ARMY RESEARCH LABCXMTORV 

by Paul Weinacht 

AFtL-TR-3015 July 2003 

? 

i 

Approved for public release; distribution is unlimited. 

Prediction of Projectile Performance, Stability, and 

Free-Flight Motion Using Computational Fluid Dynamics 

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. 

ARL-TR-3015 July 2003 

Prediction of Projectile Performance, Stability, and 

Free-Flight Motion Using Computational Fluid Dynamics 

Paul Weinacht 

Weapons and Materials Research Directorate, ARL 

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Paul Weinacht 

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14. ABSTRACT 

With the recent development of capabilities for predicting the damping derivatives, it is now possible to predict the stability 

characteristics and free-flight motion for projectiles using data that are derived solely from computational fluid dynamics (CFD). 

As a demonstration of the capability, this report presents results for a family of axisymmetric projectiles in supersonic flight. 

The particular configuration selected for this computational study has been extensively tested in aeroballistic ranges, and highquality experimental data have been obtained. Thin-layer Navier-Stokes techniques have been applied to compute the attached 

viscous flow over the forebody of the projectile and the separated flow in the projectile base region. Using the predicted 

aerodynamics coefficients, parameters that characterize the in-flight motion are subsequently evaluated, including the gyroscopic 

and dynamic stability factors, and the projectile’s fast and slow mode frequencies and damping coefficients. These parameters 

are then used to predict the free-flight motion of the projectile. In each case, the computational approach is validated by 

comparison with wtal data, and very good agreement between computation and experiment is found. It is believed that 

this demonstration represents the first known instance of a viscous CFD approach being applied to predict all the necessary data 

for perfotmance of linear aerodynamics stability and trajectory analyses. 

computational fluid dynamics, projectile aerodynamics, flight mechanics 

16. SECURITY CMSSlflCATiON OF: 

17. LIMITATION 13. NUMBER 

OF ABSTRACT OF PAGES 

19a. NAME OF RESPONSIBLE PERSON 

Paul Weinacht 

UL 32 1Sb. TELEPHONE NUMBER (hck& area co&) 

UNCLhsSIFIED UNCLASSIFIED UNCLASSIFIED (410) 278-4280 

ii 

Standard Form 293 (Rev. W93) 

Ptesahd by ANSI Std. 239.18 

L 
Contents 

List of Figures iv 

1. Introduction 1 

2. Computational Approach 2 

2.1 The PNS Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...* 2 

2.2 The Time-Marching Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*............................. 4 

3. Results 6 

3.1 Aerodynamic Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...* 6 

3.2 Aerodynamic Forces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

3.3 Stability and Free-Flight Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..,........................ 12 

4. Conclusion 19 

5. References 20 

List of Abbreviations and Symbols 22 

. . . 

111 
List of Figures 

. 

Figure 1. Schematic of Army-Navy Spinner rocket (ANSR) configuration .................................. 2 

Figure 2. Pitching moment coefficient vs. CG position, Mach = 1.8, ANSR ................................. 8 

Figure 3. Magnus moment coefficient vs. CG position, Mach = 1.8, ANSR ................................. 8 

Figure 4. Pitch-damping moment coefficient vs. CG position, Mach = 1.8, ANSR. .................... .9 

Figure 5. Roll-damping moment coefficient vs. body length, Mach = 1.8, ANSR. ...................... .9 

Figure 6. Normal force coefficient vs. body length, Mach = 1.8, ANSR ..................................... 10 

Figure 7. Magnus force coefficient vs. body length, Mach = 1.8, ANSR . ................................... 11 

Figure 8. Zero-yaw drag coefficient vs. CG position, Mach = 1.8, ANSR . ................................. 12 

Figure 9. Projectile angular motion epicycle ................................................................................ 13 

Figure 10. Comparison of computed slow mode frequency with measured slow mode 

frequency, ANSR ..................................................................................................................... 14 

Figure 11. Comparison of computed fast mode damping with measured fast mode damping, 

ANSR ....................................................................................................................................... 15 

Figure 12. Comparison of computed slow mode damping with measured slow mode 

damping, ANSR.. ....................................................................................................................15 

Figure 13. Stability plot for ANSR from range data ..................................................................... 17 

Figure 14. Stability plot for ANSR from CPD predictions ........................................................... 17 

Figure 15. In-flight angular motion, 5-caliber body, Mach 1.8. .................................................. .18 

Figure 16. In-flight angular motion, 7-caliber body, Mach 2.5 .................................................... 19 

. 

iv 
1. Introduction 

~ . 

. 

I . 

Prediction of the in-flight motion of projectiles requires the determination of the aerodynamic 

forces and moments that act on the body during the course of its trajectory. These aerodynamic 

forces and moments may be determined from free-flight aerodynamics ranges (I), wind tunnel 

experiments (2), or theoretical approaches, such as computational fluid dynamics (0). In the 

supersonic flight regime, static aerodynamic coefficients such as drag and pitching moment have 

been routinely computed using CFD for a number of years (3,4). Much less routine are 

predictions of dynamic aerodynamic coefficients, such as the pitch damping and Magnus 

coefficients. A computational capability for predicting the Magnus forces and moments for 

smooth medium- and large-caliber flight bodies has existed since the early 1980s (3,4). With 

-the recent development of capabilities for predicting the damping derivatives (5), it is now 

possible to predict the stability characteristics and free-flight motion for projectiles using data 

that are derived solely from CFD. 

