Application of Computational Fluid Dynamics to 

a Monoplane Fixed-Wing Missile With 

Elliptic Cross Sections 

by Karen Heavey and Jubaraj Sahu 

ARL-TR-3549 July 2005 

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Army Research Laboratory 

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ARL-TR-3549 July 2005 

Application of Computational Fluid Dynamics to 

a Monoplane Fixed-Wing Missile With 

Elliptic Cross Sections 

Karen Heavey and Jubaraj Sahu 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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Application of Computational Fluid Dynamics to a Monoplane Fixed-Wing 

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14. ABSTRACT 

This report describes a computational study undertaken to investigate the performance of the CFD++ flow solver for prediction 

of nonlinear aerodynamics of a complex finned missile using structured hexahedral and unstructured tetrahedral grids. A 

monoplane fixed-wing missile with elliptic cross sections provided a geometrically complex model. Numerical solutions were 

obtained for this configuration at supersonic speed for various roll orientations, angles of attack, and jaw angles. Steady-state 

solutions were obtained using a three-dimensional Reynolds-Averaged Navier-Stokes solver with a two-equation turbulence 

model. Numerical results show the qualitative features of the flow fields at various cross-sectional and streamwise positions 

along the computational model of the missile. Aerodynamic coefficients were extracted from the computed solutions and 

found to match well with the available experimental data for these configurations. These numerical results show the 

effectiveness of using computational fluid dynamics techniques to produce an accurate prediction of the aerodynamics of 

geometrically complete configurations. 

15. SUBJECT TERMS

monoplane fixed-wing, elliptic, computational fluid dynamics, experimental data 

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Karen Heavey 

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iii

Contents 

List of Figures iv

List of Tables v

Acknowledgments vi

1. Introduction 1

2. Solution Technique 3

2.1 CFD++ Flow Solver........................................................................................................3 

2.2 Numerical Technique ......................................................................................................4 

3. Model Geometry and Numerical Grids 4

3.1 Projectile Model and Geometry ......................................................................................4 

3.2 Computational Mesh .......................................................................................................5 

4. Results and Discussion 6

4.1 Qualitative Results ..........................................................................................................7 

4.2 Quantitative Results ......................................................................................................14 

4.3 Comparison of Force and Moment Data .......................................................................17 

5. Summary and Conclusions 19

6. References 20

Appendix. Computational Data 23

Distribution List 27 
iv

List of Figures 

Figure 1. Computational model for the elliptic-section missile......................................................5 

Figure 2. Expanded view of the hexahedral mesh in the aft region of the elliptic missile.............5 

Figure 3. View of the tetrahedral mesh on the aft surface of the elliptic missile. ..........................6 

Figure 4. Axial locations for experimental data..............................................................................7

Figure 5. Circumferential locations of experimental data points....................................................7 

Figure 6. Surface pressure contours for α = 5°, φ = 0°...................................................................8 

Figure 7. Mach contours for α = 0°, φ = 0°. ...................................................................................8 

Figure 8. Pressure contours at x/L = 0.95, α = 15°, and φ = 0°......................................................9 

Figure 9. Circumferential pressure contours for φ = 0°, α = 5°....................................................10 

Figure 10. Circumferential pressure contours for φ = 0°, α = 15°................................................10 

Figure 11. Circumferential pressure contours for φ = 0°, α = 20°................................................11 

Figure 12. Circumferential pressure contours for φ = 0°, α = 20°, and β = 8.6°..........................11 

Figure 13. Circumferential pressure contours for φ = 45°, α = 5°................................................12 

Figure 14. Circumferential pressure contours for φ = 45°, α = 15°..............................................12 

Figure 15. Circumferential pressure contours for φ = 45°, α = 20°..............................................13 

Figure 16. Circumferential pressure contours for φ = 45°, α = 20°, and β = 8.6°........................13 

Figure 17. Circumferential pressure at x/L = 0.3, φ = 0°..............................................................14 

Figure 18. Circumferential pressure at x/L = 0.6, φ = 0°..............................................................15 

Figure 19. Circumferential pressure at x/L = 0.95, φ = 0°............................................................15 

Figure 20. Circumferential pressure plots for x/L = 0.3, φ = 45°. ................................................16 

Figure 21. Circumferential pressure at x/L = 0.6, φ = 45°............................................................16 

Figure 22. Circumferential pressure at x/L = 0.95, φ = 45°..........................................................17 
v

List of Tables 

Table 1. Normal force (CN) for 0° roll orientation........................................................................17 

Table 2. Force and moments, φ = 0°, α = 20°, β = 4.6°. ..............................................................18 

Table 3. Force and moments, φ = 0°, α = 20°, β = 8.6°. ..............................................................18 

Table 4. Normal force (CN) for 45° roll orientation......................................................................18 

Table 5. Force and moments, φ = 45°, α = 20°, β = 4.6°. ............................................................18 

Table 6. Force and moments, φ = 45°, α = 20°, β = 8.6°. ............................................................18 
vi

Acknowledgments 

This work was accomplished as part of the Technical Cooperation Program between the United 

States, Canada, the United Kingdom, and Australia on Weapons Technical Panel TP-2 Key 

Technology Area 2–19. The authors would like to thank Jerry Allen from the National 

Aeronautics and Space Administration (NASA) Langley Research Center and Victor Burnley 

from the U.S. Air Force Research Laboratory for providing the structured and unstructured grid 

for the NASA missile case, respectively. The authors also wish to thank Sukumar Chakravarthy 

from Metacomp Technologies for his expert technical advice. The computational support of the 

U.S. Army Research Laboratory Major Shared Resource Center is greatly appreciated.
1

1. Introduction 

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

the performance of new designs. Accurate determination of aerodynamics is critical to the lowcost development of new advanced guided projectiles, rockets, missiles, and smart munitions. 

