Numerical Investigation of Aerodynamics of 

Canard-Controlled Missile Using Planar and Grid Tail Fins, 

Part II: Subsonic and Transonic Flow 

by James DeSpirito, Milton E. Vaughn, Jr., 

and W. David Washington 

ARL-TR-3162 March 2004 

Approved for public release; distribution is unlimited. 
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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3162 March 2004 

Numerical Investigation of Aerodynamics of 

Canard-Controlled Missile Using Planar and Grid Tail Fins, 

Part II: Subsonic and Transonic Flow 

James DeSpirito 

Weapons and Materials Research Directorate, ARL 

Milton E. Vaughn, Jr. and W. David Washington 

U.S. Army Aviation and Missile Command 

Approved for public release; distribution is unlimited.
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4. TITLE AND SUBTITLE 

Numerical Investigation of Aerodynamics of Canard-Controlled Missile Using 

Planar and Grid Tail Fins, Part II: Subsonic and Transonic Flow 

5c. PROGRAM ELEMENT NUMBER 

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1L162618AH80 

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6. AUTHOR(S) 

James DeSpirito, Milton E. Vaughn, Jr.,* and W. David Washington* 

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7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

U.S. Army Research Laboratory 

ATTN: AMSRD-ARL-WM-BC 

Aberdeen Proving Ground, MD 21005-5066 

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* U.S. Army Aviation and Missile Command 

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14. ABSTRACT 

Viscous computational fluid dynamic simulations were used to predict the aerodynamic coefficients and flow field around a 

canard-controlled missile in subsonic and transonic flow. Computations were performed at Mach 0.6 and 0.9, six angles of 

attack between 0° and 10°, and with planar and grid tail fins. The computations were validated with wind tunnel data. Flow 

visualizations showed that the canard downwash produced a low-pressure region on the starboard side of the missile that 

produced a large induced side force. The canard trailing vortices interacted with the tail fins until α > 8°, producing a pressure 

differential on the leeward tail fin, leading to the adverse induced roll effects. Visualizations of the flow through the grid fin 

structure showed choking of the flow at Mach 0.9 and Mach 1.5. The validated simulations results showed that grid fins did 

not improve the canard roll-control effectiveness at subsonic and transonic speeds as well as they did at the low supersonic 

speed. 

15. SUBJECT TERMS

computational fluid dynamics, canard, lattice fins, grid fins, roll-reversal, missile aerodynamics 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

James DeSpirito 

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410-278-6104 

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18
iii

Contents 

List of Figures v

List of Tables viii

Acknowledgments ix

1. Introduction 1

2. Computational Approach 1

2.1 Geometry and Simulation Parameters.............................................................................1 

2.2 Solver...............................................................................................................................4 

2.3 Computational Mesh and Boundary Conditions .............................................................4 

2.4 Solution Methodology.....................................................................................................6 

3. Results and Discussion 7

3.1 Aerodynamic Coefficients – CFD Validation.................................................................7 

3.1.1 Supersonic Flow....................................................................................................7 

3.1.2 Subsonic and Transonic Flow ...............................................................................9 

3.2 Flow-Field Visualizations .............................................................................................18 

3.3 Control Surface Forces..................................................................................................26 

3.4 Flow-Through Grid Fins ...............................................................................................28 

4. Summary and Conclusions 35

5. References 37

Appendix A. Aerodynamic Coefficients for Planar Fin Case 39

Appendix B. Aerodynamic Coefficients for Grid Fin Case 49

Appendix C. Force Coefficients on Canards 59

Appendix D. Force Coefficients on Planar Fins 65
iv

Appendix E. Force Coefficients on Grid Fins 71

Appendix F. Components of Aerodynamic Coefficients 77

List of Abbreviations and Symbols 86

Distribution List 88 
v

List of Figures 

Figure 1. Generic canard-controlled missile with planar fins.........................................................2 

Figure 2. Generic canard-controlled missile with grid fins. ...........................................................3 

Figure 3. (a) Missile nose with canards at δ = 0° and (b) front view of missile.............................3 

Figure 4. Surface mesh near body–grid fin base interface..............................................................5 

Figure 5. Mesh on missile surface and axial plane in tail region for planar fin subsonic and 

transonic cases. ..........................................................................................................................6 

Figure 6. Computed and experimental rolling moment coefficient for (a) planar fin and (b) 

grid fin cases in supersonic flow................................................................................................8 

Figure 7. Computed and experimental side force coefficient for (a) planar fin and (b) grid 

fin cases in supersonic flow. ......................................................................................................9 

Figure 8. Computed and experimental normal force coefficient for (a) planar fin and (b) grid 

fin cases at Mach 0.6................................................................................................................10 

Figure 9. Computed and experimental pitching moment coefficient about the missile nose 

for (a) planar fin and (b) grid fin cases at Mach 0.6. ...............................................................12 

Figure 10. Computed and experimental center of pressure location from nose for (a) planar 

fin and (b) grid fin cases at Mach 0.6. .....................................................................................13 

Figure 11. Computed and experimental axial force coefficient for (a) planar fin and (b) grid 

fin cases at Mach 0.6................................................................................................................14 

Figure 12. Computed and experimental side force coefficient for (a) planar fin and (b) grid 

fin cases at Mach 0.6................................................................................................................15 

Figure 13. Computed and experimental side force coefficient for (a) planar fin and (b) grid 

fin cases at Mach 0.9................................................................................................................16 

Figure 14. Computed and experimental rolling moment coefficient for (a) planar fin and (b) 

grid fin cases at Mach 0.6. .......................................................................................................17 

Figure 15. Computed and experimental rolling moment coefficient for (a) planar fin and (b) 

grid fin cases at Mach 0.9. .......................................................................................................18 

Figure 16. Computed rolling moment coefficient for (a) planar fin and (b) grid fin cases at 

several Mach numbers. ............................................................................................................19 

Figure 17. Cp contours on planar finned missile surfaces at M = 0.6 and (a) α = 0°, (b) α 

= 4°, and (c) α = 10°. ...............................................................................................................20 

Figure 18. Cp contours on planar finned missile surfaces at M = 0.9 and (a) α = 0°, (b) α 

= 4°, and (c) α = 10°. ..............................................................................................................21 

Figure 19. Cp contours on grid finned missile surfaces at M = 0.6 and (a) α = 0°, (b) α = 4°, 

and (c) α = 10°.........................................................................................................................22 

Figure 20. Cp contours on grid finned missile surfaces at M = 0.9 and (a) α = 0°, (b) α = 4°, 

and (c) α = 10°.........................................................................................................................23 
vi

Figure 21. Cp contours on (a) port side and (b) starboard side of grid finned missile surfaces 

at M = 0.6 and α = 10°.............................................................................................................24 

Figure 22. Cp surface contours and vorticity magnitude contours on cross planes at x/d = 2, 

4, ..., 22 for the (a) planar finned and (b) grid finned missile at M = 0.6 and α = 4°. ...........25 

Figure 23. Cp surface contours and vorticity magnitude contours on cross planes at x/d = 2, 

4, ..., 22 for the (a) planar finned and (b) grid finned missile at M = 0.6 and α = 10°. ..........26 

Figure 24. Side force coefficient on tail fins for (a) planar fin and (b) grid fin case for δ 

= 10° and M = 0.6. ...................................................................................................................27 

Figure 25. Normal force coefficient on tail fins for (a) planar fin and (b) grid fin case for δ 

= 10° and M = 0.6. ...................................................................................................................28 

Figure 26. Fin normal force slope values as a function of Mach number (6)...............................29 

Figure 27. Grid fin flow field proposed in reference (6). .............................................................30 

Figure 28. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with leeward 

grid fin geometry removed, α = 0° and M = 0.6......................................................................31 

Figure 29. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with leeward 

grid fin geometry removed, α = 0° and M = 0.9......................................................................32 

Figure 30. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with leeward 

grid fin geometry removed, α = 0° and M = 1.5......................................................................33 

Figure 31. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with leeward 

grid fin geometry removed, α = 0° and M = 3.0......................................................................34 

Figure A-1. Normal force for the planar fin case at Mach 0.6......................................................40 

Figure A-2. Normal force for the planar fin case at Mach 0.9......................................................40 

Figure A-3. Side force for the planar fin case at Mach 0.6...........................................................41 

Figure A-4. Side force for the planar fin case at Mach 0.9...........................................................41 

Figure A-5. Axial force for the planar fin case at Mach 0.6.........................................................42 

Figure A-6. Axial force for the planar fin case at Mach 0.9........................................................42 

Figure A-7. Pitching moment about the nose for the planar fin case at Mach 0.6. ......................43 

Figure A-8. Pitching moment about the nose for the planar fin case at Mach 0.9. ......................43 

Figure A-9. Pitching moment about the moment reference point (MRP) for the planar fin 

case at Mach 0.6.......................................................................................................................44 

Figure A-10. Pitching moment about the MRP for the planar fin case at Mach 0.9. ...................44 

Figure A-11. Rolling moment for the planar fin case at Mach 0.6...............................................45 

Figure A-12. Rolling moment for the planar fin case at Mach 0.9...............................................45 

Figure A-13. Yawing moment about the nose for the planar fin case at Mach 0.6. .....................46 

Figure A-14. Yawing moment about the nose for the planar fin case at Mach 0.9. .....................46 

Figure A-15. Center of pressure location from the nose for the planar fin case at Mach 0.6.......47 

Figure A-16. Center of pressure location from the nose for the planar fin case at Mach 0.9.......47 
vii

Figure B-1. Normal force for the grid fin case at Mach 0.6. ........................................................50 

Figure B-2. Normal force for the grid fin case at Mach 0.9. ........................................................50 

Figure B-3. Side force for the grid fin case at Mach 0.6. .............................................................51 

Figure B-4. Side force for the grid fin case at Mach 0.9. .............................................................51 

Figure B-5. Axial force for the grid fin case at Mach 0.6.............................................................52 

Figure B-6. Axial force for the grid fin case at Mach 0.9.............................................................52 

Figure B-7. Pitching moment about the nose for the grid fin case at Mach 0.6. ..........................53 

Figure B-8. Pitching moment about the nose for the grid fin case at Mach 0.9. ..........................53 

Figure B-9. Pitching moment about the moment reference point (MRP) for the grid fin case 

at Mach 0.6...............................................................................................................................54 

Figure B-10. Pitching moment about the MRP for the grid fin case at Mach 0.9. .......................54 

Figure B-11. Rolling moment for the grid fin case at Mach 0.6...................................................55 

Figure B-12. Rolling moment for the grid fin case at Mach 0.9...................................................55 

Figure B-13. Yawing moment about the nose for the grid fin case at Mach 0.6..........................56 

Figure B-14. Yawing moment about the nose for the grid fin case at Mach 0.9..........................56 

Figure B-15. Center of pressure location from the nose for the grid fin case at Mach 0.6...........57 

Figure B-16. Center of pressure location from the nose for the grid fin case at Mach 0.9...........57 

Figure C-1. Canard normal force, δ = 10°, Mach 0.6. ..................................................................60 

Figure C-2. Canard normal force, δ = 10°, Mach 0.9. ..................................................................60 

Figure C-3. Canard side force, δ = 10°, Mach 0.6........................................................................61 

Figure C-4. Canard side force, δ = 10°, Mach 0.9........................................................................61 

Figure C-5. Canard axial force, δ = 10°, Mach 0.6.......................................................................62 

Figure C-6. Canard axial force, δ = 10°, Mach 0.9.......................................................................62 

Figure D-1. Planar fin normal force, δ = 10°, Mach 0.6...............................................................66 