While previous research work has focused on predictions of the individual aerodynamic 

coefficients, this report focuses on the predictions of aerodynamic performance, in particular, 

aerodynamic stability. This report attempts to demonstrate, through a particular case study, the 

suitability and accuracy of Navier-Stokes computational techniques for the prediction of 

projectile performance and stability. Until recently, this was not possible since methods had not 

been implemented to predict all the necessary coefficients. 

As a demonstration of the capability, this report will present results for a family of axisymmetric 

projectiles in supersonic flight. The particular configuration selected for this computational 

study has been extensively tested in aeroballistic ranges, and high-quality experimental data have 

been obtained (I). The configuration consists of 5-,7-, and g-caliber length versions. By 

varying the mass distribution, three different centers of gravity (CG) were tested for each body 

length. A schematic of the configuration is shown in Figure 1. 

The computational results were obtained by applying a parabolized Navier-Stokes (PNS) 

approach to compute the attached viscous flow over the forebody of the projectile. Aerodynamic 

forces and moments were then extracted from the computed flow fields by integration of the 

predicted pressure and shear stresses acting on the projectile surface. To compute the 

recirculating flow at the base of the projectile and subsequently the base drag, a time-marching 

Navier-Stokes approach was applied. 

Using the predicted aerodynamics coefficients, parameters that characterize the in-flight motion 

are subsequently evaluated, including the gyroscopic and dynamic stability factors and the 

projectile’s fast and slow mode frequencies and damping coefficients. These parameters are then 
ALL DIMENSIONS IN CALIBERS (ONE CALIBER = 20 mm) 

Figure 1. Schematic of Army-Navy Spinner rocket (ANSR) configuration. 

used to predict the free-flight motion of the projectile. In each case, the computational approach 

is validated by comparison with experimental data. 

2. Computational Approach 

The supersonic viscous flow field about the projectile configuration was obtained by applying a 

parabolized Navier-Stokes technique to determine the attached flow field over the projectile 

forebody and a time-marching Navier-Stokes technique to determine the recirculating flow in the 

base region. Both of these numerical techniques are briefly described in the following 

subsections. 

2.1 The PNS Approach 

Computation of the attached viscous flow field about the projectile forebody was accomplished 

by solving the thin-layer Navier-Stokes equations using the PNS technique of Schiff and Steger 

(6). Using the PNS technique, computational results are obtained by marching through the grid 

from the projectile nose to the base. This technique is applicable in the supersonic flow regime 

and requires that the flow field contain no regions of flow separation in the axial direction. 

Because the computational approach requires only a single sweep through the computational 

grid, it is very efficient, compared with time-marching approaches that require many sweeps 

through the grid. 

As is standard practice, prediction of the static aerodynamic coefficients, such as pitching 

moment and normal force coefficients, are performed by computing the flow field about the 

projectile at a constant angle of attack. The Magnus coefficients are obtained from a 

2 
computation of the flow field at a constant angle of attack and spin rate. Since the static 

aerodynamics for spinning axisymmetric bodies are generally invariant with spin rate, it is 

possible to obtain the static aerodynamic coefficients and the Magnus coefficients Tom a single 

computation. The pitch-damping coefficients are determined from the side force and moment 

acting on the projectile undergoing steady coning motion. Steady coning motion is defined as 

the motion performed by a missile flying at a constant angle with respect to the freestream 

velocity vector and undergoing a rotation at a constant angular velocity about a line parallel to 

the freestream velocity vector and coincident with the projectile’s CG. Coning motion is, in fact, 

a specific combination of two orthogonal planar pitching motions plus a spinning motion. From 

linear flight mechanics theory, it can be shown that the pitch-damping coefficients are related to 

the side force and moment due to steady coning motion (5). 

The flow field predictions of the projectile in steady coning motion have been performed in a 

novel rotating coordinate frame that rotates at the roll rate or coning rate of the projectile. The 

fluid flow relative to the rotating coordinate frame does not vary with time, allowing the steady 

(non-time varying) Navier-Stokes equations to be applied. To implement the rotating coordinate 

frame, the governing equations have been modified to include the effect of centrifugal and 

Coriolis forces. The steady, thin-layer Navier-Stokes equations are shown in equation 1: 

Here, I?, F, and 6 are the inviscid flux vectors, S is the viscous flux vector, and H is the 

source term containing the Coriolis and centrifugal force terms, which result from the rotating 

coordinate frame. Each of these matrices is a function of the dependent variables represented by 

the vector q(p, pu, pv, pw, e), where p and e are the density and the total energy per unit volume, 

and u, v, and w, are the velocity components in x, y, and z directions. Further details of the flux 

terms and the source term containing the Coriolis and centrifugal forces are given by Schiff and 

Steger (6) and Weinacht et al. (5), respectively. 

The thin-layer equations are solved using the PNS technique of Schiff and Steger (6). Following 

the approach of Schiff and Steger, the governing equations, which have been modified here to 

include the Coriolis and centrifugal force terms, are solved using a conservative, approximately 

factored, implicit finite-difference numerical algorithm as formulated by Beam and Warming (7). 

