AD-754 098

BODY ALONE AERODYNAMICS OF GUIDED ANDUNGUIDED PROJECTILES AT SUB S--ONIC, TRAN-SONIC AND. SUPERSONIC MACH NUMBERS

Frank G. Moore

Naval Weapons LaboratoryDahlgren, Virginia

November 1972

'A_

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% -NAVAL WEAPONS LABORATORY

"" DAHLGREN. VA. 22448 IN REPLY 111 IPER TO

GBJ FGM:bal8800

JAN 2 3 973

"•iK

Fr'n: Commander, Naval Weapons LaboratoryDahlgren, Virginia 22448

To: Distribution

•kJ, •j: NWL Technical Report TR-27915, "Body Alone Aerodynamics ofGuided and Unguided Projectiles at Subsonic, Transonic and

S)Supersonic Mach Numbers" by Frank G. Moore, correction to

1.. Please change equation (6) on page 5 of TR-2796 to read:

S( ( z +,( U 2 ...+ v 2 ,+v 2 Y - 1

LEE A. CLAYBERGBy direcfton

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NWL Technical Report TR72796November 1972

BODY ALONE AERODYNAMICS OF GUIDED

AND UNGUIDED PROJECTILES AT SUBSONIC,

TRANSONIC AND SUPERSONIC MACH NUMBEPS

~ vy

Frank G. Moore

4$Surface Warfare Department -fr

Distribution approved for public release; distribution unlimited.

> 4

K • • FORE.WORD

This work was performed in support of the guided projectileand 5"/54 ammunition improvement programs. In additon to theabove programs, support for this -work was provided by the NavalOrdnance Systems Command under ORDTASK 35A-501/090-I/UF 32-323-505.

This repurt was reviewed by Mr4 D. A. Jones, III, Head of theAeroballistics Group, Mr. L. M. Williams, III, Head of the BallisticsDiviýion and Mr. W. R. Chadwick, Research Aerodynamicist.

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UNCLASSIFIED

DOCUMENT CONTROL DATA-R &D D,Secruttv classilfcation of title. bodt of abstract and indexing annotation must be entered when the ove.all report j., classatheJ,

I. ORIGINATING ACTIVITY (Corporal& author) ,.• REPORT SECURITY CLASSIFICATION

.•,- i|Dahlgren, Virginia 22448 Sb. GROUPKI•.• •"; 3. REPORT TITLE

BODY ALONE AERODYNAMICS OF GUIDED AND UNGUIDED PROJECTILES AT SUBSONIC,ZRANSONIC AND SUPERSONIC MACH NUMBERS

". 4. DESCRIPTIVE NOTES (Typo of report arnd inclusive dates)

5 AU THOR(S) (Fitst name, Ajiddle Initial, last name)

Frank G. Moore

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November 1972 I8a. CONTRACT OR G3RANT NO ga. ORIGINATOR'S REPOP- NUMBER(S)

b. PROJECT 1'0. '1•.--2796

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10 DISTRIBUTION STATEMENT

* ' Distribution approved for public release, distribution unlimited.

11 SUPPLEMENTARY NOTES 12 SPONSORING MILITARY AC TIVITY

13 ABITRACT

.got" Several theoretical and empirical mer1.--2., are combined into a single computerprogram to predict litt, drag, and center of pressure on bodies of revolutiorn atsuosonic, transonic, and supersonic Mach numbers. The body geometries can be quitegeneral in that pointed, spherically blunt, or truncated noses are allowed as wellsa d!.,-ontinuities in nose shape. Pa-'ticular emphasis is placed on methods which

yield accuraci.aes of ninety percenL or better for most configurations but yet are- '•" • computationally faat. Theoretical and experimental results are presented for several

projectiles and a computer program listing is included as an appendix.

SDD4.'3 ,

FR.(PAGE 1)_ ___

S....NO.. ,17 0o -UCLASSIFIED-/N 010;Pý07 6S Securit, tVClaAIfIc.atI,,lI

CONTENTS

N Page

FOREWORD .. . . . . . . . . . . . . . . . . . . . . . .

ABSTRACT. . . . . . . . . . . ... . . .. . 1 .

INTRODUCTION . . . . . . . . . . . . . .. .. .. .. .1

A. WAVE DRAG . . . . . . . . . . . . . . . . . . 7

B. SKIN FRICTION DRAG . . . . . . . . . . . .. . 1..C. BASE DRAG . . . . . . . . . . . ...... 13

D. VISCOUS SEPARATION AND ROTAT'ING BAND DRAG ... 16E. INV LCD LIFTING PRQnERTIES . . . . . . . . . . 17F. VISCOUS LIFTING PROPERTIES. .. .......... . . . 19

RESULfS AND DISCUSSION . . . . . . . . . . . . . . . . 2

A.. NUMERICAL SOLUTIONS .. ........B. COMPARISON WITH EXPERIMENT .. . . . . . . . . . 21

~0CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . .2

.lk- REFERENCES .. . . . . . . . . . . . . . . . . . . . 27

APPENDIZE.S

A. GLOSSARY. . . . . . . . . . . . . . A-1

B. COMPUTER PROGRAM. . . . . . . . . . . .-

c. DiIs'RIBUTION . . . . . . . . . . . . . . . . . . C-i

z- %-¾

LIST OF FIGURES

1 Typical Body Geometry

2 !oundaries of Perturbation and Newtonian Theory

3 Transonic Wave Drag of Tangent Ogives

4 Mean Base Pressure Curve

5 Viscous Separation and RotatiDg Band Drag

6 Constants to Determine (CN)n for M. < 1.2

7 Increase in CN at Subsonic and Transonic Mach Numbers Due

to Afterbody

8 Decrease in CN Due to Boattail

9 Center of Pressure of Afterbody Lift for M, < 1.2

10 ,Drag Proportionality Factor and Crossflow D-ag Coefficient

11 Methods Used to Compute Body Alone Aerodynamics

12 Comparison of Theory and Experiment for Blunted Cone;In/TB - 0.35, M1 = 1.5, a = 80

13 Comparison of Theory and Experiment for Blunted Cone;rn/TB = 0.35, M, = 2.96, a = 8*

"14 Comparison of Theory and Experiment for Blunted Cone:MW0 = 1.S, Oc = 100

15 Comparison of Theory and Experiment for Blunted Cone;.c = 100, rn/rB = 0 2

16 Comparison of Theory and Experiment for Blunted Cone;.c 10%, rn/rB = 0.4

17 Compaxlson of Theory and Experiment for Tangent Ogive-Cylinder, k = 14 calibers

18 Comparison of Theory and Experiment for Cones of VariousLpngths

19 Comparison of Present fheo.'y with Experiment as a Functionof Afterbody Length (2.83 Caliber Tangent Ogive Nose)

iv

t9j

4, I

LIST OF FIGURES (Cont'd)

20 Zero Lift Drag Cruve for 5"/38 RAP Projectile

21 Comparison of Theory and Test Data for 5"/54 RAP Projectile

22 Comparison of Theory and Test Data for Improved 5"/54Projectile

23 Comparison of Theory and Test Data for 175mm XM437Projectile

24 Comparison of Theory and Test Data for 155w-m Projectile2S Aerodynamics of 5-inch Guided ProJzcti.le Body

-I

4i."

k0 V

INTRODUCTION

In the past, designers have relied on wind tunnel and ballisticrange tests to predict static forces "nd moments on projectiles.This is not only very expensive but also quite time consuming becauseof the man hours required in scheduling and performing a test of theabove nature. At most test facilities, theii is also a backlog ofwork of about three to six months.

It is believed that a large pcrtion of wind tunnel and ballisticrange tests could be eliminated (particularly for preliminary and'ntermediate design) if an.accurato theoretical method were availableto compute static forces ai• moments throughout the Mach nunberrange. More impcrtant th•,.;gh, is the practical use of such a methodto the design engineer - determining the configuration which isthe most optimum fror, : liet, drag, and pitching moment standpoint forhis given design go, is. Quite often, due to the lack of such amethod or the furds for wind tunnel tosting, less than optimumaerodynamic configurations are used to accomplish a given task.A typical example is the external shape of the 5t1/38 projectile.According to the work of reference 1, the range of that projectilecould have been increased by more than fifty percent with properdesign.

It is the purpose then of the present work to develop a general1rogram which can be applied to the body of the guided or unguidedprojectile to predict lift, drag, pitching moment, and centei efpressure over the Mach number range of current interest, 0 < M, , - 3.The methods used in the development of the program should beaccurate enough to replace preliminary and intermediate wind tunneltests (accuracy of ninety percent or bettei formost configurations)but yet should be computationally fast enough so it can be used asan efficient design tool.

