Computational Fluid Dynamic (CFD) 

Analysis of a Generic Missile 

With Grid Fins 

by James DeSpirito, Harris L. Edge, Paul Weinacht, 

Jubaraj Sahu, and Surya Dinavahi 

ARL,-‘R-23 18 

L 

. 

September 2000 

Approved for public release; distribution is unlimited. 
The findings in this report are not to be construed as an official 

Department of the Army position unless so designated by other 

authorized documents. 

Citation of manufacturer’s or tradenames does not constitute an 

official endorsement or approval of the use thereof. 

Destroy this report when it is no longer needed. Do not return 

it to the originator. 

I 

, 
Abstract 

ii 

This report presents the results of a study demonstrating an approach for using viscous 

computational fluid dynamic simulations to calculate the flow field and aerodynamic coefficients 

for a missile with grid fins. A grid fm is an unconventional lifting and control surface that 

consists of an outer frame supporting an inner grid of intersecting planar surfaces of small chord. 

The calculations were made at a Mach number of 2.5 and several angles of attack for a missile 

without fins, with planar fins, and with grid fins. The results were validated by comparing the 

computed aerodynamic coefficients for the missile and individual grid fins against wind tunnel 

measurement data. Very good agreement with the measured data was observed for all 

configurations investigated. For the grid fm case, the aerodynamic coefficients were within 2.8- 

6.5% of the wind tunnel data. The normal force coefficients on the individual grid fins were 

within 11% of the test data. The simulations were also successful in calculating the flow 

structure around the fin in the separated-flow region at the higher angles of attack. This was 

evident in the successful calculation of the nonlinear behavior for that fin, which showed 

negative normal force at the higher angles of attack. The effective angle of attack is negative on 

either part of or all of the top grid fm for the higher angles of attack. 
L 

Acknowledgments 

The authors would like to thank Graham Simpson and Anthony Sadler of the Defence 

Evaluation and Research Agency, United Kingdom, for providing the comprehensive database of 

wind tunnel test data. Thanks are also due to support engineers at Fluent, Inc., for insight on 

approaches for mesh generation of the grid fm configuration. 

This work was supported in part by a grant of high performance computing time from the 

Department of Defense High Performance Computing Center at Aberdeen Proving Ground, 

Maryland. 

. . . 

111 
Y 

iv 
Table of Contents 

. . . 

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

2. Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

3. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

3.1 Aerodynamic Coefficients ................................................................................... 11 

3.1.1 Body Alone Case (BIA) ................................................................................. 11 

3.1.2 Planar Fin Case (BlAC2R) ........................................................................... 12 

3.1.3 Grid Fin Case (BIAL2R) ............................................................................... 15 

3.1.4 Forces on Fins ............................................................................................... 18 

3.2 Grid Fin Flow Field ............................................................................................. 20 

4. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

Distribustion List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 

Report Documentation Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 

1 

3 

11 

24 

V 
J 

d 

vi 
List of Figures 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

Figure 

1. Basic Missile Shape With No Fins (BlA, Top), Planar Fins (BlAC2R, Middle), 

and Grid Fins (B lAL2R, Bottom) ............................................................................. 

Unstructured Mesh for Basic Missile Shape Without Fins ....................................... 

Unstructured Mesh for Basic Missile Shape With Planar Fins ................................. 

Unstructured Mesh in Tail Region of Planar Fin Case .............................................. 

Geometry for Grid Fin Case (B lAL2R) .................................................................... 

Unstructured Mesh for Basic Missile Shape With Grid Fins .................................... 

Unstructured Mesh in Tail Region of Grid Fin Case ................................................. 

Surface Mesh on Grid Fin Cells ................................................................................ 

Axes and Sign Convention for Force and Moment Coefficients ............................... 

Pitching Moment Coefficient vs. a for B 1 A and B 1 AC2R Cases ............................ 

Normal Force Coefficient vs. a for B 1A and B lAC2R Cases .................................. 

Axial Force Coefficient vs. a for BlA and B lAC2R Cases ..................................... 

Pitching Moment Coefficient vs. a for B lAL2R Case ............................................. 

Normal Force Coefficient vs. a for B lAL2R Case ................................................... 

Axial Force Coefficient vs. a for B 1 AL2R Case ...................................................... 

Normal Force Coefficient on Individual Grid Fins vs. a .......................................... 

Mach Contours on Symmetry Plane for Grid Fin Case, a = 10” .............................. 

Mach Contours on Symmetry Plane for Grid Fin Case, a = 20” .............................. 

Pressure Coefficient Contours on Symmetry Plane Through Bottom Fin (Fin 2) .... 

vii 

4 

6 

6 

7 

8 

8 

9 

9 

12 

13 

13 

14 

16 

16 

17 

19 

20 

21 

22 
Figure 

20. 

21. 

Pressure Coefficient Contours on Symmetry Plane Through Top Fin (Fin 4) . . . . . . . . . . 

& 

22 

23 

22. 

23 

23. 

Mach Number Contours on Symmetry Plane Through Bottom Fin (Fin 2) 

ata= 12” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Mach Number Contours on Symmetry Plane Through Top Fin (Fin 4) 

ata= 12” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Mach Number Contours on Symmetry Plane Through Bottom Fin (Fin 2) 

at a = 20” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Mach Number Contours on Symmetry Plane Through Top Fin (Fin 4) 

at a = 20” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Turbulent Viscosity Contours on Symmetry Plane of Fin Region at a = lo”........... 

24 

24. 

25 

25. 25 

26. Turbulent Viscosity Contours on Symmetry Plane of Fin Region at a = 20”........... 26 

4 

5 

. . . 

Vlll 
c 

List of Tables 

Table m 

1. Aerodynamic Coefficients vs. a (Mach No. 2.5, Reynolds No. 1.2 x lo6 . . . . . . . . . . . . . . . . 14 

2. Axial Force Coefficient on Fins (B lAC2R and B lAL2R Cases) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

Y 

ix 
I~E~I~NALLY Lwr BLANK. 

X 
1. Introduction 

0 

: 

The U.S. Army Research Laboratory (ARL) is using computational fluid dynamics (CID) to 

investigate the aerodynamic characteristics and flow field structure of grid fins, also known as 

lattice controls. 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. Grid fins 

have been employed on some Soviet missiles, but have not been included on any missile systems 

in the West (Simpson and Sadler 1998). Interest in grid fins is primarily geared toward their 

potential use on highly maneuverable munitions due to their advantages over conventional planar 

controls at high angles of attack and high Mach number. 