A fully turbulent boundary layer has been modeled using the Baldwin-Lomax turbulence model 

(8X 

The computations presented here were performed using a shock-fitting procedure reported by 

Rai and Chaussee (9). This procedure solves the five Rankine-Hugoniot jump conditions, two 

geometric shock-propagation conditions, and one compatibility equation to determine the values 

of the five dependent variables immediately behind the shock, as well as the position of the 

3 
shock. The shock-fitting procedure was appropriately modified when coning motion was 

employed (5). 

The computational results presented here were obtained using a grid that consisted of 60 points 

between the body and the shock. In the circumferential direction, gridding was performed over 

the entire body 360” using 36 points spaced at 10’ increments. Approximately 75 axial 

marching steps were used for each caliber of body length. Because several of the aerodynamic 

coefficients are strongly influenced by viscous effects, the boundary layer was adapted by 

monitoring the nondimensional boundary layer coordinate, y”, along the leeside of the body and 

adjusting the normal grid spacing so that the y’ was -3. This approach has been successfully 

applied in previous studies (3). 

The computations have been performed on a variety of computer platforms. Typical solutions 

for a single set of conditions (Mach number, coning rate, angle of attack, etc.) required 5-10 min 

total CPU time on a Cray SVl computer. 

The selection of grid parameters for the current study is consistent with those used in the 

previous studies that focused on the individual aerodynamics coefficients (3-5). The grid 

requirements shown in these previous studies indicate that accurate results for the complete set of 

aerodynamics coefficient can be obtained with a single grid. As part of the current effort, a grid 

refinement study for the length-to-diameter-ratio ([LJD] = 9) body, middle CG position was also 

performed. A more refined grid with 90 points between the body and the shock, 72 

circumferential points and 150 axial marching steps for each caliber of body length was utilized. 

A comparison of the results obtained with the baseline and refined grids indicated a maximum 

difference of 4.5% between the two complete sets of aerodynamicoefficients. 

2.2 The Time-Marching Approach 

For the recirculating flow in the base region of the flow field, the PNS technique can no longer 

be used, and a more generally applicable approach such as the time-marching technique must be 

applied. The time-marching technique is based on the unsteady or time-dependent NavierStokes equations. For steady flow applications, such as those discussed here, the solution is 

obtained by marching an initial “guessed” solution to a final converged solution in a timeiterative manner. This requires the entire flow field to be updated at each time step and 

numerous time steps are required as the solution evolves to the converged steady-state result. 

Because of this, the time-marching approach is much more computationally intensive in terms of 

its run-times and computer memory requirements than the PNS technique. 

For the predictions presented here, the time-dependent approach was applied to compute the 

axisymmetric (zero angle of attack) base flow as a means of evaluating the zero-yaw drag. The 

small, but measurable, effect of yaw on drag was not considered in this study. The effect of the 

base flow on the other coefficients was also assumed to be small. 

4 
The particular time-marching technique applied here is the implicit, partially flux-split, upwind 

numerical scheme developed by Steger and his colleagues (10, II). The technique is based on 

the flux-splitting approach of Steger and Warming (12) and is often referred to as the F3D 

technique. The upwind scheme provides natural numerical dissipation, better stability properties 

and better computational efficiency compared with implicit central differencing schemes 

(l&12). 

The technique is based on the time-dependent thin-layer Navier-Stokes equations shown in 

equation 2: 

The flux-splitting algorithm of the F3D approach reduces the computational requirements by 

eliminating one of the three implicit inversions associated with the central difference BeamWarming algorithm (7). The two-factor implicit algorithm involves two sweeps through the grid 

at each time step. The first sweep involves inverting the block tridiagonal system of equations in 

the r-direction along grid lines of constant 4 and 77 to determine the intermediate solution 

variable. During the second sweep, a second block tridiagonal system of equations is inverted in 

the v-direction along grid lines of constant c and <to determine the dependent variable. 

The particular version of the code applied here has a multizone capability that allows the 

computational grid to be decomposed into smaller blocks. This allows greater flexibility in 

gridding of flight bodies, particularly when it is desirable to preserve sharp comers. The 

multizone capability also allows the computer memory requirement for the computation to be 

reduced since only one zone needs to be in memory at a time and the other zones can be stored 

on disk. This capability was implemented by Sahu and Steger (II). 

The zonal gridding approach was used to compute the recirculating viscous flow in the base 

region. A large grid zone was located in the base region and extended axially from the projectile 

base 6.35 calibers down stream. In the radial direction, the grid extended outward from the axis 

of symmetry to a radial location 3.5 calibers above the base comer. 

To allow the flow to properly expand around the base comer, a small zone, one body diameter in 

length, was placed on the forebody upstream from the base comer. This grid had a one-plane 

overlap with the base grid and a one-to-one mapping of the grid points in the overlap region. 

The dimensions of the base grid were 120 axial points, 50 radial points along the base, and 50 

radial points above the base comer. The forebody grid had 30 axial points and 50 radial points. 

Clustering of the grid points in the radial direction was performed to resolve the boundary layer 

on the forebody and the shear layer in the base region. The algebraic turbulence model of Chow 

(13) was applied in the base region. 

