. 

Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-2445 April 2001 

Effect of Body Cross Section on 

Projectile Aerodynamic Performance 

With Application to 

Electromagnetic (EM) Guns 

Paul Weinacht 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
Abstract 

The most effective bore cross-section geometry for electromagnetic (EM) 

guns may differ significantly from the circular cross section of conventional 

guns. The geometry of the bore cross section will influence the shape of the 

armature/subproject launch package. For certain applications, a noncircular 

cross section may provide payload packaging benefits which have not been 

considered previously for conventional circular cross-section guns. Whether 

these benefits can be fully realized depends, in part, on the aerodynamic 

performance of these geometries in free flight. The current study considers the 

aerodynamics of chemical energy (CE)-type munitions to determine whether 

there are potential aerodynamic benefits for noncircular cross-section geometries 

compared with conventional circular cross-section bodies. The study compares 

the aerodynamic drag and static stability of both circular and noncircular 

geometries using sophisticated computational aerodynamic predictive tools to 

determine the potential aerodynamic benefits. 

ii 
Contents 

List of Figures v 

1. Introduction 1 

2. Computational Approach 3 

3. Results 4 

3.1 Aerodynamics of the Baseline Circular Cross-Section Geometry.. ........ . 

3.2 Comparison of Aerodynamics Between Circular and Noncircular 

Geometries ............................................................................................................ 7 

3.3 Effect of Roll Orientation on Aerodynamic Performance .................... .13 

4. Conclusion 16 

5. References 

Distribution List 

Report Documentation Page 

17 

19 

25 

. . . 

111 
iv 
List of Figures 

. 

Figure 1. Schematic illustration of the baseline circular cross-section flight 

- body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 

. 

. 

Figure 2. Schematic illustration of the noncircular cross-section geometries.........2 

Figure 3. Development of the drag coefficient over the baseline circular 

cross-section geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 

Figure 4. Development of the normal force coefficient over the baseline 

circular cross-section geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

Figure 5. Development of the pitching moment coefficient over the baseline 

circular cross-section geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

Figure 6. Development of the drag coefficient over the baseline circular 

cross-section and noncircular cross-section geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 

Figure 7. Development of the drag coefficient over the noncircular crosssection geometries and equivalent cross-sectional area circular 

geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 

Figure 8. Development of the pitching moment coefficient over the circular 

cross-section geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

Figure 9. Development of the normal force coefficient over the circular crosssection geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..I............................. . . . . . . . . . . . . . . 10 

Figure 10. Development of the pitching moment coefficient over the baseline 

circular cross-section and noncircular cross-section geometries . . . . . . . . . . . . . . . . . . . . . . 11 

Figure 11. Development of the normal force coefficient over the baseline 

circular cross-section and noncircular cross-section geometries. . . . . . . . . . . . . . . . . . . . . .ll 

Figure 12. Crossflow particle traces at X/D=7.5, circular cross-section body, 

Mach 3,2” angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 

Figure 13. Crossflow particle traces at X/D=7.5, 43-mm noncircular crosssection body, Mach 3,2” angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 

Figure 14. Effect of roll orientation on the development of the normal force 

coefficient over the baseline circular cross-section geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

Figure 15. Effect of roll orientation on the development of the pitching 

moment coefficient over the baseline circular cross-section geometry. . . . . . . . ...14 

Figure 16. Effect of roll orientation on the development of the normal force 

coefficient over the 43-mm noncircular cross-section geometry. . . . . . . . . . . . . . . . . . . . . . 15 

Figure 17. Effect of roll orientation on the development of the pitching 

moment coefficient over the baseline circular cross-section geometry. . . . . . . . ...15 

V 
INTENTIONALLY um BLANK. 

vi 
1. Introduction 

The U.S. Army is currently investigating electromagnetic (EM) guns as a means 

of increasing the lethality of its weapon systems. Currently, the geometric 

configuration of EM guns is a subject of research. In particular, the bore cross 

section of these guns may not be circular as with most conventional powder 

guns, but rather may possess a square or rectangular cross section. The 

geometry of the bore cross section will directly influence the shape of the 

armature/subprojectile launch package. For certain applications, the noncircular 

cross section may provide payload packaging benefits which have not been 

considered previously for conventional circular cross-section guns. Whether 

these benefits can be fully realized depends on a number of factors. One of the 

considerations is the aerodynamic performance of these geometries in free flight. 

