.AFATL-T-" 6

AERO.YNAMIC CHARACTERiSTICS

BASIC N OSE-CY LIND ER BODIES

FOR

LARGERANGES OF ANGLE OF ATTACK

WEAPONS EFFECTS DIVISION

TECHNICAL REPORT AFATL-TR-41-67

SDDD CJUNE 1971 -

:• ' R~9 19l i!

C

7Approved for public release; distribution unlimited.itRoptoducod by

NATIONAL TECHNICALINFORMATION SERVICE

1.p nghold, Va 22151

AIR FORCE ARMAMENT LABORATORYAIR FORCE SYSTEMS COMMAND* UNITED STATES AIR FORCE

EGLIN AIR FORCE BASE, FLORIDA

UNCLASSIFIED .SSeCIIu* t"'- I aion"' . . . -•"SIecunity'ClOSSW llfc

DOCUMENT CONTROL DATA R & D(SeWiy.ca.s... 0,,ii of #it#*. be* of 8,,,Iu. t .iif •e.e.,,,w9 or.• .4 I. must be Vns .o . 9. h .,. ,e . .ll .i II em.sited,)

1. 0RIGINATIPM6 ACTIVITY fCO"Pl,"4014e) 'AD. VICIPORT SECURITY -CLASSIP ICA&TIOW, Weapons Effects Division 1. 'UNCLASSIFIFD -

'Air Force Armament Laboratory M GROUP " /

Eglin Air Force Base, :Florida 325423. REPORT TITLE

AERODYNAMIC CHARACTERISTICS OF BASIC NOSE--CYLINDER BODIES ,FOR LARGE RANGESOF ANGLE OF ATTACK

4. DESCRIPTIVE N;OTCS (7p),V olropmt a" Smel nchi, dffe.)'Final Report - Qctober 1969 to' June 1971.-

5. AUTNORIS) (FiOOT Mass. mldelbiftfl. lst feeneW)

Robert E. Kellock, CaptAt'n, USAF'Percy H. RITler, PhD

6, REPORT DATE 7,v. TOTALF.NO, OPFAES jb NO. OF REFS'

June 1971 . ' . 59 17Ba CONTRACT OR GRANT NO. to, ORIGINATONRS.REPORT NUMVERIS "

F08635-70-C-0053b. PsdoECT MO. AFATL-TR-71-67

' 9. ,b. OTHER REPORT NOIS, (A,? .olhexue.a. &at m ,,.y be a.sged

d. __

10. DISTRIOUTION STATEMENT

Approved for public release; distribution unlimited, '

II. SUPPLEMENTARY NOTES "12. SPONSORING MILITARY-ACTIVITY

Air Force Armament LaboratcryAvailable in DDC Air-Force Systems Command

i < ! Eglin Air Force Base, Fldld .3. _ 4a 195 .13. AD TRACT

"•The existing methods of predicting aerodynamic characteristics for basicblunt-based bodies of revolution are reviewed and assessed. Utilizing themore accurate methods, a digital computer programwas-developed which uses,numerical techniques, thus avoiding less accurate simplifying assumptions.This new approach provides ,reasonable-accuracy in predicting the normal forceand moment coefficients-of basic model combinations ,for angles of attack ranging'from zero to 90 degrees. This accuracy in prognosis also includes' freestreamMach numbers ranging from subsonic to hypersonic.

DD Nov_ J473 tassification"- •-icy-Classification ._

~UNCLASSOTrED.- ____ _____ &

Key WODSLIKA ACý

"OLs, WY NCLe -:WT. mwoust WT-

Bl unt-bMted bodies of revol'ution-Aerodyhnamif chiracterlstict'Normal1 force and moment 'coefficientsý

[, DDigital' cciputer- program

UNI .4TJ

Aerody oismi, Caracteristics,of

- Basi~ ose-Cylinder Bodiesfor

Large, {7 ,"nges of Angle of Attack

tibert E. K*llocic, Captain, USAFPercy H. Miller, Ph.D.

Approved for public release; distribution unllmlted.1

FOREWORD

This report is based gn a master's degree 'thesis pre!entet ,by t,author to the graduate faculty of the Louifiana State rU;dvers'tv- in 1971.The Air ;Force Armament Laboratory recommended that it be prepared',and pub-lishedas a technical report si'nte all research was perf3ý,ned by an AirForce student at the cited educatinna.ainstitution. The'researchwasaccomplished during the October 1969 z- June.i971 period, ah,' was identifiedand encouraged through'contracts F08635-68-C-0107 and F086135-70-C-0053;also,, related project humber 2543, "Aerodynamics of Bodies at High AngleofAttack". Acknowledgement for assistanrce and guidance is extended toMrs. V. B. Harvey. and-Mr. W. E. Stant (DLOS)..