Figure D-2. Planar fin normal force, δ = 10°, Mach 0.9...............................................................66 

Figure D-3. Planar fin side force, δ = 10°, Mach 0.6....................................................................67 

Figure D-4. Planar fin side force, δ = 10°, Mach 0.9....................................................................67 

Figure D-5. Planar fin axial force, δ = 10°, Mach 0.6. .................................................................68 

Figure D-6. Planar fin axial force, δ = 10°, Mach 0.9. .................................................................68 

Figure E-1. Grid fin normal force, δ = 10°, Mach 0.6. .................................................................72 

Figure E-2. Grid fin normal force, δ = 10°, Mach 0.9. .................................................................72 

Figure E-3. Grid fin side force, δ = 10°, Mach 0.6.......................................................................73 

Figure E-4. Grid fin side force, δ = 10°, Mach 0.9.......................................................................73 

Figure E-5. Grid fin axial force, δ = 10°, Mach 0.6......................................................................74 
viii

Figure E-6. Grid fin axial force, δ = 10°, Mach 0.9......................................................................74 

Figure F-1. Components of normal force for the planar fin case, δ = 10°, Mach 0.6...................78

Figure F-2. Components of normal force for the planar fin case, δ = 10°, Mach 0.9...................78 

Figure F-3. Components of normal force for the grid fin case, δ = 10°, Mach 0.6. .....................79 

Figure F-4. Components of normal force for the grid fin case, δ = 10°, Mach 0.9. .....................79 

Figure F-5. Components of side force for the planar fin case, δ = 10°, Mach 0.6. ......................80 

Figure F-6. Components of side force for the planar fin case, δ = 10°, Mach 0.9. ......................80 

Figure F-7. Components of side force for the grid fin case, δ = 10°, Mach 0.6. ..........................81 

Figure F-8. Components of side force for the grid fin case, δ = 10°, Mach 0.9. ..........................81 

Figure F-9. Components of axial force for the planar fin case, δ = 10°, Mach 0.6. .....................82 

Figure F-10. Components of axial force for the planar fin case, δ = 10°, Mach 0.9. ...................82 

Figure F-11. Components of axial force for the grid fin case, δ = 10°, Mach 0.6........................83 

Figure F-12. Components of axial force for the grid fin case, δ = 10°, Mach 0.9........................83 

List of Tables 

Table 1. Canard and fin characteristics...........................................................................................3 

Table C-1. Aerodynamic coefficients on canards, planar fin case, δ = 10°, Mach 0.6. ...............63 

Table C-2. Aerodynamic coefficients on canards, planar fin case, δ = 10°, Mach 0.9. ...............63 

Table D-1. Aerodynamic coefficients on planar fins, δ= 10°, Mach 0.6......................................69 

Table D-2. Aerodynamic coefficients on planar fins, δ = 10°, Mach 0.9.....................................69 

Table E-1. Aerodynamic coefficients on grid fins, δ = 10°, Mach 0.6.........................................75 

Table E-2. Aerodynamic coefficients on grid fins, δ = 10°, Mach 0.9.........................................75 

Table F-1. Components of aerodynamic coefficients, planar fin case, δ = 10°, Mach 0.6...........84 

Table F-2. Components of aerodynamic coefficients, planar fin case, δ = 10°, Mach 0.9...........84 

Table F-3. Components of aerodynamic coefficients, grid fin case, δ = 10°, Mach 0.6. .............85 

Table F-4. Components of aerodynamic coefficients, grid fin case, δ = 10°, Mach 0.9. .............85 
ix

Acknowledgments 

The authors thank Richard Angelini, U.S. Army Research Laboratory (ARL), for providing 

scientific visualizations of the flow fields, and Eric Fournier, Defence Research and 

Development Canada–Valcartier, for providing discussions on the wind tunnel data. This work 

was supported in part by a grant of high-performance computing time from the U.S. Department 

of Defense High Performance Computing Modernization program at the ARL Major Shared 

Resource Center, Aberdeen Proving Ground, MD. 
x

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

Missile concepts with forward control fins, or canards, have been used for many years. 

However, previous studies have shown that concepts with canards can suffer from adverse 

induced rolling moments (1–4). The use of grid fins, or “lattice controls,” for the tail control 

surfaces instead of conventional planar fins was proposed by the U.S. Army Aviation and 

Missile Command (AMCOM) personnel as a possible remedy for the roll control problems (5). 

A grid fin is an unconventional lifting and control surface that consists of an outer frame 

supporting an inner grid of intersecting planar surfaces of small chord (6). Computational fluid 

dynamics (CFD) techniques to calculate the viscous flow around a missile with grid fins were 

recently demonstrated (7, 8). 

The present study extends an earlier CFD investigation of the adverse roll effects of canards on a 

missile with both conventional planar fins and grid fins in supersonic flow (9, 10). The 

supersonic CFD investigation, validated with data from an earlier wind tunnel investigation (5), 

confirmed that grid fins alleviate the adverse roll effects at low supersonic speed. The CFD 

compliments the wind tunnel data by providing the ability to visualize the flow field to aid in 

understanding the flow physics responsible for the adverse forces and moments. The CFD 

calculations also provide the forces on each individual canard and fin, which were not measured 

in the wind tunnel. The present work extends the CFD database to include subsonic and 

transonic flow. Wind tunnel results showed that grid fins are not as effective in alleviating the 

adverse rolling moments at subsonic and transonic speeds. The CFD provides insight into the 

flow physics responsible for the difference in effectiveness and may help in designing a grid fin 

that will be more effective for roll control at lower speeds. 

2. Computational Approach 

2.1 Geometry and Simulation Parameters 

The investigation used CFD to determine the flow field and aerodynamic coefficients on a 

16-cal., four-finned, generic canard-controlled missile. The study followed an experimental 

wind tunnel investigation performed by AMCOM and the Defence Research and Development–

Valcartier, formerly the Defence Research Establishment–Valcartier (DREV), Canada (5). The 

DREV wind tunnel is an intermittent, in-draft wind tunnel with a 0.6- × 0.6-m test section. In 

this type of tunnel, the air flows from an atmospheric pressure tank to a vacuum tank, and the 

Reynolds number is lower than free-flight values at high Mach numbers. The wind tunnel 

Reynolds number ranges from ~1.56 × 107

 m−1 at M = 1.15 to 4.7 × 106 m−1 at M = 4.0.
2

The wind tunnel model geometry was used in the CFD study. Four canards on the ogive were 

in-line with the fins. Two fin types were investigated: conventional planar fins and grid fins. 

Figures 1–2 show the geometry for the planar fin and grid fin cases, respectively. The missile 

has a 3.7-cal.-long truncated tangent ogive with a hemispherical nose and a 12.3-cal. long body. 

The canard midchord is located 0.96 cal. from the missile nose, and the fin midchord is located 

1.5 cal. from the missile base. The canards (figure 3a) have a double-wedge, trapezoidal 

planform with a span of 0.37 cal., a root chord of 0.36 cal., a tip chord of 0.13 cal., a midchord 

root thickness of 0.03 cal., and a taper ratio of 1.48. The planar fins have a double wedge, 

rectangular planform with a span of 0.78 cal., a chord of 0.65 cal., and a mid-chord thickness of 

0.03 cal. The grid fins (figure 3b) consist of 23 cubic and 12 prismatic cells with a span of 

0.74 cal., a chord of 0.10 cal., and a thickness of 0.46 cal. The web thickness between the grid 

fin cells is 0.003 cal. The canard and fin characteristics are summarized in table 1. 

Figure 1. Generic canard-controlled missile with planar fins. 

The analyses were performed at two Mach numbers: M = 0.6 and 0.9, with one canard 

deflection: δ = 10°, and at six angles of attack: α = 0, 2, 4, 6, 8, and 10°. The DREV wind 

tunnel conditions were used in this study. For M = 0.6, the freestream conditions were a 

Reynolds number of 1.01 × 107 m−1, a static temperature of 284 K, and a static pressure of 

7.66 × 104

 Pa. For M = 0.9, the freestream conditions were a Reynolds number of 

1.40 × 107 m−1, a static temperature of 255 K, and a static pressure of 5.62 × 104 Pa. The model 

reference diameter (D) was 30 mm, and the moment reference point (MRP) was 10.63 cal. aft of 

the missile nose. The simulations were performed with the missile in the cruciform (+) 

configuration. The DREV wind tunnel data ranged from 4 to +15° angle of attack. In the 10° 

canard deflection case, all four canards were deflected in the same direction, intended to give a 

positive roll, which by convention was clockwise when viewed from the rear of the missile. 
3

Figure 2. Generic canard-controlled missile with grid fins. 

 

 (a) (b) 

Figure 3. (a) Missile nose with canards at δ = 0° and (b) front view of missile. 

Table 1. Canard and fin characteristics. 

Control Type Span 

(cal.) 

Root Chord

(cal.) 

Tip Chord

(cal.) 

Root Thickness

(cal.) 

Taper Ratio Web 

(cal.) 

Canard 0.37 0.36 0.13 0.03 1.48 — 

Planar fin 0.78 0.65 0.65 0.03 1.0 — 

Grid fin 0.74 0.10 0.10 0.46 1.0 0.003 
4

2.2 Solver 

Steady-state calculations were used to compute the flow field using the commercial CFD code, 

FLUENT (v6.0) (11). The implicit, compressible, unstructured-mesh solver was used. The 

three-dimensional (3-D), time-dependent, Reynolds-Averaged Navier-Stokes (RANS) equations 

are solved using the finite volume method: 

∫[ ] = ∫ + − ⋅ ∫ ∂

∂

V V

dV dA dV

t

W F G H , (1) 

where 

.

0

, ,





































+

=





































+

+

+

+

=





































=

τ q

τ

τ

τ

G

v v

v k

v j

v i

v

W F

ij j

zi

yi

xi

E p v

w p

v p

u p

E

w

v

u

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

 (2) 

The inviscid flux vector F is evaluated by a standard upwind flux-difference splitting. In the 

implicit solver, each equation in the coupled set of governing equations is linearized implicitly 

with respect to all dependent variables in the set, resulting in a block system of equations. A 

block Gauss-Seidel, point implicit linear equation solver is used with an algebraic multigrid 

method to solve the resultant block system of equations. The coupled set of governing equations 

is discretized in time, and time marching proceeds until a steady-state solution is reached. In the 

implicit scheme, which was used in this study, an Euler-implicit discretization in time is 

combined with a Newton-type linearization of the fluxes. 

A modified form of the k-ε two-equation turbulence model was used in this study. Called the 

“realizable” k-ε model in FLUENT, it differs from the standard k-ε model in that it contains a 

new formulation for the turbulent viscosity and a new transport equation for the dissipation rate, 

which was derived from an exact equation for the transport of the mean-square vorticity 

fluctuation (12). The term “realizable” means that the model satisfies certain mathematical 

constraints on the Reynolds stresses consistent with turbulent flow physics. The realizable 

k-ε model has shown substantial improvements over the standard k-ε model where flow features 

include strong streamline curvature, vortices, and rotation (11). 

2.3 Computational Mesh and Boundary Conditions 

The geometry and unstructured mesh were generated using the preprocessor, GAMBIT, supplied 

in the FLUENT software suite. Canard deflection and angle of attack precluded the use of 

symmetry or periodicity, so a full 3-D mesh was required. In generating the meshes, boundary 

layer mesh spacing was used near the missile body and fin surfaces. The enhanced wall 

treatment option, new in FLUENT v6.0, was used. The enhanced wall treatment option uses the 

two-layer model when the viscous layer is resolved by the grid and enhanced wall functions 
5

when the grid is too coarse to resolve the viscous layer. The first point off the surface (cell 

center) was ~7.0 × 10–5 cal., chosen to give a y+ value of ~1.0. This could not be achieved on the 

surfaces of the grid fins, as will be discussed, but with the new enhanced wall treatment option, 

the near wall turbulence should still be modeled appropriately. All mesh stretching ratios were 

uniform and were kept below 1.25. At M = 3.0 (thinnest boundary layer), there were ~32 cells in 

the boundary layer, with ~11 cells in the viscous sublayer. About 144 cells were used on the 

missile body in the circumferential direction, with this value increased in the grid fin region 

(figure 4). 