Fins, canards, and jets can be used to provide control for maneuvering projectiles and missiles. 

The flow fields associated with these control mechanisms for the modern weapons are complex, 

involving three-dimensional (3-D) shock-boundary layer interactions and highly viscous 

dominated separated flow regions (1–5). Computational fluid dynamics (CFD) has emerged as a 

critical technology for the aerodynamic design and assessment of weapons. Improved computer 

technology and state-of-the-art numerical procedures enable solutions to complex, 3-D problems 

associated with projectile and missile aerodynamics. In general, these techniques produce 

accurate and reliable numerical results for simple projectile and missile configurations at small 

angles of attack. 

Modern weapons and the associated flow fields have increased in complexity as a result of 

changing military requirements. An increasing number of new weapon configurations are 

beyond the scope of traditional engineering prediction methods. Modern projectiles and missiles 

are expected to experience moderate to large angles of attack during flight. The flow field 

surrounding a slender missile at incidence features flow separations both from smooth bodies 

and sharp-edged lifting and control surfaces. Body and wing vortices (6, 7) can dominate the 

flow field and interact with one another resulting in a very complex flow field even for simple 

geometries. Earlier work (8) on modeling of high angle of attack flow fields for finned missile 

configurations was initiated as part of the Technical Cooperation Program (TTCP) effort with 

participants from Canada, the United Kingdom, Australia, and the United States. This study 

addressed the application of Navier-Stokes methods to the prediction of flows around finned 

missile configurations and assessed the capabilities of Navier-Stokes solvers for the prediction of 

missile flow field at moderate to high angles of attack. It promoted technical interchange 

between aerodynamics specialists in the areas of grid generation, algorithms, turbulence 

modeling, and flow field visualization. The results of this study were reported in a special 

session of the American Institute of Aeronautics and Astronautics (AIAA) Applied 

Aerodynamics Conference in August 2000 (8–10). The study has shown that all Navier-Stokes 

methods employed were capable of providing accurate predictions of the overall normal force 

and pitching moment characteristics, even at very high angles of attack. However, some 

significant discrepancies in the prediction of the axial force were evident. In general, the 

agreement between computed and measured flow fields was good. Higher order turbulence 

models such as the k-ε models did not provide better accuracy compared to simpler algebraic 
2

and one-equation models. The next phase of the research was to extend the scope of that study 

and to undertake a follow-on study to apply CFD technology to the prediction of aerodynamic 

flow fields for nonaxisymmetric missile configurations. 

Maneuvering nonaxisymmetric missile configurations can be used for increased lethality and are 

being explored for future weapon systems in various scenarios. Additionally, nonaxisymmetric 

projectiles have been shown to generate large amounts of lift compared to conventional 

projectiles with circular cross sections (11). The aerodynamics of such configurations at high 

speeds are quite complex due to strong cross-flow influences and flow separation. Of particular 

interest was the accurate determination of supersonic and hypersonic flow over elliptic 

projectiles at moderate angles of attack. The flow field for such projectiles with 

nonaxisymmetric cross sections is complex, especially in the presence of jets used to maneuver 

these projectiles. This work was also initiated as part of another TTCP effort with international 

participants, and was aimed at assessing the capabilities of the both Euler and Navier-Stokes 

solvers currently available to research scientists for supersonic flow over elliptic projectiles for 

both jet-off and jet-on conditions (12, 13). This research effort also focused on the wind-tunnel 

testing as well as free-flight testing of these projectiles. Various aspects of computational 

techniques, such as grid generation, algorithms, turbulence modeling, and flow field 

visualization, were addressed by the group. The results of this study were reported in a special 

session of the AIAA Atmospheric Flight Mechanics Conference in August 2001. CFD 

approaches that included both Euler and Navier-Stokes schemes, structured and unstructured 

techniques, and a variety of turbulence models were used in the numerical simulations (10–15). 

In general, very good agreement of the computed aerodynamic coefficients with the 

experimental data was achieved at all angles of attack investigated for the jet-off uncontrolled 

conditions. Accurate prediction of the jet-on cases was found to be a little more challenging than 

the jet-off cases (16).

The TTCP efforts have continued recently with a focus on the application of CFD to missile 

aerodynamics which takes it to the next level of difficulty, that of multiple sets of fins and 

nonaxisymmetric configurations. Again, the objective was to compare the results obtained from 

Navier-Stokes computations against experimental data, evaluate the accuracy of the predictive 

technologies, and identify priorities for future development. The research focused on the 

application of advanced CFD techniques for accurate numerical prediction of supersonic flow 

over two nonaxisymmetric missile configurations, one with a square cross section and the other 

with an elliptic cross section; both have multiple sets of complex fins. Numerical computations 

for these complex configurations have been performed at supersonic speeds, for two roll 

orientations, for a range of angles of attack. Several cases were also computed with yaw angles. 