'There are many methods available in any particular Mach numberregion to compute static forces and moments on various body shapes.These methods range in complexity from exact numerical to semi-

-A empirical and the body shapes vary from simple pointed cones tocomplex multi-stage launch vehicles. However, attempts at com.-bining the various methods above into an accurate and computationallyfast computer program have been scarce. Saffell, et a12 developeda mathod for predicting static aerodynamic characteristic. for

, typical missile configurations with emphasis placed on large anglesof attack. However, the drag was computed by hanidbook techpiques 3

and slender body theory was used for the lift and pitching moment.As a result, limited accuracy for body alone aerodynamics waso -ained using this method.

Another method which computes forces and noments throughoutthe Mach number range is the GE "Spinner" program4 designedspecifically for projectiles. This program, which is based onempirical correlationF as a fihnction of nose length, boattail

j length, and overall length, gives very good accuracy for moststandard shaped projectiles. However, its use as a design toolis somewhat limited in that the drag of a given length nose isthe same no matter what ogive is present or if there are dis-continuities present along the nose. The same statement appliesto the boattail since a conical boattail of from So to 90 isassumed no matter what the boattail shape is. Moreover, nopressures can be computed by the GE program and no attempt hasbeen made to include nonlinear angle of attack effects.

It is apparent then, from the above discussion, that there isa definite need for an analytical method which can take intoaccount nose bluntness and ogive shape, discontinuities alongthe body surface, as well as nonlinear angle of attack effects.The method presented herein for accomplishing this task reliesheavily on analytical work and to a lesser degree on empiricaldata. As such it is believed to be the first such program withmajor emphasis on analytical as opposed to empirical procedures.

The body shapes which the program can handle should be generalenough so that most projectile and missile configurations couldbe handled in detail. This means that the nose may be pointed,truncated, or blunted with a spherical cap and that che nose mayhave two ogives present. For example, on a typical p-ojt.ctilethe fuze has one contour and the ogive between the fuze :ndshoulder has a different contoiir with a discontinuity i.'' between.The afterbody should consist of a cylinder followed by a boattailor flare. A typical body shape along with the coordinate systemsused is shown in Figure 1.

k 2

-K•

"0 ANALYSIS

A. Wave Drag

Wave drag results from the expansion and compression of the airis it 'lows over the body surface. Compression of the air is seenin the form of shock waves which first occur around Mach number 0.7to 0.9 depending on the body shape. The methodc used to calculatethis form of drag differ significantly in transonic and supersonicflow and thus will be discussed individually below.

Supersonic Flow

There are several methods available for calculating the super-sonic pressure distribution but only two of these metbods holdpromise of meeting our requirements on speed of computation andaccuracy as set forth in the introduction. These methods are thesecond order perturbation theory of Van Dyke 5 , 6 and the second ordershock expansion theory 7 modified for bluit bodies in reference 8.Since the major portion of t'he flight of most projectiles is in thelower supersonic speed regime the perturbation approach is chosenbecause it is more accurate than shock expansion theory at theseMach numbers. However, Van Dyke's theory can only be applieddirectly to bodies where the slope is less than the slope of thefree-stream Mach lines. Thus for blunt-nosed configura-,ions, theperturbation theory is combined with the modified Newtonian Theory(the means for combining the two will be discussed shortly).

Before discussing the combined perturbation Newtonian approacha brief discussion of Van Dyke's cheory is helpful.

"The general first order perturbation problem is: (see reference9 for the details of the derivation):

F rr + ýr/r %e6/r 2 (M2 -1) qxx 0

p (O,r,0) = 4 x (O,r,e) = 0 (1)

v 4r (x,R,O) + sin a cos e = R' [-os a + ýx (x,R,e)i

3

where -Le subscripts indicate partial aiffere.atiation. The firstorder problem is satisfied exactly by

c (xrO) = " (x:r) cos a + • (x,r) sin a cos 6 (2)

where the ffrst term corresponds to the axial flow solution andthe second tarm to the cross flow solution. The first order problemeq. (1) can then be separated into an axial problem:

hPrr + r/r - 0 xx 0

ip (O,r) ýx (o,r) 0 0 (3)

lr (x,R) = [' I. + (x (x,R)]

and a cross flow problem:

,+ C/r -/r 2 " a2xx 0~rr r

o,r) =•x (o,r) 0 o(4)

1+ ýr (x,R) : R' 4x (x,R)

4

I� Without going into the details, suffice it to say that the solutionsof eqs. (3) and (4) are found numerically by placing a .. Z1tributionof sources and doublets respectively along the x..axis.

* IVan Dyke then discovered a secotd order axial solution in termsof tb, r'irst order solution • Further, since disturbances in thecroz- rloW plane do not affect the pressure as much as disturbancesin the axial flow, Van Dyke reasoned that it would be quite legiti-

mate physically to combine this second order 1 i~xal solution withthe first order cross flow solution of Tsion to form a hybrid theory.

- Once the perturbation velocities ;x, Or (second-order axial)!, and tx,6 and ýr (first-order crossflow) are computed at each point along the

body surface the local velocity components are:

u (cos a) (1 + *x) + (sin a cos 0) (Cx) (5a)

V = (COS •) a •r) + (sin a cos 0) C] + 4r) (Sb)

w (- sin a sin 0) (1 + ý/r) (Sc)

The pressure doefficient at each body station is then:

2° + U2+V2÷•

C .,). I+ tj2 (x, -1 (6)

cco

Finally the force-coefficients are:

CA 2 7 t Cp (xo) rdr do dx (7)

irR2J

05

C r2 Cp (x,e) cos 6 r d6 dx (8)

R3 = 1 % (X,6) Cos ( x r dO dx()

•: !"It should be pointed out that in the actual numerical integration of

)• :,,eqs, (7), (8) and (9) the integration must be carried out in segments

of the body between each discontinui-ty due to the discontinuouspressure distribution.

nIf the nose is pointed, one need go no further. But if the

nose is truncated or is blunted with a spherical cap then some otherImethod must be used to determine the pressure distribution over the

truncated portion. The method used herein is modified Newtonian

ntheoryse Although this theory is derived assuming a very large

Mach number, reasoiable values for the pressure :oefficient can beobtained over a portion of the' nose even at low supersonic Machnumbers. The modified Newtonian pressure coefficient is

O p CPo sin2 6

where 6 is the angle between a tangent to the local body surfaceand the freestream direction and where the stagnaticon pressurebehind a normal shock is:

6

-9.

_y_ Y+ -1(2It Y 22 2yNE - (y- 1)

•, ~According to reference 12, if the nose is truncated then the pressure i: on the truncated portion is only abovt nincty percent of the stagnation

value given by eq. (12) so that for a truncated nose: :

S-(-_-1 _ _3

,'m•!:'If the nose has a spherical cap then it can ýe shown that:

}I

6 sin - sin cos a {cos cos l sin a (14)

Accowhere tan ref dr/dx.

Then combining eqs. (11) and (14) one obtains for a spherical nose

S• r •cap:

.•",: •': Cp(x,e) =Cp (O sin2coa - sin2a sino cosa cose +

PY

Cos2l COS2 Sip2a

Cohere CPo is given by eq. (12).

7

'Cv

-IN

The only question that remains now so far as the supersonicMach number region is concerned is where does the modified Newtoniantheory ena on the body and where does the perturbation theory begin.To determine this match point recall that the slope of the bodysurface must be less than the Mach angle I:o apply perturbationtheory, that is

6 <_sin l- i ._ ) (16)

Thus, the upper limit of the perturbation theory is S= sin - (I/R 0 ).

Using, this relation in eq. (14) and assuming a spherical nose capthere is obtained for the coordinates of the point below whichNewtonian tneory must be applied:

ru0 cos a + sin

(17)

xu ru tan + rn (l- 1 )

IL is important to note here that if x - xu Newtonian theory may0/ still be applied but if x < xu perturbation theory cannot be

.pplied.

The limiting angle of eq. (16) corresponding to the coordinatesof eq. (17) is shown in Figure 2 as the upper curve. Note that very

large cone half angles can be computed using the perturbation theoryaL the lower Mach numbers. However, as shown by lan Dyre 5 the lossin accuracy of perturbation theory increases rapidly as the angle 6is increased. Realistically, since at an angle of 250 - 30* theerror is still slight the maximum angle 6 for which perturbationtheory is applied should not exceed these values. Based on these

8

accuracy considerations, the Newtonian theory should be applied for6 values outside the solid line boundary of Figure 2 and perturba-tion the.-or" within the boundary, Now the match point, which forthe present work will be defined as the point where the pressurecoefficients of the Newtonian theory and the perturbation theoryare equal, can be determined as the solution proceeds downstream.For body stations downstream of the match point, perturbationpressures are used in the force coefficient calculations of eqs.(7), (8) and (9) whereas for x values along the surface less thanthat at the match point Newtonian pressures must be used.

