CFD Prediction of Magnus Effect in Subsonic to 

Supersonic Flight 

by James DeSpirito 

ARL-TR-4929 September 2009 

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Army Research Laboratory 

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ARL-TR-4929 September 2009 

CFD Prediction of Magnus Effect in Subsonic to 

Supersonic Flight 

James DeSpirito 

Weapons and Materials Research Directorate, ARL 

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CFD Prediction of Magnus Effect in Subsonic to Supersonic Flight 

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James DeSpirito 

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14. ABSTRACT 

The aerodynamic coefficients of the 7-cal. U.S. Army-Navy Spinner Rocket were characterized using computational fluid 

dynamic (CFD) calculations and validated using archival experimental data. The static aerodynamic coefficients, roll-damping, 

and pitch-damping moments were accurately predicted by steady-state Reynolds-averaged Navier-Stokes (RANS) as well as 

unsteady hybrid RANS/large-eddy simulation (LES) CFD. The Magnus moment was overpredicted in the subsonic and 

transonic regime. Unsteady RANS/LES computations did not improve the prediction of Magnus moment at the lower Mach 

numbers. Both steady-state RANS and unsteady RANS/LES simulations resulted in similar predictions of all aerodynamic 

coefficients. Distributions of Magnus moment along the projectile body showed that the largest difference in Magnus moment 

between configurations and Mach numbers was in the last caliber of the projectile body. 

15. SUBJECT TERMS

computational fluid dynamics, pitch damping, Magnus, roll damping, aerodynamic 

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James DeSpirito 

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iii

Contents 

List of Figures iv

List of Tables v

Acknowledgments vi

1. Introduction 1

2. Numerical Approach 2

2.1 Computational Geometry and Mesh ................................................................................2

2.2 Navier-Stokes CFD .........................................................................................................6

3. Results and Discussion 7

3.1 Grid Resolution Study .....................................................................................................7

3.2 The 7-cal. ANSR Model Results .....................................................................................8

4. Summary and Conclusions 21

5. References 22

Appendix. Aerodynamic Coefficient Data 25

List of Symbols, Abbreviations, and Acronyms 31

Distribution List 33 
iv

List of Figures 

Figure 1. The 7-cal. Army-Navy Spinner Rocket (ANSR) model with (a) standard base and 

(b) 0.5-cal. boattail. ....................................................................................................................3

Figure 2. Computational mesh used for standard-base ANSR model: (a) RANS mesh (SB-1) 

and (b) close-up of the high-density mesh in the wake region of meshes SB-1 (left) and 

SB-2 (right). ...............................................................................................................................5

Figure 3. Drag coefficient (at  = 2°) vs. Mach number: standard-base ANSR. ..........................9

Figure 4. Roll-damping coefficient vs. Mach number: standard-base ANSR. ............................10

Figure 5. Normal force coefficient vs. Mach number: standard-base ANSR. .............................10

Figure 6. Pitching moment coefficient vs. Mach number: standard-base ANSR. .......................11

Figure 7. Normal force center-of-pressure location vs. Mach number: standard-base ANSR. ...11

Figure 8. Pitch-damping moment coefficient vs. Mach number: standard-base ANSR. .............12

Figure 9. Magnus moment coefficient (at  = 2°) vs. Mach number: standard-base ANSR. .....13

Figure 10. Magnus moment coefficient vs. Mach number: comparison of standard-base and 

boattail ANSR. .........................................................................................................................14

Figure 11. Distribution of Magnus (a) force and (b) moment coefficient (at  = 2°) on 

standard-base ANSR: comparison of RANS and RANS/LES computations. ........................15

Figure 12. Distribution of Magnus (a) force and (b) moment coefficient (at  = 2°) on 

boattail ANSR: comparison of RANS and RANS/LES computations. ..................................16

Figure 13. Distribution of Magnus (a) force and (b) moment coefficient (at α = 2°): 

comparison of effect of ANSR base shape for steady-state RANS computations. .................17

Figure 14. Instantaneous Mach number contours from unsteady RANS/LES calculations: 

standard-base and boattail ANSR at α = 2° and (a) Mach 0.7; (b) Mach 0.8; (c) Mach 1.1; 

and (d) Mach 1.8. .....................................................................................................................19

Figure 15. Instantaneous Mach number contours from unsteady RANS/LES calculations: 

standard-base and boattail ANSR at  = 2° and (a) Mach 0.7; (b) Mach 0.8; (c) Mach 1.1; 

and (d) Mach 1.8. .....................................................................................................................20

Figure 16. Instantaneous turbulent kinetic energy contours from unsteady RANS/LES 

calculations: standard-base ANSR at Mach 0.7 using (a) RANS/LES and (b) RANS 

mesh. ........................................................................................................................................21
v

List of Tables 

Table 1. Computational mesh characteristics of 7-cal. ANSR model. ...........................................3

Table 2. Percent differences of aerodynamic coefficients from medium-sized mesh in grid 

resolution study. .........................................................................................................................8 

Table A-1. Static aerodynamic coefficient data: standard-base configuration, 7-cal. ANSR. .....26

Table A-2. Aerodynamic derivative data: standard-base configuration, 7-cal. ANSR. ..............27

Table A-3. Static aerodynamic coefficient data: boattail configuration, 7-cal. ANSR ...............28

Table A-4. Aerodynamic derivative data: boattail configuration, 7-cal. ANSR. .........................29
vi

Acknowledgments 

The author thanks Dr. Paul Weinacht and Dr. Sidra Silton of the U.S. Army Research Laboratory 

(ARL) for many helpful discussions. This work was supported in part by a grant of highperformance computing time from the U.S. Department of Defense (DOD) High Performance 

Computing Modernization program at the ARL DOD Supercomputing Resource Center, 

Aberdeen Proving Ground, MD.
1

1. Introduction 

The computation of Magnus effects via steady-state computational fluid dynamics (CFD) in the 

supersonic and transonic flight regimes was demonstrated more than 20 years ago (1–3). The 

comparisons with experimental data in the transonic region were limited and tended to consist of 

Mach numbers >1.0. Recent work (4) directly comparing time-accurate and steady-state 

methods to predict the Magnus moment in the supersonic regime shows each method accurate to 

within the error in the experimental data. Recent studies (5–7) using the much-improved 

meshes, turbulence models, and solver methodologies have shown that the prediction of Magnus 

moment near Mach 1.0 and below does not always compare well with experimental data. Silton

(5) demonstrated limited success in predicting the Magnus moment of a 0.50-cal. (12.7-mm) 

projectile. The prediction of Magnus moment in the subsonic and supersonic regimes was found 

to be fair, but in the transonic regime it was not as good. Another study of the 25-mm M910 

training projectile (6) predicted the static aeroballistic coefficients plus Magnus moment and rolldamping moment over the Mach number range of 0.4 < M < 4.5. Magnus moment was predicted 

adequately via steady-state CFD for M > 2. However, for M < 2 the Magnus moment was 

overpredicted, not capturing the large negative values of nonlinear Magnus moment present in 

the M910 flight dynamics. Time-accurate, hybrid Reynolds-averaged Navier Stokes (RANS) 

and large-eddy simulation (LES) calculations appeared to perform better, accurately predicting 

the Magnus moment in the subsonic Mach number range. A more recent continuation of the 

M910 study (7), which used a more refined LES mesh in the wake region, again showed that 

time-accurate RANS/LES simulations performed better than steady-state computations in the 

subsonic range. However, the newer calculations did not predict the magnitude of the Magnus 

moment as well as in the earlier study (6). 

Today, nonlinear Magnus moments are routinely observed for nearly all spin-stabilized 

projectiles at subsonic and transonic speeds (8). Observing the distribution of Magnus moment 

along the length of a typical spin-stabilized projectile shows that the nonlinearity is confined to 

very near the base of the projectile. The shape of the projectile base has a large effect on the 

nonlinear Magnus component (4). A rounded base was shown to exhibit a large variation with 

angle of attack, while a flat (sharp-cornered) base nearly eliminates the nonlinear Magnus effect. 