As with earlier efforts, these results were presented in conjunction with the other TTCP 

participants in a special session at the AIAA Applied Aerodynamics Conference in August 2004 

(17).
3

The information presented in this U.S. Army Research Laboratory report focuses on the elliptic 

model only. Additional computational data that was not part of the TTCP effort are included in 

the report. A description of the computational techniques is presented, followed by a description 

of the applications of these techniques to the model at roll orientations of 0° and 45°. Computed 

results for these configurations are presented at Mach 2.5 at angles of attack ranging from 0° to 

20°. The computed data are compared with experimental data provided by the National 

Aeronautics and Space Administration (NASA) Langley Research Center (18).

2. Solution Technique 

2.1 CFD++ Flow Solver

A commercially available code, CFD++ (19, 20), is used for the numerical simulations. The 

basic numerical framework in the code contains unified-grid, unified-physics, and unifiedcomputing features. The user is referred to these references for details of the basic numerical 

framework. Here, only a brief synopsis of this framework and methodology is given. 

The 3-D, Reynolds-averaged Navier-Stokes (21) equations are solved using the following finite 

volume method: 

[ ] ∫ ∫ ∫ + − ⋅ =

V V

dV dA dV

t

W F G H

∂

∂ , (1) 

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

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

area of the cell face. 

The numerical framework of CFD++ is based on the following general elements: (1) unsteady 

compressible and incompressible Navier-Stokes equations with turbulence modeling (unifiedphysics), (2) unification of Cartesian, structured curvilinear, and unstructured grids, including 

hybrids (unified-grid), (3) unification of treatment of various cell shapes including hexahedral, 

tetrahedral, and triangular prism cells (3-D), quadrilateral and triangular cells (2-D), and linear 

elements (1-D) (unified-grid), (4) treatment of multiblock patched aligned (nodally connected), 

patched nonaligned and overset grids (unified-grid), (5) total Variation Diminishing 

discretization based on a new multidimensional interpolation framework, (6) Riemann solvers to 

provide proper signal propagation physics, including versions for preconditioned forms of the 

governing equations, (7) consistent and accurate discretization of viscous terms using the same 

multidimensional polynomial framework, (8) pointwise turbulence models that do not require 

knowledge of distance to walls, (9) versatile boundary condition implementation includes a rich 
4

variety of integrated boundary condition types for the various sets of equations, and (10) 

implementation on massively parallel computers based on the distributed-memory messagepassing model using native message-passing libraries or MPI, PVM, etc. (unified-computing). 

The code has brought together several ideas on convergence acceleration to yield a fast 

methodology for all flow regimes. The approach can be labeled as a “preconditioned-implicitrelaxation” scheme. It combines three basic ideas: implicit local time-stepping, relaxation, and 

preconditioning. Preconditioning the equations ideally equalizes the eigenvalues of the inviscid 

flux Jacobians and removes the stiffness arising from large discrepancies between the flow and 

sound velocities at low speeds. Use of an implicit scheme circumvents the stringent stability 

limits suffered by their explicit counterparts, and successive relaxation allows update of cells as 

information becomes available and thus aids convergence. 

2.2 Numerical Technique 

Second-order discretization was used for the flow variables and the turbulent viscosity equation. 

The turbulence closure was based on topology-parameter-free formulations. The realizable k-ε 

turbulence model (22) was used for the computation of turbulent flows for the missile 

configurations considered here. A pressure-temperature based inflow/outflow routine was 

utilized for the inflow, outflow, and farfield boundary conditions, while an adiabatic viscous wall 

condition was used on the missile surfaces – body, wings, and tailfins. A symmetry boundary 

condition was used along the x-axis in front of the missile for the hexahedral grid only. All 

calculations were performed at Mach number 2.5 using the experimental (wind tunnel) 

conditions: Re = 5.56 × 106

/m, T0 = 339 K, P0 = 81.36 kPa. This corresponds to a static 

temperature and pressure of 150.67 K and 4.76 kPa, respectively, for the CFD calculations. 

Computations were performed using the various platforms available at the Major Shared 

Resource Center, to include SGI 3800 and IBM SP3 processors. The resources used, such as 

memory and CPU time, varied based on the mesh size. Most of the runs were done utilizing 

16 or 24 processors. The next section describes the model geometries and the flow conditions 

for the elliptic cross-section NASA missile (23, 24). 

3. Model Geometry and Numerical Grids 

3.1 Projectile Model and Geometry 

The geometric model for the elliptic missile is a sharp-nosed monoplane winged missile with tail 

fins. The overall body length is 0.7112 m with a 3:1 elliptical cross section, having maximum 

body width and height of 0.1760 m and 0.0587 m, respectively. There are four tail fins on the aft 

end of the missile, aligned with the base of the missile. This model, shown in figure 1, has been 

used extensively in wind tunnel tests at the NASA Langley Research Center (18).
5

Figure 1. Computational model for the elliptic-section missile. 