This technical report has been reviewed and is approved.

CHARLES K. ARPKE, Lt Co nel, UChief, •Weapons Effects Division

ii

ABSTRACT

The existing methods of predicting aerodynamic characteristics- forbasic blunt-based bodies of revolution are reviewed and assessed.Utilizing the more accurate methods, a digital computerrogramwasdeveloped which uses numerical'.tdchniques, thus avoidihg les's accuratesimplifying assumptions. This new approach provides reasonab•e accuracyin predicting the normal force aid momeint coefficients of basic modelcombinations for angles of attack ranging fromzero to 90 degrees. Thisaccuracy in prognosis also includes freestream:i{:ch numbers ranging fromsubsonic to hypersonic.

Approved for public rellasi; distribution unlimited.

" i!i(The reverse of this page is blank)

TABLE OF CONTtNTSSection

Pagei .INTRODUCTION 1

II" REVIEW OF THEORIES 2I. Munk's Potential Theory '2

2. Allen's Work3. Kelly's Interpretation,4, Supersonic Potential Flow 75.. Boundary- LayeFCorrections

76-,, :Previous Results 7'III

DISCkSSIONI. Tiansient Effect 82. Cýo*ss-Flow Drag Coefficient 83. End Effects 1,4. Nose Radius

145. Half-Angia 14,6. DCMDA and .DCNDA, 16

IV PROGRAM DESCRIPTION17. Program Information

172. Main Program 173. Subsonic Subroutine 174. .Supersonic Subroutines5. Second Term Integral1.6. Cross-Flow Drag Coefficient 217. Transient Effects 218. :Radius 219. Tfeta

2610. Drag Correction for Finite Length 2711. -DCNDA 2712. DCMDA 2713,. Output 29

321. Su6sonic

322. Supersonic

353. Indicated Improvements 35

VI FUTURE DEVELOPMENT 42

REFERENCESAPPENDIX

45

V

LIST OF -FIGURES'

Number Title Page

Schwabe's Transient Drag Coefficient 5Model a h Angledof Attack Schematic

3 Plot of kI and k2 '(Munk) 6.

4 Limit of 'Schwabe's Results 9:

5 Approximation of Transi"r0t Effect 10

6 CrossflQw Drag Coefficient, 12

7 Goldstein's Correctif,•n for Finite Lcngth 12

8 'Gowen-and Perkins' Experimental, Results- 13

9 Nose Radius Computation Methods 15

1T0 Initial Normal Fofce Slope 16

11 Comparisons: to Experimental Data - Subsonic 33

12 Comparisons to. Experimeht#1l Data - Supersonic 36

LIST OF TABLES

Number Title Page

I Program'Information 18'

II Main Program 19'

III, Subsonic Subroutine 20

IV Supersonic 'Subroutine 22

V Second Term Integral of Normal Force and Moment 23

VI Drag Coefficient for Cylinder at Crossflow Machand Reynolds Numbers 24

VII Cylinder Cross-Flow Drag Coefficient TransientFactor 25

VIII Radius at Any Given X 26

vi

LIST OF TABLES (Concluded)

Number Title Page"

IX Theta for the Nose Model Being Considered 26

X Drag Coefficient Correction foe Finite Length 27

X1 DCNDA .28XII DCMDA 28

XIII Output (Left Half) 30

XIV Output (Right Half) 31

Iii

vii

LIST OF SYMBOLS

A =-Apparent mass factor - transverse.= 1 +,k

A- : Plarform area

AR F Refereape area-for coefficients (base area),

B = Apparent. mass factor - longitudinal = 1 + k2

-CD = Drag coefficient (D/qA)

Cd = Crossflow drag:coeff-icientC

CL = Lift coefficient (L/qA)

Cm = Moment coefficient :(M/qAd).