In the M910 study (6, 7), significant differences in the near-body wake of the projectile were 

observed between steady-state RANS and unsteady RANS/LES simulations at subsonic and 

transonic speeds. It was speculated that the unsteady base flow interacts with the asymmetric 

pressure distribution (which is responsible for the Magnus effect) on the projectile body. Some 

or all of this interaction effect is not captured by the steady-state simulations. The wake effects 

on the body forces are likely small enough not to significantly impact the normal force and 

pitching moment; however, the Magnus force is much smaller, and the small wake effect 

perturbations can significantly effect the Magnus moment. 
2

Research in the area of base flow phenomena is ongoing. Other researchers have recently used 

unsteady RANS/LES methods to investigate projectile base flows in the supersonic (9) and 

subsonic and transonic regimes without (10) and with (11) spin. Their work primarily 

concentrates on the demonstration and validation of their hybrid RANS/LES technique and 

illustrates the unsteady quantities that can be obtained with the hybrid RANS/LES methods. The 

authors have not published work investigating nonzero angles of attack; thus no Magnus effects 

have been reported. 

The capabilities of steady-state CFD to predict the static aerodynamic coefficients of munitions 

are well established. Roll damping and pitch damping have also been accurately predicted using 

steady-state CFD methods (12). The goal of this report is to further evaluate the steady-state 

RANS and unsteady RANS/LES CFD methods to predict the Magnus effects in the transonic and 

subsonic flight regimes. The accurate prediction of Magnus moment across the Mach number 

range is important because it, along with the other dynamic and static coefficients, is needed as 

input into trajectory codes for preliminary design purposes. If CFD can demonstrate highconfidence prediction of these dynamic coefficients, then it can be used with confidence to 

evaluate projectile designs in lieu of free-flight and spark range tests in the preliminary design 

phase. Full aerodynamic characterization using CFD, spark range, and wind tunnel 

investigations can then be performed on the final design concepts. 

The projectile used in this study is the U.S. Army-Navy Spinner Rocket (ANSR), for which 

extensive archival experimental data exists (both free-flight spark range and wind tunnel) 

(13–15). As such, this model has also been used as a reference for many previous semiempirical and CFD calculations. The basic interest of the ANSR test program was in the 

dynamic stability of spinning bodies of revolution. The two important ballistic coefficients for 

determining dynamic stability—Magnus and pitch-damping moments—are dependent on the 

center of gravity (c.g.). Therefore, three c.g. locations were investigated by using different 

materials for parts of the projectile body. In this report, the standard-base and boattail versions 

of the 7-cal.-long ANSR model are studied in the transonic and supersonic Mach regimes, 

0.7 < M < 2.5. 

2. Numerical Approach 

2.1 Computational Geometry and Mesh 

Two different ANSR configurations were investigated, each with a total body length of 7 cal. 

(1 cal. = 20 mm). The computational models of the standard-base and boattail versions of the 

ANSR are shown in figure 1. Each consists of a 2-cal. secant ogive followed by a 5-cal. 

cylindrical afterbody. The boattail model had a 0.5-cal., 7° boattail. The three c.g. locations 
3

(a)

(b)

Figure 1. The 7-cal. Army-Navy Spinner Rocket (ANSR) model with (a) 

standard base and (b) 0.5-cal. boattail. 

investigated in the experimental program were also studied here, 3.250, 4.036, and 4.818 cal. 

from the nose. 

The geometry and unstructured mesh were generated using GAMBIT, a geometry and grid 

preprocessor supplied with the FLUENT CFD software suite (16). A full, three-dimensional 

(3-D) mesh was required to simulate the spinning shell at angle of attack. In generating the 

meshes, boundary layer mesh spacing was used near the projectile body. The meshes used in the 

study are summarized in table 1. Three versions of the same standard-base (SB) mesh were 

generated to use in a grid resolution study. The smaller standard base and boattail (BT) meshes 

(SB-1 and BT-1) were used for all the steady-state computations. The larger meshes (SB-2 and 

BT-2) were used for the unsteady RANS/LES computations, which require a denser mesh behind 

the projectile to resolve the large eddies in the wake. All the other mesh characteristics were the 

same. Table 1 shows the number of cells in the circumferential and axial directions along the 

projectile body, in the radial direction away from the body, and on the axis to the rear and 

forward of the body. 

Table 1. Computational mesh characteristics of 7-cal. ANSR model. 

Case 

Number 

Cells 

Number 

Circumferential 

Cells 

Number 

Axial Cells 

Number 

Radial Cells 

Number Cells 

to Rear 

Number 

Cells 

Forward 

SB-1 3,455,920 144 170 106 210 102 

SB-1a 1,781,200 112 136 84 168 82 

SB-1b 5,878,112 176 200 116 252 122 

BT-1 3,731,224 144 202 106 210 102 

SB-2 11,296,728 144 170 106 492 102 

BT-2 11,754,520 144 202 106 492 102 
4

The extent of the computational domain was the same for all meshes: 50 cal. radially away, 

35 cal. forward, and 50 cal. rearward of the projectile. The y+ value was also the same for all 

meshes, 2.5 × 10-5. The y+ value is actually based on the first cell centroid, or one-half of the 

first-edge spacing. Resolving the viscous boundary layer is critical for predicting the Magnus 

and roll-damping moments; therefore, wall functions were not used. Equations were integrated 

to the wall instead. Normally, a y+ value on the order of 1.0 is adequate to resolve the boundary 

layer. However, a y+ value of 0.5 or less was needed to ensure capturing the Magnus effect of 

the spinning shell. All mesh stretching ratios were kept to 1.2 or less. 

The mesh for steady-state RANS computations, SB-1, is shown in figure 2a. It is an 

unstructured mesh consisting of hexagonal cells, with 144 cells in the circumferential direction. 

An O-grid-type mesh was generated around the projectile body to contain the dense viscous 

mesh required to resolve the boundary layer. This O-grid extended to a distance of 0.1 cal. from 

the projectile body. A C-grid-type mesh was then generated around the body to a distance of 

0.5 cal. from the projectile body. The C-grid mesh was extended from the base of the projectile 

to the rear boundary of the computational domain. The mesh stretching was kept very small 

behind the projectile to ensure the wake flow was resolved. The mesh stretching was about 1% 

for the first 3 cal., and then increased to about 4% from that point to the rear end of the 

computational domain. The region ahead of the projectile was meshed with a mapped-type 

mesh. The remainder of the computational domain was meshed with hexagons generated by 

revolving the paved quads around the axis (this is a Cooper mesh in GAMBIT). The mesh for 

the boattail ANSR, BT-1, was very similar to SB-1. Only the small region in the vicinity of the 

base was changed, with most of the mesh the same as the standard-base ANSR. 

Two additional meshes based on SB-1 were generated as part of a grid convergence 

investigation. These are listed in table 1 as SB-1a and SB-1b, respectively. These meshes were 

made to be nominally one-half and twice the size, respectively, of SB-1. To achieve this, the 

number of cells in the axial and circumferential directions was divided by and multiplied by the 

3 2 , respectively. However, in the radial direction, the first cell spacing was kept constant, and 

the number of cells was only reduced to the extent that the mesh stretching ratio did not exceed 

1.25. If the mesh in the radial direction is reduced to a level that is inappropriate for resolving 

the viscous boundary layer, then the coarse mesh is almost guaranteed to produce poor results. 

Although meshes SB-1 and BT-1 were designed for steady-state RANS simulations, the cell 

density in the wake region was made higher than might normally be used. This was done to see 

if unsteady RANS/LES simulations on this mesh could produce reasonable results, as they had in 

a previous study (6). However, two additional meshes, SB-2 and BT-2, were made with a dense 

mesh of nearly isotropic cells for about 0.75 body lengths behind the projectile. These meshes 

are more appropriate to capture the large-scale eddy structures in the LES (wake) region. A 

comparison of the mesh density in the wake between the RANS mesh and RANS/LES mesh for 
5

(a)

(b)

Figure 2. Computational mesh used for standard-base ANSR model: (a) RANS mesh (SB-1) and 

(b) close-up of the high-density mesh in the wake region of meshes SB-1 (left) and SB-2 

(right). 

the standard-base ANSR is shown in figure 2b. The surface mesh on the base of the ANSR was 

also made more uniform for the RANS/LES mesh. The mesh size increase was primarily due to 

the doubling of the circumferential mesh density, which was done to have nearly isotropic 

volume cells at the outer radial edge of the LES region of the mesh. As mentioned, the LES 

mesh in the wake only extended about 0.75 body lengths, then the mesh was stretched in the 

axial direction, as is normally done in RANS meshes. Therefore, the coherent structures present 

in the LES region will begin to be numerically dissipated by the RANS mesh upon leaving the 

LES zone. 

