Time-Accurate Numerical Prediction of Free Flight 

Aerodynamics of a Finned Projectile 

by Jubaraj Sahu 

ARL-TR-3603 September 2005 

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Army Research Laboratory 

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ARL-TR-3603 September 2005 

Time-Accurate Numerical Prediction of Free Flight 

Aerodynamics of a Finned Projectile 

Jubaraj Sahu 

Weapons and Materials Research Directorate, ARL 

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

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 Jubaraj Sahu (ARL) 

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14. ABSTRACT 

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

simultaneously predict the unsteady free flight aerodynamics of a finned projectile configuration. Actual flight trajectories are 

computed by an advanced coupled computational fluid dynamics (CFD)-rigid body dynamics technique. An advanced timeaccurate Navier-Stokes computational technique has been used in CFD to compute the unsteady aerodynamics associated with 

the free flight of the finned projectile at supersonic 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. Predicted 

aerodynamics forces and moments also compare well with the forces and moments used in the six-degree-of-freedom fits of the 

results of the same tests. Unsteady numerical results obtained from the coupled method show the aerodynamic forces and 

moments and the flight path of the projectile. 

15. SUBJECT TERMS

 coupled CFD-RBD; finned projectile; flow control; projectile aerodynamics; time accurate; 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 3

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

2.2 Grid Movement ...............................................................................................................4 

2.3 Six-Degree-of-Freedom Coupling...................................................................................5 

3. Results and Discussion 6

4. Summary and Conclusions 13

5. References 15 

Distribution List 17
iv

List of Figures 

Figure 1. Finned projectile configuration. ......................................................................................6 

Figure 2. Computational grid near the projectile............................................................................7 

Figure 3. Expanded view of the grid in the base region. ................................................................7 

Figure 4. Computed z distance versus range. .................................................................................8 

Figure 5. Computed y distance versus range. .................................................................................9 

Figure 6. Euler pitch angle distance versus range...........................................................................9 

Figure 7. Euler yaw angle versus range........................................................................................10 

Figure 8. Motion plot (a) computation, and (b) flight test............................................................11 

Figure 9. Total angle of attack versus range.................................................................................12 

Figure 10. Angle of attack and side slip versus range. .................................................................12 

Figure 11. Comparison of earth-fixed aerodynamic forces and moments....................................14 
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 the Metacomp Technologies for his technical assistance on 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 6-degree-of-freedom simulations with their software, ARFDAS (Aeroballistics 

Research Facility Data Analysis System). Thanks are also due Ms. Karen Heavey of ARL for 

her help in the grid generation. The scientific visualization and the computational support of 

ARL’s Major Shared Resource Center are greatly appreciated. 
vi

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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 (1,2,3). Advanced weapons, a major part of our armed forces’ arsenal, are vital 

to the defense of our nation. Precision guided munitions that hit targets more accurately can 

greatly increase lethality and enhance survivability. To ensure that the United States maintains a 

strong defense, the Department of Defense (DoD) must continually develop (with fewer dollars) 

more lethal and effective munitions. The munitions must stay abreast of the latest technology 

available to our adversaries. The DoD needs ways to develop these advanced weapons without 

extremely long and costly design cycles. Although actual flight testing of advanced munitions 

systems will undoubtedly be an essential ingredient in the eventual success of these Army 

programs, it is expensive and time consuming. Computer simulations can and have provided an 

effective means of determining the unsteady aerodynamics and flight mechanics of guided 

projectile systems. The use of high performance computers to model, simulate, and test alternate 

projectile and missile designs is one response to this requirement. Recent advances made in high 

performance computing and computational fluid dynamics (CFD) technologies have the potential 

for greatly reducing the design costs while providing a more detailed understanding of the 

complex aerodynamics than that achieved through experiments and actual test firings. 

Understanding the aerodynamics of projectiles, rockets, and missiles is, of course, critical to the 

design of stable configurations and contributes significantly to the overall performance of 

weapon systems. 