The aerodynamics of grid fins have been investigated since 1985 by the U.S Army Aviation 

and Missile Research and Development Center (AMRDEC), Huntsville, Alabama (Washington 

and Miller 1998, Miller and Washington 1994, and Washington and Miller 1993). These 

investigations indicated that grid fins have some advantages over conventional, planar fins. One 

advantage is the ability to maintain lift at higher angles of attack since grid fins do not have the 

same stall characteristics of planar fins. Another is the very small hinge moment, which can 

reduce the size of control actuator systems. Since curvature of the grid fins had little effect on 

their performance, folding down the fins onto the missile body is a storage design advantage. 

The main disadvantage was higher drag than that of planar fins, although techniques for 

minimizing drag by altering the grid fm frame cross-section shape were demonstrated (Miller 

and Washington 1994). These studies also showed that grid fins experience a loss in control 

effectiveness in the transonic regime due to flow choking in the individual cells. 

The Defence Evaluation and Research Agency (DERA), United Kingdom, has performed 

wind tunnel tests on grid fms and compared their aerodynamic characteristics to conventional 

planar fins (Simpson and Sadler 1998, and Simpson 1997). These studies confirmed some of the 

previous results and showed improved yaw stability due to the ability of the grid fins to generate 

1 
side force. The results also showed that the vertical fins contribute about 30% of the static 

longitudinal stability when oriented in the cruciform (+) configuration. 

Aeroballistic range flight tests have recently been conducted at the U.S. Air Force Research 

Laboratory Aeroballistic Research Facility (ARF), Eglin Air Force Base, Florida (Abate et 

al. 2000, and Abate et al. 1999). These were the first reported subscale free-flight tests of a 

missile configuration with grid fins. In addition to providing the raw data to derive the 

aerodynamic coefficients for the grid fin models, shadowgraphs of the models in flight at Mach 

numbers between 0.39 and 1.65 showed an interesting shock structure in and around the fins. 

There have also been theoretical and numerical methods used to estimate the lift 

characteristics of grid fins. Methods have been developed for the subsonic (vortex lattice 

theory), transonic (empirical methods), and supersonic (Evvard’s theory) regimes (Kretzschmar 

and Burkhalter 1998, Burkhalter 1996, Tong et al. 1996, Burkhalter and Frank 1996, and 

Burkhalter et al. 1995). Empirical extensions were used to gain data for missiles at large angles 

of attack. If body upwash terms are included, these methods provide adequate aerodynamic 

characteristics of grid fins for preliminary design purposes in the subsonic and supersonic 

regimes. 

The first reported CFD calculations made on grid fins were sponsored by the Defence 

Research Establishment, Valcartier (DREV), Canada (Chen et al. 2000, Khalid et al. 1998, Sun 

and Khalid 1998, and Lesage 1998). These studies only included inviscid (Euler) simulations, 

mainly due to a lack of high performance computing resources. The authors concluded that the 

inviscid CFD calculations provided a reliable means of studying the flows past missiles with grid 

fins. Comparisons with data from Washington and Miller (1993) showed reasonable agreement 

for the fm normal force. The Euler calculations for the grid fins compared better with the 

measured data than those for the planar fins. The authors attributed this to the shorter surfaces of 

the grid fins in the axial direction, compared to planar fins (Sun and Khalid 1998). The results of 

the viscous calculations from the present study help to explain this effect. 

2 
This report presents the results of the first viscous CFD calculations made involving a missile 

with grid fins. The results are validated by comparing them against wind tunnel data provided 

by DERA (Simpson 1997). Simulations of the missile body alone and with planar fins were also 

performed as part of the validation process. 

2. Numerical Approach 

The investigation involved using CFD to determine the flow field and aerodynamic 

coefficients on a 13-Cal., four-finned, generic missile shape (Figure 1). The missile has a 3-cal. 

tangent ogive, and the fm pitch axis is located 1.5 diameters before the aft end of the missile. 

The analysis proceeded in three steps. The first involved the missile without fins, case BlA 

(Figure 1, top); the second was with the missile with planar fins, case BlAC2R (Figure 1, 

middle); and the third was the missile with a set of grid fins, case BlAL2R (Figure 1, bottom). 

The configurations were labeled according to the designations used at DERA (Simpson 1997). 

The span and chord of the planar fm was 1 .O cal. The span of the grid fin was 1 .l cal. and the 

chord was 0.17 cal. All analyses were performed at a Mach number of 2.5 and at a minimum of 

three angles of attack (a): O”, lo”, and 20”. The simulations of the missile were made in the 

cruciform (+) configuration, and symmetry (x-z plane) was used so that only a half-plane was 

modeled. 

Steady-state calculations were made to compute the flow field for the three cases using the 

commercial CFD code, FLUENT Version 5.1. The implicit, compressible (coupled), 

unstructured-grid solver was used. The three-dimensional, time-dependent, 

Navier-Stokes (RANS) equations are solved using the finite volume method: 

Reynolds-averaged 

(1) 

where W is the vector of conservative variables, F is the inviscid flux vector, G is the viscous 

flux vector, H is the vector of source terms, V is the cell volume, and A is the surface area of the 

3 
d- ,: 

3D 

Figure 1. Basic Missile Shape With No Fins (BlA, Top), Planar Fins (BlAC2R, Middle), 

and Grid Fins (BlAL2R, Bottom). 

cell face. The Spalart-Allmaras, one-equation turbulence model (Spalart and Allmaras 1992) 

was used for these calculations. In FLUENT, the original version of the Spalart-Allmaras model 

is modified to allow the use of wall functions when the mesh resolution is not sufficiently fine to 

resolve the viscous-affected, near-wall region of the boundary layer (Fluent 1998). This 

capability was used in generating the mesh so that the computational requirements were reduced 

as much as possible. Second-order discretization was used for the flow variables and the 

turbulent viscosity equation. 