The outer boundaries in the forward and radial directions were set as far-field (characteristicsbased inflow/outflow), with standard temperature and pressure free-stream conditions 

(101.325 kPa, 288 K). The projectile wall was modeled as a no-slip, isothermal (288 K) wall 

boundary, rotating (clockwise when viewed from rear) around the x-axis at the specified spin 

rate. The projectile spin rates were determined from the muzzle exit twist, 10 cal./revolution 

(13), corresponding to a muzzle-exit nondimensional spin rate (pd/2V) of 0.315.
6

2.2 Navier-Stokes CFD 

The commercially available CFD++ code (17), version 6.1.1, was used in this study. The CFD++

code can simulate a range of fluid dynamic phenomena, ranging from incompressible to 

hypersonic flow. The 3-D, time-dependent RANS equations are solved using the finite volume 

method. The implicit solver with dual time-stepping was used. The spatial discretization was a 

second-order, multidimensional total variation diminishing (TVD) polynomial interpolation 

scheme. Solutions to semi-infinite “Riemann problems” are used in CFD++ to provide upwind 

flux information to the underlying transport scheme. Approximate Riemann solvers are used to 

determine the higher-order fluxes to avoid spurious oscillations that may become physically 

unbounded if determined via fixed-stencil interpolation. Far-field absorbing layers were also 

used on the outer boundaries for the subsonic and transonic simulations. 

The three-equation k--R model, which solves transport equations for the turbulence kinetic 

energy, k, its dissipation rate, , and the undamped eddy viscosity, R, was used in the steady-state 

computations. This model provided the best performance across the Mach number range in ref 

6. CFD++ also provides LES models and hybrid RANS/LES models, including the detachededdy simulation (DES) model. The unsteady hybrid RANS/LES approach used the BattenGoldberg model. The RANS/LES methodology of CFD++ is based on the solution of transport 

equations for the unresolved turbulence kinetic energy and its dissipation rate and incorporates 

anisotropy and low Reynolds number-damping effects in both LES and RANS modes. CFD++

reverts to a cubic k- model on RANS-type meshes and blends automatically to an anisotropic 

form of the Smagorinsky model in regions of uniformly refined mesh (17). Past unsteady 

RANS/LES simulations have been started from a steady-state solution using the k--R model 

with good results (6, 7). However, the steady-state solutions used as the starting point for the 

unsteady RANS/LES simulations in this study were performed using the cubic k- turbulence 

model to provide the cleanest transition to the unsteady calculations. As a by-product, an 

additional set of steady-state CFD data is obtained for those Mach numbers at which the 

RANS/LES simulations were performed. 

Most of the simulations were performed in parallel on an 1100-node, 4400-core Linux Networx 

Advanced Technology Cluster at the U.S. Army Research Laboratory DOD Supercomputing 

Resource Center. Each node has two dual-core, 3.0-GHz Intel Woodcrest processors. Some 

simulations were performed on an older 1024-node Linux Networx Evolocity II system with two 

3.6-GHz Intel Xeon EM64T processors per node. The number of processors used for each 

steady-state run was such that about 145,000–150,000 cells were partitioned on each processor. 

The basic RANS calculations were run until steady state was achieved. The calculations took 

about 11–15 s of CPU time per iteration (wall-clock time was nearly the same) and convergence 

was achieved in about 600–800 iterations. The solution was deemed converged when the flow 

residuals had reduced at least 3 orders of magnitude and the aerodynamic coefficients changed 

less than about 0.5% over the last 100 iterations. The aerodynamic coefficients were the 

determining factor in convergence in all cases. 
7

The double-precision solver with second-order spatial discretization was used. The maximum 

Courant-Friedrich-Lewy (CFL) number was chosen using the recommendations within the 

CFD++ solver; for the transonic flow regime (0.7 < M < 1.4), the CFL number is ramped from 1.0 

to 75.0 over the first 100 iterations. The steady-state RANS calculations were performed at 14 

Mach numbers over the range 0.6 ≤ M ≤ 2.5 and at three angles of attack:  = 0°, 2°, and 5°. The 

final solution of the appropriate steady-state case was used as the initial conditions for the 

unsteady RANS/LES calculations. 

The unsteady RANS/LES calculations were performed at  = 2° and four Mach numbers: Mach 

0.7, 0.8, 1.1, and 1.8. The time steps used were 4.0 × 10–6 s (Mach 0.7 and 0.8), 2.5 × 10–6 s 

(Mach 1.1), and 1.5 × 10–6 s (Mach 1.8). These values were determined based on having about 

70 time steps within the period of the oscillations in the wake flow, assuming a Strouhal number 

of 0.25. Five inner iterations were performed at each time step. The number of processors used 

for the RANS/LES simulations was such that about 60,000 cells were partitioned on each 

processor. The unsteady calculations took about 6 s of CPU time per iteration, or about 

30 s/time step, and were run for a total nondimensional time (based on Strouhal number) of 

85–96 (Mach 0.7, 0.8, and 1.1) or 60 (Mach 1.8). The Mach 1.8 case had a shorter period of 

transition from the steady-state solution to the unsteady wake flow, so a shorter total time was 

used. 

3. Results and Discussion 

3.1 Grid Resolution Study 

A grid resolution study was conducted for the Mach 0.7, 0.95, and 1.2 steady-state cases at 

 = 0°, 2°, and 5°. The solutions for the coarse (SB-1a), medium (SB-1), and fine (SB-1b) 

meshes were compared for differences between the aerodynamic coefficients, and the results are 

shown in table 2. The table shows the percent differences in the aerodynamics coefficients 

between either the coarse (SB-1a) or fine (SB-1b) mesh and the medium mesh (SB-1). For 

 = 0°, only the axial force and rolling moment are nonzero. 

The percent differences in coefficients between the coarse and medium meshes are mostly less 

than 1%. The larger differences, still less than 5%, are mostly for the rolling moment and 

Magnus force and moments, which tend to be relatively small-valued coefficients. Nearly all the 

coefficients show a decrease in the percent difference between the medium and fine meshes 

compared to the difference between the medium and coarse meshes, as would be expected to 

show grid independence. Where there isn’t a significant decrease, the relative change in the 

actual values is small. These results indicate that the coarse mesh was fairly well resolved. The 

medium mesh was designed based on past experience and knowledge of what type of mesh is 
8

Table 2. Percent differences of aerodynamic coefficients from medium-sized mesh in grid resolution study. 

Mach Mesh 



CX 

(%) 

CY 

(%)

CN 

(%)

Cl 

(%)

Cm 

(%)

Cn 

(%)

0.7 

Coarse–med 

0 –0.68 — — 0.20 — — 

2 –0.89 –0.50 0.50 –0.92 1.66 –0.35 

5 –0.78 –3.02 0.40 –0.51 1.26 –4.29 

Med–fine 

0 –0.67 — — –1.94 — — 

2 –0.47 –0.75 0.52 –0.43 1.76 –1.10 

5 –0.35 –1.71 0.27 –0.28 0.86 –2.58 

0.95 

Coarse–med 

0 –0.44 — — –2.24 — — 

2 –0.21 –2.27 0.19 –0.69 0.99 –3.30 

5 –0.18 –1.20 0.15 –0.38 0.55 –1.39 

Med–fine 

0 –0.14 — — –0.97 — — 

2 –0.20 –1.88 0.85 –1.19 2.62 –2.84 

5 –0.12 –0.48 0.14 –0.48 0.30 –0.55 

1.2 

Coarse–med 

0 –0.25 — — –0.67 — — 

2 –0.47 –1.09 4.50 –0.47 19.25 –1.21 

5 –0.54 –0.53 0.18 –0.74 0.83 –0.42 

Med–fine 

0 –0.09 — — 0.74 — — 

2 –0.11 0.92 –4.27 –0.22 –15.59 1.36 

5 0.02 –0.31 –0.34 –0.53 –1.34 –0.47 

required to accurately predict the aerodynamic coefficients. These meshes tend to be of 

relatively high quality and high resolution. The coarsening of this mesh did not degrade the 

quality significantly, especially since the y+

 spacing was not modified. The Mach 1.2 normalforce and pitching-moment coefficients at  = 2° appear to be anomalous, giving 4.5% and 

19.3%, respectively, between the coarse and medium meshes and –4.3% and –15.6%, 

respectively, between the medium and fine meshes. The actual values are such that those from 

the coarse and fine meshes are nearly equal. Therefore, the medium mesh, SB-1, was deemed 

appropriate for the steady-state computations, and all other meshes were based on it. 

