TECHNICAL REPORT ARBRL-TR-02535

* NUMERICAL INVESTIGATION OF THE AERODYNAMICS

AND STABILITY OF A FLARED AFTERBODY FOR

AXISYMMETRIC PROJECTILES AT

SUPERSONIC SPEEDS

Michael J. Nusca -

4 DEC 281W3

December 1983QUS ARMY ARMAM[NT R[S[ARCH AND D[VR[OPM[NJ C[NTERBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

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The findings in this report are not to be construed asan official Department of the Army position, unlessso designated by other authorized documents.

Th4 use Of trude naees or mmufaoturwrs' 'ta',. in this iwportdoes not constitugte i-.aorqwe,.t of anyCom O~3rI produce.

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. . . . '.. - ,, ". ., I . ., _. . - . ."- - .- . - .. .- ,,..

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REPOT DCUMNTATON AGEBEFORE COMPLETING FORM,. : 1. REPORT NUMBER IGOVT ACCESSION MO. 3. RECIPIENT'S

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" TECHNIC.AL REPORT ARBRL-TR-02535 P-441-3/6 Me

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

NUMERICAL INVESTIGATION OF THE AERODYNAMICS AND FinalSTABILITY OF A FLARED AFTERBODY FOR AXISYMMETRIC 6. PERFORMING ORG. REPORT NUMBER

PROJECTILES AT SUPERSONIC SPEEDS7. AUTHOR(&) S. CONTRACT OR GRANT NUMBER(e)

Michael J. Nusca

9. PERFORMING ORGANIZATION NAME AND ADDRESS t. PROGRAM, ELEMENT. PROJECT, TASKAREA & WORK UNIT NUMBERS

U.S. Army Ballistic Research Laboratory, ARDCA TTN: DRSMC-BLL (A) R T E 1 1 2 1 A SAberdeen Proving Ground, Maryland 21005II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

US Army AMCCOM, ARDC December 1983Ballistic Research Laboratory, ATTN: DRSMC-BLA-S(A) ,3. NUMBEROFPAGESAberdeen Proving Ground, Maryland 21005 564. MONITORING AGENCY NAME & AIDDRESS(II dflferent lroat Controfllin Oflice) iS, SECURITY CLASS. (ol the report)

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I$. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reveree side if necoeery and identify by block ntnbor)

Afterbody Gyroscopic Stability

Flare Dynamic StabilityParabolized Navier-Stokes Supersonic FlowProjectile Aerodynamics20. ATrACT (onthmie .v rn eIv If Neemy owd Idenify by block nontber)

A parametric study has been conducted for eight similar axisymmetric, spinstabilized projectile shapes. The aerodynamic and stability parameters of flaredprojectile afterbodies at freestream Mach numbers of 2, 3 and 4 and at an angleof attack of 2°"have been determined. The projectiles under investigation had anogive nose, a cylindrical midsection and one of three afterbodies: an extendedcylinder, a boattail (conical frustum), and a boattail-flare (two conicalfrustums end to end). Total projectile lengths of 6 and 7 calibers have been

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20. ABSTRACT (Continued)

studied. Finite difference solutions of the Navier-Stokes equations, the methodof characteristics, and semi-empirical models are used to predict normal force,static pitching moment, total drag, Magnus moment, roll damping and pitch damp-ing. Estimations of gyroscopic and dynamic stability parameters are then made.

This study determined that the boattail-flare renders the ogive nose andcylindrical midsection projectile configuration more gyroscopically and dynamic-ally stable than the conventional boattail afterbody. Results also showed thatthe boattail-flare afterbody decreases the Magnus moment but increases total dragtherefore providing a stability advantage at the expense of reduced range. Thisstability advantage can be utilized in long spin-stabilized projectiles as wellas long rod penetrators where fin stabilization has been replaced by a boattail-flare. Future work may be aimed at utilizing present computational techniques asa design tool in the use of a flared afterbody for stable, reduced range projec-tiles.

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4' TABLE OF CONTENTS"-.1

Page

LIST OF ILLUSTRATIONS ........................ ..................... 5

I. INTRODUCTION .................... .. ................................ 7

II. BACKGROUND ......................................................... 7

III. COMPUTATIONAL TECHNIQUE ................................... ......... 9

IV. PROCEDURE ..... .... * . ....... * , ... . ............. 10

- V. RESULTS AND DISCUSSION .............................. .............. 12

A. Static and Gyroscopic Stability ................................ 13

B. Dynamic Stability ... ..................... 14" CONCLUSION...

- ,VI. CO C U I N......................................................... 1REFERENCES ............................. *............................ 51

LIST OF SYMBOLS .................. . .... ................. 53

DISTRIBUTION LIST .................................................. 55

*uric

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j..--.