Transonic Flow

£ •If the flow is transonic, the available theories for the wavedrag calculations are again limited. Here the main limitationsare in body shape because there does not appear to be a theoretical"method available whjich can handle the blunted nose or the dis-continuities along the body surface. Wu and Aoyoma 13 , 1" havedeveloped a method which handles tangent-ogive-cylinder-boattailconfigurations at zero angle of attack but no general nose geom-etries can be used as is the case in supersonic flow. Thus theapproach of the present paper will be to calculate the wave dragfor tangent ogive noses of various lengths throug;hout the cransonicMach number range and to estimate the wave drag of the more com-plicated nose geometry based on these results. It is true tnatthe accuracy here is not consistent with that of the supersonicwork but it appears from the results (as will be discussed later)that this approach is justified, at least for noses with slightblunting (rn/rb < 0.3).

For transonic flow the perturbaticn -,uation (1) has an addi-tional term sc that for a = 0, eq. (1) is replaced by:

[1 - M2 = (y+l) Ne Px 3 4 xx + Or/r + rr = 0 (18)

Eq. (18) is now nonlinear as opposed to the linear eq. (1) used, in supersonic flow. Eq. ý18) is again solved numerically"? for

the velocity potential and the pressure and axial force coeffi-cients calculated by eqs. (6) and (7) for the nose of the nose-cylinder-boattail configuration. Figure 3 gives the wave dragobtained by solving eq. ý18) for tangent ogive noses of iariouslangths throughout the transonic Mach number range. For a givenMach number and nose length, the axial force coefficient can be

9

obtained from this curve by interpolation. If the pressai2coefficients along tve body surface are desired, however, thegeneral program of Wu and Aoyoma 13 must be used.

The pressure coefficient on the boattail at zero angle ofattack in transonic flow1I is given by:

•, 1/2

C~()~ (Xi -C) Il Ix -C)/ - I1- W~ 1 - (19)• , " 5 V~~/(+1) ma,.2/3 (yl Tvý1 y+)•2 \x

where x, is measired from the shoulder of the boattail and

2

C2 =2'- (y+l) MNel i(yi , +J*

2/3 [4/3 1/2

+ 2 1 M,2 3 dR/dx\ 3 dR/dxI IM,•j2/3 \(y+l) ýd 2 /y+1 I2M. VY+I1

In addition to the restriction of zero angle of attack, eq. (19)is to be applied for 1 < M < 1.2 (for 'I_ > 1.2, afterbody wavedrag is calculated using tle previous supersonic theory for the"entire body). For M. < 1, e::periment shows that the shock firstoccurs on a boattail at M, z 0.95. Accordingly,wave drag will beassumed to vary linearly from zero at M, 0.95 to its maximumvalue at M. 1.0 which is calculated using the above equation.

10

B. Skin Friction Drag

The boundary layer will generally be turbulent over about ninetypercent of the projectile body for large caliber projectiles. Sincethe laminar flow region is usually less than ten percent of thetotal surface area, it will be assumed the entire boundary layer isturbulent. Under this assumption the total or mean skin-frictioncoefficient, Cf., according to Van Driest1 5 mast be obtained from:

A C 0.242 (sin '' C + sin C 2) log1 0 (RN. Cf.)A (CfJ11-2 (Tw/,•)'/? !C)=lgl Ro f

(1 + 2n l)glo (Tw/T,) (20)

where C1 2A2 B BC2 BI +B 4) 1/2 ' (B2 * 4A2) 1/2

1/2

and A (-Y-l) B I + (y-l)/2 -4 1

2 TT/TT

The variable n of eq. (20) is the power in the power viscosity lzw:

N n11 _

T w\ (21)

g 11

and n for air is 0.76. Eq. (20) assumes a fully developed turbulent"boundary layer with zero pressure gradient and Prandtl number equalto one.

In ordez to solve eq. (20) for the mean skin friction coefficientCf,, one musi have values for Tw/T•,, RN and M.. The freestreanReynolds nuu ar is simply

RN,= Vw k (22)

To relate Tw/TO to the freestream Mach number, assume the wall isadiabatic. Defining a turbulent recovery factor RT by

/T Tw

T. 1

then

T,--, Y (23)+ Rr - Me

It has been shown that t~e recover)' factor varies as the cube rootof the Prandtl number (see reference 16) for turbulent flow so that:

"RT 3 p r (24)

12

S- -I --. --

Recall that Van Driest's Method assumes a Prandtl number of unityso if this were used then RT would also be unity. Hc;,ever, theactual value of Pr 0.73 so that the previous assumption of"Prandtl number one can be compensated for somewhat by the aboverecovery factor which for Pr = 0.73 uould be 0.90. Thus eq. (23)becomes:

TW/T. 1 +0. 9 4l -1 (25)2

;en for a given set of freestream conditions (NI, p., v., V,) onecan combine eqs. (22) and (25) with (20) to solve for Cf . Theequation must be solved numerically however, since Cf. cannot besolved for explicitly. A procedure adaptable to equations of thistype is the well known Newton-Raphson method discusszd in reference17.

Once the mean skin friction coefficient has been determined fora given set of freestream conditions, the viscous axial forcecoefficient is simply:

CAf = Cf Sw (2b)I' Sr

The wetted area S,., is the total surface area of the body which canbe integrated numerically given a set of body coordinates.

¾~ ,C. Base Drag

Much theoretical work ha- been performed to predict base pressure(references 18 - 22). There is still no 3W2isfactory theory available,however, and the standard practice has been ,• use empirical methods.This is the approach taken here. Figure - -s a mean curve of experi-mental data from references 18. 19 ana 23 29. This data assumes a

> '•- long cylindrical afterbody with fully deve.,oped turbulent boundary

13

)ayer ahead of the bAse. There could be deviations from this curvedue to low body finenes-. ratio, boattails, angle of attack, Reynoldsnumber and surface temperAure. Each of these effects will be dis-cussed below.

The minimum lengch of most projectiles is about four calibers.According to references 23 and 29 the base pressure at low super-sonic Mach numbers is essentially unaffected by changes in bodylength if the fineness ratio is greater than four. This is nottrue at high supersonic and hypersonic Mach numbers as shown byLove . But since the main interest is for M. < 3 the effect ofoverall fineness ratio on base pressure can be neglected,

In addition to the above, Love shows that the nose shape haslittle effect on base pressure for high fineness ratio bodies.Thus, for bodies of fineness ratio of four or greater the effectof nose shape and total length on base pressure can be neglected.

The base pressure is sign..ficantly altered by the presenceof a boattail so that th 4 s change must be accounted for. Probablythe most simple method co do this is an empirical equation givenby Stoney28,

3

CABA = CpBA d (27)A r

Eq. (27) can be used throughout the entire Mach dumber range whereC is the base pressure giver3 by the curve of Figure 4. An alter-nRn~ve to this procedure is to find the base pressure as afunct;ion of boattail angle and then the diameter of the base wouldbe squared instead of cubed as in equation (27). That is

21 • (28)CABA _CBA d(2

14

where C. is the base pressure coefficient for a given boattailangle. tnis requires knowing C howerer which is not alwaysavailable. Because of this, eq. (27) will be used,

It has been shown in many works 2 1 ,3 0 that the base pressureis essentially independent of Reynolds' numbers, RN, if the boundarylayer ahead of the base is fully developed turbulent flow. Aturbulent boundary layer usually occurs for RN of 500,000 to 750,000

41 depending on the roughness of the body surface. The minimum RNahead of the base one would expect tu encounter on the presentbodies would be about 1,000,000. Moreover, most projectiles havevarious intrusions and protrusiors such as on a fuve which tendsto promote boundary layer separation. in view of these practicalconsiderations, Reynolds numb,-i effects on base pressure may safelybe neglected.

The same arguments as the ones above hola for surface tempera-Li. ture as well. Thus in addition to Reynolds number effects, surfacetemperature effects on base pressure need not be accounted for.