CN = Normal force coefficient (N/qA)

d = Model diameter, one caliber

EL = Length measured from nearest end of the model

F = I.Orce

k, = Apparent mass factor transverse - Munk

k2 = Apparent mass factor longitudinal - Munk

t =.Model length

-M = Moment (about model base)

Mc = Crossflow Mach number

N = Normal flow

q = Freestream dynamic pressure

1qc = Crossflow, dynamic pressure

r = Radius of mod•el

S SB = Area of base

{..i

t 'time,

.U =,Freestream velocity

" V = Model volume

Vc = Crossfl'ow velocity

x = Axi-al position -on model= Axial position of centroid of planform area

xm = Axi`a1, position of momeht reference, (base)

a- • = Angle, of attack

n = Drag correction factor for finite length

Ili -- Local drag correction factor for finite length

6 0 = Model surface angle arctan dr/dx

Ov = Nose half-angle

p = Density

ix(The reverse of this page is blank)

SECTION I

INTRODUCTION

The efficient design of missiles requires accurate predictions ofaerodynamic characteristics. iThe early missile designerswere forced to,'test models of their design in wind tunnels to obtain these predictions.But as the flow about bodies became better described by theory, methods.of'calcul'ting the aerodynamic characteristicswere devised. However,these meth'ds, are valid only for a rather narrow range of flow Conditions.As the theo;ies were advanced and improve&, the predictions became moreaccurate., and the limitations with regard to such~parameters as angle ofattack and'-Mach number bacame less restrictive. This is not to mean thatthe Mach number'and anglerof-attack ranges •wereg large.

Recently,,_changina requirements for the purpose and general structulreof missiles require predicting the aerodynamic charaoteristics for smallerlefigth-to-diameter ratios over larger ranges of angle of attack and"Machnhumber. This study provides: advances to existing'computational techniquesand furnishes greater flexibility with .regard to these parameters. The,retIlts are limitedto basic nose-cylinder combinations featuring pointednoses and blunt bases'because no -evaluations, of blunt nose - cylindercombinations were 'ncluded.

To accomplish the objective of the'study, a computerized documontsearch provided about -J1,000 report abstracts, and about 1,400 were select-ed for more careful, eq,'mi nation. Of these, some 400 were obtained .forfurther study. A review of these reports showed that very little experbi-mental data was available in the region of interest; therefore, the effortfor the study was shifted from an empirical study toa review and morethorough examifiation of theories. An analysis of the migresuccessfulmethods indicated that a''digital computer programr-miyht be produced to-easily apply these methods to simple model combinations. By avoiding thelimiting assumptions that were dommonly, imposed, it appeared that theaccuracy could be extended to the region of interest.

This s~tudy, then, represents the first stage of an effort to producea method that 'will accurately predict aerodynamic characteristics forcomplex missile configurations.

I¶

SECTION 'II

REVIEW OF THEORIES

1. MUNK',S' POTETIAL THEORY

Most -bf the successful theoriesý,e.elated to this studyhaVe begun Withthe clas'sicd.l work of Mu.ik presented• in Reference-] and expanded byKelly ii-,'Re'Ynce 2 and 3. Munk imsed potential flow theory to show thatthe force o4istribution or. an airsh'ip IS,,:given by:

2dF k2 -kj d sin 2c dx (I)

In this expression, S is the surface area, a is the angle of attack,x is the distance along the longitudinal axis, and (k -I is theapparent mass factor for the difference in longitudingl and transverseflow. Equation (1) integrates to zero for an airship, but i'f the bodyhas a' flat Lase ofarea, SB, the result is a finite force:

B

Lift jdF pU2 (k k SB sin 2a (2)

Since Munk's research was concerned with very small angles ,of yaw,this potential fGrce is referenced to the wind vector as lift. At higherangles ofattack, 'this estimate would, yield increasing error because theforces due to visc,'us effects had not been included.

2. ALLEN'S WORK

Allen, in Reference 4, assumes that the viscous forces are independentof the potential flow, and the lift and moment coefficients ars writtenas the sum of two. terms; namely, the potential term and a viscous term.

SB cA j2a• Z~oso 3aCL (k2 " k1) R sin 2a cos S + n C AP sin (3a)

L AR 2 dC AR

Cm [VSB ( Xm)] (k2 k,) sin 2a cosaARA 2 (3b)

+ nCd A'R (Xm .x) sin 2 t

CR

2

The first term is Munk's potential theory as modified by Ward inReference 5 to show that the potential cross force is directed midwaybetween (the normals, to the logitudinal axis. and the wihd vector. The ARrepresents the reference area-for the coefficients, V is the volume ofthe model, and t is the total lefigth. 'In the second(viscous) term, Cdrefers to the cylibder steady-state two-dimensionil drag ccoefficient for two-dimensional flow. The value of n corrects this dragcoefficient for f"nite model length according to values tabulated by-Goldstein in Refer&nce 6. The xm is the moment reference point, x isthe centroid of the planform area, A , and sin 2ca comes from theresolution of dynamic pressure from the e crossfl'ow velocity U sin a tothe -free-stream velocity:..