The current study considers the aerodynamics of chemical energy (CE)-type 

munitions to determine whether there are potential aerodynamic benefits for 

noncircular cross-section geometries compared with conventional circular crosssection bodies. For instance, it may be possible to package explosively formed 

projectile (EFP) warheads more effectively in a noncircular geometry than in a 

conventional circular geometry. The study compares the aerodynamic drag and 

static stability of both circular and noncircular geometries using sophisticated 

computational aerodynamic predictive tools to determine the potential 

aerodynamic benefits. 

The geometry of the baseline circular cross-section geometry is shown in 

Figure 1. The geometry consists of a conical nose/cylinder/boattail/tail 

boom/fin assembly which approximates the geometric characteristics of modem 

CE-type munitions. Two noncircular cross-section geometries have also been 

considered. These noncircular bodies share similar geometric characteristics 

with the circular cross-section geometry. The noncircular cross-section 

geometries were obtained by extruding the baseline circular cross-section 

geometry through a square cross section of width 43 mm and 50 mm, 

respectively. A schematic of the circular and two noncircular cross sections is 

shown in Figure 2. The 43-mm geometry results in a nearly square cross section 

over the cylindrical portion of the body, while the 50-mm geometry results in a 

square cross section with rounded comers. Both noncircular cross-section 

geometries have circular cross sections on the conical nose and on the boattail 

when the local diameter of the cross section is less that 43 mm and 50 mm, 

respectively. 
3.18 -w 

ALL DIMENSIONS IN CALIEERS (ONE CALIBER = 120 mm) 

Figure 1. Schematic illustration of the baseline circular cross-section flight body. 

Figure 2. Schematic illustration of the noncircular cross-section geometries. 

A fixed fin planform is utilized for all configurations. The fins are oriented in an 

x-configuration with respect to the noncircular cross-section geometries. This 

allows the fins to have maximum span without interfering with the walls of the 

bore. The fin geometry chosen here represents only one option for fin 

stabilization. It may be possible to utilize folding fins which have a larger span 

than the bore cross section when deployed in free flight. Nevertheless, the 

results presented here using the non-folding fins should provide a good basis for 

evaluating the effects of the noncircular cross section on the aerodynamics of the 

projectile. 

Two additional circular cross-section bodies were considered for the purpose of 

comparisons. These,bodies have the same cross-sectional area as each of the two 

noncircular cross-section bodies at each axial station along the body. As a result, 

the bodies have the same total volume as the corresponding noncircular crosssection bodies. For all the geometries considered, the same tail boom and fin 

assembly were used. 
. 

The current study assumes that a functional armature could be built for the 

geometry under consideration, perhaps a discarding pusher-type armature 

which might be located between the boattail and the fins. Obviously, for an 

optimal design, the subprojectile and armature design must be considered 

simultaneously. This also implies that the characteristics of the gun are defined 

as well. However, the focus of the current investigation is on the effect of the 

noncircular cross section on the aerodynamic performance of the flight body. 

This study should be viewed as a first step for considering the viability of 

noncircular cross-section geometries for EM gun applications, thereby justifying 

assumptions regarding the launchability of the geometry under consideration. 

2. Computational Approach 

Computation of the viscous flow field about the various projectile configurations 

was accomplished by solving the thin-layer Navier-Stokes equations using the 

parabolized Navier-Stokes (PNS) technique of Schiff and Steger [l]. Using the 

PNS technique, computational results were obtained by marching through the 

grid from the projectile nose to the trailing edge of the fins. 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. The technique has been applied successfully to a number of 

projectile configurations, including axisymmetric shell [2, 31, flared projectiles 

[4], and finned projectiles [3]. The technique has also been applied previously to 

predict the aerodynamic performance of a railgun-launched noncircular 

cross-section projectile [5]. 

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

drag, pitching moment, and normal force coefficients are performed by 

computing the flow field about the projectile at a fixed angle of attack. 