Figure 4. Surface mesh near body–grid fin base interface. 

An all-hexahedral (hex) mesh was used for the planar fin case, while a hybrid hex and tetrahedral 

(tet) mesh with triangular prism layers were used for the grid fin case. O-grid type meshes were 

generated around the canards and planar fins. The first 13 cal. of the missile was meshed with 

the same type hex mesh in both tail fin cases. A tet mesh was used in the tail region (figure 4) in 

order to mesh the complicated grid fin structure. Layers of triangular prisms were used on the 

body to capture the boundary layer. The tet mesh in the tail was matched to the forward hex 

mesh via a layer of pyramidal cells. Due to meshing constraints, prism layers were not used 

around the grid fins, and the spacing of the first point off the grid fin surfaces was larger than 

desired. Postprocessing of the runs showed that the y

+

 value was in the range of 1–1.8 on the 

missile body, less than 1.0 on the canards and planar fins, and between 10 and 20 on the grid 

fins. Similar y

+

 values were observed in the supersonic cases. However, the supersonic runs 

were performed with a previous version of FLUENT, without the enhanced wall treatment. In 

those cases, some loss of accuracy of the flow calculation near the grid fin surfaces was 

expected. 
6

The subsonic cases require a mesh that extends farther from the missile surface than the 

supersonic cases. An “outer” mesh was generated around the previous meshes used for the 

supersonic cases. This mesh extended 50 cal. ahead of and behind, and 66 cal. radially from the 

missile surface. The outer mesh added ~2.0 million cells to the planar fin case and ~1.6 million 

cells to the grid fin case. The total sizes of the planar and grid tail fin meshes were ~6.9 and 17.3 

million cells, respectively. Figure 5 shows the tail region of the mesh used for the planar fin 

subsonic cases. 

Figure 5. Mesh on missile surface and axial plane in tail region for 

planar fin subsonic and transonic cases. 

A far-field pressure boundary condition was used at the downstream, upstream, and outer radial 

boundary. A no-slip, adiabatic wall boundary condition was used for all solid surfaces. 

2.4 Solution Methodology 

The simulations were performed in parallel on a Silicon Graphics, Inc. (SGI) Origin 3800 with 

R12000 processors and an IBM SMP P3 with Power3 processors. The simulations were run with 

the single precision solver, with a maximum Courant-Friedrich-Lewy (CFL) number of 7. Each 

case was started with a CFL of 1.0 and ramped up to the maximum during the first few hundred 

iterations. Mesh adaption was used in the supersonic cases (9, 10) to show mesh independence 

and no further mesh adaption was used in the subsonic cases. 

The calculations took ~300–600 μs/cell/iteration of CPU time, using 48 processors for the planar 

fin cases and 64 processors for the grid fin cases. For example, solving the grid fin case with 64 

processors took ~81–162 s of CPU time per iteration. Convergence was determined by tracking 
7

the change in the flow residuals and the aerodynamic coefficients during the solution. The 

solution was deemed converged when the flow residuals had stabilized and the aerodynamic 

coefficients were changing <0.5% after the last 100 iterations. The aerodynamic coefficients 

converged in ~2700–4600 iterations. The normalized residuals were reduced at least 3 orders of 

magnitude. 

3. Results and Discussion 

3.1 Aerodynamic Coefficients – CFD Validation 

In this section, the results of the supersonic cases from the previous study are first summarized, 

followed by the subsonic results from the current study. The axial force, normal force, side 

force, rolling moment, and pitching moment are presented in missile-based coordinates. This is 

a right-handed system with the x axis coinciding with the missile axis and oriented to the rear, 

the y axis oriented to the missile’s starboard side, and the z axis oriented upward. The forces are 

positive when coinciding with the positive coordinate axes. The rolling moment is positive when 

the roll is clockwise when looking forward from the aft end of the missile. The yawing moment 

is positive when the nose is moving right, and the pitching moment is positive when the nose is 

moving upward. The reference area is the cross-sectional area of the missile base, and the 

reference length is the diameter of the missile. The calculated coefficients are compared to 

wind-tunnel measurements performed at DREV (5). 

3.1.1 Supersonic Flow 

In the previous study, very good agreement was found between all the experimental and 

calculated aerodynamic coefficients (9, 10). The CFD accurately captured the adverse roll 

phenomenon, shown in figure 6. For the planar fin case at M = 1.5, Cl is negative (opposite the 

intended roll due to the canard deflection) at α = 0°. Cl decreases as α increases to ~6°, then Cl

increases, becoming positive at ~8°. For the grid fin case at M = 1.5, Cl is now positive at 

α = 0°, decreasing to near 0 between 4° <α <7° (where roll control is substantially reduced but 

not reversed). At the higher Mach number, Cl is always positive and is similar for both types of 

fins. Flow visualizations of the computed results showed that the canard trailing vortices interact 

with the tail fins—primarily the leeward tail fin—and affect the roll control effectiveness of the 

canards. 

The intensity of the canard trailing vortices is much lower at Mach 3.0, so there is little effect on 

the roll control effectiveness of the canards. At Mach 1.5, a pressure differential is generated on 

the leeward planar fin that counteracts the rolling moment generated by the canards. The flow 

interacts with the grid fins differently, not generating as large a side force on the leeward fin, and 

thus not reducing the roll control effectiveness of the canards as much as in the planar fin case. 
8

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α (degree)

Cl

EXP (M=1.5)

EXP (M=3.0)

CFD (M=1.5)

CFD (M=3.0)

(a) 

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α (degree)

Cl

EXP (M=1.5)

EXP (M=3.0)

CFD (M=1.5)

CFD (M=3.0)

(b) 

Figure 6. Computed and experimental rolling moment coefficient for (a) planar 

fin and (b) grid fin cases in supersonic flow. 

Figure 7 shows the side force coefficient for both tail fin types. Again, there is little difference 

between the two fin types at the higher Mach number, and the maximum Cy is not too large. 

However, at the lower supersonic Mach number, the side force acting on the planar fins is ~5 

times larger than at the higher Mach number. Flow visualizations (10) of the computed results 

showed that the flow field induced by the deflected canards generated a higher velocity, and thus 

a lower pressure, on the starboard side of the missile, thereby generating the side force. The 

effect increased with angle of attack. There is little effect of the type of tail fin on the induced 

side force.
9

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α (degree)

Cy

EXP (M=1.5)

EXP (M=3.0)

CFD (M=1.5)

CFD (M=3.0)

(a) 

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α (degree)

Cy

EXP (M=1.5)

EXP (M=3.0)

CFD (M=1.5)

CFD (M=3.0)

(b) 

Figure 7. Computed and experimental side force coefficient for (a) planar fin 

and (b) grid fin cases in supersonic flow. 

3.1.2 Subsonic and Transonic Flow 

The computed aerodynamic coefficients show very good agreement with the experimental values 

in general. Figure 8 shows the computed and experimental normal force coefficient vs. α for 

both tail fin cases at M = 0.6. The CFD predictions compare very well to the measured Cz. Also 

shown is a prediction from the AP98 aeroprediction code (13) for the planar fin case. AP98 is an 

engineering code based on empirical and theoretical methods. Grid fins are not modeled by the 

AP98 code. The comparison of the CFD and AP98 predictions is excellent up to ~α = 6°, where 

AP98 predicts a higher Cz. The difference between the computed and measured Cz at 
10

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-5 0 5 10 15

α

Cz

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

(a) 

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-5 0 5 10 15

α

Cz

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(b) 

Figure 8. Computed and experimental normal force coefficient for (a) planar fin and 

(b) grid fin cases at Mach 0.6. 

α = 10° was 9.4% in the planar fin case (figure 8a) and 12.4% in the grid fin case (figure 8b). 

Note that the CFD predicts a lower normal force for the planar fin case and a higher normal force 

for the grid fin case. The Cz predicted by the CFD is very similar for both tail fin cases (4% at 

α = 10°), which is expected since the two types of fins were designed to give the same static 

stability. However, the two experimental Cz values differ by 23% at α = 10°. 

The experimental data shown in these plots are data supplied by DREV. A second-order 

polynomial regression routine was used on the raw data to get values at regular, 1°, intervals of 
11

α. This routine did not alter the generally nonsmooth character of the original raw experimental 

data. Biases in the data were also removed in some cases by zeroing the α = 0° value. The 

sometimes “wiggly” nature of the experimental data makes it impractical to quantify the 

difference between the predicted and measured data at all values of α. In fact, one of the 

observations of the wind tunnel test data was that a significant amount of scatter exists in the 

data below M = 1.28 and that these data should be used with caution (5). Therefore, rather than 

state the maximum numerical difference, the percent difference will be stated where the 

qualitative trend of the curves have obviously diverged. For example, in figure 8a, the predicted 

values are linear in the range 0° < α < 5° while the experimental values have moved lower. The 

maximum difference between the predicted and measured values is at α = 2° and 4° (23%). 

However, the divergence of the predicted and measured values at α > 8° may be a more 

important trend, and so the difference at α = 10° was stated as previously mentioned. 

The predicted and measured pitching moment coefficients (about missile nose) at M = 0.6 are 

shown in figure 9. The trends are similar to those observed for Cz. At α = 10°, the CFD 

predictions for the two types of tail fins differ by 5.4%, while the experimental values differ by 

32.5%. The AP98 prediction (figure 9a) falls close to the experimental values, but this is due to 

the higher prediction of Cz by AP98. 

Figure 10 shows the measured and computed center of pressure location, aft of the missile nose, 

for both tail fin types at M = 0.6. At α = 10°, the CFD predictions for the two types of tail fins 

differ by 1.3%, while the experimental values differ by 10.2%. The CFD, AP98, and measured 

values for the planar fin case compare very well. The calculation of xcp is indeterminate at 

α = 0°, so the experimental and CFD data should be disregarded at this location. 

The predicted and measured total axial force coefficients at M = 0.6 are shown in figure 11. This 

value includes the force on the missile base. The CFD predicted and measured values compare 

well for the planar fin case (figure 11a), with differences of 4%–7.3%. The variation with α is 

also similar. The AP98 prediction has the opposite trend with α, but the values are within 8% of 

the CFD predictions. The CFD predicted values of Cx vary between 0.38 and 0.40, while the 

measured values vary between 0.35 and 0.38. In the grid fin case (figure 11b), the predicted 

values of Cx are 16%–18% higher than the measured values. The trend with α is predicted very 

well. The predicted values of Cx vary between 0.64 and 0.71, while the measured values vary 

between 0.55 and 0.60. The Cx in the grid fin case is ~1.5–1.7 times that in the planar fin case. 

These differences and trends are similar to those observed in the supersonic cases. The axial 

force for the grid fin case is expected to be higher than that for the planar fin case. However, as 

noted for the supersonic cases (9, 10), the drag of the grid fins on this model is larger than could 

be achieved with an “optimum” design. Primarily, because of the small size of the wind tunnel 

model, the web thickness of the fin elements could not be scaled down to the proper design 

thickness due to machining limitations. The web is ~1.5 times larger than optimum. 
12

-35

-30

-25

-20

-15

-10

-5

0

5

10

-5 0 5 10 15

α

Cm

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

(a) 

-35

-30

-25

-20

-15

-10

-5

0

5

10

-5 0 5 10 15

α

Cm

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(b) 

Figure 9. Computed and experimental pitching moment coefficient about the missile nose for 

(a) planar fin and (b) grid fin cases at Mach 0.6. 