3.2 The 7-cal. ANSR Model Results 

The aerodynamic data presented in this section is tabulated in appendix A. Figure 3 shows the 

drag coefficient as a function of Mach number, comparing the CFD predictions to the 

experimental data (13, 14). The CFD data is the drag at  = 2°, while the experimental data is at 

the yaw angle in the flight range (0.4°–4.4°). The drag is approximately equal to the axial force 

coefficient at these low angles of attack. The experimental data is plotted for each of the three 

c.g. locations. CFD data is shown for the steady-state RANS simulations on both the RANS 

(SB-1) and RANS/LES (SB-2) meshes. Data is shown for unsteady RANS/LES simulations on 

the RANS mesh (SB-1) for two Mach numbers. The steady-state RANS simulations using the 

cubic k- model do slightly better than the k--R model, which slightly overpredicts the drag over 

much of the Mach number range. The unsteady RANS/LES simulations vary, being closer to the 

k--R steady predictions at Mach 0.7 and 1.1 and closer to the cubic k- steady predictions at 

Mach 0.8 and 1.8. The steady-state RANS predictions are adequate for drag prediction.
9

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (F) BRL-MR-775

Exp (M) BRL-MR-775

Exp (R) BRL-MR-775

Exp (F) BRL-R-876

Exp (M) BRL-R-876

Exp (R) BRL-R-876

CFD SS RANS (SB-1)

CFD SS RANS (cke, SB-2)

CFD RANS/LES (SB-2)

CFD RANS/LES (SB-1)

D C

Figure 3. Drag coefficient (at  = 2°) vs. Mach number: standard-base ANSR. 

Figure 4 shows the roll-damping coefficient, which compares reasonably well with the 

experimental data. The roll-damping values were obtained at  = 2°; however, Clp was relatively 

insensitive to angle of attack up to the 5° investigated in this study. Over the Mach number 

range shown, the CFD values of Clp are underpredicted (i.e., larger negative, so damping is 

overpredicted) by about 8%–15%. The trend of decreasing roll damping with Mach number is 

predicted very well by the CFD. The steady-state predictions with the cubic k- model and the 

unsteady RANS/LES predictions performed better than the steady-state calculations using the k-

-R model. This is a similar result found for the M910 projectile (7). There is little difference 

between the RANS predictions using the cubic k- model and RANS/LES predictions, so the 

better prediction is solely due to the RANS region of the near-body grid. Even with these 

differences, the steady-state RANS simulations are adequate for Clp predictions. 

Figures 5 and 6 show the normal-force coefficient and pitching-moment coefficients at the three 

c.g. locations: forward (F), middle (M), and rear (R). Excellent agreement with the 

experimental data was found for all CFD predictions. There was very little difference between 

the steady-state RANS and unsteady RANS/LES predictions showing that the former 

computations are adequate for these coefficients. Figure 7 shows the normal force center of 

pressure,   m N xcp  xcg  C /C , which is also predicted very well by all computational methods 

since it is based on m C and N C . 
10

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (F) BRL-MR-775

Exp (M) BRL-MR-775

Exp (R) BRL-MR-775

Exp (F) BRL-R-876

Exp (M) BRL-R-876

Exp (R) BRL-R-876

CFD SS RANS (SB-1)

CFD SS RANS (cke, SB-2)

CFD RANS/LES (SB-2)

CFD RANS/LES (SB-1)

pl C

Figure 4. Roll-damping coefficient vs. Mach number: standard-base ANSR. 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (F) BRL-MR-775

Exp (M) BRL-MR-775

Exp (R) BRL-MR-775

Exp (F) BRL-R-876

Exp (M) BRL-R-876

Exp (R) BRL-R-876

CFD SS RANS (SB-1)

CFD SS RANS (cke, SB-2)

CFD RANS/LES (SB-2)

CFD RANS/LES (SB-1)

 N C

Figure 5. Normal force coefficient vs. Mach number: standard-base ANSR. 
11

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (F) BRL-MR-775

Exp (M) BRL-MR-775

Exp (R) BRL-MR-775

Exp (F) BRL-R-876

Exp (M) BRL-R-876

Exp (R) BRL-R-876

CFD SS RANS (SB-1)

CFD SS RANS (cke, SB-2)

CFD RANS/LES (SB-2)

CFD RANS/LES (SB-1)

 m C

Figure 6. Pitching moment coefficient vs. Mach number: standard-base ANSR. 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (F) BRL-MR-775

Exp (M) BRL-MR-775

Exp (R) BRL-MR-775

Exp (F) BRL-R-876

Exp (M) BRL-R-876

Exp (R) BRL-R-876

CFD SS RANS (SB-1)

CFD SS RANS (cke, SB-2)

CFD RANS/LES (SB-2)

CFD RANS/LES (SB-1)

cp

x

Figure 7. Normal force center-of-pressure location vs. Mach number: standard-base ANSR. 

Figure 8 shows the pitch-damping coefficients at the three c.g. locations. The pitch-damping 

calculations were performed separate from the other calculations using a steady lunar coning 

motion described by Weinacht et al. (18). These are still steady-state computations but entail 

using a rotating reference frame to put the projectile body in a coning motion. The procedure 
12

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (F) BRL-MR-775

Exp (M) BRL-MR-775

Exp (R) BRL-MR-775

Exp (F) BRL-R-876

Exp (M) BRL-R-876

Exp (R) BRL-R-876

CFD SSC (F) CFD++

CFD SSC (M) CFD++

CFD SSC (R) CFD++

a q m m C

C  

Figure 8. Pitch-damping moment coefficient vs. Mach number: standard-base ANSR. 

was recently re-evaluated and more details can be found in DeSpirito et al. (12). (No unsteady 

RANS/LES computations were performed to obtain pitch damping.) The predicted pitchdamping values compare very well with the experimental data (13, 14), especially above the 

sonic velocity. The results at subsonic velocities show some degree of underprediction of the 

pitch damping, although there is significantly more scatter in the experimental data at these 

Mach numbers. Still, the CFD predictions fall within the error bounds, albeit at the ends of the 

error bounds at the lowest Mach numbers. 

The primary goal of this report is to evaluate the prediction of the Magnus moment. Figure 9 

shows the comparison of predicted and experimental Magnus moment for the three c.g. locations 

investigated. The predicted values fall within the data very well for M ≥ 1. However, the 

experimental Magnus moment data shows a larger downward trend for M < 1 than the predicted 

values. Note there is an extremely large amount of scatter in the experimental data at all c.g. 

locations. Still, there is definitely a larger negative trend in the experimental data than there is in 

the CFD predictions. It is interesting to note that the steady-state RANS computations using the 

cubic k- model perform poorly at the lower Mach numbers. Therefore, this model was not 

chosen for the basic steady-state RANS computations in this and earlier studies (6, 7). 

Time-accurate RANS/LES simulations do not improve the predictions, showing little change 

from the steady-state RANS predictions. This result differs from that found for the M910 

projectile (6, 7), which showed an improvement in the Magnus moment prediction at the lower 

Mach numbers using unsteady RANS/LES computations. The ANSR does not exhibit the very 

large negative Magnus moment indicative of nonlinear effects, as did the M910. Weinacht (4) 

demonstrated that the shape of the projectile base (boattailed or not) is likely the dominant cause
13

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (F) BRL-MR-775

Exp (M) BRL-MR-775

Exp (R) BRL-MR-775

Exp (F) BRL-R-876

Exp (M) BRL-R-876

Exp (R) BRL-R-876

CFD SS RANS (SB-1)

CFD SS RANS (cke, SB-2)

CFD RANS/LES (SB-2)

CFD RANS/LES (SB-1)

p n C

Figure 9. Magnus moment coefficient (at  = 2°) vs. Mach number: standard-base ANSR. 

of nonlinear Magnus moment. Weinacht showed that replacing a rounded base with a sharp 

corner nearly eliminated the nonlinear Magnus of a small-arms projectile. While there is 

definitely more scatter in the experimental data at Mach numbers <1, the variation of Magnus 

moment with angle of attack is still fairly linear (13, 14). It is reasonable that the nonlinear 

Magnus effects are captured better with the unsteady RANS/LES computations. Since there was 

not a large degree of nonlinearity in the ANSR Magnus moment, the steady-state computations 

performed adequately. However, the reason for the overpredicting of Magnus moment at the 

lower Mach numbers in this and other studies (5–7, 12) is unknown. 