Assuming a time-invariant flow field, the steady thin-layer Navier-Stokes 

equations are applied as shown in equation 1: 

(1) 

The thin-layer Navier-Stokes equations are obtained by eliminating from the full 

Navier-Stokes equations, all the viscous terms except for those containing 

derivatives in the direction nearly normal to the projectile body. For high 

Reynolds number flows with no axial flow separation, the thin-layer Navier 

3 
Stokes equations are very good approximation to the full Navier-Stokes 

equations, and can be efficiently solved using available numerical algorithms. 

Here, 6, l?, and 6 are the inviscid flux vectors, and i is the viscous flux vector. 

Each of these matrices are functions of the dependent variables represented by 

the vector s(p, pu, pv, p-5 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. 

The pressure, p, which appears in the flux terms, can be related to the 

dependent variables by applying the ideal gas law: 

p=(y-1) e-“(u2 +v2 +w2 

[ 

. 

2 )I (2) 

A turbulent boundary layer has been simulated over the projectile body using 

the turbulence model of Baldwin and Lomax [6]. 

The thin-layer equations are solved using the PNS technique of Schiff and Steger 

[l]. Following the approach of Schiff and Steger, the governing equations are 

solved using a conservative, approximately factored, implicit finite-difference 

numerical algorithm as formulated by Beam and Warming [71. The 

computations presented here were performed using a shock-fitting procedure 

reported by Rai and Chaussee [S]. This procedure solves the five RankineHugoniot 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 shock. 

For the computational results presented here, the grid consisted of 60 points from 

the body to the shock. On the forebody, 72 grid points were utilized around the 

body in the circumferential direction. To capture the details of the fin geometry, 

280 grid points were used in the circumferential direction. 

In the marching (axial) direction, 470 marching planes were required for each 

caliber of body length. Over the axisymmetric portion of the body, the grid was 

generated algebraically. On the non-conical portion and finned portion of the 

body, the grid was obtained using an elliptic grid generator [9]. A complete 

computation for each configuration required approximately 2 hr of computer 

processor unit (CPU) time on a Cray C-90 supercomputer. 

3. Results 

Predictions of the aerodynamic performance of the flight bodies were made at 

flight velocity of Mach 3 at standard atmospheric launch conditions (ambient 

4 
temperature of 70 “F). A constant wall temperature of 70 “F on the body surface 

was used in all the simulations. A constant angle of attack of 2” was used to 

generate the transverse forces and moments. 

3.1 Aerodynamics of the Baseline Circular Cross-Section Geometry 

Each of the geometries considered here have similar aerodynamic characteristics. 

Before comparing the aerodynamic performance of each of the designs, it is 

useful to examine the aerodynamic characteristics of the baseline circular crossset tion configuration. 

Figure 3 shows the development of the drag over the baseline configuration. 

Most the aerodynamic drag is produced by the wave drag on the conical nose of 

the vehicle. The boattail also produced a significant amount of drag due to the 

low pressure produced by the flow expansion on the boattail. The amount of 

drag from the nose and boattail is dependent, in part, on the cross-sectional area, 

so that reductions in the cross-sectional area may result in improvements in the 

aerodynamic performance. The skin friction and fin drag account for only a 

small portion of the total drag. An estimate of the base drag indicates that the 

expected base drag is about 4% of the forebody drag. The base drag is not 

included in the plotted drag coefficient data. Computational predictions of the 

axial force coefficient at 0” and 2” showed less 1% difference. Because of the 

small differences in the axial force coefficient at small angles of attack, drag 

coefficients predictions were obtained using the axial force coefficient from the 

2” angle-of-attack solutions for the other body geometries. 

o.ou 

0.0 2.0 4.0 6.0 8.0 10.0 l2.0 

X/D 

Figure 3. Development of the drag coefficient over the baseline circular cross-section 

geometry. 
CE-type munitions are usually made aerodynamically stable using tail fins. 

When the projectile is at angle of attack, the tail fins must produce sufficient 

normal force aft of the projectile’s center of gravity to counteract the nose normal 

force produced in front of the center of gravity. By examining the distribution of 

the normal force, the relative contribution from each component of the body can 

be examined. Figure 4 shows the normal force distribution for the circular crosssection geometry. The conical nose and cylindrical portion of the body produce 

significant amounts of lift on the forward portion of the body. Some of this lift is 

lost over the boattail. The stabilizing fins produce a significant amount of lift 

which is required to balance the nose lift in order to maintain aerodynamic 

stability. 