4 
The geometry and unstructured meshes for the three configurations were generated using the 

preprocessor, GAMBIT, supplied in the FLUENT software suite. In generating the meshes, 

boundary layer mesh spacing was used near the missile body and fm surfaces. Advantage was 

taken of the wall function option of the solver in FLUENT, and the first point off the surface 

(cell center) was about 0.002 cal. All mesh stretching was kept below a ratio of 1.2. Hexahedral 

cells were used, except for a small region located ahead of and partly over the first 0.1 Cal. of the 

nose of the missile (1% of the total length). The latter region was made up of tetrahedrons and 

pyramid transition elements. Figure 2 shows the mesh on the symmetry plane for the BlA case. 

The triangular surface mesh can be observed at the upstream end. The tetrahedral mesh was 

made to cover a small part of the missile nose only to allow a transition between the two types of 

meshes near the nose. Therefore, a true boundary layer-type mesh was not covering the first 1% 

of the missile body, but this had no observable effect on the results. The BlA mesh in Figure 2 

was meshed as six separate volumes-one in the freestream region, two over the first 10 cal. of 

the missile, and two over the last 3 cal. This methodology was used so that the mesh in the first 

two regions could be used regardless of the fm type. The meshes between volumes were 

conformal, or exactly matching at the boundary surface. The total number of cells in this case 

was about 670,400. 

The mesh on the symmetry plane for the planar fm case is shown in Figure 3. The mesh was 

modified slightly so that only one volume covered the first 10 cal. of the missile. The tail region, 

shown in Figure 4, consisted of seven volumes located ahead of, behind, and between the fins 

and off the fin tips. The total number of cells for this case was about 1.2 M, and again the mesh 

was totally conformal. 

The geometry for the grid fm case is shown in Figure 5. The solid modeling capabilities of 

GAMBIT simplified the generation of the grid fm geometry. The mesh on the symmetry plane is 

shown in Figure 6. The mesh for the freestream region and the first 10 Cal. of the missile body 

was similar to that for the planar fm case, Figure 3. Due to the complexity of the mesh in the 

region around the grid fins, and to minimize the total mesh size, a nonconformal mesh interface 

5 
l 

e 

Figure 2. Unstructured Mesh for Basic Missile Shape Without Fins. 

Figure 3. Unstructured Mesh for Basic Missile Shape With Planar Fins. 

6 
Figure 4. Unstructured Mesh in Tail Region of Planar Fin Case. 

was used at the interface at 10 cal. With this type of interface, which is used in FLUENT to 

handle sliding meshes, the flow variables are averaged to determine the value at the interface. 

The total number of cells in this mesh was about 3.2 M, with 2.5 M in the fm region (10-13 Cal.). 

The mesh in the fin region is shown in Figure 7. The fins, interior cells, and the thin region 

circumferentially between the fins were meshed first. The surface mesh was then projected 

axially forward and rearward with smoothing. A close-up of the surface mesh on the grid fin 

cells is shown in Figure 8. There were one or two cells across the front and rear of the grid fin 

web and frame surface. However, this dimension is the same order of magnitude as the first cell 

spacing off the surface (0.002 Cal.). 

The base flow was not simulated in these calculations, so the mesh stopped at the end of the 

missile. An outflow boundary condition was used downstream, a pressure inflow (free-stream 

7 
i”- 

Y 3.x 

Figure 5. Geometry for Grid Fin Case (BlAL2R). 

Figure 6. Unstructured Mesh for Basic Missile Shape With Grid Fins. 

8 
Figure 7. Unstructured Mesh in Tail Region of Grid Fin Case. 

. ..- -. 

.’ ,. 

._ ..-. 

‘. / 

Figure 8. Surface Mesh on Grid Fin Cells. 

9 
conditions) boundary condition was used upstream, and a far-field pressure (nonreflecting) 

boundary condition was used for the outer boundary. A nonslip wall boundary condition was 

used for all solid surfaces. The y’ value was about 40-60 along the missile body, which is 

optimal for wall functions. The maximum value was about 150 along the ogive, and between 

100 and 140 on the grid fin surfaces. The Reynolds number was 13.1 x lo6 rn-’ (4.0 x lo6 ftt’), 

or 1.2 x lo6 for this model. The free&ream temperature and pressure were 137 K and 8325 Pa, 

respectively. 

The no fm and planar fm cases were also run with a structured-mesh, implicit, finite 

difference code ZNSFLOW, which solved the thin-layer RANS equations. A complete 

description of this code can be found in the report by Edge et al. (2000). The aerodynamic 

coefficients calculated with this code using the Baldwin-Lomax turbulence model are presented 

in the next section along with those calculated with FLUENT. Since ZNSFLOW does not use 

wall functions in the turbulence models, the first grid point off the solid surfaces was about 

5 x 10” Cal., giving a y+ value of approximately one. 

The grid fm simulations were originally to be made using ZNSFLOW and a chimera overset 

grid (Steger et al. 1983). This is a viable approach, but the FLUENT code was chosen for the 

grid fm calculations because the geometry was already created using the GAMBIT preprocessor, 

and we believed that mesh generation would be faster using GAMBIT. To have a complete 

series using FLUENT, the no fin and planar fin cases were also run with FLUENT. 

A series of inviscid calculations were also made on the grid fin configuration. These 

simulations used the same geometry as the viscous calculations, except for the fin-body interface 

where a short circular stem was used for mesh generation purposes. A Cartesian flow solver, 

TIGER (Melton 1996, and Melton et al. 1995), and automatic Cartesian mesh generation 

program, CART3D (Aftosmis et al. 1998, and Aftosmis 1997), were used for the inviscid 

simulations. The Cartesian flow solver integrates the finite volume form of the Euler equations 

to steady state using a multistage Runge-Kutta time integration procedure (Melton 1996). The 

mesh generation and solution time were much less than the time required for the viscous 

10 
. 

calculations. The Cartesian grid had about 1.3 M grid points, and the turnaround time was less 

than a day. The calculations for the flow solver were carried out on a Sun ElOK machine on a 

single CPU. The flow solver took about 20 @cell/iteration and typically needed about 1500 

iterations for convergence. The aerodynamic coefficients calculated with the inviscid 

simulations are compared to the viscous calculations in the following section. 