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ptic3 tO,,3

r- % ' " °

"

LIST OF ILLUSTRATIONS

Figure Page

.. 1 Projectile Geometry ............. ....................... 19

2 Schematic Drawings of Projectile Groups ........................ 20

3 Axial Surface Pressure Distributions; M = 3.0, a = 20.............. 21

a. SOC2.............................................................. 21

b. SOCBT2............................................................. 22

c. SOCBTF2 ................................................. 23

4 Axial Normal Force Distributions; M = 3.0, a =20, SOC2,4%SOCBT2, SOCBTF2 ..................... ..................... 24

5 Axial Surface Pressure Distributions; SOCBTF1, '=20.............. 25

a. M c . . . . . . . . . . . . . . . . . . . . . . . . . . 2

a. Mach = 2 ............ .o................................. 25

b. Mach = 3 ..... o........................................... 26

6 Axial Normal Force Distributions; SOC2, SOCBT2, SOCBTF2,aa==200............0.0.0...0.0...............a....................0........0....................0.......02 8

a. Mach =2.o.o. .....o.................. ............ ..... o... 28

b. Mach 3 3................................................................. 29

c. Mach= 4.o.oo ........ o....*oo ... o..... .... o... 30

7 Static Pitching Moment vs. Mach No ....... ......... o....o.......31

a. SOC3, SOCBT3, SOCBTF3. ............... ..... .o... .... oo...................31

b. S0C2, SOCBT2, SOCBTF2....... ............................ ............ 32

c. SOCI, SOCBT1, SOCBTF1.l........... ... .... . ... .o..................33

do SOCBTF3, SOCBTF1, SOCBTF2 .... o ... o............ 34

8 Nondimensional Spinrate (pd/V) vs. Mach No., 5g = 13.3..... .... 35

a. S0C3, SOCBT3, SOCBTF3. ooo......... ........... .. 35

b. 50C2 , SOCtBT2 , SOCBTF2. ........................... ....oo ................... .. 36

c. SOCi, SOCBT1, SOCBTF1 ............... .... ... ..... ........ 37

d. SOCBTF3, SOCBTF1, SOCBTF2...o................ o...... ... 38

5

N,~

LIST OF ILLUSTRATIONS (continued)

