Navier-Stokes Computations for a Spinning Projectile From 

Subsonic to Supersonic Speeds 

by Sidra I. Silton 

ARL-TR-2850 September 2002

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

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-2850 September 2002

Navier-Stokes Computations for a Spinning Projectile From 

Subsonic to Supersonic Speeds 

Sidra I. Silton 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited.
Acknowledgments 

The author wishes to thank J. Sahu for all the discussions and assistance in using the software 

and analyzing the data. This work was supported by a grant of computer time from the 

Department of Defense High Performance Computing Major Shared Resource Center at the U.S. 

Army Research Laboratory. 

i
Contents 

Acknowledgments i

List of Figures iii

List of Tables iv

1. Introduction 1

2. Solution Technique 2

2.1 Numerical Techniques.....................................................................................................2

2.2 Semi-Empirical Method ..................................................................................................2

3. Model Geometry and Numerical Grid 3

3.1 Numerical Geometry and Grid........................................................................................3 

3.2 Semi-Empirical Geometry...............................................................................................7

4. Results and Discussion 7

4.1 Grid Resolution Study.....................................................................................................7

4.2 Data Comparison.............................................................................................................9 

4.3 Flow Physics .................................................................................................................14 

4.3.1 Subsonic Flow.....................................................................................................14 

4.3.2 Transonic Flow....................................................................................................20 

4.3.3 Supersonic Flow..................................................................................................23 

5. Summary and Conclusions 28

6. References 29

Report Documentation Page 31 

ii 
List of Figures 

Figure 1. Sketch of .50-cal. projectile used for computational model. All dimensions 

are in calibers (1 cal. = 12.95 mm). ...........................................................................................3 

Figure 2. Three-dimensional (3-D) view of computational model.................................................3 

Figure 3. Representative surface grid where sections are indicated by different colors.................4 

Figure 4. Grid used for medium resolution supersonic CFD solutions. .........................................5 

Figure 5. Grid used for subsonic and low supersonic CFD solutions.............................................6 

Figure 6. Grid used for transonic CFD solutions............................................................................6 

Figure 7. Geometry utilized in semi-empirical method (no groove, straight boattail). ..................7 

Figure 8. Pressure coefficient contour comparison between (a) medium resolution and 

(b) fine resolution grids in the yaw angle plane.........................................................................9 

Figure 9. Mach number contour comparison between (a) medium resolution and (b) fine 

resolution grids in the yaw angle plane....................................................................................10

Figure 10. Zero-yaw drag coefficient vs. Mach number data comparison. ..................................11 

Figure 11. Drag coefficient vs. Mach number data comparison...................................................11 

Figure 12. Data comparison for lift force coefficient vs. Mach number. .....................................12 

Figure 13. Data comparison for static moment coefficient vs. Mach number..............................12 

Figure 14. Comparison of data for Magnus moment coefficient vs. Mach number 

for α = 2°. ................................................................................................................................14 

Figure 15. 155-mm artillery round (a) photo and (b) Magnus moment coefficient vs. 

Mach number for various AOAs [16]......................................................................................15 

Figure 16. Comparison of data for Magnus moment coefficient vs. Mach number 

for α = 5°. ................................................................................................................................16 

Figure 17. Mach number contours for M = 0.70, α = 0°, planar cut. ...........................................16 

Figure 18. Pressure contours for M = 0.70, α = 0°, planar cut. ....................................................17 

Figure 19. Mach number contours for M = 0.70, a = 2°, yaw plane.............................................18

Figure 20. Pressure contours for M = 0.70, α = 2°, yaw plane.....................................................18 

Figure 21. Mach number contours for M = 0.70, α = 2°, non-yaw plane. ...................................19 

Figure 22. Pressure contours for M = 0.70, α = 2°, non-yaw plane. ............................................19 

Figure 23. Pressure contours for M = 0.98, α = 0°, planar cut. ....................................................21 

Figure 24. Mach number contours for M = 0.98, α = 0°, planar cut. ...........................................21 

Figure 25. Mach number contours for M = 0.98, α = 2°, non-yaw plane. ...................................22 

Figure 26. Pressure contours for M = 0.98, α = 2°, non-yaw plane. ............................................22 

Figure 27. Pressure contours for M = 0.98, α = 2°, yaw plane.....................................................23 

iii 
Figure 28. Mach number contours for M = 0.98, α = 2°, yaw plane............................................24 

Figure 29. Mach number contours for M = 2.0, α = 0°, planar cut. .............................................25 

Figure 30. Pressure number contours for M = 2.0, α = 0°, planar cut..........................................25 

Figure 31. Mach number contours for M = 2.0, α = 2°, non-yaw plane. .....................................26 

Figure 32. Pressure contours for M = 2.0, α = 2°, non-yaw plane. ..............................................26 

Figure 33. Mach number contours for M = 2.0, α = 2°, yaw plane..............................................27 