3. KELLY'S INTERPRETATION

Kelly, in References 2 and 3, deats with the normal force coefficientwhich eliminates the factor cos a in the :9cond term of Equation (3a).This method also eliminates the need to assume that the axial drag compon-ent of the lift force-is negligible.

Allen,,in Reference 4, presents the Viscous contribution to the crossforce from a cylindrical element of length dx as:

dF = 2r Cd (p Vc2 /2) dx (4)c c

where e' is the body radius at point x (r = r(x)),, and Vc is the cross-flowvelocity. Kelly resolved the free-stream dynamic pressure, q,into cross-flow and axial components and corrected the cross-flow dragcoefficient for finite length with n. Defining the normal force and momentcoefficiefit as:

CN = N/qA (5a)

Cm = M/qAd (5b)

(where d is the body diameter), the coefficient increments due to viscouscross-flow are:

2n sin 2a r Cd dx (6a)"R AR c

3

Ft

ti 2n sin "• r • x)ACm- i Cd ARm dA R 1

These terms ar'e entirely-due to viscous effects,,since Cd will- not existfor a true potential flow. c

Allen was aware-that the viscous term drag coefficiehit;eCd 'should be

related to the transient effect as presented by Schwab6, h cReference 7. SchwabM found that ,thedrag coefficient for a circularcylinder in cross-fldw starting from rest varies "according to the ,parameterV~t/r, where, t is elapsed time (Figure 1). The steady-state value of Cdf8r Schwabe's experiment was approximately one. Kelly considered'a. clamina-,of ai7, dx.,thick', mowing along the' body axis at speed 'U cos aand across thi' body a•t speed' U si-n a. He pb6'tulated that the cross;flowsenses an impulsive flow about a circular cylinder of radius r (Figure 2);hence, the Schwabe time parameter is redefined as

'ct_, U sin a x x tan a (7)"r _,r T cosa

To aid integration, Kelly approximated Schwabe's curve (Figure 1) with a'portion of an arc tangent function, y = tan 1x, and then approximated thearc t6ngent function with a truncated Taylor series. This permitted rapidevaluation of the function without reference to a table of values.

Allen had used ý laminar value of 1.2 for Cd , based on the cross-flow Reynolds number. In Reference 8, Hill c notes 'that Kelly useda turbulent value of 0.35 when the free-stream Reynolds number indicated.the boundary Tayer to be turbulent. The observation niade was-that if theaxial boundary layer is turbulent, the cross-flow layer will behave in aturbulent manner, even"though the cross-flow Reynolds number is wellbelow that associated with transitions (References 3, 8).

To evaluate the factor (k2 - ki), Kelly replaced it with (AB/2) cos 2 0,where A = 1 + kl, B = 1 + k , and tan 0 = dr/dx (Figure 3). Thisapproach has been compared to the work presented by Upson and Klikoff inReference 9.

4

CY).

00

o > )r 4-.

C.). 0

-~~L - -- -t-

Q4)

CL)A .0

4-5-

L.0

CN.)

I /

dx

U UCos a C

' Fiure .Moel -..- d (Ical.

Figure 2. Model at Angle of Attack Schema-ti" .

k/d

1.01 2 3 4 6 8-1 2

k-2 0.8' B 1 + k2

0.6 - k : I.

0.4

0.3k1 1..

0.2 kI .5 ;(k/d -1 8 A 1 +k

0.1 -

.07 __ __ - ,_ _

.05

Figure 3. Plot of kI and k2 (Munk)

6

4. SUPERSONIC POTENTIAL FLOW

Except for vehicles with slender noses, Munk's theory is not accuratefor supersonic flow; hence, Kelly used• the second order theory presented'by Van Dyke in Reference 10 for this flow region. This changes thepotential terms of the coefficient expressions to

( dCN sin ctcosct (8a)NiW =0

( dCm\ sin a (8b)

IdAM iTUa m)a 0 sin a cosa (b

5. BOUNDARY LAYER'CORRECTIONS

Kelly, in his calculations, also included model size increases due tothickening of the boundary layer as the flow progressed along the bodylength. These relatively small corrections -(ab-ut-5% for a 14 calibermissile at a = 8 degrees, and M = 2.87, and about 4% for a 9 caliber modelat a = 30 degrees and M = 0.26) are most applicable to slender bodi'es atsmall angles of attack. As the angle of attack becomes larger, thedistance a particle flows along the body becomes smaller until at a = .90degrees there is no flow along the body,,but only cross-flow. Thesecorrections are generally small and-not significant at relatively largeangles of attack for the model} used in this, study; therefore, they willnot be considered in this report.