5 
3. Results 

As a first step in evaluating the aerodynamic performance of a flight vehicle, the aerodynamic 

force and moment coefficients must be evaluated. The aerodynamic coefficients allow forces 

and moments due to complicated three-dimensional (3-D) motions to be characterized as a 

combination of aerodynamic forces and moments due to simple specific motions. The individual 

aerodynamicoefficients can provide insight into the aerodynamic performance of the flight 

vehicles. 

3.1 Aerodynamic Moments 

For symmetric missiles, the aerodynamic moments are typically modeled using the following 

moment expansion which is cast in a non-rolling coordinate frame (14). The moment 

formulation makes use of complex variables to separate the moment components, e, and e, 

which produce rotations in the vertical and horizontal planes, respectively. The moment about 

the longitudinal axis of the vehicle, Cl, can be treated separately. 

In the moment formulation, the pitching moment coefficient, Cm*, produces a moment 

proportional to the complex angle of attack, z. For small angle-of-attack flight, the cosine of 

the total angle of attack, y , is nearly 1. This allows the pitch-damping moment coefficient, 

cq +c,&, which produces moments proportional to the angular rate z’ , to be treated as a 

single coefficient. The Magnus moment coefficient, C,,= , accounts for a side moment due to 

flow asymmetries from a combination of spin and angle of attack. 

Over the past decade and a half, CFD techniques have been developed to predict the various 

aerodynamicoefficients previously discussed. The pitching moment coefficient, C,= , has 

been routinely determined for various configurations and flight regimes for well over a decade. 

The prediction of the Magnus coefficients for axisymmetric projectiles at supersonic velocities 

was demonstrated in the early 1980s (3). The Magnus problem is still a research topic for 

complicated vehicles and at transonic velocities. More recently, techniques for predicting the 

pitch-damping coefficient, C, + C,, , based on the solution of the Navier-Stokes equations 

have been developed (5). Although the development of a capability to predict each individual 

aerodynamic coefficient can be considered an important research accomplishment, itwas not 

6 
possible to predict the stability characteristics of projectiles using CFD until a complete 

capability for predicting all the aerodynamicoefficients was developed. The prediction of the 

pitch damping coefficient represented that final step. 

Figures 2-4 show results of CFD predictions for the pitching moment, Magnus moment and 

pitch-damping moment coefficients performed as part of the current study. Results for a 

free-flight Mach number of 1.8 are shown for each of the three CG locations and body lengths. 

In general, the CFD predictions appear to be in good agreement with the experimental data 

obtained from range firings of these projectiles. Results obtained at Mach 2.5 also show similar 

agreement with the range data. 

The aerodynamic moment about the longitudinal axis of an axisymmetric vehicle can be 

modeled using a single aerodynamicoefficient. The roll-damping coefficient, C,P , represents 

the aerodynamic moment which opposes the spinning motion of the vehicle. For axisymmetric 

flight bodies, the roll-damping coefficient is produced purely by the viscous stresses acting 

circumferentially on the body surface. From a modeling standpoint, proper prediction of the rolldamping coefficient requires adequate grid resolution of the boundary layer. A comparison of 

the predicted roll-damping coefficient with range data is shown in Figure 5. Both the 

computational predictions and the range data show a nearly linear increase in the roll-damping 

coefficient with increasing body length. The predicted result shows an overprediction of -10%. 

Similar results were observed at Mach 2.5. The predictions showed a 4% variation in the rolldamping coefficient at angles of attack to 6”. 

3.2 Aerodynamic Forces 

The aerodynamic forces can also be described in a manner that is analogous to the aerodynamic 

moments. The transverse force components EY and Cz act along the “y and Z axes of the nonrolling coordinate frame and consist of components proportional to the complex yaw f and 

yawing rate F’. The third force component, C, , which acts along the axial direction, is loosely 

coupled to the transverse forces and can be treated separately. This force component consists 

predominantly of the drag force. 

(5) 

c, =-c,. (6) 

The normal force coefficient C,= acts normal to the longitudinal axis of the body in the pitchplane. The force expansion also contains a damping force coefficient, CNg + C,, , that produces 

a force proportional to the yawing rate r’. The Magnus force coefficient CvPn accounts for the 

side force produced by a combination of spin and yaw. 

7 
15.0 

10.0 

5.0 

0.0 

AFNS 

0 RANGE DATA 

I I I I I 

2.0 3.0 4.0 5.0 6.0 7.0 

CG - CALIBERS FROM NOSE 

Figure 2. Pitching moment coefficient vs. CG position, Mach = 1.8, ANSR. 

20 

C 

nPa 

10 

0.0 

-10 

0 RANGE DATA 

I I I I 1 

2.0 3.0 

CG - CALIBERS FROM NOSE 

7.0 

Figure 3. Magnus moment coefficient vs. CG position, Mach = 1.8, ANSR. 

8 
-100.0 

-80.0 

-20.0 

0.0 

20 3.0 4.0 5.0 6.0 7.0 

CG - CALIBERS FROM NOSE 

M = 1.8 

- PNS - CG EQS. 

0 RANGE DATA 

Figure 4. Pitch-damping moment coefficient vs. CG position, Mach = 1.8, ANSR. 