Figure Page

9 Drag Force Coefficient vs. Mach No ............................... 39

a. SOC3, SOCBT3, SOCBTF3 ,*.. .. .. . .. .. .. .. .................. 39

b. S0C2, SOCBT2, SOCBTF2 ......... *................ 40

c. SOCI * SOCBTL, SOCBTFI........... .......... ..... . ...... 41

d. SOCBTF3, SOCBTF1, SOCBTF2 ......................... 42

10 Magnus Moment Slope vs. Mach No ................ ........ 43

a. SOC3 * SOCBT73 SOCBTF3 .. .. . .. .. . ... . ... . . ................ 43

b. SOC2 , SOCBT2 , SOCBTF2. .. .. . .. .. . ............. . .. ...... . .. 44

c. SOC1, SOCBTI, S O C B T F 1 ....................... 45

d. SOCBTF3 , SOCBTF1 , SOCBTF2 ......... .. .. .. . .. .. .. ......... 46

11 Roll Damping Coefficient vs. Mach No ........................... 47

a. SOC3, SOCBT3, SOCBTF3. ... .. .. .. . .. .. .. .. . ... .. .. .. .. .. .. .. ... 47

* ~~~~b. S0C2 , SOCBT2 , SOCBTF2. . ... . .. .. . .... .. .. . .. , . .. .. .. .. . .. . .. 48

c. SOCL , SOCBT1 , SOCBTF1 . .. .. .. .. . ... . ... .. .. .. .. . ... ........ . 49

d. SOCBTF3 , SOCBTF1 , SOCBTF2. .. .. .. .. .. . .... .. .. .. .. .. . .. .. .. ... 50

6

I. INTRODUCTION

The long, slender, axisymmetric bodies that are conventionally used forspin-stabilized projectile designs consist of an ogive nose section followedby a constant diameter cylinder and a boattail. The ogive is similar in shapeto a cone but typically has a smaller associated wave drag and has more volumefor a payload. Secant ogives that are normally used are not tangent to thecylinder at the end of the nose section. In order to decrease base drag, aboattail is added. The boattail is a conical frustum that narrows the base totypically 75% of the cylindrical body diameter. This description character-izes the conventional spin-stabilized secant-ogive cylinder boattail (SOCBT)projectile. Afterbody designs that statically stabilize nonspinning projec-tiles are lifting surfaces of two types: thin fins and conical flares. Finspositioned as a tail generate normal force to statically stabilize the projec-tile, whereas conical flared afterbodies generate high pressure regions andmove the center of pressure rearward, behind the center of gravity.

The purpose of this study is to evaluate the ability of a boattail-flare(see Figure 1) to improve the stability of the SOCBT configuration. Computa-tional methods are used to solve the parabolized Navier-Stokes equations inthe flowfield around the projectile. Predictions of static pitching moment,drag force, and roll damping, Magnus and pitch damping moments are made todefine the effects of the boattail-flare afterbody on the aerodynamics of theogive-cylinder projectile.

II. BACKGROUND

The gyroscopic stability of a spin-stabilized projectile can be assessedby computing Sg, the gyroscopic stability factor.1

2 (pd/V)2 12

p Iyd 5 CMa

where; Ix = axial moment of inertia, kg - M3

Iy = transverse moment of inertia, kg - m3

pd/V = dimensionless axial spinrate

p = axial angular velocity, rad/sec

p = air density, kg - m

d - maximum body diameter, m

1. Murphy, C.H., "Free Flight Motion of Synnetric Missile," U.S. Ar'myBallistic Research Laboratory Report No. 1216, Aberdeen Proving Cvound,Maryland 21005, July 1963 (AD 442757).

7

.% -. . - . . . ..-. . -.. ........ ... - - - - - - - - - - -

C _; 7I . , . **- . , : .p

. . .. .

V = airspeed, m/sec

CM = static moment coefficient, per radian

If 0 < s < 1 the projectile is gyroscopically unstable. Note that sq is

inversely proportional to air density; therefore, projectiles that are gyro-scopically stable for standard atmospheric conditions may not be stable undernonstandard temperature and pressure. Due to this as well as other uncer-tainties in computin Sg, many designers have set 1.3 as a lower limit ongyroscopic stability.1

An additional requirement for spin-stabilized projectiles is that they bedynamically stable. The dynamic stability of a projectile is defined byMurphy1 as

2 (CNa - CD) + 2 kx 2 CMP(sd :_ (2)

CNa- 2 CD -ky 2 (CMq + CMa)

where; CNa = normal force coefficient slope per radian

CD = total drag force coefficient

CMPQ = Magnus moment coefficient slope per radian

CMq + CM = pitch damping coefficient

kx = axial radius of gyration

k y = transverse radius of gyration

The range of values of sd for dynamic stability are also given by Murphy1

and are found to be a function of the reciprocal of the gyroscopic stability.

l/Sg = sd (2- Sd) (3)

2. "Engineering Design Handbook - Design for Control of Projectite FlightCharacteristics,' Headquarters, US Amy Materiel Command, September 1966.

8

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% . -.. , .* - ". - . ...- .-v.,....o - . . '., , , - -'. - .-. .- . .-. * -.'. " ' '-" "i - -- -; -r.

III. COMPUTATIONAL TECHNIQUE

Evaluation of the stability of secant ogive-cylinder projectiles with aboattail, or boattail-flare afterbody, requires the calculation of the aerody-namic forces (normal, drag, and Magnus) and moments (pitching, Magnus, androll damping). These parameters were determined using three techniques: solu-tion of the parabolized Navier-Stokes equations, the method of characteris-tics, and semi-empirical models.