3.2 Computational Mesh 

The focus of this study was to investigate the performance of the flow solver CFD++ using two 

different grid structures. The first grid was received as a structured mesh consisting of 

3.8 million hexahedral cells. Figure 2 shows an expanded view of the grid in the aft region of 

the missile. This structured hexahedral mesh was provided by NASA Langley (25). The second 

grid is an unstructured mesh consisting of 2.9-million tetrahedral cells. A view of the surface 

mesh on the tail fins is shown in figure 3. This unstructured tetrahedral mesh was provided by 

the U.S. Air Force Research Laboratory (26). 

Figure 2. Expanded view of the hexahedral mesh in the aft region of the elliptic missile. 

Since CFD++ requires a code-specific unstructured format for the input geometry, both 

computational meshes were converted prior to use with the flow solver. The conversion routine 

is included in the CFD++ graphical user interface (GUI) which was also used to set up the 

numerical input parameters.
6

Figure 3. View of the tetrahedral mesh on the aft surface of the elliptic missile. 

The original grids received had a roll orientation of 0°. A tool available within the CFD++ GUI 

was used to rotate each grid 45° on the x-axis. A series of computations was performed using 

each of these four grids. 

4. Results and Discussion 

Computations using viscous Navier-Stokes methods were performed to predict the flow field and 

aerodynamic coefficients for a monoplane fixed wing missile with elliptic cross section under 

wind tunnel test conditions, using the CFD++ flow solver at a supersonic Mach number of 2.5, 

at roll orientations of 0° and 45°, for angles of attack ranging from 0° to 20°. Several cases were 

also computed with yaw angles of 4.6° and 8.6°, at 0° and 20° angles of attack. Full 3-D 

calculations were performed and no symmetry was used. Computational results obtained for 

both the hexahedral mesh and tetrahedral mesh are presented both qualitatively and 

quantitatively. The results are then compared to experimental data. 

Figures 4 and 5 are schematics of the axial and circumferential locations on the missile surface, 

normalized by the overall body length (L), at which the experimental data was collected (18). 

The numerical results are compared to available experimental data obtained at several of these 

positions.
7

Figure 4. Axial locations for experimental data. 

Figure 5. Circumferential locations of experimental data points. 

4.1 Qualitative Results 

Visualization of pressure on the missile surface appears to be the same for both the hexahedral 

and tetrahedral solutions, as seen in figure 6. However, Mach and Pressure contours show the 

flow field characteristics to be more defined in the solutions for the hexahedral mesh. The 

longitudinal Mach contours for the y = 0 pitch plane depicted in figure 7 show this to be 
8

TETRAHEDRAL TETRAHEDRAL

HEXAHEDRAL HEXAHEDRAL

Figure 6. Surface pressure contours for α = 5°, and φ = 0°. 

Figure 7. Mach contours for α = 0°, φ = 0°.

TETRAHEDRAL

HEXAHEDRAL
9

particularly noticeable in the flow behind the missile. In this region, the contours for the 

tetrahedral mesh are somewhat diffused, but may be artifacts of the contouring routine. Figure 8 

depicts a similar finding in the circumferential plane in the area of the tail fins. The details of the 

pressure contours are more highly defined for the hexahedral case, especially on the lower tail 

fins, while the contours for the tetrahedral case are less distinct. 

HEXAHEDRALTETRAHEDRAL

Figure 8. Pressure contours at x/L = 0.95, α = 15°, and φ = 0°. 

The next series of figures show circumferential pressure contours at various axial locations for 

some of the different cases computed. The color map used depicts the pressure range in Pascals, 

from 0 (blue) to 13,500 (red). Since the flow fields for both grids show very similar features, 

only the results using the hexahedral grid are presented. 

Figures 9–11 show the circumferential pressure contours for the missile with 0° roll angle at 

angles of attack of 5°, 15°, and 20°. The axial locations selected represent the flow around the 

missile on the ogive (x/L = 0.3 and 0.6), on the wings (x/L = 0.7 and 0.85), and on the tail fins 

(x/L = 0.95). As the angle of attack increased, the pressure on the lower surface of the body, 

wings, and bottom set of fins increases. The flow is symmetric on either side of the y = 0 plane. 

Figure 12 shows the change in flow field when a yaw angle of 8.6° is added to the computational 

parameters for the 20° angle of attack case. The flow field becomes asymmetric and pressure 

increases on the left side of the missile. 

Similar cases were computed with the missile at a roll angle of 45°. Figures 13–16 show 

circumferential pressure contours for the same four case studies. As the visual locations move 

down the missile, the changes in the flow field on the ogive, wings, and tail fins can be seen. As 

the angle of attack increases, the pressure on the lower surfaces of the body, fins, and wings 

increases also. At α = 15°, there is higher pressure on the inner surfaces of the bottom fins and 

the pressure involves both inner and outer surfaces at α = 20°. The addition of the yaw angle to 

the 20° case appears to cause the pressure to become more localized on the tip of the bottom fins.
10

X/L = 0.3 X/L = 0.7

X/L = 0.85 X/L = 0.95

X/L=0.6

Figure 9. Circumferential pressure contours for φ = 0°, α = 5°. 

X/L = 0.3 X/L = 0.7

X/L = 0.85 X/L = 0.95

X/L = 0.6

Figure 10. Circumferential pressure contours for φ = 0°, α = 15°.
11

X/L = 0.3 X/L = 0.7

X/L = 0.85 X/L = 0.95

X/L = 0.6

Figure 11. Circumferential pressure contours for φ = 0°, α = 20°. 