“P 

-0.025 

-0.020 

-0.015 

-0.010 

-0.005 

0.000 

tvl = 1.8 

A PNS 

0 RANGE DATA 

4.0 6.0 8.0 lo.0 

BODY L/D 

Figure 5. Roll-damping moment coefficient vs. body length, Mach = 1.8, ANSR. 
Figures 6 and 7 show comparisons of the predicted and experimentally determined normal force 

and Magnus force coefficients at Mach 1.8 as a function of body length. The predicted normal 

force coefficient is predicted to within the scatter of the range data and shows little influence of 

body length. Both the predicted and experimentally determined Magnus force coefficient 

increase with body length. The Magnus force determined from the variation of the Magnus 

moment with CG location is -30% less than the predictions for the L,/D = 7 and L,/D = 9 bodies. 

The Magnus force coefficients determined from the swerving (CG) motion for individual shots 

shows significant scatter, indicating the difficulty in accurately determining this coefficient. The 

pitch-damping force coefficient is not shown as it does not influence the projectile motion in a 

measurable manner. Comparisons of predictions with experimental data are presented by 

Weinacht et al. (5). 

A PNS 

0 RANGE - SWERVE 

l RANGE - CMa CG VARIATION 

0.0 ; I I I 

4.0 6.0 8.0 10.0 

BODY L/D 

Figure 6. Normal force coefficient vs. body length, Mach = 1.8, ANSR. 

The drag coefficient is also very important for predicting projectile flight performance as it is 

one of the primary aerodynamic coefficients affecting range. The drag coefficient is also 

required for the stability assessment, although its effect is small. The total drag on the projectile 

is composed of contributions from the forebody and the base. The forebody drag consists of 

both pressure and viscous drag components. The base drag results from the pressure acting on 

the projectile base. The forebody drag can be predicted using the PNS technique. The 

recirculating flow in the base region is predicted using the time-marching technique from which 

the base drag can be determined. 

10 
40 

cYpo, 

-0.5 

, 

0.0 

0 RANGE I SWERVE 

n RANGE - C, a CG VARIATION 

I I I 

4.0 6.0 

BODY L/r 

10.0 

Figure 7. Magnus force coefficient vs. body length, Mach = 1.8, ANSR. 

A comparison of the predicted zero-yaw drag with range results is shown in Figure 8. Drag 

predictions at Mach 1.8 and 2.5 are shown as a function of body length. Both the experimental 

results and the computational results show a slight increase in zero-yaw drag with increasing 

body length and decreasing Mach number. The computed drag coefficient is predicted to within 

the accuracy of the range data. The increase in the drag with body length is due primarily to the 

increase in the wetted surface of the longer body. The predicted results showed only minor 

variations in the base drag with body length. 

At Mach 1.8 for the LID = 9 body, the pressure and viscous forebody drag account for 39% and 

25% of the total drag with the base drag accounting for the remaining 36% of the drag. 

Predictions of the forebody pressure drag are within 1% of previous predictions using empirical 

(15) or inviscid (I) approaches. 

An estimate of the predicted skin fiction drag, CnV , can be obtained using the relation 

developed by Charters and Kent (16). 

C Dv = -4Clp. (7) 

The estimated skin friction drag determined from the predicted roll-damping coefficient 

. underestimates the predicted skin friction drag by 3%-l 0% for the predictions presented here. 

(This comparison gives an indication of the accuracy of the Charters and Kent relation.) The 

estimated skin friction obtained from the range measurements of the roll-damping coefficient is 

13%-22% less than the predicted skin friction. 

11 
- 

. 

cg ,’ l *‘08 i 

0 M = 2.5 

0.2- 

A CFD PREDICTION 

0.1 - 0 0 RANGE DATA 

0.0 I I I 

4.0 6.0 8.0 lo.0 

BODY L/D 

Figure 8. Zero-yaw drag coefficient vs. CG position, Mach = 1.8, ANSR. 

Although these estimates indicate a potential 10% over-prediction of the skin friction drag, the 

contribution to the total drag would result in an overprediction of only a few percent. 

Design codes can provide a more cost-effective alternative for predicting the base drag, thus 

eliminating the need for a time-dependent base drag computation. The semi-empirical approach 

of McCoy (15) appears to overpredict the base drag by lO%-20% compared to the CFD result. 

When the estimated base drag is added to the forebody drag CFD predictions, the total drag is 

overpredicted by 4% compared to the CFD results. For stability computations, where the drag 

coefficient plays a minor role, acceptable accuracy can be obtained using base drag estimates. 

3.3 Stability and Free-Flight Motion 

Once the aerodynamic forces and moments have been determined, the equations of motion can 

be integrated and the projectile’s motion can be determined for various initial conditions. 

Because the equations of motion are nonlinear, numerical integration is usually required to 

predict the projectile motion from launch to impact. These nonlinearities are typically weak, and 

for portions of the flight trajectory the equations can be integrated and closed-form solutions for 

the projectile motion can be obtained. 