It is somewhat disturbing that the cubic k- model does not perform well since this is the model 

that the RANS/LES computations revert to in the RANS-mesh regions. Since this near-body 

RANS region is primarily responsible for the Magnus effects, one wonders if the cubic k- model 

is adversely affecting the unsteady RANS/LES predictions also. 

Figure 10 shows a comparison of the Magnus moment of the standard-base and boattail ANSR. 

The experimental data for the boattail ANSR is from wind tunnel (WT) experiments (15), while 

the data for the standard-base ANSR is from free-flight, aeroballistic spark range experiments. 

The Magnus moment for the boattail ANSR is much higher than that for the standard-base 

ANSR at the lower Mach numbers. At the higher Mach numbers, the Magnus moment 

approaches that of the standard-base ANSR. The values of Magnus moment are again 

overpredicted; however, the peak just below Mach 1 is captured very well.
14

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mach Number

Exp (M) BRL-MR-775 (SB)

Exp (M) BRL-R-876 (SB)

Exp (M) BRL-R-1720 (BT,WT)

CFD SS RANS (SB-1)

CFD SS RANS (BT-1)

p n C

Figure 10. Magnus moment coefficient vs. Mach number: comparison of standard-base and 

boattail ANSR. 

The flow field effects on Magnus force and moment are subtle and difficult to observe from flow 

field visualizations or plots of total forces or moments. A useful tool is to plot the distribution of 

the aerodynamic forces and moments along the projectile body to determine what part of the 

body contributes to each aerodynamic coefficient. The distribution of Magnus force and moment 

on the standard-base ANSR is shown in figure 11. The location of the c.g. is illustrated on the 

profile of the projectile (only results for the middle c.g. location are shown). The Magnus force 

is negative (normal pointing to the port side of the projectile) for a right-hand twist spin and 

positive angle of attack. There is very little difference in Magnus force and moment distributions 

between the steady-state RANS and unsteady RANS/LES computations. There is a reduction in 

the absolute value of Magnus force (figure 11a) over the last caliber of the ANSR body for the 

subsonic and transonic Mach numbers. This is likely due to an interaction of the wake flow on 

the asymmetric pressure distribution that generates the Magnus effect. There is a corresponding 

decrease in the value of the Magnus moment (figure 11b) over this same region. 

The boattail ANSR (figure 12) also shows very little difference of the distribution of Magnus 

force and moment between the RANS and RANS/LES computations. There is a decrease in the 

total value of Magnus force (figure 12a) at all Mach numbers for the boattail ANSR compared to 

the standard-base ANSR. However, the shape of the distribution in the last caliber of the body is 

much different for the boattail configuration at the subsonic and transonic Mach numbers. The 

Magnus moment (figure 12b) shows similar trends.
15

C.G.

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Axial Position (m)

RANS, M=0.70

RAN/LES, M=0.70

RANS, M=1.10

RAN/LES, M=1.10

RANS, M=1.80

RAN/LES, M=1.80

p Y C

(a) 

C.G.

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Axial Position (m)

RANS, M=0.70

RAN/LES, M=0.70

RANS, M=1.10

RAN/LES, M=1.10

RANS, M=1.80

RAN/LES, M=1.80

p n C

(b) 

Figure 11. Distribution of Magnus (a) force and (b) moment coefficient (at  = 2°) 

on standard-base ANSR: comparison of RANS and RANS/LES 

computations. 
16

C.G.

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Axial Position (m)

RANS, M=0.70

RAN/LES, M=0.70

RANS, M=1.10

RAN/LES, M=1.10

RANS, M=1.80

RAN/LES, M=1.80

p Y C

(a) 

C.G.

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Axial Position (m)

RANS, M=0.70

RAN/LES, M=0.70

RANS, M=1.10

RAN/LES, M=1.10

RANS, M=1.80

RAN/LES, M=1.80 p n C

(b) 

Figure 12. Distribution of Magnus (a) force and (b) moment coefficient (at  = 2°) 

on boattail ANSR: comparison of RANS and RANS/LES computations. 

Figure 13 shows the comparison of the distributions of Magnus force and moment computed 

from steady-state RANS computations for the standard-base and boattail version of the ANSR. 

There is little difference in the Magnus force and moment distributions for the supersonic Mach 

number. At subsonic and transonic Mach numbers, the Magnus force and moment distributions 
17

C.G.

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Axial Position (m)

NBT, M=0.70

BT, M=0.70

NBT, M=1.10

BT, M=1.10

NBT, M=1.80

BT, M=1.80

p Y C

(a) 

C.G.

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Axial Position (m)

NBT, M=0.70

BT, M=0.70

NBT, M=1.10

BT, M=1.10

NBT, M=1.80

BT, M=1.80 p n C

(b) 

Figure 13. Distribution of Magnus (a) force and (b) moment coefficient (at α = 2°): 

comparison of effect of ANSR base shape for steady-state RANS 

computations. 
18

for the standard-base and boattail projectiles are nearly the same until the last caliber of the body. 

The data from figures 11–13 show that steady-state RANS computations predict the Magnus 

force and moment to the same level as unsteady RANS/LES computations for the ANSR 

configuration. Unsteady RANS/LES computations may very well be unnecessary for projectiles 

with sharp corners at the base, regardless if a boattail is present or not. Weinacht (4) found that 

the sharp-cornered base of the projectile nearly eliminated nonlinear Magnus effects. This may 

indicate that steady-state RANS simulations are adequate for projectiles that do not exhibit 

severe nonlinear Magnus moments. 

Figure 14 shows typical instantaneous flow field patterns at  = 2° for the four Mach numbers at 

which the unsteady RANS/LES simulations were performed. Contours of Mach number are 

shown to illustrate the flow field for the standard-base and boattail ANSR configurations. The 

approximate extent of the LES mesh region is also shown on the images. As the coherent flow 

structures in the projectile wake leave the LES mesh region and enter the RANS (stretched) 

region of the mesh, they begin to rapidly dissipate due to numerical diffusion. The wake 

structures are also rotating about the x-axis due to the rotation of the projectile. 

There are some small differences in the wake structure between the two base configurations, but 

they are generally the same. The differences are seen in figure 15, which shows contours of 

turbulent kinetic energy. The boattail directs the wake radially inward, generating a slightly 

smaller-diameter wake. As the Mach number increases to the supersonic regime, the unsteady 

wake begins to resemble that of a volume-average-type (steady-state RANS) flow field, which 

was illustrated previously (6, 7). 

Figure 16 shows the comparison of the instantaneous flow field for the unsteady RANS/LES 

computations on the LES mesh (figure 16a) with that performed with the RANS mesh 

(figure 16b). Two boundaries are indicated in figure 16b; the first is where the nearly isotropic 

mesh extended. After this point, the mesh was stretched in the axial direction a small amount to 

the location of the vertical black line, where the mesh was then stretched at rates typical for 

RANS simulations. The near-wake flow field is very similar, and the RANS mesh did not 

produce overly poor results (e.g., see figure 9). Performing RANS/LES computations on RANSlike meshes is a suitable way to determine simulation parameters (e.g., time steps, inner iteration 

convergence, etc.) using less computational resources before moving onto the final computations 

on the LES mesh. 
19

 

 (a) 

 (b) 

 (c) 

 (d) 

Figure 14. Instantaneous Mach number contours from unsteady RANS/LES calculations: 

standard-base and boattail ANSR at α = 2° and (a) Mach 0.7; (b) Mach 0.8; (c) Mach 

1.1; and (d) Mach 1.8. 
20

 (d) 

(b) 

(a) 

(c) 

Figure 15. Instantaneous Mach number contours from unsteady RANS/LES calculations: standard-base 

and boattail ANSR at  = 2° and (a) Mach 0.7; (b) Mach 0.8; (c) Mach 1.1; and (d) Mach 1.8.
21

 (a) (b) 

Figure 16. Instantaneous turbulent kinetic energy contours from unsteady RANS/LES calculations: standard-base 

ANSR at Mach 0.7 using (a) RANS/LES and (b) RANS mesh. 