X/L = 0.3 X/L = 0.7

X/L = 0.85 X/L = 0.95

X/L = 0.6

Figure 12. Circumferential pressure contours for φ = 0°, α = 20°, and β = 8.6°.
12

X/L = 0.3 X/L = 0.7

X/L = 0.85X/L=0.95

X/L = 0.6

Figure 13. Circumferential pressure contours for φ = 45°, α = 5°. 

X/L = 0.3 X/L = 0.7

X/L = 0.85 X/L = 0.95

X/L = 0.6

Figure 14. Circumferential pressure contours for φ = 45°, α = 15°.
13

X/L = 0.3 X/L = 0.7

X/L = 0.85 X/L = 0.95

X/L = 0.6

Figure 15. Circumferential pressure contours for φ = 45°, α = 20°. 

X/L = 0.3 X/L = 0.7

X/L = 0.85X/L=0.95

X/L=0.6

Figure 16. Circumferential pressure contours for φ = 45°, α = 20°, and β = 8.6°. 
14

4.2 Quantitative Results 

Circumferential pressure data extracted from the numerical solutions at axial locations of 

x/L = 0.3, 0.6, and 0.95 show the computed data from both grids to be similar and in good 

agreement with the experimental data. Figures 17–19 depict the comparisons for the 0° roll 

orientation. In general, there is good agreement in all the locations examined. For x/L = 0.3 and 

α = 15° and 20°, there is a slight difference in the numerical solutions in the θ = 30°–60° range, 

and again in the 300°–360° range. There is a noticeable discrepancy at x/L = 0.3, where the 

computed data does not match the experimental data at θ = 45° for the three angles of attack 

examined. As this experimental data point appears to be an outlier (i.e., it does not lie in line 

with the other data), it is possible that the experimental data is suspect here rather that the CFD 

calculation. 

Similar plots are presented for the 45° roll missile configuration. Figures 20–22 show good 

agreement for all axial locations examined. There is little noticeable difference between 

numerical solutions for the hexahedral and tetrahedral grids. With the exception of the data point 

at θ = 45° for x/L = 0.3, there is good agreement between the numerical and experimental data. 

Figure 17. Circumferential pressure at x/L = 0.3, φ = 0°. 
15

Figure 18. Circumferential pressure at x/L = 0.6, φ = 0°. 

Figure 19. Circumferential pressure at x/L = 0.95, φ = 0°.
16

Figure 20. Circumferential pressure plots for x/L = 0.3, φ = 45°. 

Figure 21. Circumferential pressure at x/L = 0.6, φ = 45°.
17

Figure 22. Circumferential pressure at x/L = 0.95, φ = 45°. 

4.3 Comparison of Force and Moment Data 

Force and moment data are compared with the available experimental data in the following 

tables. The computed axial forces (CN) for both grid solutions are similar and compare quite 

well with the available experimental data. The data presented in table 1 shows excellent 

comparison of CN for the 0° roll orientation cases. Tables 2 and 3 show data comparisons for 

cases with yaw angles of 4.6° and 8.6°, respectively. The experimental data for these cases was 

approximated visually from graphical representations found in Graves (23). Due to a difference 

in coordinate system orientation (for the x-axis), the experimental values for Cl (rolling moment) 

and Cn (side moment) were adjusted by changing the sign (*). The side force (CY) was not 

changed. With this modification, there is reasonable agreement between the two computations 

and the experimental data. 

Table 1. Normal force (CN) for 0° roll orientation. 

Angle of Attack Structured Grid Unstructured Grid Experiment 

5 1.132 1.148 1.2 

10 2.468 2.486 2.4 

15 3.978 3.984 4.0 

20 5.529 5.553 5.6 
18

Table 2. Force and moments, φ = 0°, α = 20°, β = 4.6°. 

 Structured Grid Unstructured Grid Experiment* 

(approx) 

Cl 0.300 0.288 0.28 

Cn –0.219 –0.216 –0.23 

CY –0.292 –0.317 –0.27 

Table 3. Force and moments, φ = 0°, α = 20°, β = 8.6°. 

 Structured Grid Unstructured Grid Experiment* 

(approx) 

Cl 0.495 0.495 0.48 

Cn –0.357 –0.342 –0.35 

CY –0.537 –0.575 –0.50 

There was no experimental data available for similar cases for the missile at a 45° roll 

orientation, but the computed values for CN shown in table 4 show excellent agreement between 

the two computational solutions. With the exception of Cl (table 5) and Cn (table 6), the 

computational data for yaw angles 4.6° and 8.6° show reasonable similarity for this missile 

orientation also. 

Table 4. Normal force (CN) for 45° roll orientation. 

Angle of Attack Structured Grid Unstructured Grid 

5 0.684 0.696 

10 1.472 1.491 

15 2.422 2.437 

20 3.372 3.341 

Table 5. Force and moments, φ = 45°, α = 20°, β = 4.6°. 

 Structured Grid Unstructured Grid 

Cl 0.669 0.612 

Cn –0.327 0.343 

Cy 1.414 1.426 

Table 6. Force and moments, φ = 45°, α = 20°, β = 8.6°. 