The FLUENT simulations were performed in parallel using 4-6 processors on a Silicon 

Graphics (SGI) Onyx 2 with R12000 processors. The grid fm calculations took about 4.5 to 5 

mm/iteration using six processors. The aerodynamic coefficients converged in about 600 

iterations. It took about 1,500 iterations for the turbulent viscosity to converge, with the scaled 

residual reduced to about 10e6. 

3. Results and Discussion 

3.1 Aerodynamic Coeffkients. Using the FLUENT postprocessor, the viscous and 

pressure forces were integrated along the missile body and fin surfaces to calculate the 

aerodynamic coefficients. The normal force (C,), axial force (C,), and pitching moment (C,) 

coefficients are presented in missile-based coordinates, as shown in Figure 9. The pitching 

moment is expressed about the nose of the missile. The reference area is the cross-sectional area 

of the missile base, and the reference length is the diameter of the missile (94 mm). The 

calculated coefficients are compared to wind tunnel measurements performed at DERA 

(Simpson 1997). The DERA wind tunnel data did not include the forces on the base of the 

missile in the coefficient calculation. This value, which includes only the forebody and fin 

surfaces, is appropriate to compare to CFD calculations that do not include the base flow. The 

repeatability of the DERA data was reported to be within 1% at 2.5 Mach over the range of a 

investigated (Simpson 1997). 

3.1.1 Body Alone Case @IA). The aerodynamic coefficients calculated from the FLUENT 

and ZNSFLOW solutions are shown in Figures 10-12 and Table 1. Data from DERA wind 

11 
Figure 9. Axes and Sign Convention for Force and Moment Coefficients. 

tunnel measurements are also shown. The calculated pitching moment coefficient (Figure lo), 

normal force coefficient (Figure 1 l), and axial force coefficient (Figure 12) are shown at a = O”, 

lo”, 14”, and 20” for the FLUENT calculations and at O”, 4”, lo”, 14”, and 20” for the 

ZNSFLOW calculations. The calculated results show very good agreement with the measured 

aerodynamic coefficients. The maximum difference between the calculated and measured 

pitching moment coefficient was 3.6%, with the maximum occurring at a = 14” for the FLUENT 

calculations; it was 18%, with a maximum at a = 10” for the ZNSFLOW calculations. The 

maximum difference between the calculated and measured normal force coefficient was 2.0%, 

with the maximum occurring at a = 10” for the FLUENT calculations; it was lo%, with a 

maximum at a = 10” for the ZNSFLOW calculations. The maximum difference between the 

calculated and measured axial force coefficient was 4.2%, with the maximum occurring at 

a = 20” for the FLUENT calculations; it was 12%, with a maximum at a = 20” for the 

ZNSFLOW calculations. The difference calculation was not made at a = 0” for C, and C,, since 

these parameters are close to zero at this angle of attack. 

3.1.2 PZanar Fin Case (BIAC2R). The aerodynamic coefficients calculated from the 

FLUENT and ZNSFLOW solutions are also shown in Figures 10-12 and Table 1. The 

calculated coefficients are shown at a = O”, lo”, and 20” for the FLUENT calculations and at O”, 

4”, lo”, 14”, and 20” for the ZNSFLOW calculations. The calculated normal force and pitching 

12 
lo c 

. E -10 - 

‘B ;E: 

‘tij 6 -20 - 

E 

!i -30 - o2 

P 

.- - 

5 

-40 

.z! 

n -50 

1 

-60 ’ I I 1 I I I I 

-5 0 5 10 15 20 25 30 

c% 

+ Experiment (DERA) 

$ FLUENT N-S CFD 

X ZNSFLOW N-S CFD \ 

Figure 10. Pitching Moment Coefficient vs. a for BlA and BlAC2R Cases. 

6.5 - - Experiment (DERA) 

+ FLUENT N-S CFD 

5.5 - 

x ZNSFLOW N-S CFD 

3.5 - 

BlA 

2.5 - 

-0.5 ’ I I I I I I I 

-5 0 5 10 15 20 25 30 

Figure 11. Normal Force Coefficient vs. a for BlA and BlAC2R Cases. 

13 
0.50 1 

5 0.45 : 

s 

g 0.40 : 

$ 0 0.35 : 

8 $ 0.30 1 

- Experiment (DERA) 

+ FLUENT N-S CFD 

X ZNSFLOW N-S CFD 

/-----4 B,,,f 

?a 

d 

_ 

‘x % x 

. 3 2 0.25 

0.20 

BlA 

0.15 

0.10 1 I I I I I 1 

-5.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 

a 

Figure 12. Axial Force Coefficient vs. a for BlA and BlAC2R Cases. 

Table 1. Aerodynamic Coefficients vs. a (Mach No. 2.5, Reynolds No. 1.2 x 106) 

Case a G G G 

(“) Fluent Exp. % Fluent Exp. % Fluent Exp. % 

BlA 0 0.000 -0.031 - 0.000 0.005 - 0.1837 0.1895 -3.1 

10 -4.445 -4.448 +O.l 0.961 0.98 1 -2.0 0.1919 0.1928 -0.5 

14 -10.49 -10.13 -3.6 1.919 1.897 +1.2 0.1977 0.1936 +2.1 

20 -21.36 -21.07 -1.4 3.548 3.543 +O.l 0.2098 0.2014 +4.2 

‘BlAC2R 0 -0.004 0.041 - 0.000 -0.002 - 0.2922 0.3513 -16.8 

10 -17.27 -16.97 -1.8 2.060 2.065 -0.2 0.3095 0.3746 -17.4 

20 -43.61 -43.31 -0.7 5.454 5.465 -0.2 0.3349 0.3996 -16.2 

BlAL2R 0 0.021 0.633 - -0.002 0.081 - 0.5234 0.4677 +11.8 

10 -15.96 -17.01 +6.2 1.958 2.097 -6.6 0.5152 0.4859 +6.0 

12 -19.82 -21.07 +5.9 2.503 2.650 -5.5 0.5092 0.4807 +5.9 

20 -39.28 -40.43 +2.8 5.089 5.237 -2.8 0.4853 0.4711 +3.0 

14 
moment coefficients again show very good agreement with the measured aerodynamic 

coefficients. The maximum difference between the calculated and measured pitching moment 

coefficient was 1.8%, with the maximum occurring at a = 10” for the FLUENT calculations; it 

was 3.1%, with a maximum at a = 14” for the ZNSFLOW calculations. The maximum 

difference between the calculated and measured normal force coefficient was 0.2%, with the 

maximum occurring at a = 10” for the FLUENT calculations; it was 1.5%, with a maximum at 

a = 14” for the ZNSFLOW calculations. 