4. Summary and Conclusions 

The aerodynamic coefficients of the 7-cal. U.S. Army-Navy Spinner Rocket were characterized 

using CFD calculations and validated using archival experimental data. The static aerodynamic 

coefficients, as well as the pitch-damping and roll-damping coefficients, were predicted well by 

the CFD. As expected, steady-state CFD calculations were adequate to predict these 

coefficients. 

The prediction of Magnus moment was very good at Mach numbers >1. As found for other 

projectile configurations, the Magnus moment at Mach numbers <1 was overpredicted. 

However, unlike previous studies, unsteady hybrid RANS/LES calculations did not improve the 

prediction of Magnus moment at the lower Mach numbers. A highly nonlinear Magnus moment 

was not observed with the standard-base or boattailed ANSR projectiles. Both these designs 

have a sharp corner at the radial edge of the base, which has been shown (4) to reduce the 

nonlinear Magnus moment. It is possible that unsteady RANS/LES calculations may improve 

predictions for configurations where there is a highly nonlinear Magnus moment present. This 

will be the subject of further study. 

Distributions of Magnus moment along the projectile body showed that the largest difference in 

Magnus moment between configurations and Mach numbers was in the last caliber of the body. 

Plotting these distributions is a very good way to determine the source of the contribution to each 

aerodynamic coefficient. 
22

5. References 

1. Sturek, W. B.; Schiff, L. B. Computations of the Magnus Effect for Slender Bodies in 

Supersonic Flow. AIAA Journal 1982, 20 (12), 1724–1731. 

2. Nietubicz, C. J.; Sturek, W. B.; Heavey, K. R. Computations of Projectile Magnus Effect at 

Transonic Velocities; BRL-TR-02515; U.S. Army Ballistics Research Laboratory: Aberdeen 

Proving Ground, MD, August 1983. 

3. Sahu, J. Transonic Navier-Stokes Computations for a Spinning Body of Revolution; 

BRL-TR-3265; U.S. Army Ballistics Research Laboratory: Aberdeen Proving Ground, MD, 

September 1991. 

4. Weinacht, P. Characterization of Small-Caliber Ammunition Performance Using a Virtual 

Wind Tunnel Approach; AIAA-2007-6579; Proceedings of the AIAA Atmospheric Flight 

Mechanics Conference, Hilton Head, SC, 20–23 August 2007. 

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

Supersonic Speeds; ARL-TR-2850; U.S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, September 2002. 

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

Moment of a Spinning Projectile; AIAA-2004-4713; Proceedings of the AIAA Atmospheric 

Flight Mechanics Conference, Providence, RI, 16–19 August 2004. 

7. DeSpirito, J.; Plostins, P. CFD Prediction of M910 Projectile Aerodynamics: Unsteady 

Wake Effect on Magnus Moment; AIAA-2007-6580; Proceedings of the AIAA Atmospheric 

Flight Mechanics Conference, Hilton Head, SC, 20–23 August 2007. 

8. McCoy, R. L. Modern Exterior Ballistics: The Launch and Flight Dynamics of Symmetric 

Projectiles; Schiffer Books: Atglen, PA, 1998.

9. Simon, F.; Deck, S.; Guillen, P.; Sagaut, P. Reynolds-Averaged Navier-Stokes/Large-Eddy 

Simulations of Supersonic Base Flow. AIAA Journal 2006, 44 (11), 2578–2590. 

10. Simon, F.; Deck, S.; Guillen, P.; Cayzac, R.; Merlen, A. Zonal-Detached-Eddy Simulation 

of Projectiles in the Subsonic and Transonic Regimes. AIAA Journal 2007, 45 (7), 

1606–1619. 

11. Simon, F.; Deck, S.; Guillen, P.; Cayzac, R.; Sagaut, P.; Merlen, A. RANS/LES Simulations 

of Projectiles With and Without Rotation in the Subsonic and Transonic Regimes. 

Proceedings of the 23rd International Symposium on Ballistics, Tarragona, Spain, April 

2007. 
23

12. DeSpirito, J.; Silton, S. I.; Weinacht, P. Navier-Stokes Predictions of Dynamic Stability 

Derivatives: Evaluation of Steady State Methods; AIAA-2008-0214; Proceedings of the 46th 

Aerospace Sciences Meeting, Reno, NV, 7–10 January 2008. 

13. Schmidt, L. E.; Murphy, C. H. The Aerodynamic Properties of the 7-Caliber Army-Navy 

Spinner Rocket in Transonic Flight; BRL-MR-775; U.S. Army Ballistics Research 

Laboratory: Aberdeen Proving Ground, MD, March 1954. 

14. Murphy, C. H.; Schmidt, L. E. The Effect of Length on the Aerodynamic Characteristics of 

Bodies of Revolution in Supersonic Flight; BRL-R-876; U.S. Army Ballistics Research 

Laboratory: Aberdeen Proving Ground, MD, August 1953. 

15. Nielsen, G. I. T.; Platou, A. S. The Effect of Conical Boattails on the Magnus 

Characteristics of Projectiles at Subsonic and Transonic Speeds; BRL-R-1720; U.S. Army 

Ballistics Research Laboratory: Aberdeen Proving Ground, MD, June 1974. 

16. Fluent, Inc. Fluent 6.3 User’s Guide, Vol. 2; Lebanon, NH, 2006. 

17. Metacomp Technologies, Inc. CFD++ User’s Manual; Agoura Hills, CA, 2006. 

18. Weinacht, P.; Sturek, W. B.; Schiff, L. B. Navier-Stokes Predictions of Pitch Damping for 

Axisymmetric Projectiles. AIAA J. Spacecraft & Rockets 1997, 34 (6), 753–761. 
24

INTENTIONALLY LEFT BLANK. 
25

Appendix. Aerodynamic Coefficient Data*

 *

Moment reference point is 4.036 cal. from nose in these tables. 
26

Table A-1. Static aerodynamic coefficient data: standard-base configuration, 7-cal. ANSR. 