The results at M = 0.9 were very similar to the results at M = 0.6 for the aerodynamic coefficients 

shown thus far. These results are not presented here, but they are included in appendices A and B. 

The most important objective of this study was to accurately predict the adverse induced rolling 

moment and side forces observed in the wind tunnel experiments. Validation of these 

components gives confidence to the flow visualizations obtained from the CFD. Figure 12 

compares the predicted and measured side force coefficients for the two tail fin types at M = 0.6. 
13

0

2

4

6

8

10

12

14

-5 0 5 10 15

α

Xcp

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

(a) 

0

2

4

6

8

10

12

14

-5 0 5 10 15

α

Xcp

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(b) 

Figure 10. Computed and experimental center of pressure location from nose for (a) planar 

fin and (b) grid fin cases at Mach 0.6. 

The predicted values are in relatively good agreement, considering that the experimental data are 

not very smooth. The level of induced side force is similar for both tail fin types. Flow 

visualizations showed a similar effect as in the supersonic case, with the flow past the deflected 

canards generating a low-pressure region on the starboard side of the missile. 

The comparison of the predicted and measured Cy at M = 0.9 is shown in figure 13. The 

agreement is excellent in the planar fin case (figure 13a), capturing an inflection in the data at 

~α = 4°. There was only a moderate difference (8%) between the predicted and measured values 

at α = 10°. The predicted data in the grid fin case are similar to the predicted data in the 
14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 0 5 10 15

α

CX

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

(a) 

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-5 0 5 10 15

α

CX

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(b) 

Figure 11. Computed and experimental axial force coefficient for (a) planar fin and (b) grid 

fin cases at Mach 0.6. 

planar fin case; however, the validity of the experimental data between 0° < α <4° is 

questionable. 

The predicted and measured rolling moment coefficients at M = 0.6 are compared in figure 14. 

The agreement between the predicted and measured values of Cl in the planar fin case 

(figure 14a) is very good. The agreement at α = 0° is not good, but the experimental data are 

questionable for −5° <α <0°. In the grid fin case (figure 14b), the shape of the curves match, but 

appear offset, with the predicted values lagging the measured values. However, the predicted Cl
15

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

CY

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(a) 

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

CY

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(b) 

Figure 12. Computed and experimental side force coefficient for (a) planar fin and 

(b) grid fin cases at Mach 0.6. 

indicates that the grid fins improves the roll control effectiveness of the canards (higher Cl) at 

α = 0°, whereas no improvement is shown in the measured values. The measured Cl for α >0° is 

similar for both fin types. The predicted Cl shows that the grid fins improve the canard roll 

control effectiveness for 0° <α <4°, while slightly reducing it for 5° <α <10°. 

The comparison of the predicted and measured Cl at M = 0.9 is shown in figure 15. The general 

trends are similar to those observed at M = 0.6, with the measures Cl higher than the predicted 

value for the planar fin case at α = 0° and the predicted values lagging the experimental values 
16

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

Cy

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

(a) 

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

Cy

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

(b) 

Figure 13. Computed and experimental side force coefficient for (a) planar fin and (b) grid fin 

cases at Mach 0.9. 

for the grid fin case. The predicted values for the planar fin case (figure 15a) do not follow as 

smooth a trend as in the other cases, but the trend is still similar to those other cases. 

At subsonic and transonic speed, the grid fins improves the canard roll-control effectiveness at 

low α, but the improvement is clearly not as good as achieved at the low supersonic speed. The 

predicted Cl for both tail fin types are compared in figure 16 at the four Mach numbers 

investigated in this report and references (9) and (10). The grid fins clearly have the largest 

impact on canard roll-control effectiveness at M = 1.5. The grid fins have the least impact at 

M = 3.0; but in this case, the roll control does not suffer from any adverse effects. The 
17

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(a) 

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

(b) 

Figure 14. Computed and experimental rolling moment coefficient for (a) planar fin 

and (b) grid fin cases at Mach 0.6. 

improvement in roll control at subsonic and transonic speed in the grid fin case at low α is 

evident for 0° <α <2°. While there is some improvement in roll control for α <2° for the 

subsonic and transonic cases, there is no improvement for ~2.5° <α <7°. This behavior 

effectively nullifies the small benefit of using grid fins instead of planar fins at these speeds. In 

fact, the rolling moment in the subsonic and transonic cases with grid fins is reduced from that 

with planar fins for 8° <α <10°. 
18

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

(a) 

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

(b) 

Figure 15. Computed and experimental rolling moment coefficient for (a) planar fin 

and (b) grid fin cases at Mach 0.9. 

3.2 Flow-Field Visualizations 

Visualizations of the flow field at subsonic and transonic speed show flow interaction effects that 

are similar to those observed in the supersonic cases. The downwash off the deflected canards 

produces a low-pressure region on the starboard side of the missile that in turn produces the 

induced side force. In addition, the canard trailing vortices interact with the fins until α is high 

enough so that the vortices miss the leeward fin. The adverse pressure on the leeward fin is 

primarily responsible for the adverse roll effects. 
19

-0.10

-0.05

0.00

0.05

0.10

0.15

-2 0 2 4 6 8 10 12

α

Cl

M = 0.6

M = 0.9

M = 1.5

M = 3.0

(a) 

-0.10

-0.05

0.00

0.05

0.10

0.15

-2 0 2 4 6 8 10 12

α

Cl

M = 0.6

M = 0.9

M = 1.5

M = 3.0

(b) 

Figure 16. Computed rolling moment coefficient for (a) planar fin and (b) grid fin 

cases at several Mach numbers. 

The result of the interaction of the canard trailing vortices with the tail fins is illustrated by 

contour plots of pressure coefficient on the missile surfaces in figures 17–20. In the planar fin 

case at M = 0.6, a small high-pressure region is observed on the starboard side of the leeward tail 

fin at α = 0° (figure 17a). This causes a rolling moment opposite of that induced by the canards, 

thereby lowering the canard roll-control effectiveness. This high-pressure region is much larger 

at α = 4° (figure 17b), where the adverse induced roll is nearly a maximum (figure 14a). At 

α = 10° (figure 17c), high-pressure regions are on the port side of the leeward and windward 

fins. In this case, the adverse induced roll is minimized, and the canard effectiveness is 
20

(a) 

(b) 

(c) 

Figure 17. Cp contours on planar finned missile surfaces at M = 0.6 and (a) α = 0°, (b) α = 4°, and (c) α = 10°. 

increased (figure14a). Some minor differences can be observed in the planar fin case at M = 0.9 

(figure 18), but the same induced roll effects are present. 

In the grid fin case (figure 19), the effect is a little harder to observe. As in the supersonic case, 

the induced pressure on the grid fins is distributed over the fin more than in the planar fin case. 

Although a very high pressure is not observed on the grid fin vanes (as would be indicated by a 
21

(a) 

(b) 

(c) 

Figure 18. Cp contours on planar finned missile surfaces at M = 0.9 and (a) α = 0°, (b) α = 4°, and 

(c) α = 10°. 

red color), higher pressures are observed more on the starboard sides of the leeward grid fin 

vanes than on the port sides at α = 0 and 4° (figure 19a, b). At α = 10° (figure 19c), the higher 
22

(a) 

(b) 

(c) 

Figure 19. Cp contours on grid finned missile surfaces at M = 0.6 and (a) α = 0°, (b) α = 4°, and (c) α = 10°. 

pressure has reversed sides, and canard roll-control effectiveness is improved. Again, there are 

minor differences at M = 0.9 (figure 20), but the general effect is the same. 

The induced side force is generated on the missile body. This was confirmed in the supersonic 

cases by observing the separate components of side force on each surface (10). The effect is 

shown qualitatively for the M = 0.6 case in figure 21, where a larger low-pressure region is 
23

(a) 

(b) 

(c) 

Figure 20. Cp contours on grid finned missile surfaces at M = 0.9 and (a) α = 0°, (b) α = 4°, and (c) α = 10°. 

observed on the starboard side of the missile at α = 10° (figure 21b). This generates the positive 

side force shown in figure 12. Understanding and quantifying this effect is important for 
24

(a) 

(b) 

Figure 21. Cp contours on (a) port side and (b) starboard side of grid finned missile 

surfaces at M = 0.6 and α = 10°. 

accounting for it with the proper flight control design. The effect was very similar for the 

M = 0.9 case, so it is not presented. 

The canard trailing vortices at α = 4 and 10° for the M = 0.6 cases are shown in figure 22 and 

figure 23, respectively. The effect is similar to that observed in the supersonic cases (10). The 

trailing vortex off the windward canard merges with the missile body boundary layer. The 

trailing vortices off the remaining three canards move toward the leeward side of the missile and 

will merge at different locations, depending on α. At α = 4°, the vortices interact with the tail 

fins before merging. In the planar fin case (figure 22a), the vortices pass by the leeward fin 
25

(a) 

(b) 

Figure 22. Cp surface contours and vorticity magnitude contours on cross planes at x/d = 2, 

4, ..., 22 for the (a) planar finned and (b) grid finned missile at M = 0.6 and 

α = 4°. 

apparently unperturbed. In the grid fin case (figure 22b), however, the vortices that pass through 

the leeward tail fin are broken up. At α = 10°, the vortices merge before reaching the tail fins 

and do not interact with the fins at all. The body cross-flow vortices are primarily responsible 

for the interaction with the tail fins at this α. The flow pattern of the canard trailing vortices was 

very similar at M = 0.9, but intensity of the vortices was stronger. 
26

(a) 

(b) 

Figure 23. Cp surface contours and vorticity magnitude contours on cross planes at x/d = 2, 4, 

..., 22 for the (a) planar finned and (b) grid finned missile at M = 0.6 and α = 10°. 

3.3 Control Surface Forces 

The forces on the canards and fins are summarized in tabular and graphical form in 

appendices C–F. Some results are presented in this section to quantify the trends observed in the 

flow visualizations. The forces on the leeward and windward fins resulting in the adverse rolling 

moment are shown in figure 24, which shows the side force coefficient on each fin for the planar 
27

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

(a) 

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

(b) 

Figure 24. Side force coefficient on tail fins for (a) planar fin and (b) grid fin case 

for δ = 10° and M = 0.6. 

and grid fin cases at δ = 10° and M = 0.6. In the planar fin case, the windward fin (fin 2) is at a 

nearly constant, small positive value. The leeward fin (fin 4) is negative until α >8°. As in the 

supersonic case, this force imbalance produces the adverse rolling moment. The side forces on 

the fins are of the same order as those on the canards; however, the larger moment arm of the 

fins produces a larger rolling moment. In the grid fin case (figure 24b), there is a small reduction 

in the side forces at α = 0° but not much at higher α. As in the supersonic case, the normal 

forces on the starboard and port fins also contribute to the adverse rolling moment. This is 

illustrated in figure 25, which shows the normal force coefficient on each fin for the planar and 
28

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

(a) 

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

(b) 

Figure 25. Normal force coefficient on tail fins for (a) planar fin and (b) grid fin case 

for δ = 10° and M = 0.6. 

grid fin cases at δ = 10° and M = 0.6. The normal force behavior at M = 0.9 was similar to that at 

M = 0.6. The reduction of the difference in normal force between the starboard and port fins is 

only slightly smaller when the grid fins are used. However, even for the planar fin case, the 

component of the adverse rolling moment due to the difference in normal force on the horizontal 

fins at the subsonic and transonic speeds is smaller than it was at M = 1.5 (10). 