Figure 34. Pressure contours for M = 2.0, α = 2°, yaw plane.......................................................28 

List of Tables 

Table 1. Mach numbers and resulting roll rates used in CFD calculations. ...................................8 

Table 2. Grid resolution study at M = 2.7, α = 0°. .........................................................................8 

Table 3. Grid resolution study at M = 2.7, α = 2°. .......................................................................10 

iv
1. Introduction 

Many U.S. Army projectiles are slender, spinning bodies. When flown at an angle of attack 

(AOA), these projectiles are then subject to Magnus forces and moments. Although these forces 

are generally quite small (10–100 times smaller than the normal force) and can usually be 

ignored, the moment produced by these forces is critical in determining the stability of the 

projectile. Thus, it is important to be able to accurately determine these coefficients. 

Early numerical studies that have looked at spinning projectiles [1–3] have suffered from a 

number of problems. The supersonic flow study [1] seemed to have produced fairly good results 

using the Parabolized Navier-Stokes equations. However, the grid resolution was probably 

questionable due to the computational resources available. The 1983 transonic study [2] also 

produced questionable results due to inadequate grid resolution caused again by limited

computational resources. While the 1991 study [3] was able to utilize a much better grid and 

more robust supercomputers, a very limited number of Mach number and AOA were 

investigated. As the computational fluid dynamics (CFD) software and computational resources 

improve, more complete and accurate numerical aeroballistics data can be obtained. The present 

study was undertaken in order to verify the ability of a relatively new CFD software package to 

predict this more complete aeroballistics data for a spinning projectile. This particular CFD 

software has been previously used to calculate the flow fields and resulting aeroballistics data on 

nonspinning projectiles [4], but had not yet been benchmarked for a spinning projectile and the 

associated force and moment (i.e., Magnus force and moment). Prior to using this software to 

determine aeroballistics data for new spinning projectile designs, the software must be verified 

against existing data and methods to determine its accuracy. 

The .50 cal. projectile has been in use for almost 60 years. While a limited amount of 

aerodynamic data was collected during the projectile’s development and testing [5], a relatively 

large amount of aerodynamic data has been obtained over the years for these rounds. Most of the 

data have been collected within the last 15 years at the U.S. Army Research Laboratory’s (ARL) 

Free Flight Aerodynamics Range* [6]. Parabolized Navier-Stokes numerical aeroballistics data 

have also been obtained for a similarly shaped .50-cal. training round with fairly good results [7]. 

The numerical aeroballistics data obtained in this study are compared to existing range data as 

well as semi-empirically obtained aerodynamic coefficients. The flow field of the spinning 

projectile is then examined within each flow regime investigated. 

 *

 Formerly the U.S. Army Ballistic Research Laboratory Free Flight Aerodynamics Range. 

1
2. Solution Technique 

2.1 Numerical Techniques 

CFD++ Solver Software [8] (Metacomp Technologies) was utilized for this study. CFD++ 

solves the Reynolds-averaged Navier-Stokes equations with pointwise turbulence models. The 

equations are solved within a finite volume framework using an unstructured grid topology. 

Spatial discretization is accomplished using the cell face normal at the cell face centroid, which 

is obtained by reconstructing the cell centroid values. Both implicit and explicit time integration 

schemes are available. For the present study, the point-implicit integration scheme was used to 

solve the steady-state simulation. Ramping of the Courant number allowed for quicker solution 

convergence as the flow progressed toward solution, and larger time steps had less effect on the 

solution. CFD++ also allows for turbulence modeling. For the study presented herein, the 

pointwise k-ε (epsilon) turbulence model [9] was used. 

All boundary conditions have been explicitly imposed. The far field and inflow boundaries are 

set to free stream condition. The outflow boundary is set to have free stream conditions for 

subsonic outflow and to extrapolate from the previous cell if the outflow is supersonic. Free 

stream pressure and temperature are set to 101 kPa and 288 K, respectively. Density can then be 

calculated from the perfect gas assumption. For the projectile body, the boundary condition is 

set to be a no slip, isothermal wall (at 288 K) that is rotating about the x-axis. The projectile 

rotates at a speed corresponding to the rifling twist rate of the experiments [6] of 1 turn in 

38.1 cm. 

2.2 Semi-Empirical Method 

The semi-empirical solution was obtained using AP98 code [10] (Aeroprediction Inc.). AP98 

combines a large database of experimental results and well-verified numerical results with 

theoretical methods in order to obtain aerodynamic coefficients and dynamic derivatives. The 

theoretical methods include second-order van Dyke and second-order shock expansion theory as 

well as thin wing theory and slender body assumptions. It allows coefficients to be obtained in a 

relatively short period of time within engineering accuracy as no grid is needed. 