3.0 

%x 

2.0 

1.0 

0.0 

0.0 2.0 4.0 6.0 8.0 10.0 120 

X/D 

Figure 4. Development of the normal force coefficient over the baseline circular 

cross-section geometry. 

The balance between the lift produced by all the body components is reflected in 

the pitching moment about the center of gravity of the body. A negative pitching 

moment coefficient is required for aerodynamic stability and indicates that the 

fin lift provides sufficient moment to offset the moment produced by the nose 

lift. Figure 5 shows the distribution of the pitching moment coefficient over the 

baseline circular cross-section geometry. The conical nose and the cylinder 

produce a destabilizing moment due to lift in front of the center of gravity. The 

boattail also provides a destabilizing moment by producing negative lift aft of 

the center of gravity. The tail fins produce sufficient lift to offset the 

destabilizing moment from the front of the body, resulting in aerodynamic 

6 
7 

Figure 5. Development of the pitching moment coefficient over the baseline circular 

cross-section geometry. 

stability for the complete configuration. The level of aerodynamic stability is 

comparable to conventional CE munitions. 

3.2 Comparison of Aerodynamics Between Circular and Noncircular 

Geometries 

The effect of noncircular cross sections on aerodynamic performance was 

examined by comparing the performance of the two noncircular cross-section 

geometries with the performance of the baseline configuration. Two additional 

circular cross sections that had the same local cross-sectional area as the 

nonconical geometries were also considered. 

Figure 6 shows the development of the drag coefficient over the baseline circular 

cross-section body and the two noncircular cross-section bodies. The circular 

cross-section geometry has the highest drag because of its large frontal area. The 

two noncircular cross-section geometries show progressively reduced drag as the 

frontal area is reduced. The drag reduction occurs primarily on the conical nose 

with some additional drag reduction on the boattailed portion of the body. The 

tail boom and fin assembly for all three bodies produce nearly identical 

contributions to the drag of the vehicle. 

Figure 7 shows a comparison of the drag for the noncircular cross-section 

geometries with circular cross-section geometries that possess the same local 

cross-sectional areas. In each case, the drag for the noncircular cross-section 
0.5 - 

0.4 - 

CD Oo3- 

0.2 - 

0.1 - 

. ..-.-- 

v....--.-* 

~~.____.__._.___.........-.---~- 

.~--------‘-‘-’ 

/.W.C-,-- 

- BASELINE BODY 

----.---.--. 50mm NONCIRCULAR BODY 

-.-.- 43mm NONCIRCULAR BODY 

0.0 I I I I I I I 

0.0 2.0 4.0 6.0 8.0 10.0 120 

X/D 

Figure 6. Development of the drag coefficient over the baseline circular cross-section and 

noncircular cross-section geometries. 

0.4 - 

CD 

0.3 - 

0.2 - 

0.1 - 

-‘-‘- 50mm NONCIRCULAR BODY 

- EQUIV. CIRCULAR BODY - 50mm 

--- 43mm NONCIRCULAR BODY 

-----------. EQUIV. CIRCULAR BODY - 43mm 

0.0 I I I I I 1 

0.0 20 4.0 6.0 8.0 IO.0 120 

X/D 

Figure 7. Development of the drag coefficient over the noncircular cross-section 

geometries and equivalent cross-sectional area circular geometries. 

8 
9 

body is greater that the corresponding circular cross-section geometry with the 

same local cross-sectional geometry. The drag increase occurs primarily on the 

conical nose with slight increase in drag on the boattailed portion of the body. 

The results seem to imply that if the payload is a fixed volume whose geometric 

shape is not constrained, such as high explosive, the configuration exhibiting the 

lowest drag will be a circular cross-section body. However, payloads with a 

noncircular cross-sectional geometry, such as an EFP warhead, may be 

effectively packaged in a noncircular cross-section flight body to yield improved 

drag performance relative to circular cross-sectional geometries. Most of the 

drag saving will result from the reduction of cross-sectional area on the nose of 

the flight body. 

The results for the three circular cross-section geometries show that as the crosssectional area of the nose is reduced, the pitching moment coefficient becomes 

increasingly more negative yielding more stability for the configuration, as 

shown in Figure 8. Most of the increased stability is due to the reductions in the 

nose lift produced by the reduced cross-sectional area of the nose, as shown in 

Figure 9. The configurations with the smaller cross-sectional area show a small 

increase in the stabilizing moment from the fins, which are presumably due to 

the reduction in the wake from the nose that interacts with the fins to reduce 

their effectiveness. 