The calculated axial force coefficients were lower than the measured values, but they 

predicted the same increase with a that the experimental data showed. The FLUENT 

calculations were within 17% of the experimental data, and the ZNSFLOW calculations were 

within 17-22%. The FLUENT calculations were probably closer to the experimental data 

because the one-equation turbulence model was used instead of the algebraic model used in 

ZNSFLOW. In addition, the thin-layer option used in the ZNSFLOW calculations eliminated 

the cross-stream turbulence effects in the fin region. Interestingly, in this case the predicted 

values of axial force are lower than the measured values. The opposite was found for the body 

alone case and the grid fin case. Some further investigation is warranted, and performing CFD 

calculations of the planar fin model, including part of the wind tunnel sting, has been proposed. 

Then, the CFD predictions can be directly compared to the main balance force measured in the 

wind tunnel to eliminate the experimental base pressure correction as the cause for the 

discrepancy. The viscous component of the axial force was about 40% of the total (regardless of 

angle of attack) in the FLUENT and ZNSFLOW simulations. 

3.1.3 Grid Fin Case (BlAL2R). The aerodynamic coefficients calculated from the 

FIUENT and the inviscid CART3D/TIGER solutions are shown in Figures 13-15 and Table 1. 

The calculated coefficients are shown at a = O”, lo”, 12”, and 20” for both CFD cases. The 

calculated normal force and pitching moment coefficients again show very good agreement with 

the measured aerodynamic coefficients. The maximum difference between the calculated and 

measured pitching moment coefficient was 6.2%, with the maximum occurring at a = 10” for the 

15 
10 

0 

cJE 

. E -10 

.- 0 iF 

‘iii s -20 

E 

ii -30 

s 

F 

.- 

5 

-40 

.t: h 

-50 

- Experiment (DERA) 

+ FLUENT N-S CFD 

X CART3D/TIGER lnviscid CFD 

-60 L I I I I I I 

-5 0 5 10 15 20 25 30 

a 

Figure 13. Pitching Moment Coeffkient vs. a for BlAL2R Case. 

6.5 - 

2.5 - 

1.5 - 

0.5 - 

- Experiment (DERA) 

+ FLUENT N-S CFD 

x CART3D/TIGER lnviscid CFD 

-0.5 ’ 

-5 

I 

0 

I 

5 

I I I I 

10 15 20 25 30 

a 

Figure 14. Normal Force Coefficient vs. a for BlAL2R Case. 

16 
0.60 L 

0.55 7 

0.50 : 

: 0.45 : 

E 

:g 0.40 : 

z 

$ 0 0.35 : 

8 

5 0.30 y LL 

3 20.25 : 

0.20 : 

0.15 : 

(t-0 Experiment (DERA) 

$ FLUENT N-S CFD 

x CART3D/TIGER lnviscid CFD 

X 

0.10 t I I I I 

-5 0 5 10 15 

a 

20 25 30 

Figure 15. Axial Force Coefficient vs. a for BlAL2R Case. 

FLUENT calculations; it was 18%, with a maximum at a = 12” for the CART3D/TIGER 

calculations. The maximum difference between the calculated and measured normal force 

coefficient was 6.6%, with the maximum occurring at a = 10” for the FLUENT calculations; it 

was 13%, with a maximum at a = 12” for the CART3D/TIGER calculations. 

The FLUENT axial force calculations were a little higher than the measured values-a 3-6% 

difference for a > 0” and an 11% difference at a = 0”. The inviscid CART3D/TIGER 

calculations underpredicted the axial force due to the absence of skin friction. However, the 

difference (about 60%) was larger than expected. If the viscous component is removed from the 

axial force coefficient calculated in the FLUENT simulations, the value is between 0.34 and 

0.38; the axial force coefficients from the inviscid calculations are about one-half that (0.17- 

0.18). Some further investigation of this discrepancy is warranted, since other calculations of 

missiles in supersonic flow have resulted in more accurate results (Aftosmis 2000). Regardless 

of the angle of attack, the viscous component of the axial force was 29% of the total in the 

FLUENT simulations. 

17 
The inviscid calculations show that the total normal force and pitching moment data can be 

predicted to within 18% of the experimental data. If accurate axial force or drag information is 

not required, then the inviscid calculations may provide the information needed to check out 

multiple design approaches. Whether using CART3D/TIGER or an inviscid solution with 

FLUENT, the reduction in computing time is substantial. 

3.1.4 Forces on Fins. The normal force coefficients on the individual grid fins from the 

FLUENT calculations are shown in Figure 16, along with the measured wind tunnel data 

(Simpson 1997) as a function of a. The fins are numbered 1-4, with fin 1 in the 3 o’clock 

position and fm 4 in the 12 o’clock position if looking forward from the rear of the missile in the 

“+” configuration. In the simulations, fins 1 and 3 are the same due to symmetry. The normal 

force on the fins was predicted very well, with the largest difference at about 11%. As expected, 

the normal force is greatest on the horizontal fins. The windward fin (fin 2, bottom) also 

provides a substantial normal force. The leeward fm (fin 4, top) provides a similar normal force 

as fin 2 up to about a 4” angle of attack, but then goes nonlinear and negative at higher angles of 

attack. As described by Simpson (1997), the nonlinear shape of the normal force vs. a curve for 

the leeward fm is caused by its location in the separated flow region at higher angles of attack. 

As shown later in plots of the flow field, the local angle of attack varies over the leeward grid 

fin. Some parts of the fin will be at an effective negative angle of attack, while other parts are at 

an effective positive angle of attack. 

. 