2
3. Model Geometry and Numerical Grid 

3.1 Numerical Geometry and Grid 

The computational model is a .50-cal. (1 cal. = 12.95 mm) projectile, 4.46 cal. in length with 

a .16-cal.-long, 0.02-cal.-deep groove, and a 9° filleted boattail [6] (see Figures 1 and 2). 

A center of gravity location of 2.68 cal. (34.706 mm) from the nose tip was utilized. 

Figure 1. Sketch of .50-cal. projectile used for computational model. All dimensions are in

calibers (1 cal. = 12.95 mm). 

Figure 2. Three-dimensional (3-D) view of computational model.

3
All grids used in this study were 3-D hexahedral meshes created using GRIDGEN Version 13 

software [11] (Pointwise Inc.). The grid is created in a multiblock format in order to allow for a 

better quality mesh and then written in an unstructured format. While GRIDGEN Version 13 

does not output meshes in CFD++ format, FLUENT Version 5 software [12] (Fluent Inc.) output 

format is available, which CFD++ is able to import. 

Four separate meshes were constructed over the course of this study: one for subsonic and low 

supersonic cases, one for a near-sonic case, and two for supersonic cases. Two different 

computational grids were created for the supersonic cases in order to verify grid independence. 

In each case, the far field boundary was created so that it would not interfere with the flow field 

in the regime investigated (subsonic, transonic, and/or supersonic). 

The mesh is composed of 10 structured blocks. The surface grid (Figure 3) was generated first 

and consists of four sections: two small squares (pink) projected on the nose and base to 

eliminate singularities on the symmetry axis, the groove (red), and the remainder of the body 

(blue). After constructing a companion far field grid, radial planes were constructed between 

these two surface grids in order to create 3-D blocks. Two of these blocks are small rectangular 

blocks beginning at the small projected squares on the nose and base of the body and extending 

to the outer boundary (each 30 H 30 cells on the body), with the axial dimension equal to the 

radial dimension of the remaining flow field. The rest of the body is divided into quadrants 

based on the corners of the surface squares, with the groove blocked separately. Each quadrant 

of the groove contains 30 radial cells, 30 circumferential cells, and 18 axial cells, while each 

quadrant for the remainder of the flow field varied with the position of the far field boundary. In 

order to accurately capture the flow in the turbulent boundary layer, the first radial surface 

spacing was always set to 1.07 μm.

Figure 3. Representative surface grid where sections are indicated by different colors. 

4
For both supersonic grids, the outflow boundary was two body lengths behind the base of the 

model, the inflow boundary was just over one body length in front of the model, and the 

circumferential boundary was also just over one body length away from the model (Figure 4). 

The medium resolution grid contained 30 circumferential cells, 80 radial cells (away from

the body), and 279 cells along the body surface and far field boundary in one quadrant 

(2,737,658 cells total) while the finer resolution grid contained 30 circumferential cells, 

150 radial cells, and 296 cells along the body surface in one quadrant (5,413,314 cells total). 

The additional radial cells were added near the surface so that the boundary layer cell thickness 

grew more slowly. The additional axial cells were added in the nose and base regions. The 

greater number of cells in the radial and axial directions for the finer grid allowed finer details of 

the flow field to be studied. 

Figure 4. Grid used for medium resolution supersonic CFD solutions. 

The mesh created for the subsonic and low supersonic cases was similar in structure to that 

created for the supersonic case. The far field boundary was moved farther away from the 

projectile body (Figure 5) to accommodate the weaker shocks, expansions, and longer recovery 

distances of flow field features. The new inflow boundary was located just under three body 

lengths away, the outflow boundary four body lengths away, and the circumferential boundary 

almost five body lengths away. While the circumferential dimension remained constant at 

30 cells in one quadrant, the radial dimension was increased to 100 cells and the axial dimension 

to 289 cells in order to accommodate the new position of the far field boundary. The cells were 

again added on the nose and the base of the projectile. The complete mesh consisted of 

3,712,800 cells. 

5
Figure 5. Grid used for subsonic and low supersonic CFD solutions. 

Due to the near-normal shock and large recovery distances of a near-sonic flow, a special grid 

was constructed for this case. The grid further distanced the far field boundary from the body 

and added additional cells in the radial direction (Figure 6). Specifically, the outflow boundary 

was expanded to eight body lengths downstream and the circumferential boundary was expanded 

to almost 10 body lengths away. The resulting grid contained 30 circumferential cells, 289 axial 

cells, and 109 radial cells in each quarter mesh. This increased the total mesh size to 4,041,120 

cells. 

Figure 6. Grid used for transonic CFD solutions. 

6
3.2 Semi-Empirical Geometry 

The assumptions within the semi-empirical solution all pertained to the afterbody geometry due 

to limitations of the software. First, the groove was eliminated as a decrease in body radius 

followed by a subsequent increase could not be handled by the software. Additionally, the 

filleted boattail was eliminated and replace with a standard 9° angle boattail. These 

modifications to the geometry produced a model with the cross section in Figure 7 and allowed 

for a solution to be calculated. The center of gravity for the semi-empirical solution was 

assumed to be that of the experimental model with the groove and filleted boattail. 