6.0 1 

BASELINE CIRCULAR BODY’\: 

-4.0 ----------.. EQUIV. CIRCULAR BODY - %mm 

-.-.- 

-6.0 ’ 1 

EQIJJ. CIRCULAR BODY - 43mm 

I I I I I 

0.0 2.0 4.0 6.0 8.0 10.0 li0 

Figure 8. Development of the pitching moment coefficient over the circular cross-section 

geometries. 
- BASELINE CIRCULAR BODY 

-----------. EQUIV. CIRCULAR BODY - 50mm 

--I- EQUIV. CIRCULAR BODY - 43mm 

/ I I I I I I 1 

0.0 2.0 4.0 6.0 8.0 10.0 12.0 

X/D 

Figure 9. Development of the normal force coefficient over the circular cross-section 

geometries. 

The relative stability of the baseline circular cross-sectional geometry and the 

noncircular cross-section geometries is compared in Figures 10 and 11. The 

results show that as the cross-sectional geometry of the nose is reduced, the 

moment contribution from the nose lift is reduced yielding more stability. This is 

a similar trend as that observed for the circular cross-section geometries. 

However, the moment contribution from the fins appears to be significantly 

affected by the noncircular cross section of the nose. The circular cross-section 

geometry produces a more stabilizing moment from the fin than the noncircular 

geometries with the square cross-section geometry producing the least 

stabilizing moment. This appears to be due to an interaction of crossflow 

separation vortices produced at the corners of the noncircular cross section with 

the fins. Figures 12 and 13 show crossflow particle traces at an axial station 

7.5 cal. from the nose and immediately in front of the fins for the circular and 

43-mm noncircular cross-section bodies, respectively. The crossflow particle 

traces show the presence of two lee-side crossflow vortices for the circular crosssection body (Figure 12). Lee-side crossflow vortices are fairly common on 

slender flight vehicles and produced by a crossflow separation from the 

crossflow wake. The noncircular cross-section body also has two lee-side 

vortices as well as two vortices on the windside of the body (Figure 13). 

10 
. 

. 

6.0 

4.0 

%x 

20 

0.0 

-20 

-4.0 

-6.0 

- BASELINE CIRCULAR BODY 

- -------.---. 50mm NONCIRCULAR BODY 

--- 43mm NONCIRCULAR BODY 

I I I I I I 

0.0 2.0 4.0 6.0 8.0 10.0 l2.0 

X/D 

Figure 10. Development of the pitching moment coefficient over the baseline circular 

cross-section and noncircular cross-section geometries. 

1.0 

ii 

- BASELINE CIRCULAR BODY 

-----------. 50mm NONCIRCULAR BODY 

-.-.- 43mm NONCIRCULAR BODY 

0.0 b- 1 / I 

I I I I I 

0.0 2.0 4.0 6.0 8.0 ‘IO.0 120 

X/D 

Figure 11. Development of the normal force coefficient over the baseline circular crosssection and noncircular cross-section geometries. 

13 
Figure 12. Crossflow particle traces at X/D=7.5, circular cross-section body, Mach 3, 2” 

angle of attack. 

Figure 13. Crossflow particle traces at X/D=7.5, 43-mm noncircular cross-section body, 

Mach 3, 2” angle of attack. 

12 
3.3 Effect of Roll Orientation on Aerodynamic Performance 

For completely axisymmetric bodies, the aerodynamic performance of the flight 

body will be independent of roll orientation. (This behavior is limited to 

small-to-moderate angles of attack where the lee-side vortices are symmetric.) 

For noncircular cross sections, the aerodynamics may exhibit some dependence 

roll orientation. If the body’s roll rate is above the pitching frequency, the effect 

of roll orientation is often averaged out. If the body is not rolling, the effect of 

roll orientation on the aerodynamics may be more important. Computations 

were performed to assess the effect of roll orientation. The computations 

presented previously were performed with the body oriented in the 

“x orientation” with respect to the pitch plane. In other words, the pitch plane 

was oriented between the fins and the comers of the noncircular cross section. 