6. PREVIOUS RESULTS

Kelly was able to achieve excellent results up to about 30 degreesangle of attack for subsonic flow, and up to 10 degrees for supersonicflow. With minor computational changes, he also has produced reasonablygood predictions up to 40 or 45 degrees for both cases.

7

SECTION III

DISCUSSION

The success achieved by Kelly shows that the-basic approach is sound,ýand further extension is possible. A simple modular digital computer

-program allows for more precise evaluation of the various parameters. More,specfficaly, such a program permits the rapid calculation of the aero-dynamic characteristics even though the parameters may varyzover the largeranges.

1. TRANSIENT EFFECT

The transient evaluated by SchWabe's experiments limited Kelly'ks methodto a time parameter value of nine (x/r tan ct = 9)-, since Schwabe's -experi-ments did not define the transient effect beyond this value. This limitis expressed as angle of attack versus model length in Figure 4. Schwabe'slimiting factor was the size of his equipment. He showed a maximum Cof 2.07 at the maximum time parameter. The data points show a leveliNgoff of the drag coefficient, but no indication of how it returns to asteady-state value.

The first computations of the computer program indicated a slightover-prediction of normal force coefficient near the parameter value ofnine for subsonic cases. The supersonic cases showed an extensive over-prediction at nearly all time parameter values in the transient region.

The drag buildup for a cylinder normal to impulsive flow is attributedto the formation of vorticies. As movement begins, an upstream displace--ment of the fluid occurs over a broad front. The flow field quickly forms,and begins to narrow as vorticies are developed. As velocity increases,the vortices begin to be shed, and the wake narrows to near steady-stateconditions. The narrowing of the wake results in a cross-flow drag reduction.

Schwabe's experiments were conducted in a water tank of a length thatprevented his experiment from reaching steady-state conditions. It is areasonable assumption that the upstream effect was influenced by the con-fines of the tank length and resulted in an erroneous drag increase and,delayed the onset of the reduction towards steady-state flow. Based on a'knowledge of these limitations, a reduction of the transient maximum to1.97 at a time parameter of seven is used. A linear decline to the steady-state drag coefficient at a time parameter of ten is used to provide con-tinuity of the function (Figure 5). Since upstream effects provide amajor part of the transient phenomena, the supersonic case does not utilizeany transient effect.

j8

- 1 1- I

-- -- 2 k tan• = 3- 2t tan• - 6

30- : 2 k t h•,n =9 -

20 -

F- -i ,

1011,0 - *- +, .- . - -

- -. . . ... . .-

0 10 20 30 40 50 60a, Degrees

Figure 4. Limit of Schwabe's Results

9

CD

4 J

t 4~

r_ "- e= ' *-i) Q

o °0

4-5-% 4-.)4-

*r"

xExis- 0

C.C.

10

Schwabe's experiments were conducted at a Reynolds number of "580,forwhich the steady-state drag coefficient (Reference 11) Is about 1.15.Schwabe'sd curve is therefore normalized by this value and is used as atransieht factor to apply to the actual two-dimensional drag coefficientfor the model and flow conditions being considered.

2. CROSS•LFOW DRAG, COEFFICIENT

Thecross-;fl'ow drag coefficient is a function of the cross-flow Machnumber and'-Reynolds number. To develop this function, two-dimensionaldata 'from--',veral sources are dombine6. In Reference 11, Hoerner showsan assortment of data at various Mach and Reynolds numbers. His plots ofC versus ',jh number agree with most other reports, but they are not

conti''u6us where'the magnitude of the Reynolds number causes the,transiti,,-from laminar to turbulent flow to occur below a Mach number ofapproximately0.55. Transition causes a flat-bottomed dip in the dragcoefficient (Reference 12)'for Reynolds numbers between approximately300,000 aind-700,000. Above Mach 0.55, compressibility effects appearto dominate the flow and suppress any 'Reynolds number effects.

To 'ompute this dip in the cross-flow drag coefficient curve, aReynold humber is computed. The characteristic length used in thiscomputation is' that which the freestream senses, and it is expressed as£ = 2r sina,'+ x cos a. Freestream velocity is used in the computation,based bn 'evidence that the cross-flow boundary layer appears to assumethe character of the axial boundary layer.

The drag curves obtained from Reference 11 are approximated bystraight line segments :for the computer function. Another straight linesegment provides transition from the laminar curve to the turbulent curveas the Reynolds number increases from 400,000 to 500,000 (Figure 6).

