the jet obtained with the hybrid RANS-LES (or LNS) model is small. As seen 

here, the computed data with and without flow control are very similar. There is very little change 

in the frequency. However, small changes between the jet-off and jet-on results can be observed 

in the amplitude variations. Figure 13 shows the axial force (Fx), side force (Fy), and the lift or 

normal force (Fz) obtained with the LNS model as a function of the range. The effect of the jet is 

seen to increase the side force; however, the effect of the jet on the lift force is almost negligible. 

There is some effect on the axial force as well. Our primary interest is the effect of the unsteady 

microjet on the side force. Although only a small change is observed between the jet-off and jeton results, it is expected that the cumulative effect of these changes over many spin cycles of the 

spinning body, i.e., across a larger x distance can be sufficient to provide the required control force 

for changing the course of the trajectory. 
15

-10.00

-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Range (m)

Theta (degree)

CFD - Jetoff

CFD - Jeton

Figure 12. Comparison of the computed Euler pitch angle, LNS, jet off and jet on (jet velocity 

= 110 m/s). 

-6.00

-4.00

-2.00

0.00

2.00

4.00

40.0 45.0 50.0 55.0 60.0

Range, m

Forces

Fx - Jetoff Fy - Jetoff

Fz - Jetoff Fx - Jeton

Fy _ Jeton Fz - Jeton

Figure 13. Comparison of the computed side force and the normal force, LNS, jet off and jet on 

(jet velocity = 110 m/s). 
16

Figure 14 shows the variation of y distance as function of the range for both jet-off and jet-on 

conditions. Here, the jet-on results shown were obtained with the LNS approach. In addition, 

the jet-on computations were performed for many more spin cycles or for a greater range. One 

can clearly see the effect of the jet in the y distance; it increases with an increase in the range. 

These computed results strongly indicate that applying the jet in the positive y direction moves 

the projectile in the same positive direction with little or no effect on the other aerodynamic 

forces. 

0.25

0.26

0.27

0.28

0.29

0.30

0.31

0.32

0.33

0.34

0.35

40.0 42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0

Range (X), m

Y, m

CFD - Jetoff

CFD - Jeton

Figure 14. Comparison of the computed y distance, LNS, jet off and jet on (jet velocity = 110 m/s). 

4. Summary and Conclusions 

This report describes a new coupled CFD-RBD computational study undertaken to simultaneously 

determine the flight trajectory and the associated unsteady free-flight aerodynamics of a spinning 

projectile with and without flow control. A 3-D unsteady Navier-Stokes solver is employed to 

compute the time-accurate aerodynamics associated with the free flight of the spinning projectile 

at a low subsonic velocitiy. Computed results have been obtained at an initial low subsonic speed, 

M = 0.39 and α = 2 degrees for the spinning projectile with the use of the time-accurate NavierStokes computational technique and advanced turbulence models. Computed positions and orientations of the projectile have been compared with actual data measured from free-flight tests and 

are found to be generally in good agreement. The same advanced coupled procedure has been 

used to compute the aerodynamics of the spinning projectile with flow control using a synthetic 
17

microjet. These computations with flow control clearly show the effect of the unsteady microjet 

on the side aerodynamic force. 

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

aerodynamics of a spinning projectile. Computational results obtained with flow control show 

the potential of tiny microjets as a means to provide the control authority to maneuver a spinning 

projectile at low subsonic speeds. Additional work is needed to continue the validation of the 

computed results with the data and results from other techniques and extraction of the aerodynamic 

coefficients from the simulations at hand. The present CFD-RBD simulations clearly show the 

potential capability of the coupled approach and form the basis for future multidisciplinary, timedependent computations of advanced maneuvering munitions. 
18

5. 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. Sixth 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. Physics of Fluids

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. 

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), pp. 159–167. 
19

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. 

20. Sahu, J. Unsteady CFD Modeling of Aerodynamic Flow Control over a Spinning Body with 

Synthetic Jet; AIAA Paper 2004-0747, Reno, NV, 5-8 January 2004. 

21. Sahu, J. Time-Accurate Numerical Prediction of Free Flight Aerodynamics of a Finned 

Projectile; ARL-TR-3603; U.S. Army Research Laboratory: Aberdeen Proving Ground, 

MD, September 2005. 

22. Arrow Tech Associates. ARFDAS Technical Manual. South Burlington, VT, 2001. 
20

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