 Structured Grid Unstructured Grid 

Cl 0.533 0.529 

Cn –0.431 0.494 

Cy 0.716 0.714 
19

The computational data used in tables 1–6, along with additional computational data not 

examined as part of this study, is presented in the appendix. 

5. Summary and Conclusions 

Numerical computations using viscous Navier-Stokes methods were performed to predict the 

flow field and aerodynamic coefficients on a geometrically complex missile configuration under 

wind tunnel test conditions. Full 3-D computations were performed and no symmetry was used. 

Computational results were obtained for a complex NASA missile configuration with elliptic 

cross section and multiple fins, at supersonic speed for various roll orientations and angles of 

attack, using a two-equation k-ε turbulence model. Numerical results show the qualitative 

features (vortices and cross-flow separation regions) of flow field at various axial and 

streamwise positions along the model configurations. Aerodynamic coefficients have been 

obtained from the computed solutions and found to match well with the available experimental 

data for this configuration. These numerical results show that modern CFD techniques such as 

the one used in the present computational study are capable of accurately predicting the 

aerodynamics of complex missile configurations. 
20

6. References 

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

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

Symposium, Los Alamos, NM, October 1996. 

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

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

Dynamics, Lake Tahoe, NV, September 1995. 

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

4. Sahu, J.; Heavey, K. R.; Edge, H. L. Numerical Computations of Supersonic Flow Over 

Elliptic Projectiles; ARL-TR-2589; U.S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, December 2001. 

5. Sahu, J.; DeSpirito, J.; Edge, H.; Silton, S.; Heavey, K. Recent Applications of Structured 

and Unstructured Grid Techniques to Complex Projectile and Missile Configurations. 

Proceedings of the Eighth International Grid Generation and Computational Field 

Simulations, Honolulu, HI, June 2002. 

6. Pagan, D.; Molton, P.; Delery, J. Basic Experiment on a Supersonic Vortex Flow Around a 

Missile Body. Journal of Spacecraft and Rockets 1992, 29 (3), 373–378. 

7. Grasso, F.; Iaccarino, G. Influence of Crossflow and Turbulence on Vortex Flow Around a 

Supersonic Missile. Journal of Spacecraft and Rockets 1998, 35 (1), 37–45. 

8. Sahu, J.; Heavey, K. R. Application of CFD to High Angle of Attack Missile Flowfields, 

18th Applied Aerodynamics Meeting; AIAA Paper No. 2000-4210, Denver, CO, August 

2000. 

9. Birch, T.; Wrisdale, I. E.; Prince, S. CFD Predictions of Missile Flowfields, 18th Applied 

Aerodynamics Meeting; AIAA Paper No. 2000-4211, Denver, CO, August 2000. 

10. Birch, T.; Allen, J.; Wilcox, F. J. Force, Surface Pressure and Flowfield Measurements on 

Slender Missiles at Supersonic Speeds, 18th Applied Aerodynamics Meeting; AIAA Paper 

No. 2000-4207, Denver, CO, August 2000. 

11. Edwards, J. A.; Roper, J. J. A Computational Assessment of Static and Dynamic 

Coefficients for the H3 Hypervelocity Projectile, AIAA Paper 97-0640, January 1997.
21

12. Orchard, D.; Fournier, E.; Dupuis, A.; Edwards, J. Wind Tunnel Tests on the H3P78, Power 

Law, Elliptic Section Flared Projectile From Mach 2.5 to 4, AIAA Paper No. 2001-4321, 

Montreal, Canada, August 2001. 

13. Fournier, E.; Orchard, D. Testing of a Novel Maneuvering Projectile in the DREV Trisonic 

Wind Tunnel; Technical Memorandum Report, DREV TM-2000-108, Canada, July 2000. 

14. Sahu, J.; Heavey, K. R.; Edge, H. L. Numerical Computations of Supersonic and 

Hypersonic Flow Over Elliptical Projectiles, Atmospheric Flight Mechanics Conference; 

AIAA Paper No. 2001-4320, Montreal, Canada, August 2001. 

15. Orchard, D.; Fournier, E.; Dupuis, A.; Edwards, J. Wind Tunnel Tests on the H3P78, Power 

Law, Elliptic Section Flared Projectile With Jet Interaction, AIAA Paper No. 2001-4322, 

Montreal, Canada, August 2001. 

16. Sahu, J. Numerical Simulations of Supersonic Flow Over an Elliptic-Section Projectile With 

Jet Interaction, AIAA Applied Aerodynamics Conference; AIAA Paper No. 2002-3260, 

St. Louis, MO, June 2002. 

17. Sahu, J.; Silton, S. I.; Heavey, K. R. Numerical Computations of Supersonic Flow Over 

Nonaxisymmetric Configurations, Applied Aerodynmacis Conference; AIAA Paper No. 

2004-5456, Providence, RI, 2004. 

18. Allen, J. M.; Hernandez, G.; Lamb, M. Body-Surface Pressure Data on Two MonoplaneWing Missile Configurations With Elliptical Cross-Sections at Mach 2.50; NASA Technical 

Memorandum 85645; NASA Langley Research Center: Hampton, VA, 1983. 