Figure 7. Geometry utilized in semi-empirical method (no groove, straight boattail). 

4. Results and Discussion 

A number of numerical cases were studied in order to cover the range of experimental data 

available. The Mach numbers and corresponding roll rates can be found in Table 1. A CFD 

calculation was completed for each Mach number at 0° and 2° AOA. The dynamic aerodynamic 

coefficients were calculated assuming that the values at 0° AOA were zero. Only CD0 was 

obtained directly from the 0° AOA calculation. A limited number of CFD calculations were also 

completed at 5° AOA in order to determine how the coefficients changed. Solutions for the 

semi-empirical code were obtained at 2° AOA for the static coefficients. The dynamic 

derivatives were determined at 5° AOA. All results are presented in the convention put forth by 

Murphy [13]. 

4.1 Grid Resolution Study 

A grid resolution study was conducted for the supersonic, M = 2.70, case at both 0° and 2° AOA. 

This case has the thinnest boundary layers and the most distinct flow physics for comparison. 

The solutions on the 2.7 million cell (medium) and 5.4 million cell (fine) grids were compared 

for differences in flow physics and aerodynamics coefficients, as well as for agreement with 

experimental data. 

7
Table 1. Mach numbers and resulting roll rates used in

CFD calculations. 

Mach No. Roll Rate 

(rad/s) 

0.70 3927.9

0.85 4768.5

0.90 5050.3

0.94 5274.7

0.98 5499.2

1.05 5892.0

1.10 6172.5

1.25 7014.2

1.50 8417.0

2.00 11222.8

2.70 15150.6

At 0° AOA, the medium and fine grids gave nearly identical results with values very near zero 

for all coefficients, except the drag coefficient, CD, as expected (Table 2). CD differed by less 

than 1% and the flow fields were very similar. Experimental data are only available for CD, as 

all other coefficients are 0 at 0° AOA. 

Table 2. Grid resolution study at M = 2.7, α = 0°. 

Grid/Experiment CD CL CN CMp CM

Medium grid 0.2935 −1.33e-5 1.49e-4 −0.0011 2.45e-4

Fine grid 0.2964 2.80e-5 2.23e-5 0.0004 8.33e-5

M33 experiment [6] 0.279 — — — —

When the AOA is increased to 2°, the flow gets more complicated. The flow is no longer 

axisymmetric. The flow around the body was nearly identical between the two grids (Figures 8 

and 9), although some of the flow features were better defined on the fine grid. Most of the 

aerodynamic coefficients, including Magnus moment coefficient, differed by no more than 0.5% 

(Table 3). Only CD and the normal force coefficient, CNα, differed to a greater extent. CD

increased by just over 1% from the medium grid to the fine grid and CNα increased by ~10%. 

Both sets of numerical results agreed quite well with the experimental data. While the agreement 

was slightly better with the fine grid, it was not worth the nearly doubled computational time that 

was required in order to obtain a converged solution on the fine grid as compared to that for the 

medium grid. As such, the medium grid was utilized for the remainder of the supersonic cases, 

and comparable grids were used for the transonic and subsonic cases. 

8
(a) 

(b) 

Figure 8. Pressure coefficient contour comparison between (a) medium

resolution and (b) fine resolution grids in the yaw angle plane. 

4.2 Data Comparison 

All experimental data used in this data comparison were obtained in the ARL Aerodynamics 

Range [6]. Although experimental data were obtained for three different rounds, only the data 

for the M33 ball projectile (M33 experiment) and the M8 API projectile (M8 experiment) were 

used where they were available. Plotted experimental data are limited to 1° < α ≤ 5° because 

smaller yaw angles were unavailable and in order to eliminate large yaw angle nonlinearities in 

the analysis as the CFD results were obtained at small α [7]. The zero-yaw drag force 

coefficient, CD0, was compared first. As expected, very good agreement was obtained between 

the computational results and the experimental results (Figure 10). It is interesting to note that

while the numerical solution overpredicts the experimental CD0 results, the semi-empirical 

9
(a) 

(b) 

Figure 9. Mach number contour comparison between (a) medium

resolution and (b) fine resolution grids in the yaw angle 

plane. 

Table 3. Grid resolution study at M = 2.7, α = 2°. 

Grid/Experiment CD CLα CNα CMpα CMα 

Medium grid 0.3027 2.23 0.0637 0.21 2.79

Fine grid 0.3062 2.23 0.0710 0.21 2.80

M33 experiment [6] 0.2813 2.21 — 0.15 3.01

M8 experiment [6] 0.2991 2.42 — 0.24 2.85 

Note: Experimental data given at closest data point.

solution underpredicts them. Actual drag coefficient, CD, was also compared (Figure 11) and 

shown to be in good agreement. Numerical drag appears to be a bit high, but could be accounted 

for in that the AOAs are not quite the same. Lift force coefficient, CLα (Figure 12), and static 

moment coefficient, CMα (Figure 13), were also directly compared to experimental data with 

much success. This indicates that the CFD is correctly predicting the aerodynamic coefficients. 