Recent papers 3'4 '5'6'7 describe the application of the thin-layer para-bolized Navier-Stokes (PNS) computational technique to predict the flow aboutslender bodies of revolution with sharp noses at supersonic velocities. ThePNS technique was used in this study to yield the aerodynamic coefficientsnecessary for the evaluation of static, gyroscopic and dynamic stability.

Schiff and Sturek4 have employed the PNS computational technique to solvefor the flowfield about a sharp body of revolution and have made comparisonswith experimental data. Computed and measured pressure data in both the axialand circumferential directions were consistent.

The PNS solutions do not compute the base flow region. Therefore, calcu-lation of the base drag was performed using the method described by Roache andMueller.8 The base flow computation was started upstream of the afterbody

3. Schiff, L.B., and Steger, J.L., "Numerical Simulation of Steady SupersonicViscous Flow," AIAA Journal. Vol. 18, No. 12, December 1980, pp. 1421-1430.

4. Schiff, L.B., and Sturek, W.B., "Numerical Simulation of Steady SupersonicFlow over an Ogive-Cylinder-Boattail Body," ARBRL-TR-02363, US ArmyBallistic Research Laboratory, Aberdeen Proving Ground, Maryland 21005,September 1981 (AD A106060).

5. Sturek, W.B., and Schiff, L.B., "Computations of the Magnus Effect forSlender Bodies in Supersonic Flow," ARBRL-TR-02384, US Army BallisticResearch Laboratory, Aberdeen Proving Ground, Maryland 21005, December1981 (AD A110016).

6. Sturek, W.B., and Mylin, D.C., "Computational Study of the Magnus Effecton Boattailed Shell," AIAA Journal Vol. 20, No. 10, October 1982, pp.1462-1464.

7. Sturek, W.B., Guidos, B., and Nietubicz, C.J., 'Wavier-Stokes Computation-al Study of the Influence of Shell Geometry on the Magnus Effect at Super-sonic Speeds," ARBRL-TR-02501, US Army Ballistic Research Laboratory,Aberdeen Proving Ground, Maryland 21005, June 1983 (AD A130630).

8. Mueller, T.J., and Kayser, L.D., "A Method of Determining the TurbulentBase Pressure in Uniform and Non-Uniform Supersonic Flows," ARBRL-TR-02374, US Army Ballistic Research Laboratory, Aberdeen Proving Ground,Maryland 21005, October 1981 (AD A107318).

* [9

using local velocity profiles from the PNS computation and yielded the ratioof base to freestream pressure from a method of characteristics solution. Forthe cylindrical and boattailed afterbodies, the solution was started at theend of the cylindrical portion of the projectile, whereas the solution for theboattail-flare afterbody was started at the end of the flare with the charac-teristic lines extending into the base region.

A second auxiliary calculation was required for the pitch damping coeffi-cients used in determining the dynamic stability factor, sd . (See Equation 2.)

A semi-empirical technique developed by Devan and Mason 9 computes the pitchdamping coefficient based on trends in wind tunnel and flight test data forthis class of axisymmetric bodies.

IV. PROCEDURE

To adequately evaluate the effectiveness of different projectile desigfa parametric study was undertaken using the eight different shapes shownFigure 1 and described in Table 1. Figure 2 is a schematic of the projectishapes studied grouped according to geometric similarity. All data generatin the investigation are also grouped in this fashion. The projectile sharhad the same nose section (ogive) and differed only in the length of tcylinder and afterbody sections. Included in study were both the conventionu.boattail and unconventional boattail-flare afterbody.

TABLE 1. PROJECTILE PHYSICAL PROPERTIES. MATERIAL DENSITY ASSUMED2.77 x 10-6 kg/mm 3 (ALUMINUM)

(mm) (Kg) (Kg- mm2 )Li L2 L3 Mass Ix ly

SOC1/2 171.6 0 0 1.74 6.38 x 102 1.08 x 104

SOC3 228.8 0 0 2.14 8.04 x 102 2.23 x 104

SOCBTI 114.4 57.2 0 1.65 5.75 x 102 9.51 x 103

SOCBT2 57.2 114.4 0 1.40 4.38 x 102 7.28 x 103

SOCBT3 114.4 114.4 0 1.81 6.03 x 102 1.39 x 104

9. Devan, L., and Mason, L.A., "Aerodynamics of Tactical Weapons to MachNumber 8 and Angle of Attack 1800: Part II, Computer Program and UsersGuide," N.S.W.C. TR 81-358, September 1981.

10

J...

,,-;- .. r" . i , , . r. . .r... -.,.- ..-.--.. -,- .. . - - " . - . - " - . " . -- - "

TABLE 1. (continued)

SOCBTF1 114.4 28.6 28.6 1.69 6.02 x 102 1.02 x i04

SOCBTF2 57.2 57.2 57.2 1.56 5.11 x 102 9.48 x 103

SOCBTF3 114.4 57.2 57.2 1.96 6.77 x 102 1.86 x IOh

Because of these geometric similarities the PNS code was advanced overthe 3 caliber ogive and onto the cylindrical section for 1 caliber. Theresulting data were used to restart the computation over different after-bodies. In this way the computational nime was reduced from the time thatwould be required if the PNS solution was z.rted from the nose of each of theeight bodies. The computations were carried out for an angle of attack of 20and a nondimensional spinrate of 0.19 (used for Magnus computations).

Using the Roache-Mueller method, the plane of data supplied by the PNSsolution at the end of the body was advanced into the base region to calculatebase pressure. Base pressure data, in the form of a ratio of base to free-stream pressure, was converted to base drag coefficient form l° using:

2 d2 (1 - pP (CB B/= (4)

O M2

where d = base diameter in calibersM = freestream Mach number