The effect of angle of attack on base pressure is to lower thebase pressure and hence to increase the base drag. For bodieswithout fins, the amount of this decrease is dependent mainly onfreestream Mach numbor, Tf a is given in degrees then an empiricalrelation for thi- changv in base pressure coefficient due to angleof attack is given by

PBAI] -(.012 -.0O36NIco) o. (29)

Eq. (29) was derived from a compilation of experimental data pre-sented in Figures 7 through IS of reference 23. The base dragcoefficient thus becomes, in light of eqs. (27) and (29):

CA = -IC - (.012 - .0036%,,) a]dB (30)BA L PBA C

0 ,,$

>1

0i

i .-.. .. "..., ... - • - •"~ - ' •.. .• • ,, •- -'- . t. -r

D. Viscous Separation and Rotating Band Drag

Figure 5A is a plot of forebody drag coefficient as a functionof cone half angle from data taken from reference 31. Since theskin friction drag coefficient is about 0.02 for this case, it canbe subtracted from the curve of Figure SA to yield the pressuredrag coefficient. Note that the freestream Mach number is 0.4,low enough so that no appreciable compressibility effects occur.The question therefore arises as to the origin of this type ofdrag, since it is not compressibility or skin friction drag. Itis in fact viscous separation drag. For very large cone half angles,ac, the flow over the cone, instead of remaining attached, separatesdue to the very strong adverse pressure gradient and reattachesdownstream. This separation prevents the pressure from decreasingas much as it would in inviscid flow and produces a drag. Oddlyenough, this phenomenon does not occur on ogives or or. spherical"surfaces, apparently due to body curvature effects on the boundarylayer. As a result, one can derive an empirical expression forthis viscous separation drag, where the important parameter is theangle * which the nose makes with the shoulder of the afterbody.Based on Figure 5A this relation is

CAvis = .012 (6* - 100); 6*_> 100(31)

= 0; 6* < i0

with P* in degrees and with P* = c for a conical nose.

Reference 1 gives tine measured effect of a rotating band on drag.The particular rotating band used in those wind tunnel tests had amean height of about 0.024 calibers. An expression which functionalizesthe above results ior drag increment due to a rotating band is givenby:

CARB . (ACA) (H)/.Ol (32)

16

where H is the mean height of the band in calibers and ACA is theincrement in axial !Grce for an H of 0.01 caliber given in Figure5B. Although the eq. "32) was derived for a particular band, itchecks well with the results of Charters3" for a different bandgeometry.

E. Inviscid Lifting Properties

At supersonic Mach numbers the inviscid lift, pitching moment,and center of pressure are calculated using Tsien's first ordercross flow theory which was discussed earlier in conjunction withVan Dyke's second order axial solution. This method is adequatefor small angles of attack where viscous effects are negligible.

At subsonic and transonic Mach numbers the lifting propertiesare more difficult to obtain. For subsonic velocities the liftcould be calcul2ted by perturbation theory33 but since projectilesrarely fly at Mach numbers less than 0.7, a formulati~n on thisbas, was not justified. An alternative would be slender body

A-• t• •ry but the accuracy of this approach is inadequate. In lightof the above reasoning, a semi-empirical method for no.mal forcecharacteristics was derived based on nose length, afterbody length,and boattail shape. This metnod was then extended thrcugh thetransonic Mach number range since the state-o.--the-art in transonic"flow does not allow one to handle the gener.l body shapes or flowconditions.

The total inviscid normal force acting on the body may bewritten

CN = (CN ) + (CN ) + (CN) (33)a n a a a B

where the subscripts n, a, and B stand for nose, afterbody, andboz..tail respectively. The first term of eq. (33) can be approxi-mated by

(CNO)d= C1 tan 6* + (Z4)

17

* where C1 and C2 are given in Figure 6 as a function of Mach number.This relationship was determined empirically from the cone resultsof Owens 31 . It is approximately correct for kn > 1.5, cone blunt-ness up to 0.5, and N6 < 1.2. Note that the angle P* in eq. (341is the same as that discussed previously in eq. (31).

The normal force coefficients of the afterbody and boattail canbe obtained from Figures 7 and 8 respectively. Figure 7 was derivedanalytically in the transonic Mach range from the method of Wu and

13 . 34Aoyoma and in subsonic flow from the experimental data of Springand Gwin~s. In the work of Spring and Gwin above, the normal forceof the nose plus afterbody was given but the nose comDonent can besubtracted off by the use of eq. (34). The boattail normal forcecoefficient was given by Washington 36 but he stated that there wasnot enough data available in subsonic and transonic flow. Hencethe data of Washington was supplemented by the 175mm Army pro-ectile3 7

and Improved 5"/54 Navy projectile 3 8 data to derive the general curveof Figure 8.

Although slender body theury may not be adequate for predictingthe normal force coefficient it appears to predict the center ofpressure of the nose and boattail lift components quite adequately.According to slender body theory the center of pressure of the

0. nose is

(xcp) = Zn (Vol)n (35)

n iT Rr

and of the boattai I

(Xcp) = n +a +ZB - (V-lB iT R'

r

18

a5

or

S~ (VOI)B

0 (Xcp) = £ - (36)C"P: B iT R 2r

The center of pressure of the afterbody normal force was calculatedanalytically by the method of Wu and Aoyoma in transonic flow andassumed to have the same value in subsonic flow. Figure 9 is a plotof (Xc ) /Z versus afterbody length measured at the point wherethe afeAbo"y begins. Now knowing the individual lift componentsand their cet:ter of pressure locations, one can compute the pitchingmoment about the nose as:

C = - (CN) (Xcp)n + (CNa)a (Xcp)1 + (CNa2B (Xcp)B (37)Ma )n B

F. Viscous Lifting Properties

Strictly speaking, the previous discussion on inviscid liftingproperties gave CN and CNI at a = 0 only. If a > 0 then there is

a nonlinear contrigution to lift and hence pitching moment due to0° the viscous crossflow of velocity V = V, sin a. Allen and Perkins"

list these contributions as:

S2(ACN)vis n cd a (38)

.cSr

19

(ACM)vis= -1 Cdc(S2)(p) a 2 (39)

where n and cd are given in Figure 10. Note that the cross flowdrag coefficient is here taken to be a function of Mach number onlyand the cross flow Reynolds number dependence is not accounted for.The center of pressure of the entire configuration should then be:

= CM + (ACM) €is (40)• .. ~Xcp =.

o.•' CN + (ACN)vis

G. Summary

Figure 11 gives a summary of the various methods used in each0°• particular Mach number region to compute the static aerodynamics.

As may be seen, major emphasis has been placed on analytical asopposed to empirical proceaures.

20

45

RESULTS AND DISCUSSION

v iA. Numerical Solutions

A computer program was written in Fortran IV for the CDC 6700computer to solve the various equations discussed in the analysissection by numerical means. The various methods used for eachindividual equation are the same as those discussed in the refer-ences pertaining to the particular equation and will not berepeated here. However, mention should be made of the fact thatthe step size used in the hybrid theory of Van Dyke was considerablysmaller than he sug, .sted, particularly for a blunt nosed body orbehind a discontinu..ty. For example, for the most complicated bodyshapes as many as 200 points were placed along the body surface.Also slight oscillations in the second order solution were foundbehind a corner although Van Dyke does not mention these details,

Quite often, it was necessary to evaluate an integral numericallyor to compute the value of a function and its derivative at a given

- point. The integration was carried out using Simpson's rule; theinterpolation and differentiation using a five point Lagrange scheme 1 .Both methods have truncation errors which are consistent with theaccuracy of the governing set of flow field equations.

The computational times depend on how complicat,'d the bodyshapes are and the particular Mach number of interest. The longestcomputational time for the most general body shape computed wasless than half a minute for one Mach number. For most configurationsthe average time is about fifteen seconds per Mach number for Mw >1.2 and about five seconds per Mach number for M, < 1.2. Thisassumes of cou.sse that a table look-up procedure is used 4n thetransonic region where the curves of Figure 3 are input as data

4• sets as opposed to solving the nonlinear partial differentialequation (18) for each Mach number. If the aerodynamic coefficientsof a given configuration are desired throughout the entire Machnumber range, an average execution time of two minutes is requiredfor most configurations (ten Mach numbers).

A detailed discussion of che computer program is included asAppendix A. The various input and output parametels are definedand a listing of the program along with a sample output are alsoincluded for the reader's convenience.