19. Peroomian, O.; Chakravarthy, S.; Goldberg, U. A Grid-Transparent Methodology for CFD; 

AIAA Paper 97-07245, 1997. 

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

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

98-0116, 1998. 

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

22. Goldberg, U. C.; Peroomian, O.; Chakravarthy, S. A Wall-Distance-Free k-ε Model With 

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

23. Graves, E. B. Aerodynamic Characteristics of a Monoplanar Missile Concept With Bodies 

of Circular and Elliptical Cross-Sections; NASA Technical Memorandum 74079; NASA 

Langley Research Center: Hampton, VA, 1977. 
22

24. Graves, E. B.; Fournier, R. H. Effect of Nose Bluntness and Afterbody Shape on 

Aerodynamic Characteristics of a Monoplanar Missile Concept With Bodies of Circular and 

Elliptical Cross-Sections at a Mach Number of 2.50; NASA Technical Memorandum 80055; 

NASA Langley Research Center: Hampton, VA, 1979. 

25. Allen, J. NASA Langley Research Center: Hampton, VA. Private communications, 2003. 

26. Burnley, V.; Hoke, C.; Schwabacher, G. Aerodynamic Analysis of Complex Missile 

Configurations Using AVUS (Air Vehicles Unstructured Flow Solver), Applied 

Aerodynamics Meeting; AIAA Paper No. 2004-5452, Providence, RI, 2004.
23

Appendix. Computational Data 

A.1 Missile With Elliptic Cross Section, Mach = 2.5, Yaw Angle = 0° 

Aerodynamic Coefficients 

Case Alpha CX CY CN Cm Cn Cl

 

I 0 0.30946 0.00010 0.00047 0.00056 0.00020 –0.00015 

I 5 0.30339 0.00002 1.14776 –0.09334 –0.00005 –0.00003 

I 10 0.30496 0.00004 2.48607 –0.08904 –0.00068 –0.00029 

I 15 0.32164 –0.00009 3.98372 –0.06929 –0.00007 0.00014 

I 20 0.34078 0.00002 5.53302 –23.41887 0.00012 0.00001 

 

II 0 0.30940 0.00052 0.00050 0.00006 –0.00012 –0.00010 

II 5 0.30740 0.43594 0.69628 –0.16889 –0.07070 0.07114 

II 10 0.30763 0.92257 1.49089 –0.31743 –0.15739 0.26721 

II 15 0.30703 1.50634 2.43654 –0.49145 –0.26965 0.52974 

II 20 0.35109 2.22719 3.34118 –0.50319 –0.29161 0.65374 

 

III 0 0.29565 –0.00012 –0.00002 0.00007 –0.31208 –0.00009 

III 5 0.29018 –0.00003 1.13152 –0.09817 –0.00004 0.00003 

III 10 0.29009 0.00008 2.46824 –0.07974 0.00013 0.00008 

III 15 0.30479 –0.00028 3.97936 –0.07300 0.00006 –0.01035 

III 20 0.32900 0.00007 5.52991 0.10791 0.00006 0.00003 

 

IV 0 0.29673 –0.00001 0.00000 0.00000 –0.00001 –0.00004 

IV 5 0.29504 0.43123 0.68381 –0.16375 –0.06396 0.07318 

IV 10 0.29504 0.91835 1.47230 –0.30868 –0.13460 0.27076 

IV 15 0.30064 1.51329 2.42211 0.38195 –0.81232 0.54848 

IV 20 0.32266 2.18825 3.37210 –0.52519 –0.28050 0.65656 

Pressure 4761.81 Pa 

Density 0.1098 kg/s 

Velocity 616.0 

Spin, p 8944 

Reference diameter 10.16 cg 

Reference area 0.008107 m2

Speed, sound 2.45.4 m/s 

Center of gravity 4.2 cal 

qS 168.89 

qS=dynamic press* reference area 

 Orientation Grid Type 

Case I 0° roll Tetrahedral 

Case II 45° roll Tetrahedral 

Case III 0° roll Hexahedral 

Case IV 45° roll Hexahedral 
24

 CFD++ Forces and Moments 

Case Alpha FX FY FZ MX MY MZ 

 

I 0 5.23E + 01 1.61E - 02 7.87E - 02 –2.55E - 03 9.56E - 03 3.39E - 03 

I 5 5.12E + 01 3.30E - 03 1.94E + 02 –4.99E - 04 –1.60E + 00 –8.91E - 04 

I 10 5.15E + 01 6.42E - 03 4.20E + 02 –4.89E - 03 –1.53E + 00 –1.17E - 02 

I 15 5.43E + 01 –1.53E - 02 6.73E + 02 2.34E - 03 –1.19E + 00 –1.15E - 03 

I 20 5.76E + 01 2.76E - 03 9.34E + 02 1.96E - 04 –4.02E + 02 2.10E - 03 

 

II 0 5.23E + 01 8.72E - 02 8.48E - 02 –1.66E - 03 1.10E - 03 –2.02E - 03 

II 5 5.19E + 01 7.36E + 01 1.18E + 02 1.22E + 00 –2.90E + 00 –1.21E + 00 

II 10 5.20E + 01 1.56E + 02 2.52E + 02 4.59E + 00 –5.45E + 00 –2.70E + 00 

II 15 5.19E + 01 2.54E + 02 4.12E + 02 9.09E + 00 –8.43E + 00 –4.63E + 00 

II 20 5.93E + 01 3.76E + 02 5.64E + 02 1.12E + 01 –8.63E + 00 –5.00E + 00 

 