The semi-empirical method of Devan and Mason was used to compute pitchdamping for the ogive-cylinder (SOC1, 2, 3) and ogive-cylinder-boattail(SOCBT1, 2, 3) bodies, the data base did not allow ac'urate computations ofthe boattail-flare bodies (SOCBTF1, 2, 3). The pitching moment of a boattail-flare afterbody was assumed to be the same as that of the boattailed body ofequal length (e.g., SOCBTF1 and SOCBT1). Flares in general have been shown to

10. McCoy, R.L., '"corag - A Computer Program for Estimating the Drag Coeffi-cients of Projectiles," ARBRL-TR-02293, US Army Ballistic ResearchLaboratory, Aberdeen Proving Ground, Maryland 21005, February 1981

- (AD A098110).

-P.

increase a projectile's pitch damping; predictions of this coefficient aretherefore conservative.

The computed data were reduced to the following aerodynamic coefficients.Total drag was obtained from the summation of viscous, pressure, and base dr-Igcoefficients. Normal force slope, CN. (assumed to be linear), was obtained hy

dividing the normal force coefficieqt by .0349 radian (20). Pitching momentslope, CM. and Magnus moment slope, CMp, were obtained by referencing the

moment coefficient to a point 3.6 calibers from the projectile nose in allcases, and dividing by .0349 radian. Roll damping coefficient, CX was

obtained directly from the PNS computation. Variations of dimensionless spin-rate, pd/V, were evaluated from Equation 1 assuming Sg = 1.3, while dynamicstability was evaluated from Equation 2.

V. RESULTS AND DISCUSSION

The aerodynamic stability of the projectile configurations was studied inthree ways: static, gyroscopic, and dynamic stability. Static stability isdefined in terms of the static pitching moment, gyroscopic stability in termsof sg and dimensionless spinrate, and dynamic stability as defined in Equation2.

Typical wall pressure distributions at Mach 3 (angle of attack 20) areshown in Figure 3 for one group of configurations, SOC2, SOCBT2, SOCBTF2. Theeffect of the boattail-flare afterbody is immediately apparent from the drama-tic pressure recovery that occurs over the flared portion (Figure 3c). Thehigher pressure on the windward side of the projectile is due to the angle ofattack and is responsible for the projectile's normal force. Higher leewardpressure on the boattail detracts from the total normal force of the projec-tile (Figure 3b). A large pressure recovery for the boattail-flare is shownin Figure 3c.

The axial distribution of normal force for this group of configurationsis shown in Figure 4. The local normal force rises over the 3-caliber ogiveand 1-caliber cylinder, decreases sharply over the 2-caliber boattail but ispartially recovered when one caliber of the boattail is flared.

The wall pressure and normal force recovery effect of the boattail-flareafterbody depends largely on the value of freestream Mach number. Using theSOCBTF1 configuration as an example, Figures 5a-c show the dramatic change inwall pressure recovery over the .5-caliber flare. At Mach 2 (Figure 5a) wallpressure recovery is minimal and the pressure difference from windward to lee-ward side is negligible. For Mach 3 and 4 (Figures 5b and 5c) the effect ofthe flare is significant, producing a large difference between windward and

11. Robinson, M.L., "Boundary Layer Effects in Supersonic Flow Over Cylinder-Flare Bodies," Australian Defense Scientific Service Weapons ResearchEstablishment Report No. 1238, July 1974.

12

b * . * . ** . - -

.

- .. .

leeward wall pressures. When these pressure differences are converted to nor-mal force distributions, the effect of Mach number is again observed (Figures6a-c).

A. Static and Gyroscopic Stability

The ability of the boattail-flare afterbody to augment the normal force. of the secant ogive cylinder projectile radically changes the static pitching

moment. Integration of the normal force distribution yields a total normalforce located at the center of pressure. The static pitching moment isdefined by this force and the difference between the center of pressure andcenter of gravity locations. Comparisons can be made between the SOC, SOCBTand SOCBTF configurations on the basis of static pitching moment because thismoment is referenced to the same point for each body (3.6 calibers from thenose as described in the Section IV). Figure 7a, b, and c compare the pitch-ing moment characteristics of the cylinder, boattial and boattail-flare after-bodies in the three groups of configurations. (Recall Figure 2.) Consistent

%%" with the axial distribution of normal force, the SOC and SOCBTF configurationsin each group exhibit the lowest pitching moment coefficient (significantlylower than the SOCBT). Comparing all flared bodies, SOCBTF, Figure 7d showsthat the SOCTF1 has the smallest pitching moment coefficient for all but Mach4.