B. Comparison with Experiment

The only new method presented in the current work ir tý,, com-bined perturbation - Ne.tonian theory for blunt bodies, •t isthus of interest to r:eC ho;_ the pressure coefficients along the

21

surface compare with experimental data, Figures 12 and 13 presenttwo typical comparisons at M = 1.5 and 2.96. The experimentaldata is taken from reference 8 which combined modified Newtoniantheory with shock expansion theory to compute forces on bluntedcones. The asymptotes of the pressure coefficient in each of theplanes computed by the method of reference 8 are also indicatedon the figures. As seen in the figures the present theory predictsthe aerodynamics much better than shock expansion theory at M, =1.5 and is about the same as the shock expansion approach at M. =2.96. The reason for this is that the basic perturbation theorywas derived assumping shock free flow with entropy changes slight;hence the theory should be most accurate in the lower supersonicspeed regime. On the other hand, shock expansion theory wasderived assuming a shock present and so one would expect thismethod to be better than perturbation theory as M. is increased.Apparently, the crossover point is around '41 = 2.5 to 3.0 so thatfor the major portion of the supersonic speed range of interestin the present analysis, p~rturbation theory is more accurate.

Another inr.eresting point in Figure 12 is the discontinuityin slope of the pressure coefficient curve which occurs at thematch point. This is because in the expansion region on thespherical nose the perturbation pressure decreases much more rapidlythan the Newtonian theory and as a result the overexpansion region,which occurs at low supersonic Mach numbers, is accounted for quitewell. Note that the match point is different in each plane aroundthe surface (x = 0.11 to 0.14).

One of the questions which arises in the development of a generalprediction method pertains to accuracy. To answer this question,force coefficients for several cases were computed embracingvariations in nose bluntness, Mach number, angle of attack,nose length, and afteibody length. These cases are presented inFigures 14 through 19 along with experimental data.

The first of these cases (Figure 14) gives the axial forcecoefficient, normal force coefficient derivative, and pitchingmoment coefficient derivative as a function of nose bluntness for

a simple blunted cone configuration. Note that the axial forcecoefficient includes only the wave plus skin friction componentsbecause the base drag was subtracted out of the given set ofexperimental data. An important point here is that very good.ccuracy is obtained- even for large bluntness ratios. For example,with bluntness rn/rB = 0.6, the force coefficients are in error byless than fifteen percent (this is assuming of course there is no

oj error associated with the experimental data which is not exactlyI,• correct). This tends to verify that a combined perturbation-

Newtonian theory can be used successfully for blunt configurationsji• even at low supersonic Mach numbers.

22

The next two figures, Figures 15 and 16, compare the theoreticalstatic aerodynamic coefficients with experiment as a function ofMach number for blunted cones with bluntness ratios of 0.2 and 0.406respectively. Also included in Figure 15 is the slender body theory.As seen by the error comparisons at the lower part of Figure 15,accuracies of better than 90 percent can be obtained throughout

IL the supersonic Mach number range for the force coefficients. Figure16 gives the aerodynamic data throughout the Mach number range ofinterest. Avain the comparison is favorable even though the tran-sonic wave drag was computed for a tangent ogive having a lengthequal to that of the blunted cone. The bluntness causes the tran-sonic drag rise to start at a lower Mach number and to be lessabrupt than for the pointed tangent ogive.

The third vairable of interest is angle of attack. Figure 17'I presents the results for a tangent ogive cylinder of nose lengthý,q six calibers and total length fourteen calibers. Two Mach numbers

are considered,, = 1.5 and MO = 2.5. Again the results are quite4 good, except at very large angles of attack.

Figure 18 compares the force coefficients of the present theorywith experiment for a pointed cone of various lengths. Also shownfor comparison with the Mo = 1.5 case is the slender body theory.Although perturbation theory is usually associated with nose slender-ness ratios of two and greater, it may, nevertheless, be seen thatfair accuracy is obtained for lengths as low as one. This correspondsto a cone half angle of about twenty-five degrees which is thelimiting angle used in the combined perturbation - Newtonian theoryas shown in Figure 2. For the M,, = 0.5 case eq. (31) is used tocalculate the viscous separation drag which is added to the skinfriction drag to get the total forebody drag coefficient. Using thissimple formula, excellent agreement with experimental data isobtained.

The final variable of interest, afterbody length, is examinedin Figure 19. The nose of the body is a 2.83 caliber tangent

A •ogive. For zero afterbody length, the theory agrees with experimentvery well. However, as the afterbody length increases the theoryunderestimates the afterbody lift at the lower supersonic Machnumbers for short afterbody lengths and at the higher Mach numbersfor long afterbody lengths. This loss in lift predicted by the,inviscid theory was also found by Buford4 0 and he attributed i-t toboundary layer displacement effects. Even so, the present theoryis superior to slender body theory which gives zero lift clue -to anafterbody.

To summarize the previous five figures, one could say in generalthat accuracies of ninety percent or better can be obtained for forcecoefficients of most configurations. However, for extreme cases,

3 23

such as very large nose bluntness or angle of attack, the accuracywill be decreased and the amount of this decrease can be approxi-mated from Figures 14 through 19.

The next several figures compare theory with experiment forseveral spin stabilized projectiles. Figures 20, 21 and 22 areNavy projectiles: the 5"/38 RAP (Rocket Assisted Projectile)41,the 51t/54 projectile 42, and the improved 5"/54 projectile"3,which has a longer nose and boattail than the standard 5"/S4.Figures 23 and 24 ire Army shapes: the 175mm 3 7 and 155mm4 3 projec-tiles respectively. For the detailed drawings and other aerodynamicsof these shapes the reader is referred to the references cited above.

The theoretical zero lift drag curve of the 5"/58 RAP projectilealong with three sets of experimental data' and an NWL empiricallyderived curve are shown in Figure 20. Note that the experimental

4 data varies by about thirty percent for M,, < 1 and by ten percentfor MN > 1. The theoretical curve tends to support the BRL datasubsonically and the NOL and NWC dikta supersonically. The numbersin parenthesis are the factors by which the drag curves must bemultiplied throughout the flight of the projectile to match actualrange firings. The NWL empirical curve is the curve which isactually used in range predictions due to the failure of experimentaldata to predict an adequate drag curve. This empirical curve wasderived from actual range firings. It should be,therefore, slightlyhigh because of yaw induced effects. The important point here isthat for this particular shell, the theory agrees better withactual range firings than any of the sets of experimental data.

Figures 21 and 22 give the static aecodynamic coefficientb forthe 5"/54 RAP and the improved 511/54 projectiles. The S"/54 RAPhas a nose length of about 2.5 calibers and a boattail (if 0oScalibers whereas the improved round has a 2.75 calibec nose and a1.0 caliber boattail. Also the 5"/5V RAP has a rotating bandwhereas the other shell does not. For both shells, excellentagreement with experimental data is obtained for the drag coeffi-V • cient throughout the entire Mach number range. Fair agreement isobtained for normal force coefficient and hence pitching momentand center of pressare. The comparison for the lifting propertiesis Mach number dependent: in the low supersonic region the theoryis consistently about ten percent low on normal force whereas at"high supersonic speeds it compares very well with experiment.The reason, as already mentioned, is the failure of the inviscid

. theory to predict afterbody jift correctly at low supevsonic Machnumbers. At subsonic and transonic Mach numbers, the theory doesabout as well as could be expected cqnsidering that there was aconsiderable amoun, of empirical work 4n that region.

"r24

For boattailed configurations, such as the 5"/54 RAP and theImproved 5"/54, it was found necessary to account approximatelyfor the thick boundary layer on the boattail. This was done byviewing the unpublished shadow graphs obtained in conjunctionwith the work of reference 38. Apparently, a maximum boattailangle of six degrees can be allowed before boundary layer separa-

* tion takes place. In addition, the boundary layer displacementthickness accounts for another Pbout 1/4 - 1/2 degree decrease inthe effective boattail angle. ,'hese two results were used todetermine effective boattail angles on all boattai]ed configura-tions. Without this approximate accounting of the boundary layereffect on the boattail shape, the lifting properties would havebeen in error by an additional ten percent for boattailcd con-figurations.

The final two shells, the 175 and 155mm, are considered inFigures 23 and 24. Again, excellent drag predictions are made bythe theory and good prediccions are made for normal force andcenter of pressure. IntuitiVely, one would expect the axialforce to agree better with experimenz than the lift because asecond order approach is used in supersonic flow fbr the axialforces whereas a first order cross-flow theory is used for thenormal forces,

Figure 25 presents theo:retical results for the five-inchguided projectile. The nose, is about sixty percent blunt withtwo different ogive sections. The overall length is 10.58calibers with a 0.66 caliber boattail, 7.24 caliber afterbodyand 2.68 caliber nose. Altheugh no experimental data is currenclyavailable for this extreme case, it is expected that the theoryis accurate to within ten percent on axial force and twentypercent on lifting properties.