Additional computations were performed with the body rolled at 22.5” and 45” 

from the x orientation. The 45” roll orientation is often referred to as the 

“+ orientation” since the fins and comers aligned with the pitch plane. 

The baseline circular cross-section body showed some rolI orientation 

dependence in the normal force and pitching moment over the finned portion of 

the body as shown in Figures 14 and 15. This is not completely unexpected 

since the fin lift should be affected by the orientation of the fins with respect to 

the lee-side vortices. With the fins oriented in the + orientation, the horizontal 

(lift producing) fins are below the lee-side vortices. In the x orientation, the two 

lee-side fins are located within the lee-side vortices. As a result, the normal force 

coefficient increases by 3.4% compared with the x orientation. Because of the 

long moment arm from the center of gravity (CG) to the fins, the pitching 

moment increases by approximateiy 30%. 

The normal force and pitching moment coefficients for the noncircular 

cross-section body displays dependence on roll angle over both the forebody 

and the fins as shown in Figures 16 and 17. It is interesting to note that the roll 

dependence in the transverse force and moment on the forebody occurs 

downstream of the noncircular cross section, rather than on the portion of the 

body with the noncircular cross section. This is likely due to differences in the 

wake and vortex patterns between the various roll orientations. The variation of 

the normal force and pitching moment coefficient with roll angle is actually less 

than that of the baseline circular cross-section body. 

The simulations showed very little effect of roll orientation on the axial force at 

2’ angle of attack (less than 0.2%) for either the baseline circular cross-section 

body or the 43-mm noncircular cross-section body. 

13 
0.0 2.0 4.0 6.0 8.0 10.0 120 

x/D 

Figure 14. Effect ok roll orientation on the development of the normal force coefficient 

over the baseline circular cross-section geometry. 

6.0 

4.0 

-20 

-4.0 

-.-.- 22S” ROLL ORIENTATION 

/ I I I I I I 1 

0.0 2.0 4.0 6.0 8.0 10.0 a.0 

X/D 

Figure 15. Effect of roll orientation on the development of the pitching moment 

coefficient over the baseline circular cross-section geometry. 

14 
. 

4.0 

3.0 .i 

cN* 

20 

1 /‘i/- 

to - - X ORIENTATION 

.----e-----e + ORIENTATION 

-‘-.- 22.5° ORIENTATION 

0.0 / I I I I I I 1 

0.0 2.0 4.0 6.0 8.0 IO.0 120 

X/D 

Figure 16. Effect of roll orientation on the development of the normal force coefficient 

over the &mm noncircular cross-section geometry. 

6.0 l 

4.0 

%x 

2.0 

0.0 

-2.0 *, 

0.0 2.0 4.0 6.0 8.0 10.0 120 

X/D 

Figure 17. Effect of roll orientation on the development of the pitching moment 

coefficient over the baseline circular cross-section geometry. 

15 
4. Conclusion 

The study presented here demonstrates the potential for aerodynamic benefits of 

noncircular cross-section bodies when compared with circular cross-section 

bodies. The results show that significant reduction in drag is possible through 

the use of a noncircular cross section under certain constraints. In particular, if 

the geometric constraints of the payload conform more readily to a noncircular 

cross-section body, there is a potential for aerodynamic benefits such as 

improved stability and drag reduction. If the payload is not geometrically 

constrained and can fit in a fixed volume, the results indicate that a circular cross 

section provides optimal aerodynamic performance. 

For the particular configurations examined here, the transverse aerodynamic 

force and moments for the noncircular cross-section bodies did not display a 

significant dependence on roll orientation. In fact, the roll dependence produced 

by the roll orientation of fins produced a more significant effect than the roll 

orientation effects of the noncircular body cross section. 

Finally, this particular study was performed using sophisticated computational 

tools. The results indicate that some of the aerodynamic effects demonstrated 

here can be predicted using simpler aerodynamic models although others 

cannot. Thus the use of sophisticated tools, even in preliminary or conceptual 

aerodynamic design, may be justified, particularly when the configurations of 

interest are sufficiently different than those of typical interest. 

16 
5. References 

1. Schiff, L. B., and J. L. Steger. “Numerical Simulation of Steady Supersonic 

Viscous Flow.” AIAA Journal, vol. 18, no. 12, pp. 1421-1430, December 1980. 