3.4 Flow-Through Grid Fins 

Another important aspect to the performance of grid fins is the state of the flow as it passes 

through the grid fin cells. An earlier study on grid fins (6) showed that a “bucket” exists in the 
29

normal force vs. Mach number curve through the transonic and lower supersonic range. This is 

illustrated in figure 26, which was excerpted from figure 6 of reference (6). The figure presents 

the zero angle of attack fin normal force slope (CNFα) as a function of Mach number for a fin 

configuration, similar to the one used in this study. The data follow trends similar to those 

exhibited by conventional fins at subsonic (M <0.75) and higher supersonic (M >1.60) Mach 

numbers. 

Figure 26. Fin normal force slope values as a function of Mach number (6). 

Washington and Miller (6) attributed this “bucket” to two separate flow phenomena and stated it 

was a function of the grid fin internal cell geometry. For analysis purposes, they viewed a grid 

fin as a collection of individual cells acting as separate inlets. Washington and Miller proposed a 

grid fin flow field, indicating choking in the grid fin cells. This is illustrated in figure 27, which 

is excerpted from figure 7 of reference (6). At freestream Mach numbers <1.0, the reduction in 

the inlet cross-sectional area caused by the presence of the cell structural members and the buildup of the boundary layer on the cell walls causes the flow through the cell to accelerate to sonic 

(choked) conditions (figure 27a). The cell remains choked as the freestream Mach number 

increases past Mach 1.0 until a normal shock forms ahead of the grid fin cell itself (figure 27b). 

The reduction in flow through the grid fin cells causes a reduction in the normal force generated 

by the fin. 
30

Figure 27. Grid fin flow field proposed in reference (6). 

Increases of the freestream Mach number well beyond 1.0 eventually enable the cell to 

“swallow” the shock, resulting in a shock wave attaching to the leading edge of the cell. At 

lower supersonic Mach numbers, the leading edge shock wave may reflect within the cell 

structure, as shown in figure 27c. Increasing the freestream Mach number further will eventually 

lead to the shock wave passing through the cell without reflecting off the grid fin cell structure 

(figure 27d). Increases in the freestream Mach number further should have no qualitative effect 

on the grid fin flow field. Washington and Miller (6) state that the point where the shock wave 

first passes through the grid fin cell structure undisturbed is where the grid fin begins to exhibit 

supersonic normal force characteristics similar to conventional fins. In figure 26, the onset of 

choked flow is indicated at M = 0.75, leading edge shock attachment is indicated at M = 1.35, 

and the point of no-shock reflection is indicated at M = 1.60. 

The flow-field solutions obtained via CFD computations allow us to qualitatively compare the 

flow field through the grid fin cell structure with that proposed by Washington and Miller (6). 

Figures 28–31 present the flow field on the symmetry plane of the grid fin missile for cases at 

δ = 10°, α = 0°, and M = 0.6, 0.9, 1.5, and 3.0. Contours of Cp are shown in the first two parts of 

each figure, while contours of the Mach number are shown in the third part. The leeward grid fin 

geometry is not shown in the figures so that the flow field inside the grid fin cell structure can be 

observed. Part “a” of each figure shows the flow field over the whole missile. Strong vortices 

are observed trailing the canards at M = 0.6, 0.9, and 1.5. The vortices were weaker at M = 3.0 

and could not be observed with the contour scale used in figure 31. Typical oblique shock wave 
31

(a) 

(b) 

(c) 

Figure 28. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with 

leeward grid fin geometry removed, α = 0° and M = 0.6. 
32

(a) 

(b) 

(c) 

Figure 29. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with 

leeward grid fin geometry removed, α = 0° and M = 0.9.
33

(a) 

(b) 

(c) 

Figure 30. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with 

leeward grid fin geometry removed, α = 0° and M = 1.5.
34

(a) 

(b) 

(c) 

Figure 31. Cp (a, b) and Mach number (c) contours on vertical symmetry plane with leeward 

grid fin geometry removed, α = 0° and M = 3.0. 
35

structures are observed at the supersonic Mach numbers. For the M = 0.9 case, probing of the 

flow field showed that weak shock waves form near the nose and at the tips of the canards and 

fins. 

Parts “b” and “c” of each figure show the Cp and Mach number contours, respectively, in the 

region of the leeward grid fin. It must be noted that these cases are with the canards deflected 

(δ = 10°), so vortices from the leeward, port, and starboard canards will be impacting the flow 

field near the leeward grid fin. However, general conclusions on the flow-field structure can still 

be surmised. At M = 0.6 (figure 28), the flow accelerates only to ~0.70 <M <0.75, so no flow 

choking is observed. At M = 0.9 (figure 29), a shock forms inside the grid fin cell, ~0.75 chord 

lengths from the leading edge of the fin. This shock cannot be seen with the contour scales 

shown in the figure; it was determined by interactively probing the data in the post-processing 

software. This choked flow case compares to case “a” of figure 27. 

At M = 1.5 (figure 30), a normal shock forms within 1 chord length ahead of the grid fin cells. 

Normal shock relations give M2 = 0.7 for M1 = 1.5, and interactive probing of the flow field gave 

Mach number values of 0.7–0.75 in the region behind this shock wave (M2). Some of the 

jaggedness in the contours is believed to be at least in part due to the contouring programs. 

Some may also be due to effects of the canard-trailing vortices impacting the flow field in this 

region. The mesh in this region is made up of fine tetrahedral cells (figure 4) and is adequate for 

capturing the shock waves. The flow then accelerates through the grid fin cells and chokes 

within the cells. The flow is rather complex, with oblique shocks forming on the web between 

the bottom two cells. It is likely that the impact of the canard-trailing vortices is affecting the 

flow-field structure. The flow is definitely choked, with a combination of cases “a–c” of 

figure 27. At M = 3.0 (figure 31), there are oblique shocks at the leading edges of the grid fin 

cells. These shocks reflect off each other, but do not reflect off the cell walls. This flow 

structure corresponds to case “d” of figure 27. 

It is difficult to correlate the grid fin choking with the effectiveness of the grid fins at reducing 

the roll-reversal effect. The greatest reduction of roll-reversal was at M = 1.5, while there was 

only a small effect at low α for M = 0.9, and there was fin choking at both these Mach numbers. 

The roll reversal effect at M = 0.6 (no choking) was similar to that at M = 0.9. 

4. Summary and Conclusions 

Viscous CFD calculations were used to predict the aerodynamic coefficients and flow field 

around a generic canard-controlled missile configuration in subsonic (M = 0.6) and transonic 

(M = 0.9) flow. Validation of the computed results was demonstrated by the very good 

agreement between the computed aerodynamic coefficients and those obtained from wind tunnel 

measurements. 
36

Visualizations of the flow field showed that the canard downwash produced a low-pressure 

region on the starboard side of the missile that, in turn, produced a large induced side force. This 

was very similar to the effect observed at low supersonic speed (M = 1.5) (10). Visualizations 

also showed that the canard-trailing vortices interact with the tail fins in the same way as at 

supersonic speed. This interaction takes place until α is high enough so that the vortices miss the 

leeward fin. The pressure differential on the leeward tail fin, produced by this interaction, is 

primarily responsible for the adverse induced roll effects. 

Visualizations of the flow field through the grid fin structure showed choking of the flow at 

M = 0.9 and M = 1.5. The grid fin choking phenomena could not be correlated with the 

effectiveness of the grid fin to alleviate the roll-reversal problem. 

The validated CFD results show that grid fins do not improve the canard roll-control 

effectiveness at subsonic and transonic speeds as well as they do at the low supersonic speed. 

While there is some improvement for α <2°, it is negated by reversed roll above α = 2°. This 

behavior makes it difficult for the control system to maintain a stabilized roll attitude, thereby 

making it impractical to employ grid fin tail surfaces on canard-controlled missiles at subsonic 

and transonic speeds. 
37

5. References 

1. Allen, J. M.; Blair, A. B., Jr. Comparison of Analytical and Experimental Supersonic 

Aerodynamic Characteristics of a Forward Control Missile. Journal of Spacecraft and 

Rockets 1982, 19 (2), 155–159. 

2. Blair, A. B., Jr.; Dillon, J. L., Jr.; Watson, C. B. Experimental Study of Tail-Span Effects on 

a Canard-Controlled Missile. Journal of Spacecraft and Rockets 1993, 30 (5), 635–640. 

3. Blair, A. B., Jr. Supersonic Aerodynamic Characteristics of a Maneuvering CanardControlled Missile With Fixed and Free-Rolling Tail Fins. SAE Paper 90–1993; Society of 

Automotive Engineers: Warrendale, PA, October 1990. 

4. Burt, J. R., Jr. The Effectiveness of Canards for Roll Control; RD-77-8; U.S. Army Missile 

Command: Redstone Arsenal, AL, 1976. 

5. Miller, M. Aerodynamic Investigation of a Canard Controlled Missile Using Planar and 

Grid Fin Tail. Dynetics, Inc.: Huntsville, AL, unpublished work. 

6. Washington, W. D.; Miller, M. S. Grid Fins—A New Concept for Missile Stability and 

Control; AIAA Paper 93–0035; American Institute of Aeronautics and Astronautics: 

Reston, VA, January 1993. 

7. DeSpirito, J.; Edge, H. L.; Weinacht, P.; Sahu, J.; Dinavahi, S. P. G. Computational Fluid 

Dynamics Analysis of a Missile With Grid Fins. Journal of Spacecraft and Rockets 2001, 38

(5), 711–718. 

8. DeSpirito, J.; Sahu, J. Viscous CFD Calculations of Grid Fin Missile Aerodynamics in the 

Supersonic Flow Regime; AIAA Paper 2001–2057; American Institute of Aeronautics and 

Astronautics: Reston, VA, January 2001. 

9. DeSpirito, J.; Vaughn, M. E., Jr.; Washington, W. D. CFD Investigation of CanardControlled Missile With Planar and Grid Fins in Supersonic Flow; AIAA Paper 2002-4509; 

American Institute of Aeronautics and Astronautics: Reston, VA, August 2002. 

10. DeSpirito, J.; Vaughn, M. E., Jr.; Washington, W. D. Numerical Investigation of 

Aerodynamics of Canard-Controlled Missile Using Planar and Grid Tail Fins, Part I: 

Supersonic Flow; ARL-TR-2848; U. S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, 2002. 

11. Fluent, Inc. “Fluent 6.0 User’s Guide.” Vol. 2, Lebanon, NH, 2001.
38

12 Shih, T. H.; Liou, W. W.; Shabbir, A.; Yang, Z.; Zhu, J. A New k-ε Eddy-Viscosity Model 

for High Reynolds Number Turbulent Flows—Model Development and Validation. 

Computers and Fluids 1995, 24 (3), 227-238. 

13. Moore, F. G.; McInville, R. M.; Hymer, T. C. Application of the 1998 Version of the 

Aeroprediction Code. Journal of Spacecraft and Rockets 1999, 36 (5), 633–645.
39

Appendix A. Aerodynamic Coefficients for Planar Fin Case 
40

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-5 0 5 10 15

α

Cz

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

Figure A-1. Normal force for the planar fin case at Mach 0.6. 

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-5 0 5 10 15

α

Cz

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

AP98 (M=0.9; Canard=0°)

Figure A-2. Normal force for the planar fin case at Mach 0.9. 
41

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

CY

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure A-3. Side force for the planar fin case at Mach 0.6. 