Of the three flared configurations the SOCBTF1 has the shortest boattail-flare afterbody length; the pitching moment performance of this configurationrelative to configurations with longer flares (e.g., SOCBTF3) must thereforebe explained. Wall pressure distributions reveal that the shorter flare main-tains a large difference between windward and leeward pressure over its entirelength whereas the longer flares allow the flow to expand over their lengththus losing normal force. (Compare Figure 3c between 5 and 6 calibers andFigure 5b between 5.5 and 6 calibers.) The SOCBTF1 configuration is unique inthat it has the shortest boattail and therefore does not generate the loss innormal force characteristic of long boattails.(Review Figures 3b and 5.) Asthe freestream Mach number increases to Mach 4, the long flare of the SOCBTF3generates increasingly more normal force and finally overcomes the loss asso-ciated with its long boattail. However, the SOCBTF3 is the greatest normalforce and the smallest moment arm (cg-cp) and is therefore subjected to thesmallest pitching moment at Mach 4.

From computed pitching moment data and an assumed s of 1.3, dimension-less spinrates (pd/V) can be determined from Equation 1 is described in Sec-tion IV. Figure 8a, b and c show that pd/V decreases with freestream Machnumber for all configurations with the lowest pd/V associated with the SOC andSOCBTF. In each j.se the boattail-flare afterbody reduced pd/V below thatrequired of the boattail afterbody but not as low as the configuration withoutan afterbody (SOC). The shortest boattail-flare afterbody (SOCBTF1), however,shows a pd/V comparable to the corresponding SOC configuration (Figure 8c).In addition the SOCBTF1 shows the lowest required pd/V of all flared after-bodies (Figure 8d).

13

B. Dynamic Stability

Dynamic stability as defined in Equation 2 requires the aerodynamic coef-ficients of drag, normal force, Magnus moment, and pitch damping. Evaluationof these coefficients will precede discussion of the dynamic stability factor,sd•

Figures 9a-d compare the drag coefficients of all configurations. Asmentioned in the proceedure section, drag coefficient is the algebraic sum ofviscous drag and pressure drag (from PNS computation) along with base drag(from Roache-Mueller). The boattail afterbody configurations (SOCBT) have theleast drag due to the narrowing of the base. This narrowing causes a flowexpansion that decreases the base pressure (PB/P. decreases) but also geomet-

rically decreases the base diameter (dB). The combined effect decreases basedrag. See Equation 4.)

The boattail-flare afterbody exhibits a significantly larger drag coeffi-cient in all cases but for the SOCBTF1. The source of this drag penalty istwo-fold: the pressure drag associated with the oblique shock formed at theboattail-flare junction and the higher base drag associated with the largerdiameter base. Of all flared configurations, however, the SOCBTF1 shows thesmallest drag rise (Figure 9d).

Figures 10a-b compare the Magnus moment coefficient slope of all config-urations. Consider a spinning axisymmetric body at angle of attack. Thecomponent of freestream velocity perpendicular to the body (cross flow) isdecelerated on the side that opposes spin and is accelerated on the side thatis coincident with spin. This difference in velocity across the body givesrise to a difference in pressure and a side force develops, the Magnus force.This force becomes more significant as the boundary layer thickens on the aftend of the projectile. This is especially true of boattail afterbodies wherethe expanded flow accelerates the boundary layer growth. In contrast a flaregenerates an adverse pressure gradient and thins the boundary layer.'' Inthis study it was expected that the SOCBT configuration would have the highestMagnus moment with the SOCBTF configuration having the lowest. This is sup-ported in Figure 10. Comparing all of the SOCBTF configurations the shortestflare, SOCBTF1, had the smallest Magnus moment (Figure lOd).

The roll damping coefficient data are illustrated in Figures 11a-b. Dueto the viscous boundary layer a shearing force is generated on the projectilesurface in the direction opposite the spinrate. (Hence, the coefficient isusually negative.) Because the roll damping coefficient arises from viscousshear, it is directly proportional to the velocity component tangent to the

body surface. Using an aeroballistic coordinate system, the component ofvelocity in the x-direction (along the body axis, positive rearward) is desiq-nated u, the component perpendicular to the body axis is v, and the third com-ponent to form a right-handed axis system is w. Roll damping coefficient isthen proportional to dw/dy. As the radius of the spinning body decreases,surface velocity decreases also (w = or). As Figure 11 illustrates the SOCdesign has the largest roll damping because it has the greatest body diameterfrom nose to base. When a boattail afterbody is used (SOCBT) the roll dampingdecreases slightly as the body radius decreases. The boattail-flare afterbodydesign increases roll damping to a small degree because it has an increased

14

body slope-- but the SOCBTF has less roll damping than the SOC with aconstant diameter shape.