2. Sturek, W. B., D. C. Myhn, B. J. Guidos, and C. J. Nietubicz. “Navier-Stokes 

ComputationaI Study of the Influence of SheII Geometry on the Magnus 

Effect at Supersonic Speeds.” BRL-TR-2501, U.S. Army Ballistic Research 

Laboratory, Aberdeen Proving Ground, MD, June 1983. 

3. Weinacht, P., B. J. Guidos, L. D. Kayser, and W. B. Sturek. “PNS 

Computations for Spinning and Fin-Stabilized Projectiles at Supersonic 

Velocities.” BRL-MR-3464, U.S. Army Ballistic Research Laboratory, 

Aberdeen Proving Ground, MD, September 1985. 

4. Danberg, J. E., A. S&al, and I. Cehnins. “Prediction and Comparison With 

Measurements of the Aerodynamic Characteristics of Flare-Stabilized XM910 

Prototypes.” BRL-MR-3752, U.S. Army Ballistic Research Laboratory, 

Aberdeen Proving Ground, MD, May 1989. 

5. Weinacht, P. “Application of Computational Fluid Dynamics to the Analysis 

of the Aerodynamics of a Railgun Projectile.” BRL-MR-3742, U.S. Army 

BaIIistic Research Laboratory, Aberdeen Proving Ground, MD, March 1989. 

6. Baldwin, B. S., and H. Lomax. “Thin Layer Approximation and Algebraic 

Model for Separated Turbulent Flows.” AIAA Paper 78-257, AIAA 16th 

Aerospace Sciences Meeting, HuntsviIIe, AL, January 1978. 

7. Beam, R., and R. F. Warming. “An Implicit Factored Scheme for the 

Compressible Navier-Stokes Equations.” AIAA JoumaZ, vol. 16, no. 4, 

pp. 393-402,1978. 

8. Rai, M. M., and D. S. Chaussee. “New Implicit Boundary Procedure: Theory 

and Applications.” AIAA Paper 83-0123, AIAA 23rd Aerospace Sciences 

Meeting, Reno, NV, January 1983. 

9. Rai, M. M., Chaussee, D. S., and Rizk, Y. M. “Calculation of Viscous 

Supersonic Flows Over Finned Bodies.” AIAA Paper 83-1667, July 1983. 

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i 

c 

REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

oss Section on Projectile Aerodynamic Performance With 

Paul Weinacht 

AT-IN: AMSRL-WM-BC 

Aberdeen Proving Ground, MD 2 10055066 

AGENCY REPORT NUMBER 

11. SUPPLEMENTARY NOTES 

12a. DlSTRlBUTlONlAVAlLABlLlTY STATEMENT 

Approved for public release; distribution is unlimited. 

12b. DISTRIBUTION CODE 

13. ABSTRACT(Maximum 200 words) 

The most effective bore cross-section geometry for electromagnetic (EM) guns may differ significantly from tht 

circular cross section of conventional guns. The geometry of the bore cross section will influence the shape of tlx 

annature/subprojectile launch package. For certain applications, a noncircular cross section may provide payloac 

packaging benefits which have not been considered previously for conventional circular cross-section guns. Whethe! 

these benefits can be fully realized depends, in part, on the aerodynamic performance of these geometries in free flight. 

The current study considers the aerodynamics of chemical energy (CE)-type munitions to determine whether there an 

potential aerodynamic benefits for noncircular cross-section geometries compared with conventional circula: 

cross-section bodies. The study compares the aerodynamic drag and static stability of both circular and noncircula 

geometries using sophisticated computational aerodynamic predictive tools to determine the potential aerodynamic 

benefits. 

14. SUBJECT TERMS 

projectile aerodynamics, EM guns, computational fluid dynamics 

15. NUMBER OF PAGES 

27 

16. PRICE CODE 

17. SECURITY CLASSlFlCATlON 

OF REPORT 

UNCLASSIFIED 

hlcu 7GlnA,-,IuLKinfl 

18. SECURITY CLASSIFICATION 

OF THIS PAGE 

UNCLASSIFIED 

19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT 

OF ABSTRACT 

UNCLASSIFIED UL 

, .-. . , -“-., . ..“” --“- Standard Form 298 IRev. 2-89) 

25 Presaibed by ANSI std. 239-li) 298-102 
Y 

26