Mach No.  CX CXb CXf CY CN Cl Cm Cn xcp

0.60 0 0.2837 0.1815 0.1022 –0.0009 –0.0007 –0.0179 0.0020 0.0023 — 

 2 0.2866 0.1853 0.1013 –0.0114 0.0837 –0.0179 0.1772 –0.0043 1.9183 

 5 0.3000 0.2040 0.0960 –0.0379 0.2147 –0.0179 0.4344 0.0039 2.0132 

0.70 0 0.2830 0.1838 0.0992 –0.0011 –0.0004 –0.0173 0.0011 0.0027 — 

 2 0.2855 0.1873 0.0982 –0.0118 0.0841 –0.0174 0.1781 –0.0027 1.9180 

 5 0.2986 0.2057 0.0929 –0.0381 0.2157 –0.0174 0.4376 0.0057 2.0076 

0.80 0 0.2900 0.1892 0.1008 0.0000 –0.0002 –0.0167 0.0006 0.0000 — 

 2 0.2929 0.1932 0.0997 –0.0120 0.0860 –0.0170 0.1782 –0.0021 1.9632 

 5 0.3051 0.2111 0.0940 –0.0394 0.2194 –0.0168 0.4415 0.0094 2.0235 

0.85 0 0.3007 0.1910 0.1097 0.0001 0.0000 –0.0163 0.0001 –0.0002 — 

 2 0.3035 0.1949 0.1086 –0.0123 0.0861 –0.0166 0.1792 –0.0012 1.9545 

 5 0.3155 0.2129 0.1026 –0.0392 0.2210 –0.0165 0.4413 0.0095 2.0388 

0.90 0 0.3282 0.2009 0.1273 0.0003 –0.0001 –0.0162 0.0003 –0.0008 — 

 2 0.3309 0.2046 0.1263 –0.0123 0.0885 –0.0164 0.1789 –0.0009 2.0149 

 5 0.3426 0.2224 0.1201 –0.0400 0.2267 –0.0163 0.4419 0.0117 2.0870 

0.95 0 0.3718 0.2167 0.1551 0.0000 0.0000 –0.0158 0.0001 0.0000 — 

 2 0.3751 0.2212 0.1540 –0.0125 0.0912 –0.0161 0.1790 0.0001 2.0727 

 5 0.3868 0.2388 0.1479 –0.0395 0.2325 –0.0159 0.4467 0.0118 2.1146 

0.98 0 0.4251 0.2428 0.1823 0.0001 –0.0001 –0.0159 0.0001 –0.0002 — 

 2 0.4278 0.2466 0.1811 –0.0118 0.0922 –0.0160 0.1890 –0.0025 1.9864 

 5 0.4401 0.2648 0.1753 –0.0385 0.2363 –0.0157 0.4694 0.0076 2.0492 

1.02 0 0.4911 0.2638 0.2272 0.0000 –0.0001 –0.0157 0.0003 0.0000 — 

 2 0.4928 0.2666 0.2262 –0.0118 0.0930 –0.0156 0.1924 –0.0023 1.9665 

 5 0.5060 0.2849 0.2212 –0.0386 0.2375 –0.0154 0.4790 0.0085 2.0188 

1.05 0 0.5225 0.2689 0.2536 0.0000 0.0000 –0.0152 0.0000 –0.0001 — 

 2 0.5249 0.2725 0.2523 –0.0122 0.0920 –0.0154 0.1983 –0.0013 1.8798 

 5 0.5396 0.2923 0.2472 –0.0395 0.2366 –0.0152 0.4923 0.0111 1.9550 

1.10 0 0.5272 0.2606 0.2666 0.0001 0.0000 –0.0150 0.0001 –0.0002 — 

 2 0.5306 0.2645 0.2661 –0.0133 0.0932 –0.0150 0.1997 0.0024 1.8946 

 5 0.5467 0.2844 0.2623 –0.0417 0.2380 –0.0149 0.4977 0.0189 1.9450 

1.20 0 0.5181 0.2498 0.2683 0.0000 –0.0001 –0.0143 0.0003 0.0000 — 

 2 0.5211 0.2535 0.2675 –0.0142 0.0949 –0.0143 0.2008 0.0056 1.9189 

 5 0.5357 0.2720 0.2638 –0.0442 0.2363 –0.0143 0.5243 0.0271 1.8172 

1.40 0 0.4817 0.2266 0.2551 0.0000 0.0000 –0.0135 0.0000 0.0000 — 

 2 0.4875 0.2300 0.2575 –0.0156 0.0956 –0.0139 0.2226 0.0096 1.7073 

 5 0.5029 0.2476 0.2553 –0.0479 0.2449 –0.0138 0.5615 0.0370 1.7429 

1.80 0 0.4209 0.1848 0.2361 0.0000 0.0000 –0.0124 0.0000 0.0000 — 

 2 0.4248 0.1888 0.2359 –0.0158 0.1022 –0.0124 0.2390 0.0126 1.6976 

 5 0.4410 0.2061 0.2349 –0.0483 0.2705 –0.0123 0.5929 0.0458 1.8444 

2.50 0 0.3426 0.1334 0.2092 0.0000 0.0000 –0.0102 0.0000 0.0000 — 

 2 0.3460 0.1368 0.2093 –0.0141 0.1152 –0.0102 0.2298 0.0142 2.0409 

 5 0.3603 0.1503 0.2100 –0.0421 0.3080 –0.0100 –0.6840 0.0478 2.2210 
27

Table A-2. Aerodynamic derivative data: standard-base configuration, 7-cal. ANSR. 

Mach No.  N C m C p Cl Yp C np C m m C C q 

 0 — — –0.0567 — — 

0.60 2 2.3975 5.0772 –0.0570 –1.0337 –0.3926 –30.06 

 5 2.4639 4.9839 –0.0570 –1.3790 0.1438 

 0 — — –0.0549 — — 

0.70 2 2.4094 5.1031 –0.0551 –1.0768 –0.2438 –30.89 

 5 2.4753 5.0208 –0.0551 –1.3868 0.2080 

 0 — — –0.0530 — — 

0.80 2 2.4638 5.1069 –0.0539 –1.0952 –0.1892 –32.11 

 5 2.5169 5.0653 –0.0535 –1.4360 0.3426 

 0 — — –0.0519 — — 

0.85 2 2.4664 5.1339 –0.0526 –1.1157 -0.1093 –33.17 

 5 2.5352 5.0635 –0.0525 –1.4288 0.3447 

 0 — — –0.0515 — — 

0.90 2 2.5362 5.1260 –0.0521 –1.1231 –0.0800 –34.70 

 5 2.6014 5.0701 –0.0516 –1.4563 0.4269 

 0 — — –0.0502 — — 

0.95 2 2.6123 5.1286 –0.0512 –1.1377 0.0083 –37.19 

 5 2.6677 5.1257 –0.0506 –1.4378 0.4314 

 0 — — –0.0504 — — 

0.98 2 2.6421 5.4152 –0.0507 –1.0725 –0.2288 –39.38 

 5 2.7110 5.3863 –0.0499 –1.4034 0.2769 

 0 — — –0.0498 — — 

1.02 2 2.6634 5.5120 –0.0496 –1.0771 -0.2081 –34.92 

 5 2.7247 5.4964 –0.0488 –1.4054 0.3100 

 0 — — –0.0482 — — 

1.05 2 2.6355 5.6827 –0.0488 –1.1072 -0.1185 –41.26 

 5 2.7143 5.6483 –0.0481 –1.4378 0.4054 

 0 — — –0.0477 — — 

1.10 2 2.6717 5.7213 –0.0475 –1.2120 0.2194 –41.73 

 5 2.7311 5.7107 –0.0473 –1.5204 0.6897 

 0 — — –0.0453 — — 

1.20 2 2.7180 5.7544 –0.0455 –1.2945 0.5053 –39.80 

 5 2.7114 6.0159 –0.0454 –1.6106 0.9863 

 0 — — –0.0428 — — 

1.40 2 2.7394 6.3794 –0.0442 –1.4233 0.8701 –52.25 

 5 2.8093 6.4420 –0.0438 –1.7437 1.3464 

 0 — — –0.0395 — — 

1.80 2 2.9289 6.8489 –0.0394 –1.4381 1.1502 –62.74 

 5 3.1038 6.8023 –0.0390 –1.7585 1.6674 

 0 — — –0.0323 — — 

2.50 2 3.3004 6.5845 –0.0323 –1.2867 1.2931 –66.54 

 5 3.5335 6.4132 –0.0319 –1.5347 1.7403 
28

Table A-3. Static aerodynamic coefficient data: boattail configuration, 7-cal. ANSR. 