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

Cy

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

Figure A-4. Side force for the planar fin case at Mach 0.9. 
42

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 0 5 10 15

α

CX

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

Figure A-5. Axial force for the planar fin case at Mach 0.6. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 0 5 10 15

α

CX

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

AP98 (M=0.9; Canard=0°)

Figure A-6. Axial force for the planar fin case at Mach 0.9. 
43

-35

-30

-25

-20

-15

-10

-5

0

5

10

-5 0 5 10 15

α

Cm

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

Figure A-7. Pitching moment about the nose for the planar fin case at Mach 0.6. 

-35

-30

-25

-20

-15

-10

-5

0

5

10

-5 0 5 10 15

α

Cm

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

AP98 (M=0.9; Canard=0°)

Figure A-8. Pitching moment about the nose for the planar fin case at Mach 0.9. 
44

-2

-1

0

1

2

3

4

5

-5 0 5 10 15

α

Cm

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure A-9. Pitching moment about the moment reference point (MRP) for the planar fin 

case at Mach 0.6. 

-2

-1

0

1

2

3

4

5

-5 0 5 10 15

α

Cm

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

Figure A-10. Pitching moment about the MRP for the planar fin case at Mach 0.9. 
45

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure A-11. Rolling moment for the planar fin case at Mach 0.6. 

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

Figure A-12. Rolling moment for the planar fin case at Mach 0.9. 
46

-7

-6

-5

-4

-3

-2

-1

0

1

2

-5 0 5 10 15

α

Cn

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure A-13. Yawing moment about the nose for the planar fin case at Mach 0.6. 

-7

-6

-5

-4

-3

-2

-1

0

1

2

-5 0 5 10 15

α

Cn

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

Figure A-14. Yawing moment about the nose for the planar fin case at Mach 0.9. 
47

0

2

4

6

8

10

12

14

-5 0 5 10 15

α

Xcp

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

AP98 (M=0.6; Canard=0°)

Figure A-15. Center of pressure location from the nose for the planar fin case at Mach 0.6. 

0

2

4

6

8

10

12

14

-5 0 5 10 15

α

Xcp

EXP (M=0.9 / Canard=10°)

CFD (M=0.9; Canard=10°)

AP98 (M=0.9; Canard=0°)

Figure A-16. Center of pressure location from the nose for the planar fin case at Mach 0.9.
48

INTENTIONALLY LEFT BLANK. 
49

Appendix B. Aerodynamic Coefficients for Grid Fin Case 
50

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-5 0 5 10 15

α

Cz

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-1. Normal force for the grid fin case at Mach 0.6. 

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-5 0 5 10 15

α

Cz

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-2. Normal force for the grid fin case at Mach 0.9. 
51

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

CY

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-3. Side force for the grid fin case at Mach 0.6. 

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-5 0 5 10 15

α

Cy

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-4. Side force for the grid fin case at Mach 0.9. 
52

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-5 0 5 10 15

α

CX

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-5. Axial force for the grid fin case at Mach 0.6. 

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-5 0 5 10 15

α

CX

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-6. Axial force for the grid fin case at Mach 0.9. 
53

-35

-30

-25

-20

-15

-10

-5

0

5

10

-5 0 5 10 15

α

Cm

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-7. Pitching moment about the nose for the grid fin case at Mach 0.6. 

-35

-30

-25

-20

-15

-10

-5

0

5

10

-5 0 5 10 15

α

Cm

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-8. Pitching moment about the nose for the grid fin case at Mach 0.9. 
54

-2

-1

0

1

2

3

4

5

-5 0 5 10 15

α

Cm

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-9. Pitching moment about the moment reference point (MRP) for the grid fin 

case at Mach 0.6. 

-2

-1

0

1

2

3

4

5

-5 0 5 10 15

α

Cm

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-10. Pitching moment about the MRP for the grid fin case at Mach 0.9. 
55

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-11. Rolling moment for the grid fin case at Mach 0.6. 

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 0 5 10 15

α

Cl

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-12. Rolling moment for the grid fin case at Mach 0.9. 
56

-7

-6

-5

-4

-3

-2

-1

0

1

2

-5 0 5 10 15

α

Cn

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-13. Yawing moment about the nose for the grid fin case at Mach 0.6. 

-7

-6

-5

-4

-3

-2

-1

0

1

2

-5 0 5 10 15

α

Cn

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-14. Yawing moment about the nose for the grid fin case at Mach 0.9. 
57

0

2

4

6

8

10

12

14

-5 0 5 10 15

α

Xcp

EXP (M=0.6; Canard=10°)

CFD (M=0.6; Canard=10°)

Figure B-15. Center of pressure location from the nose for the grid fin case at Mach 0.6. 

0

2

4

6

8

10

12

14

-5 0 5 10 15

α

Xcp

EXP (M=0.9; Canard=10°)

CFD (M=0.9; Canard=10°)

Figure B-16. Center of pressure location from the nose for the grid fin case at Mach 0.9.
58

INTENTIONALLY LEFT BLANK. 
59

Appendix C. Force Coefficients on Canards 
60

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

canard1 (S)

canard2 (W)

canard3 (P)

canard4 (L)

total canard

Figure C-1. Canard normal force, δ = 10°, Mach 0.6. 

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

canard1 (S)

canard2 (W)

canard3 (P)

canard4 (L)

total canard

Figure C-2. Canard normal force, δ = 10°, Mach 0.9. 
61

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

canard1 (S)

canard2 (W)

canard3 (P)

canard4 (L)

total canard

Figure C-3. Canard side force, δ = 10°, Mach 0.6. 

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

canard1 (S)

canard2 (W)

canard3 (P)

canard4 (L)

total canard

Figure C-4. Canard side force, δ = 10°, Mach 0.9. 
62

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cx

canard1 (S)

canard2 (W)

canard3 (P)

canard4 (L)

Figure C-5. Canard axial force, δ = 10°, Mach 0.6. 

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cx

canard1 (S)

canard2 (W)

canard3 (P)

canard4 (L)

Figure C-6. Canard axial force, δ = 10°, Mach 0.9. 
63

Table C-1. Aerodynamic coefficients on canards, planar fin case, δ = 10°, Mach 0.6. 

α 0 2 4 6 8 10 

Axial Force 

Canard 1 (S) 0.0082 0.0064 0.0044 0.0019 −0.0013 –0.0050 

Canard 2 (W) 0.0082 0.0081 0.0080 0.0079 0.0077 0.0076 

Canard 3 (P) 0.0082 0.0104 0.0125 0.0140 0.0150 0.0156 

Canard 4 (L) 0.0082 0.0083 0.0085 0.0088 0.0091 0.0094 

Total canard 0.0326 0.0332 0.0334 0.0326 0.0305 0.0276 

Side Force 

Canard 1 (S) 0.0020 0.0014 0.0010 0.0008 0.0010 0.0016 

Canard 2 (W) −0.0436 −0.0427 −0.0418 −0.0410 −0.0401 −0.0390 

Canard 3 (P) −0.0020 −0.0024 −0.0025 −0.0025 −0.0025 −0.0024 

Canard 4 (L) 0.0436 0.0449 0.0465 0.0483 0.0501 0.0519 

Total canard 0.0000 0.0014 0.0032 0.0056 0.0085 0.0120 

Normal Force 

Canard 1 (S) −0.0436 −0.0286 −0.0124 0.0039 0.0201 0.0360 

Canard 2 (W) −0.0020 −0.0018 −0.0015 −0.0013 −0.0011 −0.0008 

Canard 3 (P) 0.0436 0.0545 0.0623 0.0659 0.0695 0.0723 

Canard 4 (L) 0.0020 0.0022 0.0024 0.0027 0.0029 0.0031 

Total canard 0.0000 0.0263 0.0509 0.0711 0.0914 0.1106 

Notes: S = starboard; W = windward; P = port; L = leeward. 

Table C-2. Aerodynamic coefficients on canards, planar fin case, δ = 10°, Mach 0.9. 

α 0 2 4 6 8 10 

Axial Force 

Canard 1 (S) 0.0105 0.0084 0.0064 0.0042 0.0015 −0.0022 

Canard 2 (W) 0.0105 0.0102 0.0100 0.0097 0.0094 0.0091 

Canard 3 (P) 0.0105 0.0133 0.0150 0.0168 0.0180 0.0187 

Canard 4 (L) 0.0105 0.0110 0.0116 0.0129 0.0142 0.0159 

Total canard 0.0419 0.0429 0.0430 0.0436 0.0432 0.0414 

Side Force 

Canard 1 (S) 0.0024 0.0022 0.0016 0.0015 0.0016 0.0021 

Canard 2 (W) −0.0500 −0.0500 −0.0497 −0.0476 −0.0452 −0.0423 

Canard 3 (P) −0.0024 −0.0032 −0.0024 −0.0024 −0.0023 −0.0022 

Canard 4 (L) 0.0500 0.0483 0.0501 0.0540 0.0589 0.0652 

Total canard 0.0000 −0.0027 −0.0004 0.0054 0.0130 0.0229 

Normal Force 

Canard 1 (S) −0.0500 −0.0295 −0.0106 0.0061 0.0218 0.0394 

Canard 2 (W) −0.0024 −0.0023 −0.0019 −0.0016 –0.0012 −0.0008 

Canard 3 (P) 0.0500 0.0686 0.0668 0.0750 0.0817 0.0869 

Canard 4 (L) 0.0024 0.0028 0.0031 0.0040 0.0044 0.0048 

Total canard 0.0000 0.0397 0.0573 0.0836 0.1067 0.1303 

Notes: S = starboard; W = windward; P = port; L = leeward. 
64

INTENTIONALLY LEFT BLANK. 
65

Appendix D. Force Coefficients on Planar Fins 
66

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure D-1. Planar fin normal force, δ = 10°, Mach 0.6. 

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure D-2. Planar fin normal force, δ = 10°, Mach 0.9. 
67

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure D-3. Planar fin side force, δ = 10°, Mach 0.6. 

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure D-4. Planar fin side force, δ = 10°, Mach 0.9. 
68

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cx

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

Figure D-5. Planar fin axial force, δ = 10°, Mach 0.6. 

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cx

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

Figure D-6. Planar fin axial force, δ = 10°, Mach 0.9. 
69

Table D-1. Aerodynamic coefficients on planar fins, δ = 10°, Mach 0.6. 

α 0 2 4 6 8 10 

Axial Force 

Fin 1 (S) 0.0105 0.0086 0.0053 0.0053 0.0089 0.0103 

Fin 2 (W) 0.0105 0.0107 0.0107 0.0107 0.0109 0.0110 

Fin 3 (P) 0.0105 0.0095 0.0073 0.0041 0.0059 0.0091 

Fin 4 (L) 0.0105 0.0093 0.0083 0.0074 0.0087 0.0086 

Total fin 0.0420 0.0381 0.0316 0.0275 0.0344 0.0391 

Side Force 

Fin 1 (S) 0.0009 0.0019 0.0045 0.0076 0.0101 0.0121 

Fin 2 (W) 0.0192 0.0059 0.0099 0.0162 0.0200 0.0188 

Fin 3 (P) −0.0009 −0.0020 −0.0036 −0.0061 −0.0087 −0.0111 

Fin 4 (L) −0.0192 −0.0409 −0.0675 −0.0675 −0.0308 0.0524 

Total fin 0.0000 −0.0351 −0.0568 −0.0498 −0.0094 0.0721 

Normal Force 

Fin 1 (S) 0.0192 0.1049 0.1976 0.2799 0.3181 0.3463 

Fin 2 (W) −0.0009 −0.0006 −0.0003 0.0001 0.0003 0.0006 

Fin 3 (P) −0.0192 0.0555 0.1407 0.2328 0.3010 0.3355 

Fin 4 (L) 0.0009 0.0012 0.0017 0.0029 0.0034 0.0024 

Total fin 0.0000 0.1610 0.3397 0.5156 0.6228 0.6849 

Notes: S = starboard; W = windward; P = port; L = leeward. 