For a symmetric missile, the closed-form solution for the angular response as a function of the 

distance downrange, s, takes the following form (14): 

12 
F=Fs+K1+Kz=fl+ia 

K, = 60 exPH&J + #is)+ 44 (8) 

K, = ho exp[i(#20 + f%s)+ 44 

The complex variable, F, consists of the vertical and horizontal components of the angle of 

attack, c1! and fl The angular motion contains a gravity term, Fg, which is small and, for the 

remainder of the discussion, this term will be ignored. The second and third terms of equation 8 

are expressions of the amplitudes of two components of the angular motion, referred to as the 

fast and slow modes, respectively. The complex angle of attack, F, can be displayed 

graphically along with the fast and slow mode arms, Kt and K2. Graphically, the solution is 

shown by Figure 9 and is referred to as a damped epicycle. It is helpful to regard this figure as 

the path that the nose of the projectile would follow as it flies downrange. The fast and slow 

mode frequencies of the motion, 4; and &, and the damping rates, Ai and ;12 ,are functions of 

the aerodynamic coefficients, projectile spin rate, and inertial properties of the body. Generally 

speaking, the frequencies are influenced by the pitching moment coefficient, while the damping 

is influenced by the Magnus and pitch damping coefficients. The equation also contains the 

initial amplitudes, Kto and K~o, and phase angles, #io and ho, which result from applying the 

initial conditions during the integration of the equations. 

0.15 

0.10 

0.05 

-0.10 

-0.15 I, 

-o.lso.lo-0.05 0.00 0.05 0.10 ox 

“v/V 

Figure 9. Projectile angular motion epicycle. 

13 
Using the predicted aerodynamicoefficients, the fast and slow mode damping and frequencies 

were computed for the various body lengths, CG positions, launch Mach numbers, and spin rates. 

Figure 10 shows the predicted slow mode frequency plotted against the experimentally 

determined value. (The results are plotted in this fashion due to the large number of parameters 

considered here.) Points that fall close to the solid line indicate that the predicted value is close 

to the experimentally determined value. There is good agreement between prediction and 

experiment. The slow mode frequency represents the critical comparison for the frequencies 

because the fast mode frequency is dominated by the spin rate, which is essentially an input 

parameter. 

0.0 

0.0 20 4.0 s.0 

MEASURED $; qO-3 

Figure 10. Comparison of computed slow mode frequency with measured slow mode 

frequency, ANSR. 

Figures 11 and 12 show comparisons of the predicted slow and fast mode damping of each of the 

bodies. The data are scattered about the solid line, indicating that the prediction is generally in 

good agreement with the experiment. No consistent bias between the computation and 

experimental data is seen. This indicates that there is no consistent under-prediction or 

over-prediction of the damping rates. Much of the scatter in the data is attributed to the 

uncertainty in the experimental data rather than inaccuracies in the computational approach. 

Generally, from a design standpoint, predicting and controlling damping rates, rather than the 

fkquencies, is a primary consideration. One desirable requirement is that the damping rates, hr 

and 12, be negative so that the angular motion does not grow during the projectile’s flight. 

14 
6.0 

4.0 

20 

0.0 

-20 

-4.0 

-6.0 

0 

cl 

A 

/ 

I I I I I 

-6.0 -4.0 -20 0.0 20 4.0 6.0 

MEASURED hl Vo” 

Figure 11. Comparison of computed fast mode damping with measured fast mode 

damping, ANSR. 

PO k 20 

t-7 

0.0 

Ei -20 

I5 -4.0 

0 

o -6.0 

cl 

A 

-8.0 -6.0 -4.0 -20 0.0 

MEASURED X2 

Figure 12. Comparison of computed slow mode damping with measured slow mode 

damping, ANSR. 

15 
By algebraically manipulating the equations for the damping rates, three necessary conditions 

can be derived, which will ensure that the damping rates are negative. Two of these conditions 

are shown in equation 9: 

sg >I; sg =f(p,C,) 

‘g cs,(2-s,); sd =f(Cnpn,Cmq +c,)- (9) 

The first condition states that the gyroscopic stability factor, ss, must be >l for the projectile to 

be gyroscopically stable. The second condition also places an additional constraint on the 

gyroscopic stability factor through the dynamic stability factor, sd. As shown in these equations, 

the gyroscopic stability and dynamic stability factors are functions of the aerodynamic 

coefficients. Additionally, the gyroscopic stability factor is also a function of the projectile spin 

rate, p. For typical spin-stabilized projectiles that are statically unstable, the two stability criteria 

can be satisfied by adjusting the spin rate of the projectile, as long as the dynamic stability factor 

is between 0 and 2. If the dynamic stability factor is outside this bound, the projectile cannot be 

stabilized using spin. A third condition for stability essentially requires that the pitch damping 

coefficient, C, + C,, to be negative. This condition is rarely violated in practice. 

The constraints on the gyroscopic stability and dynamic stability factors discussed previously can 

be shown graphically on what is often termed a “stability plot.” Figures 13 and 14 show plots of 

the results as measured in the aerodynamics range and obtained from the computation. Shown 

on the plot are results for the each of the three CG locations for the three different length bodies, 

various spin rates, and for Mach numbers of 1.8 and 2.5. For each firing, experimental and 

computational values of the gyroscopic stability factor, ss, and the dynamic stability factor, Q, 

were computed. It is important to realize that for a given set of conditions (body length, spin rate, 

launch Mach number), the computational results will yield a unique value. However, because of 

the scatter associated with the experimental data, each firing yields a slightly different value. 