From the discussion above it is also clear that roll damping is propor-tional to surface area -- the larger a body is the more viscous shear is gen-erated. Figure 11a, that represents the longest group of projectiles (theS0C3, SOCBT3, SOCBTF3 are 7 calibers long), also shows the largest roll damp-ing. Figures 11b, and c represent the two groups of projectiles that are 6calilers in length and therefore show less roll damping. The increase in rolldamping predicted for the boattail-flare afterbody could be viewed as a disad-vantage to the design if it were not for the small magnitude of that increase.Figure 11d illustrates that the boattail-flare afterbody design that leads tothe smallest increase in roll damping is not the shortest flare (which hasbeen the pattern up to this point) but the one with the shortest cylindricalsection, SOCBTF2. Reviewing Figure 2, it would seem that the SOCBTF2 has thesmallest overall amount of surface area.

With the aerodynamic coefficients of drag and Magnus moment, the dynamic

stability factor, Sd, as given in Equation 2, can be evaluated. Table 2 lists

the values of sd grouped by geometric similarity. Dynamic stability was

increased for all boattail-flare projectiles when compared to the straightcylinder bodies. The boattail-flare afterbody increased sd in only a few

cases when compared to the boattail bodies. At Mach 4, the boattail-flareconsistently led to a more dynamically stable projectile. For the firstgroup, the SOCBTF1 increased the stability of the projectiles for all of Machnumbers.

TABLE 2. DYNAMIC STABILITY FACTORS (sd)*

MACH =2 MACH =3 MACH= 4

SOCi 22.9SOCBT1 [ i[n3SOCBTFI 36'37

SOC2 .273 .292SOCBT2 .349 .367 .365SOCBTF2 .341 .362 .384

SOC3 .307 .321SOCBT3 .427 .419SOCBTF3 .414 .398

*BOXES INDICATE THOSE CONFIGURATIONS AND MACH NUMBERS AT WHICH THEBOATTAIL-FLARE AFTERBOCYHAS INCREASED DYNAMIC STABILITY.

15.4

V

4°e.

'.2

A. The dynamic stability factor involves many aerodynamic coefficients, butis defined in a complicated manner. Platou 12 has attempted to determine therelative importance of the individual aerodynamic coefficients used with

dynamic stability. Examining sd in a parametric manner, Platou concluded:

(1) CNa and CD have only a small influence on sd except when CNa -CO

is maximum (i.e., CNa is maximum and CD is minimum) sd is maximum.

(2) sd is maximum when the transverse moment of inertia, Iy, is minimum.

(3) sd is maximum when Magnus moment slope, CMPa, is minimum.

(4) If pitch damping, CMq + CM , is minimum then sd is maximum. Also,

if pitch damping is maximum then the relative influence of CMpa on

sd is increased.

The data computed by this study showed the following trends for the flareshapes:

ly CNa CD CMpa CMq + CMa

smallest SOCBTF2 SOCBTF2 SOCBTF1 SOCBTF1 SOCBTFHSOCBTF1 SOCBTF3 SOCBTF2 SOCBTF2 SOCBTF2

largest SOCBTF3 SOCBTF1 SOCBTF3 SOCBTF3 SOCBTF3

Platou's conclusions can be used to explain the findings illustrated inTable 2. As mentioned, all projectiles showed an increase in the dynamicstability at Mach 4. Examining the data generated in this study, it has beenshown that:

(1) Magnus moment slope was minimum at Mach 4 for all projectile designsstudied.

(2) Pitch damping was essentially independent of Mach number.

(3) At Mach 4 the term CN. - 2CD is a maximum.

Also discussed was the increase in dynamic stability generated by theSOCBTF1 for the first group of configurations at all Mach numbers. Examiningthe flare data generated in this study it was shown that:

12. Platou, A.S., "The Influence of the Magnus Moment on the Dynamic Stabil-ity of a Projectile," ARBRL-MR-2155, US Arnmy Ballistic ResearchLaboratory, Aberdeen Proving Ground, Maryland 2005, January 1972(AD 738016).

16

(1) For all Mach numbers the SOCBTF1 has the highest CN. and the lowest

CD, and therefore has the greatest value for the term CNa - 2C9 .

(2) For all Mach numbers the SOCBTF1 has the slnallest Magnus momentslope.

(3) For all Mach numbers the SOCBTF1 has the smallest pitch damping.

VI. CONCLUSION

An investigation of the aerodynamics and stability of a flared afterbodyfor axisymmetric projectiles at supersonic speeds has been completed usingcomputational fluid dynamic methods to predict the aerodynamic coefficients ofinterest. The parametric nature of the study has lead to the evaluation ofthe characteristics of the boattail-flare relative to the conventional SOCBTdesign. Although all of the three SOCBTF designs studied can be characterizedby similar conclusions, the SOCBTF1 (having the shortest boattail and flare)has been shown to yield:

(1) The smallest static pitching moment coefficient and required spin-rate, therefore the greatest degree of static stability, of alldesigns with afterbodies (excluding the SOC designs).

(2) The greatest degree of dynamic stability enhancement of all designgroups studied (independent of Mach number).

(3) The least Magnus moment of all designs studied and the smallestrise in total drag and roll damping of all designs studied withboattail-flare afterbodies.

With the reduction in the static instability of the spin-stabilized pro-jectile provided by the boattail-flare afterbody, the projectile designer isafforded a greater degree of freedom in choosing values of spinrate, (or gyro-scopic stability factor), axial and transverse moments of inertia. Eventhough the boattail-flare afterbody will allow the designer to reduce spinrateand still achieve gyroscopic stability (or hold spinrate constant and gainmore gyroscopic stability) these degrees of freedom are not usually available.Rather, the projectile designer is given a minimum value of sg and axial spin-