Table D-2. Aerodynamic coefficients on planar fins, δ = 10°, Mach 0.9. 

α 0 2 4 6 8 10 

Axial Force 

Fin 1 (S) 0.0126 0.0107 0.0066 0.0125 0.0140 0.0143 

Fin 2 (W) 0.0126 0.0128 0.0128 0.0131 0.0132 0.0129 

Fin 3 (P) 0.0126 0.0110 0.0096 0.0058 0.0123 0.0139 

Fin 4 (L) 0.0126 0.0106 0.0089 0.0080 0.0111 0.0123 

Total fin 0.0502 0.0451 0.0379 0.0394 0.0506 0.0533 

Side Force 

Fin 1 (S) 0.0018 0.0031 0.0070 0.0089 0.0106 0.0118 

Fin 2 (W) 0.0260 0.0048 0.0079 0.0189 0.0172 0.0161 

Fin 3 (P) −0.0018 −0.0031 −0.0055 −0.0089 −0.0102 −0.0112 

Fin 4 (L) −0.0260 −0.0439 −0.0672 −0.0810 −0.0215 0.0917 

Total fin 0.0000 −0.0392 −0.0578 −0.0621 −0.0039 0.1084 

Normal Force 

Fin 1 (S) 0.0260 0.1208 0.2412 0.2707 0.3286 0.3812 

Fin 2 (W) −0.0018 −0.0014 −0.0009 −0.0005 −0.0002 0.0003 

Fin 3 (P) −0.0260 0.0608 0.1662 0.2835 0.3055 0.3601 

Fin 4 (L) 0.0018 0.0021 0.0031 0.0045 0.0052 0.0036 

Total fin 0.0000 0.1823 0.4095 0.5582 0.6391 0.7453 

Notes: S = starboard; W = windward; P = port; L = leeward. 
70

INTENTIONALLY LEFT BLANK. 
71

Appendix E. Force Coefficients on Grid Fins 
72

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure E-1. Grid fin normal force, δ = 10°, Mach 0.6. 

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cz

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure E-2. Grid fin normal force, δ = 10°, Mach 0.9. 
73

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure E-3. Grid fin side force, δ = 10°, Mach 0.6. 

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cy

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

total fin

Figure E-4. Grid fin side force, δ = 10°, Mach 0.9. 
74

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

0.13

0.14

0.15

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cx

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

Figure E-5. Grid fin axial force, δ = 10°, Mach 0.6. 

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

0.13

0.14

0.15

0 1 2 3 4 5 6 7 8 9 10 11 12

α

Cx

fin1 (S)

fin2 (W)

fin3 (P)

fin4 (L)

Figure E-6. Grid fin axial force, δ = 10°, Mach 0.9. 
75

Table E-1. Aerodynamic coefficients on grid fins, δ = 10°, Mach 0.6. 

α 0 2 4 6 8 10 

Axial Force 

Fin 1 (S) 0.0782 0.0807 0.0817 0.0833 0.0850 0.0869 

Fin 2 (W) 0.0782 0.0794 0.0807 0.0805 0.0807 0.0804 

Fin 3 (P) 0.0782 0.0728 0.0760 0.0803 0.0827 0.0854 

Fin 4 (L) 0.0782 0.0720 0.0659 0.0655 0.0686 0.0717 

Total fin 0.3127 0.3048 0.3043 0.3097 0.3171 0.3245 

Side Force

Fin 1 (S) 0.0314 0.0387 0.0424 0.0511 0.0619 0.0754 

Fin 2 (W) 0.0095 0.0070 0.0078 0.0103 0.0110 0.0097 

Fin 3 (P) −0.0314 −0.0481 −0.0520 −0.0586 −0.0642 −0.0722 

Fin 4 (L) −0.0095 −0.0330 −0.0619 −0.0724 −0.0449 0.0262 

Total fin 0.0000 −0.0354 −0.0637 −0.0696 −0.0362 0.0392 

Normal Force

Fin 1 (S) 0.0095 0.0755 0.1513 0.2183 0.2703 0.3099 

Fin 2 (W) −0.0314 −0.0169 0.0017 0.0225 0.0424 0.0606 

Fin 3 (P) −0.0095 0.0467 0.1096 0.1754 0.2363 0.2888 

Fin 4 (L) 0.0314 0.0422 0.0532 0.0719 0.0863 0.0714 

Total fin 0.0000 0.1475 0.3158 0.4881 0.6353 0.7307 

Notes: S = starboard; W = windward; P = port; L = leeward. 

Table E-2. Aerodynamic coefficients on grid fins, δ = 10°, Mach 0.9. 

α 0 2 4 6 8 10 

Axial Force 

Fin 1 (S) 0.1144 0.1177 0.1190 0.1210 0.1221 0.1222 

Fin 2 (W) 0.1144 0.1164 0.1174 0.1167 0.1151 0.1128 

Fin 3 (P) 0.1144 0.1104 0.1122 0.1181 0.1209 0.1213 

Fin 4 (L) 0.1144 0.1067 0.0969 0.0992 0.1039 0.1063 

Total fin 0.4575 0.4512 0.4455 0.4549 0.4620 0.4627 

Side Force 

Fin 1 (S) 0.0530 0.0637 0.0662 0.0726 0.0791 0.0855 

Fin 2 (W) 0.0064 0.0070 0.0080 0.0092 0.0088 0.0067 

Fin 3 (P) −0.0530 −0.0688 −0.0708 −0.0782 −0.0819 −0.0833 

Fin 4 (L) −0.0063 −0.0402 −0.0636 −0.0719 −0.0410 0.0621 

Total fin 0.0000 −0.0382 −0.0602 −0.0682 −0.0350 0.0710 

Normal Force 

Fin 1 (S) 0.0064 0.0729 0.1424 0.2061 0.2595 0.3032 

Fin 2 (W) −0.0530 −0.0453 −0.0297 −0.0087 0.0159 0.0382 

Fin 3 (P) −0.0063 0.0306 0.1058 0.1627 0.2214 0.2789 

Fin 4 (L) 0.0530 0.0578 0.0671 0.0883 0.1060 0.1015 

Total fin 0.0001 0.1160 0.2855 0.4483 0.6029 0.7218 

Notes: S = starboard; W = windward; P = port; L = leeward. 
76

INTENTIONALLY LEFT BLANK. 
77

Appendix F. Components of Aerodynamic Coefficients 
78

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 1 2 3 4 5 6 7 8 9 10 11

α

Cz

body

total canard

total fin

Total

Figure F-1. Components of normal force for the planar fin case, δ = 10°, Mach 0.6. 

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 1 2 3 4 5 6 7 8 9 10 11

α

Cz

body

total canard

total fin

Total

Figure F-2. Components of normal force for the planar fin case, δ = 10°, Mach 0.9. 
79

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 1 2 3 4 5 6 7 8 9 10 11

α

Cz

body

total canard

total fin

Total

Figure F-3. Components of normal force for the grid fin case, δ = 10°, Mach 0.6. 

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 1 2 3 4 5 6 7 8 9 10 11

α

Cz

body

total canard

total fin

Total

Figure F-4. Components of normal force for the grid fin case, δ = 10°, Mach 0.9. 
80

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cy

body

total canard

total fin

Total

Figure F-5. Components of side force for the planar fin case, δ = 10°, Mach 0.6. 

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cy

body

total canard

total fin

Total

Figure F-6. Components of side force for the planar fin case, δ = 10°, Mach 0.9. 
81

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cy

body

total canard

total fin

Total

Figure F-7. Components of side force for the grid fin case, δ = 10°, Mach 0.6. 

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cy

body

total canard

total fin

Total

Figure F-8. Components of side force for the grid fin case, δ = 10°, Mach 0.9. 
82

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cx

body

total canard

total fin

Total

Figure F-9. Components of axial force for the planar fin case, δ = 10°, Mach 0.6. 

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cx

body

total canard

total fin

Total

Figure F-10. Components of axial force for the planar fin case, δ = 10°, Mach 0.9. 
83

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cx

body

total canard

total fin

Total

Figure F-11. Components of axial force for the grid fin case, δ = 10°, Mach 0.6. 

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10 11

α

Cx

body

total canard

total fin

Total

Figure F-12. Components of axial force for the grid fin case, δ = 10°, Mach 0.9. 
84

Table F-1. Components of aerodynamic coefficients, planar fin case, δ = 10°, Mach 0.6. 

α 0 2 4 6 8 10 

Axial Force 

Body 0.3088 0.3075 0.3138 0.3242 0.3104 0.3326 

Total canard 0.0326 0.0332 0.0334 0.0326 0.0305 0.0276 

Total fin 0.0420 0.0381 0.0316 0.0275 0.0344 0.0391 

Total 0.3835 0.3788 0.3788 0.3843 0.3753 0.3993 

Side Force 

Body 0.0000 0.1152 0.2410 0.4078 0.5453 0.6173 

Total canard 0.0000 0.0014 0.0032 0.0056 0.0085 0.0120 

Total fin 0.0000 −0.0351 −0.0568 −0.0498 −0.0094 0.0721 

Total 0.0000 0.0814 0.1874 0.3636 0.5445 0.7014 

Normal Force 

Body 0.0012 0.1788 0.3847 0.5940 0.7606 0.9532 

Total canard 0.0000 0.0263 0.0509 0.0711 0.0914 0.1106 

Total fin 0.0000 0.1610 0.3397 0.5156 0.6228 0.6829 

Total 0.0012 0.3662 0.7752 1.1807 1.4748 1.7487 

Table F-2. Components of aerodynamic coefficients, planar fin case, δ = 10°, Mach 0.9. 

α 0 2 4 6 8 10 

Axial Force 

Body 0.3018 0.3067 0.3274 0.3337 0.3498 0.3228 

Total canard 0.0419 0.0429 0.0430 0.0436 0.0432 0.0414 

Total fin 0.0502 0.0451 0.0379 0.0394 0.0506 0.0533 

Total 0.3939 0.3947 0.4084 0.4167 0.4436 0.4176 

Side Force 

Body 0.0000 0.1232 0.2418 0.4122 0.5555 0.6255 

Total canard 0.0000 −0.0027 −0.0004 0.0054 0.0130 0.0229 

Total fin 0.0000 −0.0392 −0.0578 −0.0621 −0.0039 0.1084 

Total 0.0000 0.0813 0.1836 0.3555 0.5647 0.7568 

Normal Force 

Body 0.0000 0.1818 0.4166 0.6256 0.8091 1.0403 

Total canard 0.0000 0.0397 0.0573 0.0836 0.1067 0.1303 

Total fin 0.0000 0.1823 0.4095 0.5582 0.6391 0.7453 

Total 0.0000 0.4038 0.8835 1.2674 1.5549 1.9158 
85

Table F-3. Components of aerodynamic coefficients, grid fin case, δ = 10°, Mach 0.6. 