10
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5 3

Mach Number

CD0

M33 experiment [6]

AP98

Numerical

Figure 10. Zero-yaw drag coefficient vs. Mach number data comparison.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.5 1 1.5 2 2.5 3

Mach Number

C

D

M33 experiment [6]

M8 experiment [6]

AP98

Numerical (aoa=2°)

Numerical (aoa=5°)

Figure 11. Drag coefficient vs. Mach number data comparison. 

11
0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 0.5 1 1.5 2 2.5 3

Mach Number

CLα

M33 experiment [6]

M8 experiment [6]

Numerical (aoa=2°)

Numerical (aoa=5°)

Figure 12. Data comparison for lift force coefficient vs. Mach number. 

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 0.5 1 1.5 2 2.5 3

Mach Number

C

M

α

M33 experiment [6]

M8 experiment [6]

AP98

Numerical (aoa=2°)

Numerical (aoa=5°)

Figure 13. Data comparison for static moment coefficient vs. Mach number. 

12
The discontinuity in the AP98 results in Figure 13 at Mach 1.6 is due to the theory used to 

calculate the coefficients changing. Aerodynamic coefficients are not much affected by body 

roll, however, and were expected to be in good agreement as this CFD code had been used many 

times previously with excellent results. The question here was how well the dynamic derivatives 

would match those of the experiment. 

Unfortunately, due to the way in which the CFD was conducted (i.e., fixed boundary conditions), 

two of the dynamic derivatives—roll damping moment and pitch damping moment—could not 

be obtained. The Magnus moment coefficient, CMpα, could be obtained, as the rotating wall 

boundary condition was utilized. As a range of AOAs was investigated experimentally, it was 

important to extrapolate the experimental data to the AOAs investigated numerically. The 

CMpα was plotted vs. effective square yaw by McCoy [6] for a given Mach number range for the 

combined M33 and M8 data. This shows that even at α = 2°, CMpα has a nonlinear component. 

The M33 and M8 data have recently been re-reduced [14] using the newer range reduction 

software, ARFDAS [15]. The result of the multiple data fit using this software is also presented 

here. At α = 2° (Figure 14), agreement between the experimental and numerical data appears to 

be fair to good in the subsonic regime and the higher supersonic regime. However, agreement 

between the different experimental data sets is not very good in the transonic regime. This can 

be attributed to the differences that occur in the engraving of the rounds during rifling as the 

rounds are made of different materials. The complete lack of engraving on the numerical model 

could account for the disagreement with the experimental results in the transonic regime as well 

as in the lower supersonic regime. In fact, if one compares the shape of the Magnus moment 

coefficient curve obtained experimentally for the 155-mm artillery round [16] at α = 2° (Figure 

15) to that obtained numerically for the .50-cal. round at α = 2° (Figure 14), they are nearly 

identical. This further substantiates the hypothesis that the engraving on the projectile surface 

does affect the Magnus moment coefficient. These effects are less pronounced at α = 5° where 

the data trends in the numerical and experimental data appear to agree rather well (Figure 16). 

As only a few numerical data points were obtained, it is possible that the peak value in the 

transonic regime was not calculated. Agreement in the subsonic and supersonic regime is again 

quite good. The semi-empirical results (obtained from AP98 [10]) agree with the numerical and 

experimental data at α = 5° in the subsonic regime, appear to split the difference near Mach 1, 

and overpredict the results in the supersonic regime. Again, the data trend is the same with the 

maximum coefficient being obtained very near Mach 1. 

Based on comparison to experimental and semi-empirical data, the CFD is found to accurately 

predict the aerodynamic coefficients and limited dynamic derivatives. As such, this CFD solver 

should be able to accurately predict the flow physics. 

13
-1.5

-1

-0.5

0

0.5

1

0 0.511.522.5 3

Mach Number

CMp

α

M33/M8 experiment[6]

M33 experiment[14]

M8 experiment[14]

Numerical

Figure 14. Comparison of data for Magnus moment coefficient vs. Mach number for α = 2°. 

4.3 Flow Physics 

The Mach numbers investigated encompass three projectile flight regimes: (1) subsonic, (2) 

transonic, and (3) supersonic. Because each flight regime has different flow characteristics, they 

are discussed separately. Because the flows at α = 2° and α = 5° are similar, only the α = 0° and 

α = 2° flows are discussed. 