Thus, the experimental results give the appearance of having more data. It is important to realize 

this when comparing the two sets of plots. 

The computational and experimental results presented in these figures show that this family of 

projectiles is statically unstable (ss > 0), but has been made gyroscopically stable by appropriate 

selection of the spin rate (ss > 1). The results also show that all of the L/D = 5 projectiles are 

dynamically stable. (The region of dynamic stability is denoted by the shaded area on these 

plots.) The results indicate that several of the IJD = 7 and L/D = 9 projectiles are marginally 

stable, though these can be stabilized by selection of a higher spin rate because the dynamic 

stability factor is between 0 and 2. Several of the IJD = 9 bodies are dynamically unstable and 

cannot be stabilized by changing the spin rate. Further examination of the data revealed that 

16 
c 

20 

DY NAME 

STABILITY 

20 3.0 

Figure 13. Stability plot for ANSR from range data. 

20 

-10 

-to 0.0 10 20 3.0 4.0 

Sd 

Figure 14. Stability plot for ANSR from CFD predictions. 
L 

the IJD = 9 bodies which could not be stabilized all had the CG located in the aftmost position. 

It is important to note that the computational approach correctly predicts the stability 

characteristics of the various configurations for the conditions of dynamic stability, marginal 

dynamic stability, and dynamic instability. This demonstrates the utility of the computational 

approach for the purposes of stability prediction. 

With the prediction of the frequencies and damping rates of the angular motion, the angular 

motion itself can be predicted and compared with the motion measured in the aerodynamics 

range. Figures 15 and 16 show the angular motion of a 5-caliber-long projectile launched at 

Mach 1.8 and a 7-caliber projectile launched at Mach 2.5. These 3-D plots display the vertical 

and horizontal components of angle of attack, a and p, as a function of the distance downrange, 

Z. The thickened curve displays the 3-D motion. Two-dimensional projections of the motion in 

the vertical and horizontal planes are also shown. It is useful to regard the curves shown here as 

the path traversed by the nose of the projectile as it flies downrange. The motion predicted using 

the CFD-derived aerodynamicoefficients is denoted by the solid lines, while the motion 

obtained from the range firings is shown by the circular symbols. The agreement between the 

two motions is excellent. Comparisons of the in-flight motion at other flight conditions and for 

the other body geometries also showed good agreement with the experimental data. 

-- CFD PREDICTION 

Figure 15. In-flight angular motion, j-caliber body, Mach 1.8. 

18 
Figure 16. In-flight angular motion, 7-caliber body, Mach 2.5. 

4. Conclusion 

CFD provides an alternate means for determining the aerodynamic performance of projectiles. 

This will allow projectiles to be designed with a reduced requirement for manufacturing models 

and performing test firings. As a test case, the aerodynamic performance of a family of 

axisymmetric projectiles in supersonic flight was examined. The predictions showed good 

agreement with range data for individual aerodynamic coefficients. Using these coefficients, the 

frequencies and damping rates of the angular motion were computed. The results showed good 

correlation with the experimental data for various body lengths, center of gravity locations, Mach 

numbers and spin rates. The aerodynamic coefficients were also used to predict gyroscopic and 

dynamic stability factors. The computational approach correctly distinguished the conditions of 

dynamic instability, marginal dynamic stability, and dynamic stability. As a final comparison, 

the in-flight angular motion for individual firings was compared with range data and showed 

good agreement. The results are believed to properly representhe capability for predicting the 

linear aerodynamics for the high-speed low angle-of-attack flight of aerodynamically smooth 

axisymmetric projectiles. Additional computational effort (and, in some cases, additional 

research) may be required to establish the same capability at subsonic and transonic velocities or 

for complex geometries. 

19 
5. References 

1. 

2. 

3. 

4. 

5. 

6, 

7. 

8. 

9. 

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

Bodies of Revolution in Supersonic Flight; Report No. 876; U.S. Army Ballistic Research 

Laboratories: Aberdeen Proving Ground, MD, August 1953. 

Nietubicz, C. J.; Opalka, K. Supersonic Wind Tunnel Measurements of Static and Magnus 

Aerodynamic Coefficients for Projectile Shapes With Tangent and Secant Ogive Noses; 

ARBRL-MR-02991; U.S. Army Ballistic Research Laboratory: Aberdeen Proving Ground, 

MD, February 1980. 

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

Supersonic Flight American Institute of Aeronautics and Astronautics, Reston, VA, August 

1980; AIAA Paper 80-1586-CP. 

Sturek, W. B.; Mylin, D. C. Computational Parametric Study of the Magnus Effect on 

Boattailed Shell at Supersonic Speeds. American Institute of Aeronautics and Astronautics, 

Reston, VA, August 198 1; AJAA Paper 8 l-1 900. 

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

Axisymrnetric Projectiles. Journal of Spacecra$ and Rockets 1997,34 (6), 753-761. 

Schiff, L. B.; Steger, J. L. Numerical Simulation of Steady Supersonic Viscous Flow. AlAA 

Journal 1980,18(12),1421-1430. 

Beam, R.; Warming, R. F. An Implicit Factored Scheme for the Compressible NavierStokes Equations. ALU Journal 1978,16 (4), 85-129. 