2

•, 25

CONCLUS IONS

1. A general method has been developed consisting of soveral theore-tical _nd empirical procedures to calculate lift, drag, and pitchingmoment on bodies of revolution from Mach number zero to about threeand for angles of attack to about twenty degrees.

2. Comparison of this method with experiment for several configu-rations indicates that accuracies of ninety percent or better canbe obtained for force coefficients of most configurations. Thisis at a cost of about $30. for ten Mach numbers in the range"0 < M% < 3.

3. A second order axial perturbation solution can be combined withIf modified Newtonian theory to adequately predict pressures ongeneral shaped bodies of revolution. This is true for Mach numbersas low as 1.2 even though Newtonian theory was derived for highMach number flow.

4. A first order inviscid crcssflow solution is not sufficient topredict afterbody or boattail lift at low supersonic Mach numbers.However, when account is made for the boundary layer, markedlyimproved results for boattail lift was obtained.

5. There is still no adequate theory available in transonic flowwhich is computationally fast and accuxate and can consider bluntnosed configurations with 4discontinuities along the ugive. Thusrmore research needs to be directed along these lines.

J:22

< 26

"REFERENCES

"1". Moore, F. .: "A Study to Optimize the Aeroba'listic Design ofNaval Projectiles", NWL TR-2337, September 1969.

2. Saffell, B. F., Jr.; Howard, M. L.; Brooks, E. N., Jr.: "AMethod for Predicting the Static Aerodynamic Characteristics

te of Typical Missile Configurations for Angles of Attack to 180Degrees", NSRDC Report 3645, 1971.

3. Douglass Aircraft Co., Inc.: USAF Stability and Control DATCOM,Revisions by Wright Patterson Air Force Base, July 1963, 2 vols.

4. Whyte, R. H.: "'Spinner' - A Computer Program for Predictingthe Aerodynamic Coefficients of Spin Stabilized Projectiles",General Electric Class 2 Reports, August 1969.

S. Van Dyke, M. D.: "A Study of Second-Order Supersor.ni FlowTheory", NACA Report 1081, 1952.

6. Van Dyke, M. D.: "i'r-:tical Calculation of Second-Order Super-sonic Flow Past Nonlifting Bodies of Revolution", NACA TN-2744,July 1952.

7. Syvertson, C. A.; Dennis, D. H.: "A Second-Order Shock-ExpansionMethod Applicable to Bodies of Revolution Near Zero Lift", NACAReport 1328, 1957.

8. Jackson, C. M., Jr.; Sawyer, IV. C.; Smith, R. S.: "A Method forDetermining Surface Pressures on Blunt Bodies of Revolution at

F Small Angles of Attack in Supersonic Flow", NASA TN D-4865,A November 1968.

d9. Van Dyke, M. D.: "First and Second-O-rder Theory of SupersonicFlow Past Bodies of Revolution", JAS, Vol. 18, No. 3, March1951, pp. 161-179.

10. Tsien, H. S.: "Supersonic Flow Over an Inclined Body of Revo-

lutioal", JAS, Vol. 5, No. 12, October 1938, pp. 480-483.

ol. Truitt, R. I. : Hypersonic AeroC namics, ie Ronald Press12.Company, New York, 1959.

12. Tetervin, Neal: "Approximate Analysis of Effect or, Drag ofTruncating the Conical Nose of a Body of Revolution in Super-sonic Flow", NOL TR 62-111, December 1962.

27

I

13. Wu, J. M.; Aoyoma, K.: "Trarsonic Flow-Field Calculation AroundOgive Cylinders by Nonlinear-Linear Stretching Method", U. S.

2 Army Missile Command Technical Report No. RD-TR-70-12, April1970. Also AIAA 8th Aerospace Sciences Meeting, AIAA. Paper,70-189, January i1-70.

14. Wu, J. M.; Aoyoma, K.: "Pressure Distributions for AxisymmetricBodies with Discontinuous Curvature in Transonic Flow", U. S.Army Missile Command Technical Report No. RD-TR-70-25, November1970.

15. Vn Driest. E. R.: "Turbulent Boundary Layer in CompressibleFluids", JAS, Vol. 18, No. 3, 1951, pp. 145-160, 216.

16. Truitt, R. W.: Fundamentals of Aerodynamic Heating, The RonaldPress Company, New York, 1960.

17. Carnahan, B.; Luther, H. A.; Wilkes, J. 0.: Applied NumericalMethods, John Wiley and Sons, Inc., New York, 1969.

18. Love, E. S.: "Base Pressure at Supersonic Speeds on Two-Dimen-sional Airfoils and on Bodies of Revolution with and withoutTurbulent Boundary Layers", NACA TN 3819, 1957.

19. Chapman, D. R.: "An Analysis of Base Pressure at SupersonicVelocities and Comparison with Experiment", NACA TR 1051, 1951.

20. Cartright, E. M., Jr.; Schroeder, '%. H.: "Investigation atMach Number 1.91 of Side and Base Pressure 1,istributions OverConical Boattails Without and With Jet Flow Issuing From Base",NACA RM E51F26, 1951.

21. Kurzweg, 1I. It.: "Interrelationship Between Boundary Layer andBase Pressure", JAS, No. 11, 1951, pp. 743-748.

j 22. Crocco, L.; Lees, L.: "A Mixing Theory for the InteractionBetween Dissipative Flows and Nearly-Isentropic Streams", Report

"• I No. 187, Princeton University, Aero. Engr. Lab; 1952.

23. Bureau of Naval Weapons: "Handbook of Supersonic Aerodynamics",o j NAVWEPS Report 1488, Vol. 3, 1961.

I. 24. Reller, J. 0., Jr.; Hamaker, F. M.: "An ExDerimental Investiga-7¢ tion of the Base Pressure Characteristics of Nonlifting Bodies"of Revolution at Mach Numbers from 2.73 to 4.98", NACA TN 3393,. • 1955.

%2

25. Peck, R. F.: "Flight Measurements of Base Pressure on Bodiesof Revolution with and without Simulated Rocket Chambers',NACA TN 3372, 1955.

26. Fraenkel, L. E.: "A Note on the Estimation of the Base Pressureon Bodies of Revolution at Supersonic Speeds," Royal AircraftEstablishment TN No. AERO 2203, 1952.

27. U. S. Army Missile Command: Engineering Design Handbook: Designof Aerodynamically Stabilized Free Rockets, AMCP 706-280, 1968.

28. Stoney, W. E., Jr.: "Collection of Zero-Lif' Data on Bodies ofRevolution from Free-Flight Investigations", NASA TR R-100, 1961.

29. Kurzweg, H. H.: "New Experimental Investigations on Base Pressurein the NOL Supersonic Wind Tunnels at Mach Numberb 1.2 to 4.24",

,. •- NOL Memo 10113, 1950.

30. Krens, F. J.: "Full-Scale Transonic Wind Tunnel Test of the8-Inch Guided Projectile", NWL TR-2535, 1971.

31. Owens, R. V. : "Aerodynamic Characteristics of SphericallyBlunted Cones at Mach Numbers from 0.5 to 5.0", NASA TN D-3088,December 1965.

32. Charters, A. C.: "Some Ballistic Contributions to Aerodynamics",14th Annual Meeting of IAS, New York, January 1946.

33. Karamcheti, K.: Princip" ,f Ideal-Fluid Aerodynamics, JohnWiley and Sons, Inc., Ne& :k, 1966.

34. Spring, D. J.: "The Effect of Nose Shape and Afterbody Lengthon the Normal Force and Neutral Point Location of AxisymmetricBodies at Mach Numbers from 0.80 to 4.50", U. S. Army MissileCommand Report No. RF-TR-64-13, 1964

35. Gwin, H.; Spr'.ng, D. J.: "Stability Characteristics of a Familyof Tange.nt Ogive-Cylinder Bodies at Mach Numbers from 0.2 to 1.5",U. S. Army Missile Command, Report No. RG-TR-61-1, 196.1.

36. Washington, W. D.; Pettis, W., Jr.: "Boattail Effects on StaticStability at Small Angles of Attack", U. S. Army Missile CommandReport No. RD-TM-68-5, 1968.

2

:. 29

K

37. Whyte, R. H.,: "Effects of Boattail Angle on Aerodynamic Charac-teristics of 175mm M437 Projectile at Supersonic Mach Numbers",U. S. Army Munitions Command Technical Memorandum 1646, Septem-ber 1965.k 38. Ohlmeyer, E. J.: "Dynamic Stability of the Improved 5"/54Projectile", NIVL Technical Report in publication.