For the past couple of decades, computational capabilities have been developed and employed at 

the U.S. Army Research Laboratory (ARL) for computing the flight aerodynamics of various 

projectile and missile configurations (4,5,6,7,8). Three-dimensional (3-D) steady and unsteady 

Navier-Stokes computational techniques have been used to predict aerodynamics of spinning and 

fin-stabilized projectiles from subsonic to supersonic speeds. In many applications, 3-D NavierStokes computational techniques were employed in conjunction with the Chimera (9,10,11) 

overlapping grid method to model complex projectile and missile configurations. The Chimera 

approach was especially useful in CFD modeling of multi-body projectile and missile 

configurations. An advanced Chimera CFD technique was then extended for accurate numerical 

calculation of aerodynamics involving multiple bodies with relative motion (2,3). The 

underlying complex physics and fluid dynamics structure of the aerodynamic interference for 

multi-body problems were identified. This work showed how maximum savings of time and 

dollars could be achieved when CFD is brought into weapon system development programs early 

in the design phase. 
2

Improved computer technology and state-of-the-art numerical procedures now enable solutions 

to complex, 3-D problems associated with projectile and missile aerodynamics. In particular, our 

recent focus has been directed at the development and application of advanced predictive 

capabilities to compute unsteady (12,13) projectile aerodynamics. Accurate numerical modeling 

of the unsteady aerodynamics was found to be challenging and required the use of time-accurate 

solutions techniques. Recently, the time-accurate technique was used to obtain improved results 

for Magnus moment and roll damping moment of a spinning projectile at transonic and subsonic 

speeds (13). Other recent examples have shown the use of time-accurate Navier-Stokes 

techniques to predict dynamic derivative such as the pitch-damping coefficient (14,15). Recent 

advances made in CFD now allow one to compute static and dynamic derivatives, which, in turn, 

make it possible to predict the in-flight motion of projectiles with the numerical data derived 

solely from CFD. This has been recently demonstrated for a family of axisymmetric projectiles 

at supersonic speeds (16). The present work is focused on the coupling of CFD and rigid body 

dynamics (RBD) techniques for simultaneous prediction of the unsteady free-flight 

aerodynamics and the flight trajectory of projectiles. Our goal is to be able to perform timeaccurate multi-disciplinary coupled CFD-RBD computations for the entire flight trajectory of a 

complex guided projectile system. 

Knowledge of the detailed aerodynamics of maneuvering guided smart weapons is rather limited, 

especially during and after the maneuvers. Multi-disciplinary computations can provide detailed 

fluid dynamic understanding of the unsteady aerodynamics processes involving the maneuvering 

flight of modern guided weapon systems. Such knowledge cannot be easily obtained by any 

other means. The computational technology involving CFD and RBD is now mature and can be 

used to determine the unsteady aerodynamics associated with the entire mission trajectory of the 

munitions. These multi-disciplinary computations can lead to better experimental test designs 

and much better returns for full-scale flight tests. More importantly, they can provide physical 

insight of fluid mechanics processes that may not be gained from experimental techniques and 

flight tests. 

The advanced CFD capability used here solves the Navier-Stokes equations (17), incorporates 

unsteady boundary conditions and a special coupling procedure. The present research is a big 

step forward in that it allows “virtual fly-out” of projectiles on the super-computers and allows 

numerical prediction of the actual fight paths of a projectile and all the associated unsteady free 

flight aerodynamics with coupled CFD-RBD techniques in an integrated manner. The following 

sections describe the solution technique, coupled CFD-RBD procedure, and the computed results 

obtained for a finned projectile at supersonic speeds. 
3

2. Solution Technique 

At ARL, research efforts are continuing to perform real-time multi-disciplinary coupled CFDRBD aerodynamic computations for the entire flight trajectory of a complex guided projectile 

system. In order to save computer time, our first attempt was to use a quasi-unsteady approach 

(4). The quasi-unsteady approach relies on Navier-Stokes equations and six-degree-of-freedom 

(6-DOF) computations to compute a missile trajectory. A degree of freedom is a displacement 

quantity that defines the location and orientation of an object. In 3-D space, a rigid object has six 

degrees of freedom: three translations and three rotations. The 6-DOF code computes linear and 

angular velocities as well as the orientation of the missile, which are used as input to the CFD 

code. The quasi-unsteady approach uses the following simple procedure to compute a missile 

trajectory. 

1. A CFD solver repeatedly solves the Navier-Stokes equations to obtain missile flow field 

solutions. 

2. A sequence of average values for the forces and moments is generated from the flow field 

solutions when the consecutive maximum and minimum values are averaged to 

determine the forces and moments to be used as input in the 6-DOF computations. 

3. The 6-DOF computations are performed. 

4. With the new initial conditions from the 6-DOF computations, a true increment in time is 

taken, and the CFD solver computes a new set of solutions. These steps are repeated 

until the length of the desired trajectory is reached. 