III 0 4.99E + 01 –2.01E - 02 –4.06E - 03 –1.59E - 03 1.12E - 03 –5.36E + 00 

III 5 4.90E + 01 –5.32E - 03 1.91E + 02 5.21E - 04 –1.68E + 00 –6.73E - 04 

III 10 4.90E + 01 1.29E - 02 4.17E + 02 1.40E - 03 –1.37E + 00 2.19E - 03 

III 15 5.15E + 01 –4.80E - 02 6.72E + 02 –1.78E - 01 –1.25E + 00 1.11E - 03 

III 20 5.56E + 01 1.17E - 02 9.34E + 02 5.31E - 04 1.85E + 00 1.11E - 03 

 

IV 0 5.01E + 01 –9.34E - 04 –4.44E - 04 –6.67E - 04 4.80E - 05 –1.75E - 04 

IV 5 4.98E + 01 7.28E + 01 1.15E + 02 1.26E + 00 –2.81E + 00 –1.10E + 00 

IV 10 4.98E + 01 1.55E + 02 2.49E + 02 4.65E + 00 –5.30E + 00 –2.31E + 00 

IV 15 5.08E + 01 2.56E + 02 4.09E + 02 9.41E + 00 6.55E + 00 –1.39E + 01 

IV 20 5.45E + 01 3.70E + 02 5.70E + 02 1.13E + 01 –9.01E + 00 –4.81E + 00 
25

A.2 Missile With Elliptic Cross Section, Mach = 2.5, Yaw Angle = 4.6° 

 Aerodynamic Coefficients 

Case Alpha CX CY CN Cm Cn Cl

 

I 20 0.33458 –0.31657 5.46295 –0.18056 –0.21573 0.28930 

 

II 20 0.35480 1.42679 3.14192 –0.80731 –0.34279 0.61229 

III 20 0.32130 –0.29218 5.47801 –0.04582 –0.21996 0.30001 

IV 20 0.32275 1.41406 2.96861 –8.24658 –0.32749 0.66923 

 CFD++ Forces and Moments 

Case Alpha FX FY FZ MX MY MZ 

 

I 20 5.65E + 01 –5.35E + 01 9.23E + 02 4.96E + 00 –3.10E + 00 –3.70E + 00 

 

II 20 5.99E + 01 2.41E + 02 5.31E + 02 1.05E + 01 –1.39E + 01 –5.88E + 00 

 

III 20 5.43E + 01 –4.93E + 01 9.25E + 02 5.15E + 00 –7.86E - 01 –3.77E + 00 

 

IV 20 5.45E + 01 2.39E + 02 5.01E + 02 1.15E + 01 –1.42E + 02 –5.62E + 00 

Pressure 4761.81 Pa 

Density 0.1098 kg/s 

Velocity 616.0 

Spin,p 8944 

Reference diameter 10.16 cg 

Reference area 0.008107 m2

Speed, sound 2.45.4 m/s 

Center of gravity 4.2 cal 

qS 168.89 

qS=dynamic press* reference area 

 Orientation Grid Type 

Case I 0° roll Tetrahedral 

Case II 45° roll Tetrahedral 

Case III 0° roll Hexahedral 

Case IV 45° roll Hexahedral 
26

A.3 Missile With Elliptic Cross Section, Mach = 2.5, Yaw Angle = 8.6° 

CFD++ Forces and Moments 

Case Alpha FX FY FZ MX MY MZ 

 

I 20 5.77E + 01 –9.25E + 01 9.47E + 02 8.49E + 00 –2.65E + 00 –5.87E + 00 

 

II 20 5.90E + 01 1.21E + 02 4.39E + 02 9.08E + 00 –1.45E + 01 –8.48E + 00 

 

III 20 5.73E + 01 –9.07E + 01 9.46E + 02 8.50E + 00 –1.72E + 00 –6.13E + 00 

 

IV 20 5.56E + 01 1.21E + 02 4.31E + 02 9.15E + 00 –1.47E + 01 –7.40E + 00 

Pressure 4761.81 Pa 

Density 0.1098 kg/s 

Velocity 616.0 

Spin,p 8944 

Reference Diameter 10.16 cg 

Reference Area 0.008107 m2

Speed, sound 2.45.4 m/s 

Center of Gravity 4.2 cal 

qS 168.89 

qS=dynamic press* reference area 

 Orientation Grid Type 

Case I 0 deg roll Tetrahedral 

Case II 45 deg roll Tetrahedral 

Case III 0 deg roll Hexahedral 

Case IV 45 deg roll Hexahedral 

Aerodynamic Coefficients 

Case Alpha CX CY CN Cm Cn Cl

 

I 20 0.34159 –0.54755 5.60456 –0.15415 –0.34227 0.49468 

 

II 20 0.34942 0.71394 2.59850 –0.84507 –0.49421 0.52901 

III 20 0.33920 –0.53711 5.60275 –0.10027 –0.35737 0.49527 

IV 20 0.32911 0.71618 2.55470 –0.85667 –0.43142 0.53300 
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