3. END' EFFECTS

The'two-dimensional drag coefficient must be corrected for finitecylinder length. Goldstein's data tabulated in Reference 6 gives anoverall, or average, ratio of CD to Cdi (Figure 7).

inf

Since a portion of the conputer program developed for this study usesa strip'techni'qlie (numerical integration by trapezoid rule), the localcorrection referenced to the nearest end, rater than the average value,must be used. The drag coefficient distribution 'is not clearly indicatedin any of the referenices examined except in Gowen and Perkin's repor,t(Reference 12). Efforts to curve-fit these data (Figure 8). led toA"'hechoice of the following algebraic function to express ni:

ni =l-exp[-(Mc + I)/EL] (9)

I

'~/Cd Lam.

(Tan Hoerner's Curves

-I -- Computer Approximation

Turb.

01 , "_

0 2 3Mach Number

Figure 6. Crossflow Qrag Coefficient1. ___--"_'

0.7 ____

0.3n

0.2

0.1 _ _ _ _ _

1 2 3 4 6 8 10 20 30 40i/d

Figure 7. Goldstein's Correction for Finite Length12

-7-ý

C

-P

Cj 0~I4Jcr..

0

4O)C6

w

1-3

cU,

0 3~

d

EL is the distance from the nearest end of the model, and the quantity(M + 1) arises from the fact that the end effects diminish rapidly withincreasing Mach number. This is confirmed by Penland in Reference 13,where end effects have been determined to be virtually n6n-existai't forMach 6.86.

Since these end eff-ct corrections apply to right circular cylinders,further corrections are required for a vehicle with a pointed nose. Asthe nose becomes more slender, the end effects increase and the local' dragcoefficient returns to that for a right cylinder. Thus, the drag reductiondAue to end effects is further reduced by the oblique flow around a pointednose. If the nose half-angle is near zeif degrees, the 1.ow is essentiallythe same asýa cylinder of equal length. If the nose half-angle is near90 degrees, the end effect is so drastically reduced that almost no:dragis contributed by the nose portion of the model. These observationsindicate that the end effect change due to nose half-angl'• foli6w§ the formof the cosine of the nose half-angle. Therefore, when the str.ip beingcomputed lies on the nose portion of the model, the n. computed is reducedby multiplication with the cosine of the nose half-an~le.

4. NOSE RADIUS

Computation of the transient effect, the nose half-angle, and the two-,dimensional drag coefficient requires that the radius at ,any axial pointbe available. Since the commonly used nose shapes include cones, tangentogives, secant ogives, power series, and parabolics, a variable functionis Used to express radius variation with axial distance. Based on a-nose-type code,, the applicable function is selected for each computation(Figure 9).

A tangent ogive function is used for both ogives, since including afunction foiý a secant ogive would be more complex than necessary at thisstage. Power series and parabolic nose shapes are approximated by powerseries expression, and cones are defined by a±tahgent function. None ofthe functions employed account for any nose hIuntilng; hence, this analysisis valid'only for pointed noses. However, minor'blunting would, introduce'only minor error in the results.

5. HALF-ANGLE

The computation of the apparent mF 7 factor AB/2 cos1ev requires anaverage value of eV (=tan-' dr/dx) over the entire model. Also, thecomputation 6f n. requires '•he use of the nose half-angle. (For powerseries and parab lic noses, the local half-angle would increase accuracybut this additional c)mplicatiOn is not included at this stage.) Tocompute these factors, the average nose half-angle is computed by tenincrements of Ar/Ax, each equal to one-tenth of the nose length. In thecomputations, this value is proportioned, or weighted, according to noselength divided by the total model length in oider to obtain an averagevalue of 0 for the model.

14

ALENN ALENN

r j ocl I r .5cal

tan 0 .= .5 ALENNP

r = x tan v P.= 102.3

Code I - Cones Code 3 - Power se-ies •a•dParabol]ics

"ro;2 ALENN2 + )2

rr - Yoa

(Ol Yo= o 6r.

0

Coe2 - 2agn adScn

00SOgives

.Inscribed Cone

(,xo,0)

Figure 9. d•ose Radius Computation Aethods

15

'6. DCMDA AND DANDA

The supersonic potential flow term requires the determination, ofvalues for the derivatives (dC /da) = 0 and (dC N/da) = 0 for theparticular model being considered. a If these valugs are availablefrom experimental data or other sources, they may be read into the computerprogram on the input data cards. If they are not read in as data, twofunctions will be, used to approximate thenm.