The axial force coefficients on individual grid fins were about 2-3 times greater than those 

on the planar fins. The viscous component of the axial force on the grid fm was about 1.5 times 

greater than on the planar fin. These values are presented in Table 2. Although it was 

speculated that the smaller chord of the grid fm might impart fewer viscous effects than a planar 

fin (Sun and Khalid 1998), the summation of the viscous effects of all the lattice surfaces of the 

grid fm leads to higher viscous forces. The likely reason inviscid calculations of the 

aerodynamic coefficients on a missile with grid fins were more accurate than those for a missile 

with planar fins (Sun and Khalid 1998) is that the viscous component, as a percentage of the total 

18 
0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 

-0.1 

-0.2 

M Fin 1 (DERA) 

M Fin 2 (DERA) 

+-o Fin 3 (DERA) 

w Fin 4 (DERA) 

+ Fin l/3 (Fluent) 

A Fin 2 (Fluent) 

V Fin 4 (Fluent) 

Figure 16. Normal Force Coeffkient on Individual Grid Fins vs. a. 

Table 2. Axial Force Coefficient on Fins (BlAC2R and BlAL2R Cases) 

19 
t 

1.63e+OO 

t 1.36e+OO 

i l.O9e+OO 

8.15e-01 

I 543e-01 

2.72e-01 

L X 

O.OOe+OO 

Figure 17. Mach Contours on Symmetry Plane for Grid Fin Case, a = 10”. 

20 

axial force on the fin, is less for the grid fm case. In this study, the viscous component of the 

axial force on the individual grid fm was about 16% of the total vs. about 30% for the planar fin. 

With a larger component of the axial force in the grid fin case due to form and wave drag, an 

inviscid calculation may be expected to compare better with measured data than a planar fm 

case. 

3.2 Grid Fin Flow Field. Contour plots of Mach number on the symmetry plane are shown 

for the 10” and 20” angle of attack cases in Figures 17 and 18, respectively. A strong oblique 

shock is seen emanating from the windward side of the nose, with a weaker shock coming off of 

the leeward side. An expansion fan is seen coming off of the ogive-body interface. A large 

separated-flow region is seen coming off of the leeward side of the missile at a = 20”, enclosing 

the top grid fin. The separation zone is smaller at a = 10”. 

3.26e+OO 

2.99e+OO 

2.72e+OO 

2.44e+OO 

2.17e+OO 

1.90e+OO 
3.00e+oo 

2.67ecOO 

2.33e+OO 

2.00e+oo 

I i; 1.67e+OO 

1.330+00 

1 .ooe+oo 

6.67e-01 

3.33e-01 

45 X 

O.OOe+OO 

Figure 18. Mach Contours on Symmetry Plane for Grid Fin Case, a = 20”. 

There is a complex, three-dimensional shock structure emanating off of the grid fins. 

Figures 19 and 20 show pressure contours on the symmetry plane through the bottom and top 

fins, respectively, at a = 12”. The outline of the grid fin frame is shown in the figures, with the 

shocks emanating off of the intersection of several grid fm cells (see Figures 7 and 8). At this 

Mach number, the shock and expansion waves do not reflect off of the interior walls of the grid 

fm cells (Washington and Miller 1998). Instead, they first reflect off of one another inside the 

grid fin cell, setting up several more reflections in the wake of the fin. In Figures 21 and 22, 

contours of Mach number are shown on the bottom and top fins for a = 12”. For the bottom fm 

(Figure 21), the entire fin is at a positive relative angle of attack so that there is an expansion fan 

emanating from the lower part of each cell and a shock wave emanating off of the upper part of 

each cell. Part of the top fm (Figure 22) is at a negative relative angle of attack with respect to 

the incident flow; this is due to the vortices in the separation zone on the leeward side of the 

missile. The top cell is at a positive angle of attack, with the shock wave emanating from the top 

of the cell. The cell second from the top is nearly at a 0” angle of attack, with shock waves 

21 
R 2.10 

i 

1.91 

1.72 

1.53 

1.34 

1.15 

j 0.96 

; 0.77 

0.58 

! 0.20 

0.01 

-0.18 

L X 

-A : : 

‘\: “, 

1 

,, ‘..:. ‘,. ,, ; :, ;:.; ,‘, : 

i ;: 

-_ 

‘;,:,.;T 8, I , :.,’ .’ 

.::’ 

,.; _\ 

,.. t, .-.: ‘, \. ‘\. \ iJ I 

Figure 19. Pressure Coefficient Contours on Symmetry Plane Through Bottom Fin (Fin 2). 

a 2.10 

_' 

1.91 

1.72 

1.53 

I 0.96 

0.58 

Figure 20. Pressure Coefficient Contours on Symmetry Plane Through Top Fin (Fin 4). 

22 

. 
3.26e+00. --. . I,, ., 

I 

,. 

2,72e+OO 0 

2.45e+OO 

2.1&3+00 

1.91e+OO 

t 1.64e+OO 

! 1,38e+OO 

1.11e+OO 

2.99e-01 

3,00e-02 

b. 

.” 

“, I,,, 

“’ “, 

: _: 

Figure 21. Mach Contours on Symmetry Plane Through Bottom Fin (Fin 2) at a = 12’. 

3.268+00 

2,99e+OO 

2,72e+OO 

._ ,.- :. * ;- 

2,45e+OO 

2.18e+OO 

1.910+00 

1.64e+OO 

! i 1,38e+OO 

/ 

: l.lle+OO 

8,37e-01 \'~'y,~ 

; 568e-01 

2.99e-01 

3.00e-02 

Figure 22. Mach Contours on Symmetry Plane Through Top Fin (Fin 4) at a = 12”. 

23 
emanating from the top and bottom of the cell. The third and fourth cells from the top are at an 

effective negative angle of attack, with the shock wave emanating from the bottom of the cell. 

At a = 20” (Figures 23 and 24), the entire top fm was at a negative relative angle of attack. This 

flow structure is responsible for the negative normal forces observed on the top fm in wind 

tunnel tests and in these calculations (see Figure 16). 

3.98e+OO ‘.- 

3,32e+OO‘ 

2.990+00 

2.66e+OO 

2.33e+OO 

2.00e+oo 

1"..'1.67e+OO 

1 

1.34e+OO 

l.O2e+OO 

[ .6.85&01 

3.568-01 

2.57e-02 

. 

Figure 23. Mach Contours on Symmetry Plane Through Bottom Fin (Fin 2) at a = 20°. 

Contours of turbulent viscosity in the fm region are shown in Figures 25 and 26 for a = 10” 

and a = 20”, respectively. The turbulent wakes downstream from each grid fm vane are evident 

in the figures. Note that the scale in Figure 26 is larger than the one in Figure 25. At a = 20”, 

there is more interaction of the separated flow coming off of the missile body with the top grid 

fin. 