Mach No.  CX CXb CXf CY CN Cl Cm Cn xcp

 0 0.1981 0.0686 0.1295 –0.0005 0.0008 –0.0155 –0.0019 0.0014 — 

0.60 2 0.2018 0.0724 0.1294 –0.0171 0.0672 –0.0159 0.2167 0.0098 0.8109 

 5 0.2150 0.0884 0.1266 –0.0498 0.1775 –0.0158 0.5240 0.0334 1.0831 

 0 0.1940 0.0654 0.1286 0.0004 0.0004 –0.0152 –0.0010 –0.0009 — 

0.70 2 0.1976 0.0692 0.1284 –0.0175 0.0674 –0.0154 0.2191 0.0114 0.7869 

 5 0.2104 0.0849 0.1255 –0.0501 0.1767 –0.0153 0.5340 0.0357 1.0133 

 0 0.1938 0.0610 0.1328 0.0000 0.0002 –0.0147 –0.0004 0.0001 — 

0.80 2 0.1971 0.0646 0.1325 –0.0179 0.0671 –0.0150 0.2247 0.0129 0.6877 

 5 0.2094 0.0800 0.1294 –0.0513 0.1758 –0.0148 0.5494 0.0396 0.9106 

 0 0.2028 0.0576 0.1452 0.0001 0.0007 –0.0144 –0.0018 –0.0002 — 

0.85 2 0.2062 0.0610 0.1452 –0.0181 0.0665 –0.0147 0.2304 0.0138 0.5703 

 5 0.2187 0.0756 0.1431 –0.0524 0.1749 –0.0146 0.5620 0.0435 0.8236 

 0 0.2275 0.0492 0.1784 0.0000 0.0002 –0.0142 –0.0006 0.0000 — 

0.90 2 0.2302 0.0519 0.1783 –0.0203 0.0637 –0.0144 0.2448 0.0205 0.1946 

 5 0.2399 0.0637 0.1762 –0.0574 0.1680 –0.0143 0.5991 0.0586 0.4691 

 0 0.2818 0.0413 0.2405 0.0000 0.0000 –0.0140 0.0000 –0.0001 — 

0.95 2 0.2852 0.0450 0.2402 –0.0222 0.0722 –0.0142 0.2369 0.0286 0.7556 

 5 0.3002 0.0631 0.2372 –0.0600 0.1945 –0.0141 0.5652 0.0720 1.1302 

 0 0.3463 0.0833 0.2630 0.0000 –0.0001 –0.0139 0.0002 0.0000 — 

0.98 2 0.3495 0.0870 0.2626 –0.0192 0.0791 –0.0142 0.2290 0.0201 1.1398 

 5 0.3646 0.1054 0.2592 –0.0532 0.2082 –0.0140 0.5554 0.0529 1.3679 

 0 0.4374 0.1347 0.3028 0.0000 0.0000 –0.0136 0.0000 0.0000 — 

1.02 2 0.4410 0.1388 0.3022 –0.0171 0.0833 –0.0139 0.2230 0.0140 1.3585 

 5 0.4584 0.1597 0.2987 –0.0493 0.2166 –0.0138 0.5461 0.0415 1.5145 

 0 0.4555 0.1372 0.3184 0.0000 0.0000 –0.0135 0.0000 0.0000 — 

1.05 2 0.4602 0.1417 0.3185 –0.0166 0.0834 –0.0137 0.2218 0.0126 1.3784 

 5 0.4778 0.1626 0.3152 –0.0479 0.2180 –0.0136 0.5432 0.0381 1.5442 

 0 0.4736 0.1414 0.3322 0.0000 0.0000 –0.0133 0.0000 0.0000 — 

1.10 2 0.4784 0.1460 0.3324 –0.0163 0.0849 –0.0133 0.2220 0.0118 1.4206 

 5 0.4961 0.1669 0.3292 –0.0472 0.2203 –0.0134 0.5458 0.0364 1.5582 

 0 0.4649 0.1426 0.3223 –0.0001 0.0000 –0.0129 0.0000 0.0001 — 

1.20 2 0.4692 0.1468 0.3223 –0.0164 0.0853 –0.0129 0.2289 0.0121 1.3529 

 5 0.4878 0.1678 0.3200 –0.0481 0.2242 –0.0129 0.5612 0.0388 1.5332 

 0 0.4360 0.1400 0.2960 0.0000 0.0001 –0.0123 –0.0001 0.0001 — 

1.40 2 0.4420 0.1439 0.2981 –0.0167 0.0902 –0.0126 0.2375 0.0127 1.4010 

 5 0.4604 0.1629 0.2975 –0.0503 0.2355 –0.0126 0.5886 0.0438 1.5365 

 0 0.3872 0.1236 0.2636 0.0000 0.0000 –0.0114 0.0000 0.0000 — 

1.80 2 0.3908 0.1270 0.2639 –0.0168 0.1029 –0.0114 0.2379 0.0151 1.7240 

 5 0.4071 0.1432 0.2640 –0.0501 0.2599 –0.0113 0.6205 0.0503 1.6486 

 0 0.3217 0.0953 0.2264 0.0000 0.0000 –0.0095 0.0000 –0.0001 — 

2.50 2 0.3256 0.0984 0.2272 –0.0150 0.1120 –0.0096 0.2383 0.0162 1.9086 

 5 0.3399 0.1111 0.2288 –0.0430 0.3005 –0.0094 0.5790 0.0498 2.1094 
29

Table A-4. Aerodynamic derivative data: boattail configuration, 7-cal. ANSR. 

Mach No.  N C m C p Cl Yp C np C

 0 — — –0.0494 — — 

0.60 2 1.9257 6.2107 –0.0506 –1.5561 0.8947 

 5 2.0362 6.0128 –0.0503 –1.8153 1.2168 

 0 — — –0.0481 — — 

0.70 2 1.9318 6.2767 –0.0490 –1.5891 1.0389 

 5 2.0268 6.1265 –0.0486 –1.8264 1.3021 

 0 — — –0.0468 — — 

0.80 2 1.9227 6.4379 –0.0475 –1.6252 1.1707 

 5 2.0168 6.3032 –0.0471 –1.8680 1.4424 

 0 — — –0.0458 — — 

0.85 2 1.9048 6.6015 –0.0467 –1.6440 1.2571 

 5 2.0073 6.4482 –0.0463 –1.9101 1.5829 

 0 — — –0.0450 — — 

0.90 2 1.8262 7.0153 –0.0457 –1.8447 1.8650 

 5 1.9270 6.8735 –0.0454 –2.0898 2.1342 

 0 — — –0.0445 — — 

0.95 2 2.0689 6.7869 –0.0452 –2.0232 2.5990 

 5 2.2317 6.4848 –0.0447 –2.1862 2.6235 

 0 — — –0.0442 — — 

0.98 2 2.2660 6.5626 –0.0450 –1.7494 1.8252 

 5 2.3884 6.3727 –0.0444 –1.9393 1.9255 

 0 — — –0.0431 — — 

1.02 2 2.3869 6.3909 –0.0441 –1.5576 1.2705 

 5 2.4847 6.2653 –0.0437 –1.7970 1.5128 

 0 — — –0.0427 — — 

1.05 2 2.3911 6.3545 –0.0436 –1.5116 1.1472 

 5 2.5010 6.2320 –0.0431 –1.7454 1.3868 

 0 — — –0.0421 — — 

1.10 2 2.4323 6.3614 –0.0423 –1.4829 1.0766 

 5 2.5276 6.2628 –0.0425 –1.7188 1.3267 

 0 — — –0.0409 — — 

1.20 2 2.4445 6.5588 –0.0409 –1.4911 1.0988 

 5 2.5730 6.4395 –0.0410 –1.7515 1.4126 

 0 — — –0.0390 — — 

1.40 2 2.5831 6.8066 –0.0400 –1.5211 1.1521 

 5 2.7021 6.7539 –0.0400 –1.8326 1.5951 

 0 — — –0.0363 — — 

1.80 2 2.9489 6.8177 –0.0362 –1.5254 1.3773 

 5 2.9819 7.1189 –0.0360 –1.8263 1.8322 

 0 — — –0.0301 — — 

2.50 2 3.2099 6.8287 –0.0304 –1.3610 1.4719 

 5 3.4481 6.6433 –0.0300 –1.5663 1.8129 
30

INTENTIONALLY LEFT BLANK. 
31

List of Symbols, Abbreviations, and Acronyms 

ANSR Army-Navy Spinner Rocket 

CD drag coefficient 

CFD computational fluid dynamic 

CFL Courant-Friedrich-Lewy 

c.g. center of gravity 

Cl roll moment coefficient 

p Cl roll-damping coefficient 

Cm pitching moment coefficient 

m C slope of pitching moment coefficient with angle of attack 

m m C C q  pitch-damping moment coefficient sum 

CN normal force coefficient 

N C slope of normal force coefficient with angle of attack 

Cn side moment coefficient 

np C Magnus moment coefficient 

Cp pressure coefficient 

CX total axial force coefficient 

CXb axial force coefficient on projectile base 

CXf axial force coefficient on projectile forebody 

CY side force coefficient 

Yp C Magnus force coefficient 

d missile base diameter, m 

DES detached-eddy simulation 

k turbulence kinetic energy, m2/s2 

LES large-eddy simulation 

M Mach number 
32

p pressure, N/m2; projectile spin rate, radians/s 

q∞ dynamic pressure, 1⁄2  V2, N/m2 

R undamped eddy viscosity, m2/s 

RANS Reynolds-averaged Navier-Stokes 

S projectile cross-sectional area, m2

3-D three dimensional 

TVD total variation diminishing 

V free stream velocity, m/s 

WT wind tunnel 

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

xcg center of gravity location 

xcp normal force center of pressure location,   m N xcp  xcg  C /C , cal. 

 angle of attack 

 turbulence dissipation rate, m2/s3

 density, kg/m3
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INTENTIONALLY LEFT BLANK.