39. Allen, J. H.; Perkins, E. W.: "Characteristics of Flow OverInclined Bodies of Revolution", NACA RM A 50L07, 1965.

40. Buford, W. D.: "The Effects of Afterbody Length and Mach Numberon the Normal Force and Center of Pressure of Conical and OgivalNose Bodies", JAS, No. , 1958, pp. 103-108.

41. Chadwick, W. R.: Sylvester, J. F.: "Dynamic Stability of the"5-Inch/38 Rocket Assisted Projectile", NWL Technical MemorandumNo. K-63/66.

42. Donovan, W. F.; MacAllister, L. C.: "Transonic Range Tests of5-Inch/54 Rocket Asissted Projectile (Inert), BRL MR 2107,July 1971.

43. Karpov, B. G.; Schmidt, L. E.; Krial, K.; MacAllister, L. C.:"The Aerodynamic Properties of the 155mm Shell MlOl from FreeFlight Range Tests of Full Scale and 1/12 Scale Models", BRLMR 1582, June 1964.

30

79

2

-I

h.-��1�

a

LaC,

C �q�Ib�w

Ux

g

§ _______L.0.

C�k '��c'

K�. 31

C, --

0l)

w 6

0

0 50SUPERSONIC FLaTN(M 0 ,I.2)

40 8= SIN'(,/M.)

o NEWTONIAN ITHEORYri-w 30.-

iiJOF

Io • 20+Z

w PERTURBATION 'THEORY

Co

1.0 1.5 2.0 2.5 3.0 3.5MACH NUMBER- Moo

FIGURE2 BOUNDARIES OF PERTURBA1ON AND"NEWTONIAN THEORY

32

0 ul

z

U)

z

o CDsom

" C , I IIn

33

Cr C1 0

w

CM/

clicSt-~

4 0~ 34

0.4, CAq, 0 .02

0.0 0. ,0 10 20 40 40 50

CONE HALF ANGLE-Gc (DEGREES)

FIGURE 5A. VISCOUS SEPARATION DRAG, Moo= 0.4

.0101

11CA

.0%I. ' ;' / CA.(ACA )W()/O.I

0.5 1.0 1.5 2.0 Z5

MACH NUtJR - I',.z

FIGURE 58, ROTATING BAND DRAG -CAR8

FIGURE 5. VISCOUS SEPARATION AND ROTATING BANDDRAG

3S

3

C2/rod N

(cN°)' c1 TANS+ C2

•::,•,• o EXPERIMENT ( REF.311

Z -I

,o-0.25 0.50 0.75 1.00 1.251

•'¢: MACH MU33ER- Moo

,-, ~FIGURE 6. CONSTANTS TO DETERMIINE ýClNj FOR Moo< .

316

-

VI 2

6 a 0

I ir4 Z

I) 0vi g~ o

c; C

iv ~ ~V av *N) 1 1 ci -J

37 C

w 1

1w

00

00

'I r -r

.c/

CL I

LID(n w

(I)

Old 0

wK~~~ c'I 1 fr--&

38

S •. LL

o i --°_________________ - ---

1I;

V V.

K'• 0

CLL

x 039 I,

LI,

La..

IL.I

"' P

4',g

0.7

0.6-

0.5-

.0.0 L 4 6 8 10

BODY FINENESS RATIO- .•

•' FIGURE IO-A. DRAG PROPORTIONALITY FACTOR-7

f" 2 .0

Cdc

1.6 -

0.81I0. 0 0.5 .0 1.5 2.0 2.5

CROSSFLOW MACH NUMBER-.MWo Sin aFIGURE lOB. CROSSFLOW DRAG COEFFICIENT

FIGURE 10. DRAG PROPORTION•ALITY FACTOR AN•DCROSSFLOV, DRAG COEF'FICIENT

40

. .

IISM

-

W~ 0(/0 >0wU) ~ 010r-c

0 wl

r LUa: Waz

41 z z0 4 0

a: cn

z a:c)~

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- -D

0- L <WLF

w 0 U0 0- < 5z a: co !E 0 Z > a- L

'II

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a: 0

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42

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1 Ii43

.50-- PRESENT THEORY

0 & EXPERIMENT(Ref.31)

CAW+CAf-• 0

.30

i.IC I _I__

-- •2,0 - - 4

CN.(a= 0°) -CN.CM(a-0o)

rI 2

S0 2 .4 .6 .8I 20 -r, AB8

ri 0o. -20k-

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FIGURE 14 COMPARISON OF TMRY AD EXPERIM NT FORA BLLNTED COME; M01 l.5,GcuIOo.

44

K

ont

• \ PR~ESEMY" THEORY. 6 EXP•RENT(Ref.31)

- SLENDER BODY THEORY

.10.

2-0 -3.0i• •cN.,(a=Ol oo -cma(a==o

L8 -2?8

1B6 L I ?.10 1.5 2.0 2.5 3.0

X 01 MACH NUMBER

w- 10

So oI_•

0 1

Cre

o -o FIGURE 15 00i*FARISON OFF TE~.m AjpD' E~mrFOR BLUNhT COAS;Go~ 100, A,. m0 ~.2.

4S

0.2r0 -

C +CA0

0.1 EXPERIMENT (REF.31)- THEOR"

2.01

CN

9 1.5

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522

06

• 2r-

I I

0.5 1.0 1.5 2.0 2.5MAC MANJMPFR- M

FIGURE 16. COM5PARISON OF THEORY AND EXPERIMENTFOR BLUNTED CONE; tO• 1.00 rn/rA 0.4.

46

C N

ot

lE

- - - -o -o- --

o 0.4-

'••Mo PRESENT SLENDER EXN.REE 39)06 0.6- TH Y BODY0 THEORY TO

1.01

• 10n4 a 12 16 20

.,C M ,, ,| / ....

• ANGLE OF ATTACK (DEGREE.S)

4- C

S• oFIGURE 17 COM1PARISOM OF THEORY WITH EXPERIM1•'• FOR TANGENT OGIVE.- CYL.INDER.

• •lu14 C3ALIBERS

"-j 4

0.80M=.5d

'•" Af 0 -, -•

0.4 - / ,

S0.0c I , I- -- .- , -4i

S2.0--

1.0..0 PRESENT SLENDER............................. . ......................... T H O Y T EO..D X P R f 1

00 0 0.5 THEORY0o- ~1.5

•000 .0 1 1... . - J ,,

5.0

• C M,(t=-O )

2.5-

K 3 I' _ _ _ __O_ _ ,,__ _ _i_ _.1

0Lc 18I I

0 1 2 3 4 5NOSE LENGTH (CALIBERS)-,[

FIGURE 18 COMPARISON OF THEORY AND EXPERIMENTFOR CONES OF VARIOUS LENGTHS.

48

"C.

M THEORY EXPERIMENT(Re.40)2.75 0

1

C~ ~ Na17 090 -a" •' 2~~~~~.0 "'-.. .. , , _

2.0 I I

T.0 - C3

CM(=O @JI -= -.

5.0 /

Iii3.0

-'"s0 0

~I0

0 2 4 6 8 10

AFTERBODY LENGTH (CALIBERS)

C F-IGUR 19 COMPARISON OF PRESENT THEORY WITHEXPERIMENT AS A FUNCTION OF AFTERBODY LG.

(2.83 CALIBER TANGENT OGIVE NOSE).

"49

z z.

101zO

0 -~WN

050

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2 0 BRL BALLISTIC RANGE

THEORY W

00.5 10 1.5 2.0 2.5M~P J~E-Moo

FIGUR9E 21 COMPARISON TMERY AA*D ITEST DATAFOR 5'Y54 RAP PROJECnLE.

c51

404-404

S0.0D ,

oo4.0

CN4c=01

0.0 I I I I !

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• .)

~ ANOL WIND TWNEL

1.0 I

0.0 0.5 1.0 1.5 2.0 2.5MACH 5JY8ER-Moc

FIGURE 22 COMPARISON OF Th"LMRY AND TEST DATA FORIMPROVED 5?54 PROJECTILE.

10 4, jr2

0.4-

0.0-

4.0-

00

0 BRL WND TUXXEL0o

J--THEORY

0 05 10.5.0 2.5MACH NUMSER - Mw

FIGURE23 COMPARISON OF THE0RY AND TEST DATA FOR 175?MMXM437 PROJECTILE

53

0.4

CDO

0.2

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CNaGcO)

2.0

C w< 5-

0 4

c~2- o BRi. BAkLLISTIC RANGE0. -THEORY

x

0.0 0.5 1.0 1.5 20 2.5

M1ACH NUM1BER- Moo

FIGURE 24. COMVPARIS-ON OF THEORY ANID TEST DATAFOR 155WM PROJECTILE

54

cDo

0.2

0.0

4.