α 0 2 4 6 8 10 

Axial Force 

Body 0.2931 0.3011 0.3078 0.3336 0.3496 0.3336 

Total Canard 0.0327 0.0332 0.0334 0.0326 0.0305 0.0277 

Total Fin 0.3127 0.3048 0.3043 0.3097 0.3171 0.3245 

Total 0.6385 0.6391 0.6454 0.6758 0.6971 0.6857 

Side Force 

Body 0.0000 0.1240 0.2626 0.4082 0.5274 0.5874 

Total Canard 0.0000 0.0014 0.0032 0.0057 0.0086 0.0119 

Total Fin 0.0000 −0.0354 −0.0637 −0.0696 −0.0362 0.0392 

Total 0.0000 0.0900 0.2021 0.3442 0.4998 0.6385 

Normal Force 

Body 0.0000 0.1528 0.3086 0.4802 0.6675 0.8325 

Total Canard 0.0000 0.0263 0.0509 0.0712 0.0915 0.1104 

Total Fin 0.0000 0.1475 0.3158 0.4881 0.6353 0.7307 

Total 0.0000 0.3266 0.6753 1.0396 1.3943 1.6735 

Table F-4. Components of aerodynamic coefficients, grid fin case, δ = 10°, Mach 0.9. 

α 0 2 4 6 8 10 

Axial Force 

Body 0.2826 0.3033 0.3026 0.3442 0.3422 0.3396 

Total canard 0.0417 0.0425 0.0428 0.0432 0.0429 0.0411 

Total fin 0.4575 0.4512 0.4455 0.4549 0.4620 0.4627 

Total 0.7817 0.7970 0.7909 0.8423 0.8471 0.8434 

Side Force 

Body 0.0000 0.1220 0.2680 0.4078 0.5327 0.5557 

Total canard 0.0000 −0.0032 −0.0004 0.0054 0.0132 0.0225 

Total fin 0.0000 −0.0382 −0.0602 −0.0682 −0.0350 0.0710 

Total 0.0000 0.0805 0.2073 0.3450 0.5109 0.6492 

Normal Force 

Body 0.0000 0.1627 0.3263 0.4968 0.6836 0.8639 

Total canard 0.0000 0.0409 0.0577 0.0848 0.1073 0.1315 

Total fin 0.0001 0.1160 0.2855 0.4483 0.6029 0.7218 

Total 0.0001 0.3197 0.6695 1.0299 1.3938 1.7172 
86

List of Abbreviations and Symbols 

A cell face area, m2

cal. caliber (1 caliber = D) 

Cl rolling moment coefficient 

Cm pitching moment coefficient 

Cn yawing moment coefficient 

CNFα zero angle-of-attack fin normal force slope 

Cp pressure coefficient 

Cx axial force coefficient 

Cy side force coefficient 

Cz normal force coefficient 

D missile base diameter, m 

E total energy, J 

F inviscid flux vector 

G viscous flux vector 

H vector of source terms 

i, j, k Cartesian unit vectors 

M Mach number 

MRP moment reference point 

p pressure, N/m2 

q heat flux vector 

u, v, w velocity components in x, y, z directions, m/s 

V cell volume, m3 

v velocity vector (= ui + vj + wk) 

xcp location of center of pressure 
87

W vector of conservative variables 

x, y, z axial, horizontal, and vertical body axes 

α angle of attack, degree 

δ canard deflection angle, degree 

ν kinematic viscosity, m2/s

ρ density, kg/m3 

τ viscous stress tensor 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

88

 1 DEFENSE TECHNICAL 

 (PDF INFORMATION CENTER 

 Only) DTIC OCA 

 8725 JOHN J KINGMAN RD 

 STE 0944 

 FT BELVOIR VA 22060-6218 

 1 COMMANDING GENERAL 

 US ARMY MATERIEL CMD 

 AMCRDA TF 

 5001 EISENHOWER AVE 

 ALEXANDRIA VA 22333-0001 

 1 INST FOR ADVNCD TCHNLGY 

 THE UNIV OF TEXAS 

 AT AUSTIN 

 3925 W BRAKER LN STE 400 

 AUSTIN TX 78759-5316 

 1 US MILITARY ACADEMY 

 MATH SCI CTR EXCELLENCE 

 MADN MATH 

 THAYER HALL 

 WEST POINT NY 10996-1786 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL D 

 DR D SMITH 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CS IS R 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 3 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CI OK TL 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 3 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CS IS T 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

ABERDEEN PROVING GROUND

 1 DIR USARL 

 AMSRD ARL CI OK TP (BLDG 4600) 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

89

 3 USAF WRIGHT AERONAUTICAL 

 LABORATORIES 

 AFWAL FIMG 

 J SHANG 

 N E SCAGGS 

 WPAFB OH 45433-6553 

 1 COMMANDER 

 NSWC 

 CODE B40 

 W YANTA 

 DAHLGREN VA 22448-5100 

 1 COMMANDER 

 NSWC 

 CODE 420 

 A WARDLAW 

 INDIAN HEAD MD 20640-5035 

 4 DIRECTOR 

 NASA 

 LANGLEY RESEARCH CTR 

 TECH LIBRARY 

 D M BUSHNELL 

 M J HEMSCH 

 J SOUTH 

 LANGLEY STATION 

 HAMPTON VA 23665 

 2 DARPA 

 P KEMMEY 

 J RICHARDSON 

 3701 N FAIRFAX DR 

 ARLINGTON VA 22203-1714 

 7 DIRECTOR 

 NASA 

 AMES RESEARCH CTR 

 T 27B 1 L SCHIFF 

 T 27B 1 T HOLST 

 MS 237 2 D CHAUSSEE 

 MS 258 M RAI 

 MS 258 1 B MEAKIN 

 MS T27B 2 M AFTOSMIS 

 MS T27B 2 J MELTON 

 MOFFETT FIELD CA 94035 

 3 AIR FORCE ARMAMENT LAB 

 AFATL/FXA 

 S C KORN 

 B SIMPSON 

 D BELK 

 EGLIN AFB FL 32542-5434 

 4 COMMANDER 

 US ARMY TACOM ARDEC 

 AMSTA AR FSF T 

 BLDG 382 

 C NG 

 H HUDGINS 

 J GRAU 

 W KOENIG 

 PICATINNY ARSENAL NJ 

 07806-5000 

 1 DEFENSE INTELLIGENCE AGENCY 

 MISSILE AND SPACE INT CTR 

 MSA 1 

 A NICHOLSON 

 BLDG 4545 FOWLER RD 

 REDSTONE ARSENAL AL 35898-5500 

 2 AFRL/MNAV 

 G ABATE 

 M VALENTINO 

 101 W EGLIN BLVD STE 219 

 EGLIN AIR FORCE BASE FL 32542 

 1 COMMANDER 

 US ARMY TACOM 

 AMSTA AR CCH B 

 P VALENTI 

 BLDG 65 S 

 PICATINNY ARSENAL NJ 07806-5001 

 1 COMMANDER 

 US ARMY ARDEC 

 ATTN SFAE FAS SD 

 M DEVINE 

 PICATINNY ARSENAL NJ 07806-5001 

 1 AEROPREDICTION INC 

 F MOORE 

 9449 GROVER DR STE 201 

 KING GEORGE VA 22485 

 1 AEROJET ELECTRONICS PLANT 

 D W PILLASCH 

 B170 DEPT 5311 

 PO BOX 296 

 1100 W HOLLYVALE ST 

 AZUSA CA 91702 

 1 MASSACHUSETTS INSTITUTE OF 

 TECHNOLOGY 

 TECH LIBRARY 

 77 MASSACHUSETTS AVE 

 CAMBRIDGE MA 02139
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

90

 1 LOS ALAMOS NATIONAL LAB 

 B HOGAN 

 MS G770 

 LOS ALAMOS NM 87545 

 3 DIRECTOR 

 SANDIA NATIONAL LABS 

 DIV 1554 W OBERKAMPF 

 DIV 1554 F BLOTTNER 

 DIV 1636 W WOLFE 

 ALBUQUERQUE NM 87185 

 1 NAWC 

 D FINDLAY 

 MS 3 BLDG 2187 

 PATUXENT RIVER MD 20670 

 

 1 METACOMP TECHNOLOGIES INC 

 S R CHAKRAVARTHY 

 28632-B ROADSIDE DR 

 SUITE 255 

 AGOURA HILLS CA 91301 

 1 ADVANCED TECHNOLOGY CTR 

 ARVIN/CALSPAN 

 AERODYNAMICS RESEARCH DEPT 

 M S HOLDEN 

 PO BOX 400 

 BUFFALO NY 14225 

 1 UNIV OF ILLINOIS AT URBANA 

 CHAMPAIGN 

 DEPT OF MECHANICAL AND 

 INDUSTRIAL ENGRG 

 J C DUTTON 

 URBANA IL 61801 

 1 UNIVERSITY OF MARYLAND 

 DEPT OF AEROSPACE ENGRG 

 J D ANDERSON JR 

 COLLEGE PARK MD 20742 

 1 UNIVERSITY OF TEXAS 

 DEPT OF AEROSPACE ENGRG 

 MECHANICS 

 D S DOLLING 

 AUSTIN TX 78712-1055 

 

 1 COMMANDER 

 US ARMY TACOM ARDEC 

 BLDG 162S 

 AMCPM DS MO 

 P J BURKE 

 PICATINNY ARSENAL NJ 07806-5000 

 1 COMMANDER USAAMCOM 

 AMSAM RD SS 

 G LANDINGHAM 

 REDSTONE ARSENAL AL 35898-5252 

 10 COMMANDER USAAMCOM 

 AMSAM RD SS AT 

 R KRETZSCHMAR 

 L AUMAN 

 E KREEGER 

 C MIKKELSON 

 E VAUGHN (5 CPS) 

 W WASHINGTON 

 REDSTONE ARSENAL AL 35898-5252 

 4 FLUENT INC 

 G STUCKERT 

 T SCHEIDEGGER 

 D LYTLE 

 D LEE 

 10 CAVENDISH CT 

 CENTERRA RESOURCE PARK 

 LEBANON NH 03766-1442 

ABERDEEN PROVING GROUND 

 27 DIR USARL 

 AMSRD ARL WM 

 J SMITH 

 T ROSENBERGER 

 AMSRD ARL WM B 

 A HORST 

 W CIEPIELA 

 AMSRD ARL WM BA 

 D LYON 

 T BROWN 

 D HEPNER 

 AMSRD ARL WM BD 

 B FORCH 

 M NUSCA 

 AMSRD ARL WM BC 

 P PLOSTINS 

 J DESPIRITO (5 CPS) 

 B GUIDOS 

 K HEAVEY 

 J NEWILL 

 J SAHU 

 S SILTON 

 AMSRD ARL WM BC 

 P WEINACHT 

 AMSRD ARL WM BF 

 S WILKERSON 

 H EDGE 
NO. OF 

COPIES ORGANIZATION 

91

ABERDEEN PROVING GROUND (CONT’D) 

 AMSRD ARL SL BE 

 A MIKHAIL 

 AMSRD ARL CI 

 C NIETUBICZ 

 AMSRD ARL CI HC 

 R NOAK 

 R ANGELINI 
NO. OF 

COPIES ORGANIZATION

92

 1 DSTL BEDFORD 

 T BIRCH 

 BEDFORD TECHNOLOGY PARK 

 BEDFORD 

 MK44 2FQ 

 1 DSTL FORT HALSTEAD 

 J EDWARDS 

 SEVENOAKS KENT 

 TN14 7BP 

 3 DEFENCE RESEARCH AND 

 DEVELOPMENT CANADA 

 VALCARTIER 

 F LESAGE 

 E FOURNIER 

 A DUPUIS 

 2459 PIE XI BLVD NORTH 

 VAL BELAIR QC G3J1X5 

 CANADA 

 1 INSTITUTE SAINT LOUIS 

 C BERNER 

 PO BOX 34 

 68301 SAINT LOUIS CEDEX 

 FRANCE