The approximation.of Grimminger, Williams, and Young (Reference 14)ýfor ogives and cones and afterbodies, (Figure 10) is used as a starting,point for these functions. (For purposes of these functions, power seriesand parabolic noses have been grouped together with ogives.) These curvesare further approximated by straight line segments, with corrections forMach number. Through logic switching, the result produced is an approximatefunction subroutine. The application is limited to cone half-length(inscribedcone for noses not true cones) from five to fifteen degrees.Comparisons with other data (Reference 2) indicate reasonable agreement.Since potential force is a minor part of the total normal force in theregion-of interest, the approximation is adequate.

The dC /dc is computed by multiplying the dCN/da by the moment arm,measured fr&m the model center of pressure to the model base. Accuracy ofthe results has suffered somewhat since .no particular effort has been madeto accurately predict the moment of the center of pressure for varyingmodel dimensions or flight conditions. The inclusion of these factors wouldsignificantly compl~icate the method of this study. When additional experi-mental data are available, further study of these refinements would be inorder.

0. - CONICAL NOSES

S...... OCIVAL NOSES0.06 -G•' OT" NOM{

S30 MACH

0 hum#"

~ ).0

0 _ 0,04 . .

rr 0.01 -- 10 0 4 6 1 0 It 14

AFTCASOY fINME(SS NATIOA/d,

Fig~ure 10. Initial Normal Force Slope (From Reference 14)

16

HVI~rON IV

PROGRAM DESCkI P!TION

The computer program resultirng from this study is written for theiIBM360 Model 65 as installd in the Louisiana State Universi-ty"ComputerResearch Center. The WATFIV compiler, which gives complete syntax and'logic analysis with explicit error descriptions, was used to,- insuue-anaccurate and rapid transition from one phase of the program deveiopment'to the next. Some of the specfal capabilities of this load-and-go compilerwere utilized; consequently, changes, in the program might be required ifanother type of computer was used.

1. PROGRAM INFORMATION

The computer program information is provided by a series -of commentcards which define the variables, c6des, and terms used. in the program(Table I).

2. MAIN PROGRAM

The main program controls the overall operation of' the computer jirogrIm,(Table II). This program provides a means of data input, terminatesexecution, writes output headings (which include input data), and exerciseslogic to differentiate between subsonic and supersonic computation techniques.

The input data for each case are read from a single card. They includea nose-type code, the nose length and model length (both in calibers), thefreestream Mach number, the Reynolds number per caliber divided by the Machnumber, and the constants (dC /da) = 0 and (dC /da) = 0 if experimentalor other source data are to bu useg. The progr& tekmination is based ona nose-type code of zero, which is supplied by a blank card at the end ofthe data stack. The next statements call the proper subroutine for the casebeing considered, and then returns to the input step for either the nextcase or termination.

3. SUBSONIC SUBROUTINE

The subroutine that directs the computation for the subsonic case isshown in Table III. It computes constants, initializes incremental variables,increments the angle of attack, computes the first (potential) term, callsanother subroutine for the second (viscous) term, and prints out the com-puted values. All dimensional constants are computed in calibers. Someconstants are computed through the use of additional subprograms, which willbe described.

17

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20

4. SUPERSONIC SUBROUTINES

The subroutine that directs the computation for the supersonic caseis shown in Table IV. It differs from the subsonic subroutine in thecomputation of the first (potential) term. Italso provides values for(dCd Nda) 0 and '(dC /da)a = 0 when they have not been supplied as datain theýa main prograM. a

5. SECOND TERM INTEGRAL

The subroutine that computes the integral portion of the second(viscous) terms of the normal force and moment -coefficients i,s shown inTable Y. It initializes incremented variables, sets the increment fortrapezoid integration for the nose portion, and sets the initial value ofx as one-half an increment ahead of the nose. After incrementing x, itcomputes the.-radius, selects the drag coefficient, corrects it for transienteffect and finite length, then computes the incremental value. When thevalue of x reaches the last increment on the nose portion, it is increaseda half-increment to equal the nose length, the increment size is changed,x is reset one-half increment forward of the beginning of the afterbody,and the integration is continued. For the afterbody portion, the radiusremains constant and is defined outside the integration loop.

6. CROSS-FLOW DRAG COEFFICIENT

The function that computes the drag coefficient for the integrationsubroutine is shown in Table VI. Using the cross-flow Mach number, thefunction selects one of eight equations. Then, if the cross-flow Machnumber is below 0.55, the free-stream Reynolds number is Computed. If itis in the transition region, the drag coefficient is adjusted by one oftwo functions.

7. TRANSIENT EFFECTS

The function that corrects the cross-flow drag coefficient for thetransient effect is shown in Table VII. If the cross-flow-Mach number isgreater than one, no correction is made. Otherwise, the time parameter iscalculated, and one of six functions is selected to compute the correctionfactor.