2.0

0.0- 1

:i n 12' r .6

2 10 70131

I 8

-i

7240 o.02

0.0 0.5 1.0 1.5 2.0 2.5MACH NUMBER- Moo

FIGURE 25 AERODYNAMICS OF 5 INCH GUIDED- PROJECTILE BODYK,5

GLOSSARY

CA Total axial force coefficient

" CABA Axial force coefficient contribution from base pressure

" f " C.f Axial force coefficient contribution from skin friction

CARB Axial force coefficient contribution from rotatin.g band

"CAv Axial force coefficient contribution from viscous separationfIS on nose at subsonic Mach numbers

CAWq Axial force coefficient contribution from expansion andshock waves

CDo Zero lift drag coefficient; CD = CA

C d Crossflow drag coefficientSc

CfO Mean skin friction coefficient based on freestream Reynoldsnumber

SCM Pitching woment coefficient about nose unless otherwise"specified (positive nose-up)

CMot Pitching mcment coefficient derivative - dCM/dc

CN Normal force r'oefficieazL

CN O Normal force coefficient derivative - dCN/dc

"C Pressure coefficient; Cp =P-Pco)/I/2p.V. 2

p 1

d Diameter (calibers)

dB Base diameter

H Mean height of rotating band in calibers

' • Body Length (calibers)

M Mach number

SPr Prandtl number

R Body Radius (calibers)

-dR/dx

A-1

RN Reynolds number CV2)i

- - RT Turbulent boundary layer recovery factor

Sw Wetted surface area of body

S p Planform area of body

I w Wall temperature

u,v,w Velocity components in cylindrical coordinate system

V Total velocity - V = U2 +V

2+W 2

Vol Volume of body

x,r,O Cylindrical coordinates with x along axis of symmetry andin calibers

x,y,z Rectangular coordinates with x along axis of symmetry andin calibers

Ycp Center of pressure in calibers from nose unless otherwise

specified

xP, X Distance to centroid of planform area in calibers from nose

xu,ru Coordinates of point below which perturbation theory cannotbe applied

a xI Distance measured relative to shoulder of boattail

a Angle of Attack

Angle between tangent to body surface and axis of symmetry

y Ratio of specific heats (y = 1.4)

6 Angle between a tangent to the body surface and freestream

direction

6* Angle wHizh the nose makes with the shoulder of the body (degrees)

Velocity potential in cross flow direction

_n Ratio of drag coefficient of a circular cylinder of finitelength to that of a circular cylinder of infinite length

Cylindrical coordinate measured with e - 0 in leeward plane

A-2

!! Oc Cone hal4. angle

[ Coefficient of absolute viscosity

* p Density2

Total velc .•y potential which is made up of axial andcrossflow velozity potentials

Velocity potential for axial flow

Subscripts

O " ,Freestream conditions

a Afterbody

B Boattail

BA Base

n Nose

o Stagnationi r Reference conditions (reference length is the afterbody• diameter =dr)

A-3

COMPUTER PROGRAM TO DETERMINE PRESSURE DISTRIBUTIONS AND FORCES ON

"UNGUIDED PROJECTILES OR THE BODY ALONE OF THE GUIDED PROJECTILE

and The met'iods described in the report to obtain surface pressuresand orce coefficients have been programmed for high-speed digitalcomputation. The purpose of this appendix is to provide a generaldescription of the program including a listing of the program anda sample of the required input and resulting output.

A. DESCRIPTION OF PROGRAM

The program reads in the body geometry with x = 0 as shown inFigure 1. If the nose is truncated, a conical nose of angle givenby Figure 2 is automatically placed on the truncated portion toget the perturbation solution started; but the pressure integrationbegins at x = 0, which is the location of the first point read in.If the nose has v spherical cap, then the program automaticallyccmputes this and again the first point read into the computer isat x = 0. However, the pressure integration begins at x = -rn.

It is suggested tha- the description of the body be read in toat least three decimal places if possible because the resulting solutionwill not be as accurate as it could be otherwise. For example, ifthe ogive has a formula it is suggested a desk calculator be usedto compute the body coordinates as opposed to a slide rule.

Once the coordinates of the body are read in (the various bodygeometry options are discussed below). the program then computesa ne• set of body coordinates where the flow field solution willactually be founu. These points are unequally spaced along thebody to conserve computational time but are also snaced closelyenough so an accurate solution can be assured. Once the .ody

- ogeometry has been found, the program checks to see whether the Machnumber is subsonic (M• < 0.8), transonic (0.8 < Mw < 1.2), orsupersonic (K[• > 1.2) and then proceeds to numerically calculatethe force coefficients for that particular Mach number.

B. INPUT DATA CkRDS

CARD NUMBER PARAMETERS READ FORMAT

1 M 13

2 AT,, DIA, HB, AINF, RHOINF, AMUINF, (4FIO.4,2F15.12,IPRINT 15)

4 AM (I) 16F5.3I = 1,2,3,..., %t.1, MN

S N, NSHAPE, NI, N2, N3, NBLUNT NFL, (815, 4Fl0.S)OP°4 NN!A, C2, C4, F, RR

L7)•

"CARD NUMBER P'ARAMETERS READ FORMAT

6 X(I), R(l) 2F15.10I = 1,2,3,..., N-I, N

SN+S5

C. DEFINITION OF PARAMETERS

PARAMETER USE

-M Specifies number of cases to be run. If M > 1, thenonly one data card needs to be included for M butcards 2 through (N+S) are included for each additionalcase

AL Angle of attack (degrees)

* DIA Body reference diameter (feet)

HB Mean height of rot:.±ng band in calibers

AINF Freestrearm speed of sound (ft/sec)

RHOINF Freestream density (slugs/ft 3 )

AMUINF Freestream absolute viscosity (lb-sec/ft2)

IPRINT IPRINT = 1; pressure coefficients are to be printed2; no pressure coefficients printed

MN Number of Mach numbers where solutions are computed(MN < 16)

AM(1) Mach number where solution is computed

N Total number of points read in along body surface(N < 30)

B-2

PARAMETER USE

NSHAPE Parameter which describes body shape

1. Pointed Body

NSHAPE = 1; nose only2; nose plus afterbody3; nose with discontinuity in it.

There may or may not be anafterbody present.

4; nose plus aftcrbody plus boattail5; nose with discontinuity in it plus

afterbody plus boattail

2. Bluinted or Truncated Nose

NSHAPE = 3; nose w'ith or without discontinuity.There may or may not be an after-

body present.5; same as above except afterbodv and

boattail are present,

If NSHAPE = 3 or 5, at least S points must be read

in along each of the ogives even if the ogive is a

straight line.

NI Number of points read in along first ogive

N2 Number of points read in up through the secondogive (includeF 4irst ogive)

N3 N3 1; conical boattail2; ogival boattail (at least 5 points must

be given along boattail if N3 = 2)

NBLUNT NBLUNT = 1; pointed body2; truncated or spherical cap

NFL NFL = 1; spherical cap2; truncated nose

• NNIA NNlA = 1; Blunted nose with no discontinuitiesSh . present other than the intersection

"Irk f, of the nose cap with the ogive (Nl :=1and N2 > 5).

2; Blunted-nose with a discontinuity inthe ogive so there are two ogives

(•'1 present (Nl > 5 and N2 > 9).

K B- 3

PARAMETER USE

C2,C4 Parameters which specify mesh spacing. If noseis pointed,C2 = 0.9 and C2 = 20 are nominal values.For other nose shapes,C2 = .05 and C4 = 1.0 arenominal values.

SF Constant which determines limiting body slope fora given Mach niumber. F < L.0 with F u 0.95 recom-mended.

RR Radius of spherical cap or truncated meplat incalibers.

X(I),R(I) Body coordinates (in calibers) where I < 30.

9: D. PROGRAMf LISTING

The Fortran listing of the source desk currently being used atthe Naval Weapons Laboratory is as follows:

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- ~The samnple ca.se below is the iPnput data cards for the improved5t1/54 proj ecti~ie 3 8 . The Formats and paramueter locations and de"initionshave been discussed previously.

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rk F. SAMPLE OUTPUT

The resulting output for the above input data case is shown below.The reference conditions are printed first followed by the input bodycoordinates. The last output quantity is a table listn theindivi-

- dual axial force cont'ribution and a table listing Vhe force coeffici-ents.

B-52

ANL OF A--.X, --5O ~ LFRNEDAEEt .1F

-~MM - .0023 *ua769' SLUG3/FT3

ABSOLUTE VISCI9SITY .900342 B-E/T

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