Baldwin, B. S.; Lomax, H. Thin Layer Approximation and Algebraic Model for Separated 

Turbulent Flows. American Institute of Aeronautics and Astronautics, Reston, VA, January 

1978; AIAA Paper 78-257. 

Rai, M. M.; Chaussee, D. S.’ New Implicit Boundary Procedure: Theory and Applications. 

American Institute of Aeronautics and Astronautics, Reston, VA, January 1983; AIAA 

Paper 83-0123. 

10. Ying, S. X.; Steger, J. L.; Schiff, L. B.; Baganoff, D. Numerical Simulation of Unsteady, 

Viscous, High-Angle-of-Attack Flows Using a Partially Flux-Split Algorithm. American 

Institute of Aeronautics and Astronautics, Reston, VA, August 1986; AI&4 Paper 86-2179. 

11. Sahu, J.; Steger, J. L. Numerical Simulation of Three-Dimensional Transonic Flows. 

International Joumalfor Numerical Methods in Fluids 1990, IO, 855-873. 

20 
12. Steger, J. L.; Warming, R. F. Flux Vector Splitting of the Inviscid Gasdynamic Equations 

With Application to Finite-Difference Methods. Journal of Computational Physics 1981, 

40,263-293. 

13. Chow, W. L. Improvement on Numerical Computation of the Thin-Layer Navier-Stokes 

Equation With Special Emphasis on the Turbulent Base Pressure of a Projectile in Transonic 

FZight Condition; Contract Report No. DAAG29-8 1 -D-0100; University of Illinois, UrbanaChampaign: Urbana, IL, November 1985. 

14. Murphy, C. H. Free Flight Motion of Symmetric Missiles; Report No. 1216; U.S. Army 

Ballistic Research Laboratories: Aberdeen Proving Ground, MD, July 1963. 

15. McCoy, R. L. MC DRAG-A Computer Program for Estimating the Drag Coeflcients of 

Projectiles; ARBRL-TR-02293; U.S. Army Ballistic Research Laboratory: Aberdeen 

Proving Ground, MD, February 198 1. 

16. Charters, A. C.; Kent, R. H. The Relation Between the Skin Friction and the Spin Reducing 

Torque; Report 287; U.S. Army Ballistic Research Laboratories: Aberdeen Proving Ground, 

MD, July 1942. 
List of Abbreviations and Symbols 

aLa 

CD 

C Do 

% 

Cl 

clP 

cm 

C ma 

C mci 

% 

‘% +cmk 

C” 

C 

“pa 

GJ 

Freestream speed of sound 

Drag coefficient 

Zero-yaw drag coefficient 

Viscous drag coefficient 

Net roll moment coefficient 

Roll-damping moment coeffCnt, ac, 

Pitching moment coefficient, M 

+ pV2S,,D 

ac Pitching moment coefficient slope with respect to angle of attack, 2 

aa 

Pitch-damping moment coefficient slope, acm 

Pitch-damping moment coefficient slope, acm 

Pitch-damping moment coefficient 

Side moment coefficient 

Magnus moment coefficient, a2 c, 

aa 

Normal force coefficient, F 

fpV2S,, 

22 
CNa 

CN& 

aCN Normal force coefficient slope with respecto angle of attack, - 

aa 

Pitch-damping force coefficient slope, a CN 

&D 

ai-1 V 

CNq Pitch-damping force slope, a CN 

CNq +cN, Pitch-damping force coefficient - 

CX Axial force coefficient 

CY& Transverse force coefficient in y and z direction 

C Yw Magnus force coefficient 

D Reference diameter 

e Total energy per unit volume 

B,P,6 Flux vectors in transformed coordinates 

F Force 

a Source term in Navier-Stokes equations due to rotating coordinate Came 

K, 9K2 Amplitude of fast and slow yaw arms 

K,o YK20 Initial amplitude of fast and slow yaw arms 

L Body length 

M Moment I 

M Freestream Mach number 

P Spin rate, as used in roll equations 

9 Transverse angular rate of body, as used in flight mechanics equations 

9 

Re 

S 

Vector of dependent variables, as used in Navier-Stokes equations 

Reynolds number, a,p,D / p, 

Distance downrange normalized by reference length 

23 
sd Dynamic stability factor 

% Gyroscopic stability factor 

S ZD2 

ref Reference area, S,, = - 4 

!2 Viscous flux vector 

t Time 

u,v,w Velocity components in the x, y, z directions 

V Freestream velocity 

X,Y,Z Axial, horizontal, and vertical coordinates 

Y+ Nondimensional boundary layer coordinate 

Z Dimension downrange distance 

Greek Svmbols 

Angle of attack 

Angular rate associated with angle of attack 

Yaw angle 

Cosine of total angle of attack 

Fast and slow mode damping rates 

Viscosity 

Transformed coordinates in the Navier-Stokes equations 

Complex angle of attack 

Complex angle-of-attack component induced by gravity 

Density 

Phase angle of fast and slow mode arms 

Fast and slow mode frequencies 

24 
SuperscriDts 

(‘1 Rate of change with respect to time 

, 0 

(-) 

Rate of change with respect to space 

Referenced to non-rolling coordinate frame 

Subscripts 

co _ Quantity evaluated at fi-eestream conditions 

25 
INTENTIONALLYLEFTBLANK. 

26