4. Summary and Conclusions 

Calculations of the viscous flow past a missile with grid fins were made using CFD 

simulations. The calculations were made at a Mach number of 2.5 and several angles of attack. 

24 
3.98e+OO 

3.651~00 "'^ ,. 

3.328+00 '> 

'i 

., 

, '2 

2.990+00 :..w '$C 

2.66e+OO 

2,33e+OO 

1; 2.00e+oo 

‘L, 

t 

'., 

-*i 

l 1.67e+OO Y.. -.._ 

. . . . 

/ 

~ 1.34e+OO 

1.02e+OO 

j 6.85e-01 

3.56e-01 

k 

2.57e-02 

‘. 

Figure 24. Mach Contours on Symmetry Plane Through Top Fin (Fin 4) at a = 20”. 

4.80e-03 

4.40e-03 

4.00e-03 

3,60e-03 

3.20e-03 

2.80e-03 

2.40e-03 

1: 2.00e-0: 

! 1,60e-0, 

1.20e-0, 

i 8.00e-0 

Figure 25. Turbulent Viscosity Contours on Symmetry Plane of Fin Region at a = 10”. 
1.37~~02' 

1.25e-02 

1.12e-02 h 

9.98e-03 

8.74e-03 k 

1: j 7.49e-03 

f+.j 6.24e-03 

[ i 

I 

4.990-03 

3.74e-03 

! 2.50e-03 

1.25e-03 

O.OOe+OO 

Figure 26. Turbulent Viscosity Contours on Symmetry Plane of Fin Region at a = 20”. 

The results were validated by comparing the computed aerodynamic coefficients for the missile 

and individual grid fins against detailed wind tunnel measurement data. The validation process 

also included calculating the flow field for the missile body alone and with conventional planar 

fins and comparing the calculations against wind tunnel data. 

Very good agreement with the measured data was observed for all configurations 

investigated. For the grid fin case, the aerodynamic coefficients were within 6.6% of the wind 

tunnel data. The normal force coefficients on the individual grid fins were within 11% of the test 

data. The simulations were also successful in calculating the flow structure around the fin in the 

separated-flow region at the higher angles of attack. This was evident in the successful 

calculation of the nonlinear behavior for that fin, which showed negative normal force at the 

higher angles of attack. The physical nature of this effect was illustrated by presenting contours 

of Mach number on the symmetry plane through the vertical grid fins. The effective angle of 

attack is negative on part of or all of the top grid fin for the higher angles of attack. 

26 
The viscous component of the axial force on the grid fin was about 1.5 times greater than that 

on the planar fin. This contrasts previous speculation that the smaller chord of the grid fin would 

result in less viscous force than a planar fin. The total axial force on grid fin was about 2-3 

times greater than that on the planar fin. As a percentage of the total force, the viscous 

component was about 16% for the grid fin and about 30% for the planar fin. 

Results for inviscid calculations of the grid fin case were also presented. The normal force 

and pitching moment coefficients were calculated to within 18% of the experimental data. If 

axial force or drag information is not required, then inviscid calculations may provide reasonable 

design data in less time than viscous calculations. 

The investigation detailed in this report demonstrated an approach for using viscous CFD 

simulations to calculate the flow field and aerodynamic coefficients for a missile with grid fins. 

Nevertheless, even when an unstructured mesh and wall functions were used to reduce the mesh 

size and computational requirements, substantial computing resources were required. An 

alternative approach would be to use the chimera overset grid technique to generate a structured 

mesh; however, the nature of the grid fm design makes the required resources large, regardless 

of the approach used. Inviscid calculations showed that the normal force and pitching moment 

coefficients could be calculated with reasonable accuracy. 

27 
. 

I~E~I~NALLY Lwr BLANK. 

28 
5. References 

Abate, G., R. P. Duckerschein, and W. Hathaway. “Subsonic/Transonic Free-Flight Tests of a 

Generic Missile with Grid Fins.” AIAA Paper 2000-0937, January 2000. 

Abate, G., R. P. Duckerschein, and G. Winchenbach. “Free-Flight Testing of Missiles with Grid 

Fins.” Proceedings of the 50th Aeroballistic Range Association Meeting, Pleasanton, CA, 

November 1999. 

Aftosmis, M. J., Personal communication. NASA Ames Research Center, Moffett Field, CA, 

January 2000. 

Aftosmis, M. J., M. J. Berger, and J. E. Melton. “Robust and Efficient Cartesian Mesh 

Generation for Component-Based Geometry.” AZAA Journal, vol. 36, no. 6, pp. 952-960, 

1998. 

Aftosmis, M. J. “Solution Adaptive Cartesian Grid Methods for Aerodynamic Flows with 

Complex Geometries.” Computational Fluid Dynamics VKI Lectures Series 1997-05, von 

Karman Institute for Fluid Dynamics, Belgium, 1997. 

Burkhalter, J. E. “Grid Fins for Missile Applications in Supersonic Flow.” AIAA Paper 96-0194, 

January 1996. 

Burkhalter, J. E., and H. M. Frank. “Grid Fin Aerodynamics for Missile Applications in 

Subsonic Flow.” J. Spacecraft and Rockets, vol. 33, no. 1, pp. 38-44, 1996. 

Burkhalter, J. E., R. J. Hartfield, and T. M. Leleux. “Nonlinear Aerodynamic Analysis of Grid 

Fin Configurations.” J. ofAircraft, vol. 32, no. 3, pp. 547-554, 1995. 

Chen, S., M. Khalid, H. Xu, and F. Lesage. “A Comprehensive CFD Investigation of Grid Fins 

as Efficient Control Surface Devices.” AIAA Paper 2000-0987, January 2000. 

Edge, H. L., J. Sahu, W. B. Sturek, D. M. Pressel, K. R. Heavey, P. Weinacht, C. K. Zoltani, 

C. J. Nietubicz, J. Clarke, M. Behr, and P. Collins. “Common High Performance Computing 

Software Support Initiative (CHSSI) Computational Fluid Dynamics (CFD)-6 Project Final 

Report: ARL Block-Structured Gridding Zonal Navier-Stokes Flow (ZNSFLOW) Solver 

Software.” U. S. Army Research Laboratory, ARL-TR-2084, Aberdeen Proving Ground, 

MD, February 2000. 