4.3.1 Subsonic Flow

Only one case, Mach 0.70, falls completely within the subsonic regime, as the flow remains less 

than Mach 1 everywhere along the body (Figure 17). Figure 17 shows the computed Mach 

contours and indicates that the flow at α = 0° is an axisymmetric flow. As expected, a small 

stagnation region is present at the nose tip. A lower velocity bubble (less than free stream) exists 

upstream of the stagnation region because this is a subsonic flow and the flow is anticipating the 

body. The flow is turned as it moves around the body, but remains everywhere subsonic. 

Additionally, the flow in the groove is nearly stagnant. The flow then expands and separates as 

it passes the boattail. This creates a low-velocity, nominal pressure region of base flow 

(Figure 18). The pressure contours confirm the axisymmetric nature of the flow. 

14
15

(a) 

(b)

Figure 15. 155-mm artillery round (a) photo and (b) Magnus moment coefficient vs. Mach

number for various AOAs [16].
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3

Mach Number

CMp

α

M33/M8 experiment [6]

M33 experiment[14]

M8 experiment [14]

AP98

Numerical

Figure 16. Comparison of data for Magnus moment coefficient vs. Mach number for α = 5°. 

Figure 17. Mach number contours for M = 0.70, α = 0°, planar cut.

16
Figure 18. Pressure contours for M = 0.70, α = 0°, planar cut. 

At α = 2°, the flow becomes asymmetrical as can be observed from the Mach number contours 

in the yaw angle plane (Figure 19). The asymmetry is not so obvious in the nose tip region, but 

becomes much more apparent as the flow moves along the ogive. A smaller adverse pressure 

gradient on the leeward side (Figure 20) allows the Mach number to increase more quickly and 

stabilize as the air moves past the groove. As the air reaches the angled boattail, a greater 

expansion angle is necessary on the leeward side (as the flow is at a 2° angle up) and the flow 

appears to separate. On the windward side, the turning angles are not as large, and the flow is 

allowed to expand, causing a region of high subsonic flow on the windward side of the boattail. 

The asymmetry in the velocity field decreases as one progresses away from this plane, except in 

the base flow region, and reaches a minimum in the non-yaw plane (Figure 21). A close look at 

the pressure contours (Figures 20 and 22) reveals some slight asymmetry in the low-pressure 

areas near the body in both planes. In the yaw angle plane (Figure 20), the higher pressure on 

the windward side of the body and lower pressure on the leeward side of the body account for the 

lift that is generated. The slightly unbalanced pressure regions in the non-yaw plane (Figure 22) 

accounts for the non-zero, though extremely small, side force and Magnus moment. 

17
Figure 19. Mach number contours for M = 0.70, a = 2°, yaw plane. 

Figure 20. Pressure contours for M = 0.70, α = 2°, yaw plane.

18
Figure 21. Mach number contours for M = 0.70, α = 2°, non-yaw plane. 

Figure 22. Pressure contours for M = 0.70, α = 2°, non-yaw plane. 

19
4.3.2 Transonic Flow

The Mach numbers investigated that fall within this category, where the local Mach number may 

be greater than 1, include Mach 0.85, 0.90, 0.94, and 0.98. The small delta between Mach 

numbers is due to the rapidly changing flow characteristics in this regime. Although each case is 

unique, the Mach 0.98 case was the most challenging as indicated by the need for an increase in 

flow field dimensions in order to properly resolve the flow field. Thus, it is the focus of this 

section. 

As previously mentioned, the far field boundary had to be moved extremely far from the body in 

order to ensure that it would not interfere with the flow field solution. The main problem with 

utilizing the grid constructed for the subsonic, other transonic, and low supersonic cases, was 

that the flow took a great distance to recover from expansion waves and shocks that were created 

as the flow moved along the body. Once an adequate flow field grid was obtained for the M = 

0.98 flow, the solution converged fairly quickly. As expected, the flow at α = 0° is 

axisymmetric. The large expanse of the flow recovery distance is very prominent in the pressure 

contours of this flow (Figure 23). The blue-green color indicates where approximately free 

stream pressure is present. One notices the extent of the low pressure regions around the body 

and the high pressure regions upstream and downstream of the body. The regions take a 

relatively great distance to recover to nominal free stream pressure. This recovery distance for 

Mach number (F enomena still occur, 

mes slightly supersonic as it expands around the ogive nose and again at 

weak shock occurs at the groove corner. Additionally, a shock occurs 

as

ect 

hosen for observation. Beginning with the simplest (i.e., non-yaw) plane, the Mach 

number contours (Figure 25) remain almost completely symmetric around the body, and the only 

the Magnus

igure 24) is not as large. Several interesting transonic ph

however. The flow beco

the boattail junction. A

prior to the base flow region, as the flow here is nearly stagnant. The region of lower Mach 

number flow in front of the nose tip is due to the flow anticipating the presence of the body,

the flow is still subsonic. 