A second approach that could be used is the real-time accurate approach. This is the preferred 

approach and is used in the present work; however, the computations require much greater 

computer resources. The real-time accurate approach also requires that the 6-DOF body 

dynamics be computed at each repetition of the flow solver. Both the quasi-unsteady and the 

real-time accurate approach require a CFD solver. 

The CFD capability used here solves the Navier-Stokes equations and incorporates advanced 

boundary conditions and grid motion capabilities. The present numerical study is a big step 

forward and a direct extension of that research which now includes numerical simulation of the 

actual fight paths of the projectile using coupled CFD-RBD techniques with a real-time accurate 

approach. The complete set of 3-D time-dependent Navier-Stokes equations is solved in a timeaccurate 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. 
4

The 3-D, time-dependent Reynolds-averaged Navier Stokes (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. 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 higher order RANS turbulence models (20) were used for the 

computation of turbulent flows. These models are ideally suited to unstructured book-keeping 

and massively parallel processing because of their independence from constraints related to the 

placement of boundaries and/or zonal interfaces. For computations of unsteady flow fields that 

are of interest here, dual time stepping (as described next) was used to achieve the desired time 

accuracy. In addition, the projectile in the coupled CFD-RBD simulation, along with its grid, 

actually moves and rotates as it flies down range. 

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 

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 available. 

2.2 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 
5

spinning projectile. In this case, the grid speeds are assigned as if the grid were attached to the 

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

as if the grid were attached to the rigid body with 6-DOF. 

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. CFD++ satisfies this requirement. 

Another important aspect concerns how boundary conditions are affected by grid velocities. 

Consider here the two significant classes of boundary conditions: slip or no slip at a wall and 

far-field boundary. In CFD++, 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, the tangential component of the 

velocity is required to be 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 far-field 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, or 

subsonic outflow. The characteristics theory is applied to determine what and how much 

information is applied as the boundary condition for each type and the consistency of the mesh 

with respect to the physics at hand. 

2.3 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/moments and the 

dynamic response of the projectile/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 that 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 are 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 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 kinematic 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 
6

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 a steady state solution. This becomes the initial condition for 

the next step which involves adding only 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 unsteady flow fields, aerodynamic forces and moments, and the 

flight paths of a finned projectile at supersonic speeds. In all cases, full 3-D computations were 

performed and no symmetry was used. 

The supersonic projectile modeled in this study is an ogive-cylinder-finned configuration (see 

figure 1). The length of the projectile is 121 mm and the diameter is 13 mm. The ogive nose is 

98.6 mm long and the afterbody has a 22.3-mm, 2.5-degree boat-tail. Four fins are situated on 

the back end of the projectile. Each fin is 22.3 mm long and 10.16 mm thick. The computational mesh for the 25-mm projectile model is a C-grid (see figure 2) consisting of seven zones. 

The first zone encompasses the entire projectile body, from the tip of the nose to the end of the 

fins. In general, most of the grid points are clustered in the afterbody fin region. Figure 2 shows 

a 3-D view of the full projectile mesh. The total number of grid points is 4 million for the full 

computational grid. Figure 3 shows an expanded view of the grid in the base region. The first 

grid point spacing from the projectile body is chosen to achieve a y+ value of 1.0. 

Figure 1. Finned projectile configuration. 
7

Figure 2. Computational grid near the projectile. 

Figure 3. Expanded view of the grid in the base region. 

Here, the primary interest is in the development and application of coupled CFD and RBD 

techniques for accurate simulation of the free-flight aerodynamics and flight dynamics of the 

projectile in supersonic flight. The first step here was to obtain the steady state results for this 

projectile at a given initial supersonic velocity. Also imposed were the angular orientations at 

this stage. Corresponding converged steady state solution was then used as the starting condition 

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

Numerical computations have been made for these cases at initial velocities of 1037 and 
8

1034 m/s, depending on whether the simulations were started from the muzzle or a small 

distance away from it. The corresponding initial angles of attack were α = 0.5 degree or 

4.9 degrees, and initial spin rates were 2800 or 2500 radians per second, respectively. 

Figures 4 and 5 show the computed z and y distances as a function of x (or, the range). The 

computed results are shown in solid lines and are compared with the data measured from actual 

flight tests. For the computed results, the aerodynamic forces and moments were completely 

obtained through CFD. One simulation started from the gun muzzle and the other from the first 

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

4.9 m from the muzzle. Both sets of results are generally found to be in good agreement with the 

measured data, although there is a small discrepancy between the two sets of computed results. 