Fluent 5.0 Users Guide, Vol. 2. Lebanon, NH: Fluent, Inc., 1998. 

29 
Khalid, M., Y. Sun, and II. Xu. “Computation of Flows Past Grid Fin Missiles.” Proceedings of 

the NATO RTO-MP-5, Missile Aerodynamics, NATO Research and Technology 

Organization, November 1998. 

Kretzschmar, R. W., and J. E. Burkhalter. “Aerodynamic Prediction Methodology for Grid 

Fins.” Proceedings of the NATO RTO-MP-5, Missile Aerodynamics, NATO Research and 

Technology Organization, November 1998. 

Lesage, F. “Numerical Investigation of the Supersonic Flow Inside a Grid Fin Cell.” 

Proceedings of the 17*h International Symposium on Ballistics, vol. 1, American Defense 

Preparedness Association, Arlington, VA, pp. 209-216, 1998. 

Melton, J. E. “Automated Three-Dimensional Cartesian Grid Generation and Euler Flow 

Solutions for Arbitrary Geometries.” Ph.D. dissertation, University of California Davis, 

June 1996. 

Melton, J. E., M. J. Berger, M. J. Aftosmis, and M. D. Wong. “3-D Applications of a Cartesian 

Grid Euler Method.” AIAA Paper 950853, January 1995. 

Miller, M. S., and W. D. Washington. “An Experimental Investigation of Grid Fin Drag 

Reduction Techniques.” AIAA Paper 94-1914-CP, June 1994. 

Simpson, G. M., and A. J. Sadler. “Lattice Controls: A Comparison with Conventional, Planar 

Fins.” Proceedings of the NATO RTO-MP-5, Missile Aerodynamics, NATO Research and 

Technology Organization, November 1998. 

Simpson, G. “A Preliminary Analysis of the DERA Lattice Controls Database.” Defense 

Research Agency, DERA/AS/HWA/WP97 196/ 1 .O, Famborough, UK, July 1997. 

Spalart, P. R., and S. R. Allmaras. “A One-Equation Turbulence Model for Aerodynamic 

Flows.” AIAA Paper 92-0439, January 1992. 

Steger, J. L., F. C. Dougherty, and J. A. Benek. “A Chimera Grid Scheme.” Advances in Grid 

Generation, edited by K. N. Ghia and U. Ghia, American Society of Mechanical Engineers, 

ASME FED-5, New York, June 1983. 

Sun, Y., and M. Khalid. “A CFD Investigation of Grid Fin Missiles.” AIAA Paper 98-3571, 

July 1998. 

Tong, Z., 2. Lu, and X. Shen. “Calculation and Analysis of Grid Fin Configurations.” Advances 

in Astronautical Sciences, AAS 95-647, 1996. 

30 
Washington, W. D., and M. S. Miller. “Experimental Investigations of Grid Fin Aerodynamics: 

A Synopsis of Nine Wind Tunnel and Three Flight Tests. ” Proceedings of the NATO RTOMP-5, Missile Aerodynamics, NATO Research and Technology Organization, 

November 1998. 

Washington, W. D., and M. S. Miller. “Grid Fins - A New Concept for Missile Stability and 

Control.” AIAA Paper 93-0035, January 1993. 

31 
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I~ENTI~NALLY mm BLANK. 

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38 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

Public ra~wtlng burden ,orth,a eollactlon o,lnfo”-“.tlon Is estlmtisd ,o ,vqe 1 hour per r-“se, lncludlng the time lor ravlewlng In8truCtlonS, wrching Otistlng data SOUP==, 

gMwhtg l d maintalnlng the data nondad, and eompletlng and revlwtng the collection 01 Inlomwtlon. Send cornmanta regarding this burden estimate or any other aspset Of this 

collecllon of Inlorm,tlon, Including suggestIons lor reducing thl8 burden, to Washington “eadqu,“aa Sewlees, Dlrectonte lor InlormtilOn Opatilons and Reports. 1216 Jsttemon 

1 Fluid Dynamic (CFD) Analysis of a Generic Missile With Grid Fins 

PR: lL162618AH80 

pirito, Harris L. Edge, Paul Weinacht, Jubaraj Sahu, and 

Dinavahi* 

AT-IN: AMSRL-WM-BC 

Aberdeen Proving Ground, MD 2 1005-5066 ARL-TR-23 18 

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS 1O.SPONSORING/MONlTORlNG 

AGENCY REPORT NUMBER 

11. SUPPLEMENTARY NOTES 

*Mississippi State University 

12e. DISTRIBUTION/AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited. 

12b. DISTRIBUTION CODE 

13. ABSTRACT(hlaxlmum 200 words) 

This report presents the results of a study demonstrating an approach for using viscous computational fluid dynamic 

simulations to calculate the flow field and aerodynamic coefficients for a missile with grid fins. 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. The calculations were made at a Mach number of 2.5 and several angles of attack for a missile 

without fins, with planar fins, and with grid fins. The results were validated by comparing the computed aerodynamic 

coefficients for the missile and individual grid fins against wind tunnel measurement data. Very good agreement with 

the measured data was observed for all configurations investigated. For the grid fin case, the aerodynamic coefficients 

were within 2.8-6.5% of the wind tunnel data. The normal force coefficients on the individual grid fins were within 

11% of the test data. The simulations were also successful in calculating the flow structure around the fin in the 

separated-flow region at the higher angles of attack. This was evident in the successful calculation of the nonlinear 

behavior for that fin, which showed negative normal force at the higher angles of attack. The effective angle of attack is 

negative on either part of or all of the top grid fin for the higher angles of attack. 

14. SUBJECT TERMS 15. NUMBER OF PAGES 

grid fins, lattice controls, computational fluid dynamics, Navier-Stokes, missile 42 

aerodynamics 16. PRICE CODE 

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT 

OF REPORT OF THIS PAGE OF ABSTRACT 

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL 

b.IChl -?CAl-Ln-OLll--CCC-m CI”..A,..A C1.r.. on0 ,P^.. c) on\ 

39 “La,II”PI” r”llll La” \“cw. C-OJ, Prescribed by ANSI Std. 239-18 299-l 02 
hTENTIO&ULY LEFT BLANK. 

40 
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