As would be expected, the flow becomes much more interesting to study at α = 2°, if the corr

plane is c

asymmetry appears to be in the base flow. This is to be expected, as the flow is parallel as it 

approaches the body in this plane. In the base flow region, the flow is able to mix more with the

other planes as the flow separates from the spinning body contributing to the asymmetry. The

spinning body contributes to the asymmetry of the pressure in this plane as a Magnus (or side) 

force is created (Figure 26). This difference in pressure is small, but a slightly higher pressure 

on the bottom side of the body near the boattail can be seen in the figure. Although

force is small (and is usually neglected [6]), the Magnus moment is appreciable (Figure 14). 

20
21

Figure 24. Mach number contours for M = 0.98, α = 0°, planar cut.

Figure 23. Pressure contours for M = 0.98, α = 0°, planar cut. 
Figure 25. Mach number contours for M = 0.98, α = 2°, non-yaw plane. 

Figure 26. Pressure contours for M = 0.98, α = 2°, non-yaw plane. 

22
The asymmetries are much more apparent in the yaw angle plane. Many of the features are 

ose of the subsonic case. The first thing that one notices is the nose region where the 

rther downstream on the windward side of the nose than the 

leeward side of the nose (Figure 27). The pressure along the main portion of the body is nearly 

symmetric with only a slightly higher pressure on the windward side of the ogive accounting for 

t being generated for this transonic case (Figure 12). Th

ch as expected (Figure 28). The Mach number increases later on the windward side 

of the nose as the flow takes a greater distance to expand on this side. The larger supersonic 

ber region on the windward side of the boattail is due to the flow having reached the 

ber and having to expand through a lesser angle to turn

and then back parallel to the flow. These smaller turning angles allow the flow to rem

er and a higher Mach number to be achieved. 

similar to th

high pressure region extends fu

the minimum lif e Mach number contours 

look very mu

Mach num

critical Mach num parallel to the body 

ain 

attached long

4.3.3 Supersonic Flow

1.10, 1.25) and m

similar in na he 

moderate supersonic cases are fairly similar to each other. As such, discussion here will be 

limited to the Mach 2.0 case. 

Figure 27. Pressure contours for M = 0.98, α = 2°, yaw plane. 

The supersonic flow cases can easily be separated into two groups: low supersonic (Mach 1.05, 

oderate supersonic (Mach 1.5, 2.0, 2.7). The low supersonic cases are fairly 

ture and difficulty to the transonic cases and will not be discussed here. T

23
Figure 28. Mach number contours for M = 0.98, α = 2°, yaw plane. 

As with the two previous flow types, the flow at M = 2.0, α = 0° is axisymmetric. However, 

 the two previous flow types in that the flow does not anticipate the 

body. Thus, a curved bow shock exists (Figure 29) so that the flow can be turned parallel to the 

body. The bow shock also allows the flow to become subsonic along the stagnation stream

locity can continue to decrease to zero at the stagnation point. The velocity of the 

ain body is below that of the free stream velocity. The flow does not 

 subsonic, as the flow is not turned through very large angles. A sm

 edge of the groove as would be expected of any sharp corner. As the flow 

expands around the boattail, a region of high supersonic flow develops. A se

 to allow the flow to turn parallel to the free stream as well as meet th

ear layer where the pressures match (Figure 30). The regi

supersonic flow differs from

line 

so that the ve

flow along the nose and m

become all shock occurs at 

the downstream

ries of weak shocks 

then form e base flow 

region at a sh on of high pressure is 

limited to just be ited solely to area where 

the flow is expanding: ogive to cylinder, downstream edge of the groove, and around the 

e pressure field is much less responsive to the changes in the flow. 

he

in this

es

hind the bow shock. The low-pressure regions are lim

boattail. In this respect th

As always, the flow field for the α = 2° configuration is asymmetric. Looking first at the 

non-yaw angle plane, both the Mach number and pressure contours remain symmetric around t

body and only become asymmetric in the base flow regime (Figures 31 and 32). Once aga

is due to the flow in this plane being parallel to the body and only in the base flow region do

24
Figure 29. Mach number contours for M = 2.0, α = 0°, planar cut.

25

Figure 29. Mach number contours for M = 2.0, α = 0°, planar cut.

Figure 30. Pressure number contours for M = 2.0, α = 0°, planar cut.
Figure 31. Mach number contours for M = 2.0, α = 2°, non-yaw plane. 

Figure 32. Pressure contours for M = 2.0, α = 2°, non-yaw plane. 

26
the spinning of the projectile mix the planar flows. The asymmetries are again m

e. One first notices the asymmetry of the bow shock, again due to the angle 

through which the flow must turn in order to remain parallel to the body (Figure 33). The 

ain body remains fairly symmetric, becoming asymme

is turned around the boattail. The higher Mach number flow appears to 

of the body. A larger velocity gradient is also present in this region as

gles to have the flow parallel to the boattail and then parallel to 

Thus the flow on the leeward side separates earlier. The maximum lift is crea

er (Figure 12) due to the large differential in the low-pressure area on the le

the relatively higher pressure on the windward side (Figure 34). The area of higher pressure on 

the windward side of the ogive indicates that the nose is probably trying to pitch up. An 

try in the pressure field on the base flow is also noticeable for this Mach num

ost noticeable 

in the yaw angle plan

velocity field around the m tric as the flow 

be on the windward side 

 the flow turns through 

shallower an the free stream flow. 

ted at this Mach 

numb eward side and 

asymme ber. 