Both y and z distances are found to increase with increasing x distance. 

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 70.0 80.0

Range, m

Z-Coord, m

Experimental Data

CFD

CFD (Muz)

Figure 4. Computed z distance versus range. 

Figure 6 shows the variation of the Euler pitch angle with distance traveled. As seen in this 

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

the computed results and match extremely well with the data from the flight tests. Both sets of 

computations, whether they started from the muzzle or the first station away from the muzzle, 

yield essentially the same results. One can also clearly see that the amplitude damps as the 

projectile flies down range, i.e., with increasing x distance. Figure 7 shows similar behavior 

with Euler yaw angle with x distance. In this case, however, the yaw angle damps somewhat in 

the beginning to a 20-m range and then the amplitude stays within +0.5 and -0.5 degree for the 

rest of the flight. The computed results again compare very well with the measured data from 

the flight tests. 
9

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 70.0 80.0

Range, m

Y-Coord, m

Experimental Data

CFD

CFD (Muz)

Figure 5. Computed y distance versus range. 

-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0

Range, m

Theta, deg

Experimental Data

CFD (FS)

CFD (Muz)

Figure 6. Euler pitch angle distance versus range. 
10

-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0

Range, m

Psi, deg

Experimental Data

CFD (FS)

CFD (Muz)

Figure 7. Euler yaw angle versus range. 

The time histories of the pitch and yaw angles are often customarily presented as a motion plot 

where the pitch angle is plotted versus the yaw angle during the flight of the projectile. The 

motion plot represents the path traversed by the nose of the projectile during the flight trajectory 

(looking forward from the back of the projectile). Such a plot is shown in figure 8. This figure 

shows the comparison of the motion plots obtained from the numerical simulations and the 

6-DOF analysis of the flight results from ARFDAS (Aeroballistics Research Facility Data 

Analysis System) (21). Computed results match very well with the experimental flight test 

results. 

Figures 9 and 10 show the computed total angle of attack and the individual components (angle 

of attack and side slip), respectively, as a function of the range. These quantities are not directly 

measured in the actual tests. However, they can be easily derived from the “6-DOF fits” of the 

actual data. Figure 9 shows the comparison of the total angle of attack derived from such an 

analysis with the computed results. Computed results include two simulations with different 

starting conditions, and both sets of these results match quite well with the data derived from the 

usual 6-DOF fits. All results show the total angle of attack decreasing with increasing range or x 

distance. Figure 10 shows both component angles: angle of attack and the side slip. 
11

(a) 

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Psi

(b) 

Figure 8. Motion plot (a) computation, and (b) flight test. 
12

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 70.0 80.0

Range, m

Alpha (total), deg

ARFDAS

CFD

CFD (Muz)

Figure 9. Total angle of attack versus range. 

-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 70.0 80.0

Range, m

Alpha, Beta (deg)

Alpha Beta

Figure 10. Angle of attack and side slip versus range. 
13

Figure 11a and 11b, respectively, shows the comparison of the predicted aerodynamic forces and 

moments with the ARFDAS results for the same flight conditions. The aerodynamic forces are 

shown in three directions: x (down range), y (cross range, positive to the left looking from the 

gun), and z (altitude, positive up). The same sign conventions used in ARFDAS are used in the 

present study. The moments in this case are taken with respect to the center of gravity of the 

projectile. As seen in these figures, the predicted unsteady forces and moments have very 

similar behavior to that used in the 6-DOF analysis of the actual flight test data. The aerodynamic forces in y and z directions match well with their counterparts in the ARFDAS analysis. 

However, there is clearly a discrepancy in the comparison of the aerodynamic force in the x 

direction. Since the drag acts in the negative x direction, the computed drag is over-predicted by 

20% to 25%. Although not shown here, a finer grid was used in the steady state mode and the 

drag results did not change appreciably. Another source of error could come from the inaccurate 

modeling of the model itself. The actual flight model had a cavity in the base region whereas 

the computational model has a solid base. A part of the discrepancy can perhaps be attributed to 

this. Currently, additional work is being performed to find the source of the discrepancy. As 

seen in figure 11b, the aerodynamic moments are generally found to be in good agreement with 

the aerodynamic moments used in ARFDAS. 