rate suitable for the atmospheric operating conditions and the type of gun. tube that will be used for firing, and asked to meet these parameters with a. projectile that will also carry a given payload. The designer can then use

the boattail-flare afterbody to extend the range of values over which the ."axial and transverse moments of inertia may vary. (Axial moment can bedecreased and the transverse moment can be increased.)

Although the present investigation has established the aerodynamic andstability characteristics of a subcaliber flare (boattail-flare) afterbodyfor spin-stabilized projectiles, subsequent work may demonstrate further

*" application to other ballistic projectile designs.

It has already been shown that the length (transverse moment of inertia)of spin-stabilized projectiles with the boattail-flare afterbody can be

17

. . .-. ,

. . . . . . . . . .

increased without the loss of gyroscopic stability. Current research isapplying the subcaliber flare to long spin-stabilized projectiles with a high

.4 density penetrator core as well as long rod penetrator fin-stabilized projec-tiles where a long boattail followed by a subcaliber flare replaces the fins.

With the present computational technique applied to a flare, future workcould investigate the application of a flared afterbody to long projectilesthat require stable flights without spin, in addition to enhanced drag forshort range.

'N

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Figure 2. Schematic Drawings of Projectile Groups

20

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REFERENCES

1. Murphy, C.H., "Free Flight Motion of Symmetric Missles," U.S. ArmyBallistic Research Laboratory Report No. 1216, Aberdeen Proving Ground,Maryland 21005,July 1963 (AD 442757).

2. "Engineering Design Handbook - Design for Control of Projectile FlightCharacteristics," Headquarters, US Army Materiel Command, September 1966.

3. Schiff, L.B., and Steger, J.L., "Numerical Simulation of Steady Super--. sonic Viscous Flow," AIAA Journal, Vol. 18, No. 12, December 1980, pp.

1421-1430.

4. Schiff, L.B., and Sturek, W.B., "Numerical Simulation of Steady Super-sonic Flow Over an Ogive-Cylinder-Boattail Body," ARBRL-TR-02363, US ArmyBallistic Research Laboratory, Aberdeen Proving Ground, Maryland 21005,September 1981 (AD A106060).

5. Sturek, W.B., and Schiff, L.B., "Computations of the Magnus Effect forSlender Bodies in Supersonic Flow," ARBRL-TR-02384, US Army BallisticResearch Laboratory, Aberdeen Proving Ground, Maryland 21005, December1981 (AD A110016).

6. Sturek, W.B., and Mylin, D.C. "Computational Study of the Magnus EffectS.'-" on Boattailed Shell," AIAA Journal, Vol. 20, No. 10, October 1982, pp.

1462-1464.

7. Sturek, W.B., Guidos, B., and Nietubicz, C.J., "Navier-Stokes Computa-tional Study of the Influence of Shell Geometry on the Magnus Effect atSupersonic Speeds," ARBRL-TR-02501, US Army Ballistic Research Labora-tory, Aberdeen Proving Ground, Maryland 21005, June 1983 (AD A130630).

8. Mueller, T.J., and Kayser L.D., "A Method of Determining the TurbulentBase Pressure in Uniform and Non-Uniform Supersonic Flows," ARBRL-TR-02374, US Army Ballistic Research Laboratory, Aberdeen Proving Ground,Maryland 21005, October 1981 (AD A107318).

9. Devan, L., and Mason, L.A., "Aerodynamics of Tactical Weapons to MachNumber 8 and Angle of Attack 1800: Part II, Computer Program and UsersGuide," N.S.W.C. TR 81-358, September 1981.

10. McCoy, R.L., "McDrag - A Computer Program for Estimating the Drag Coeffi-cients of Projectiles," ARBRL-TR-02293, US Army Ballistic ResearchLaboratory, Aberdeen Proving Ground, Maryland 21005, Febuary 1981 (ADA098110).

11. Robinson, M.L., "Boundary Layer Effects in Supersonic Flow over Cylinder-Flare Bodies," Australian Defense Scientific Service Weapons ResearchEstablishment Report No. 1238, July 1974.

12. Platou, A.S., "The Influence of the Magnus Moment on the Dynamic Stabi-lity of a Projectile," ARBRL-MR-2155, US Army Ballistic Research Labora-tory, Aberdeen Proving Ground, Maryland 21005, January 1972 (AD 738016).

51

• 4" , o " . . . - , " .. . . . ,. . . . . . . .. . . ..

LIST OF SYMBOLS

CD = drag coefficient

CDB = base drag coefficient (Equation 4)

Cz = roll damping coefficientp

CM = slope of static pitching moment coefficienta

Cm = slope of Magnus moment coefficientpa

C M + CM pitch damping coefficientq

C N -slope of normal force coefficient

d =maximum body diameter~ m

I axial moment of inertia kg -Mi3

I= transverse moment of inertia,kg -m 3

k= axial radius of gyration

k y transverse radius of gyration

Li = cylindrical body length

VL2 = boattail afterbody length

L3 = flare afterbody length

4M., = freest ream Mach number

p = axial angular velocity, rad/sec

P ~ B/P- = ratio of base pressure to freestream pressure

pd/V = nondimensional spinrate about projectile axis

5g gyroscopic stability factor (Equation 1)

Sd= dynamic stability factor (Equation 2)

V = air speed, in/sec

X/D = projectile length in calibers

y = ratio of specific heats

p air density, kg/in3

- 53

I,°

.-. .. .p

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