Figure 33. Mach number contours for M = 2.0, α = 2°, yaw plane. 

27
28

5. 

Figure 34. Pressure contours for M = 2.0, α = 2°, yaw plane. 

Summary and Conclusions 

putational study has been undertaken to predict the aerodynami

odern unstructured Navier-Stokes flow solver. Num

have been obtained for a wide range of Mach numbers covering the subsonic, transonic, and 

es. Effects of 0° and 2° AOA have been investigated. There is a 

ental range data available for these cases. Comp

ental range data as well as semi-empirical aeroprediction code 

c coefficients shows remarkably good agreement. This agreem

able to provide reasonably good aerodynamic coefficient predictions for spinning projectiles and 

A com c coefficients of a standard 

spinning projectile using a m erical results 

supersonic flight regim

significant amount of experim arison of CFD 

results to both experim

aerodynami ent shows that CFD is 

benchmarks the ion of aerodynamic 

coefficients also implies that the flow field is being correctly predicted. This allowed for much 

asier investigation of a projectile of interest to the U.S. Army and a better understanding of flow 

haracteristics.

code for further use in this area of study. Accurate predict

e

c
6. References 

 B., and L. B. Schiff. “Computations of the Magnus Effect for Slender B

AIAA Journal, vol. 20, no. 12, pp. 1724–1731, December 1982. 

. B. Sturek, and K. R. Heavey. “Computations of Projectile Magnus 

Effect at Transonic Velocities.” BRL-TR-02515, U.S. Army Ballistic Res

Aberdeen Proving Ground, MD, August 1983. 

Sahu, J. “Transonic Navier-Stokes Computations for a Spinning B

my Ballistic Research Laboratory, Aberdeen Prov

ber 1991. 

Sahu, J., K. R. Heavey, and H. L. Edge. “Numerical Computations of Supersonic F

tical Projectiles.” ARL-TR-2589, U.S. Army Research Laboratory, Aberdeen

mber 2001. 

1. Sturek, W. odies in 

Supersonic Flow.” 

2. Nietubicz, C. J., W

earch Laboratory, 

3. ody of Revolution.” 

BRL-TR-3265, U.S. Ar ing Ground, MD, 

Septem

4. low Over 

Ellip Proving 

Ground, MD, Dece

5. Hitchcock, H BRL Report 

No. 620, U.S. Army Ballistic Research Laboratories, Aberdeen Proving Ground, MD, 

October 1947. 

6. McCoy, R. L. “The Aerodynamic Characteristics of .50 Ball, M33, API, M8, and APIT, 

M20 Ammunition.” BRL-MR-3810, U.S. Army Ballistic Research Laboratory, Aberdeen 

ng

.

c. “GRIDGEN User’s Manual.” Bedford, TX, 1998. 

12. FLUENT Inc. “FLUENT 5.0 User’s Guide.” Vol. 2, Lebanon, NH, 1998. 

13. Murphy, C. H. “Free Flight Motion of Symmetric Missiles.” BRL Report No. 1193, U.S. 

Army Ballistic Research Laboratories, Aberdeen Proving Ground, MD, March 1963. 

. P. “Aerodynamic Data for Spinning Projectiles.”

Proving Ground, MD, January 1990. 

7. Guidos, B. J., and S. K. Sung. “Computational Flight Design of .50-Cal. Limited Range 

Training Ammunition.” ARL-TR-662, U.S. Army Research Laboratory, Aberdeen Provi

Ground, MD, January 1995. 

8. Metacomp Technologies. “CFD++ User’s Manual.” Westlake Village, CA, 2000

9. Goldberg, U. C., O. Peroomian, and S. Chakravarthy. “A Wall-Distance-Free K-E Model 

With Enhanced Near-Wall Treatment.” Journal of Fluids Engineering, American Society of 

Mechanical Engineers, vol. 120, pp. 457–462. 

10. Aeroprediction Inc. “AP98 User’s Guide.” King George, VA, 1998. 

11. Pointwise In

29
14. Whyte, B. Personal communication. Arrow Tech Corporation, August 2002. 

ciates. “ARFDAS 4.1.1 User’s Guide.” South Burlington, VT, 1997. 

16. Kline, R. W., and T. D. Hoffman. “Static and Magnus Aerodynamic Characteristics of the 

15. Arrow Tech Asso

155-mm M483 Projectile and Its Modifications.” TR-4942, Picatinny Arsenal, Dover, NJ, 

August 1976. 

30
31
32

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