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 finned 

projectile. A 3-D unsteady Navier-Stokes solver is employed to compute the time-accurate 

aerodynamics associated with the free flight of the finned projectile at supersonic velocities. 

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. Predicted 

aerodynamics forces and moments also compare generally well with the forces and moments used 

in the 6-DOF fits of the results of the same tests except for the force in the x direction. This work 

demonstrates a coupled method to accurately predict the time-accurate unsteady aerodynamics and 

the flight trajectories of projectiles at various 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, time-dependent computations of advanced maneuvering munitions. 
14

-80.00

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-20.00

0.00

20.00

40.00

60.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0

Range, m

Forces

Fx

Fy

Fz

Fx (ARFDAS)

Fy (ARFDAS)

Fz (ARFDAS)

(a) aerodynamic forces 

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0

Range, m

Moments

Mx

My

Mz

Mx (ARFDAS)

My (ARFDAS)

Mz (ARFDAS)

(b) aerodynamic moments 

Figure 11. Comparison of earth-fixed aerodynamic forces and moments. 
15

5. References 

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

Configurations. Proceedings of the Third 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/Finned 

Store Configuration; American Institute of Aeronautics and Astronautics (AIAA) 92-4568-

CP, Hilton Head, SC, August 1992. 

4. Sahu, J.; Edge, H. L.; DeSpirito, J.; Heavey, K. R.; Ramakrishnan, S. V.; Dinavahi, S.P.G. 

Applications of Computational Fluid Dynamics to Advanced Guided Munitions. 39th AIAA 

Aerospace Sciences Meeting, Reno, NV, 8-12 January 2001. 

5. Sahu, J.; Edge, H. L.; Dinavahi, S.; Soni, B. Progress on Unsteady Aerodynamics of 

Maneuvering Munitions. Users Group Meeting Proceedings, Albuquerque, NM, June 2000. 

6. Silton, S. I. Navier-Stokes Computations for a Spinning Projectile for Subsonic to 

Supersonic Speeds. Journal of Spacecraft and Rockets 2005, 42 (2), 223-231. 

7. Heavey, K. R.; Sahu, J. Application of Computational Fluid Dynamics to a Monoplane 

Fixed Wing Missile with Elliptic Cross Sections; ARL-TR-3549; U.S. Army Research 

Laboratory: Aberdeen Proving Ground, MD, July 2005. 

8. Sahu, J.; Silton, S. I.; Heavey, K. R. Navier-Stokes Computations of Supersonic Flow over 

Complex Missile Configurations. AIAA Paper No. 2004-5456, AIAA 22nd Applied 

Aerodynamics Conference, Providence, RI, August 16-19, 2004. 

9. Steger, J. L.; Dougherty, F. C.; Benek, J. A. A Chimera Grid Scheme. Advances in Grid 

Generation, edited by K. N. Ghia and U. Ghia, ASME FED-5, June 1983. 

10. Meakin, R.; Gomez, R. On Adaptive Refinement and Overset Structured Grids. AIAA 

Paper No. 97-1858-CP, 1997. 

11. Meakin, R. Unsteady Simulation of the Viscous Flow About a V-22 Rotor and Wing in 

Hover. AIAA-95-3463, August, 1995. 

12. 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. 
16

13. DeSpirito, J.; Heavey, K. R. CFD Computation of Magnus Moment and Roll Damping 

Moment of a Spinning Projectile. AIAA Paper No. 2004-4713, Atmospheric Flight 

Mechanics Conference, Providence, RI, August 2004. 

14. Oktay, E.; Akay, H. U. CFD Predictions of Dynamic Derivatives for Missiles. AIAA Paper 

No. 2002-0276, January 2002. 

15. Park, S. H.; Kwon, J. H. Navier-Stokes Computations of Stability Derivatives for Symmetric 

Projectiles. AIAA Paper No. 2004-0014, January 2004. 

16. Weinacht, P. Prediction of Projectile Performance, Stability, and Free-Flight Motion Using 

Computational Fluid Dynamics; ARL-TR-3015; U.S. Army Research Laboratory: Aberdeen 

Proving Ground, MD, July 2003. 

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

CFD. AIAA Paper 97-07245, 1997. 

18. 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. 

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

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

0116, 1998. 

20. 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, pp. 457-

462. 

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

22. Costello, M. Extended Range of a Gun Launched Smart Projectile Using Controllable 

Canards. Shock and Vibration 2001, 8 (3-4), 203-213. 
17

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