8. RADIUS

The function that computes the radius at any point on the model noseis shown in Table VIII. The proper nose type is determined from the nose-type code, then the appropriate function is selected. For the power seriesand parabolic noses, a simple relation is used to generate an exponent, andthe radius is computed. The ogives are represented by the function fortangent ogives. Radii for cones are computed from the tangent of the nosehalf-angle.

21

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TABLE VIII. RADIUS AT ANY GIVEN X

FUNCTION RADIUS (NoXtALIENN)

IF (N,•O.I) GO TO 2IF (N*,E.O2) GO TO I

P=ALOG( .5)/ALOG(ALENN)RAADIUS=X**P GO TO 3

1 OR=ALENN**2+÷25Y=S(jRT( UR**2-( X-ALENN)'**2)

RADIUS=Y-OR+.5 GO TO 3

2' RADIUS=(°5/ALENN)*X3 RETURN END

9. THETA

The theta subroutine that computes the average change in radius withrespect to axial length is shown in Table IX. First, cones are considered,and the nose angle is computed directly. Next, incremented variables areinitialized, and the increment is computed. Using the radius function, theradius is computed at each increment and compared to that of the previousiicrefent. These values are summed to obtain an average nose angle. Finally,the nose angle is proportioned to the total model length to get an averageangle for the model.

TABLE IX. THETA FOR THE NOSE MODEL BEING CONSIDERED

SUEBROUTINE THETA (N*ALLNNALENMTHEN*THEM)

IF (N.EO.1.) GO TO

THEN =0. X=04 DELX=ALENN/10, RO=00

I X=X+DELXRN=RADIUS (N, X9,ALENN)

RC=RN- RUTHE T=ATAN(RC/DELX)THEN =THEN ÷.1*THET RO=RN

IF (XLTo(ALtNN-DELX/2,)) GO TO 1 0O 32 THEN =ATAN(*5/ALENN)3 THEM =IHEN *ALLNN/ALENM4 RETURN END

26

10. DRAG CORRECTION FOR FINITE LENGTH

The function that corrects the two-dimensional drag ccefficient forfinite length is shown in Table X. The length from the nearest end iscomputed first, then, thp cross-flow Mach number and the square root ofthe end length are used to conipute a correction. If the point beingconsidered is on the nose, another adjustment is made.

TABLE X. DRAG COEFFICIENT' CORRECTION FOR FINITE LENGTH

FUNCTION CoM11 (XALENM*rHENAM•.,A.oAIXNN)FL=X IF (XGTALr-NlA/2.) FL= ALt:NM - X

AMC=AM*SIN.(A) ARGQXP=(AMC+I,)*SOR1CCEL)

IF (XoLT*ALLNN) UCORR = DC.)RR*COS(THLN).RETURN END

11. DCNDA

The DCNDA function that computes the initial slope of the normalforce coefficient when plotted against the angle of attack is shown inTable XI. After the afterbody length is determined, a branch to thecorrect Mach number is taken. For each Mach number branch, a function isselected to correspond to nose, type and afterbody length. Finally, .ihevalue is converted from ACN per degree to ACN per radian.

12. DCMDA

The DCMDA function that approximates the initial slope of the momentcoefficient when plotted against angle of attack is shown in Table XII.It computes the length of the moment arm from a function which describesthe movement of the center of pressure versus Mach number. This arm isthen applied to the DCNDA supplied from the argument list to obtain DCMDA.

27

TABLE XI. DCNDArG COMPUTE DCNOA, IF NOT SUPPLIED*

rHfETA-SUC-V FROM 5 TO 15 DEGREES

FUNCTIO-N CNA' (N*ALFNN9ALENM,AM)ALENA=ALENM-~ALfaNNIF (AM,.LTod.*5) GO TO 10IF (AMeLT93.5) GO TO 6IF '(AMj.LT.4*C5) GO TO 2IF (ALENA*GT*49) GO TO ICNA.Z- e032,+ 9003*ALr-N 'A - . U2*(Ahl-5e) GO TO 14

I CNA=*043 G O12 IF (ALIGNAe,-r.4.5) GO rc 4

IF (No.GTeI) GO TO 31CNA= .03J + .004*ALt~NA D 04*(AM-4e) GC TO 14

3 CNA= .038 + *003*ALtENA -*Ob*(AM-4*) Go rO 144 IF (ALt-iNAeGT98*0) rGO TO 5CNA= .052 + &001*ALENA - *04*(AM-4e) GU TU 14

3 CNA=.0q7 GC~ TO 146 IF (ALt-NA.'.ýT*3*5) GO TO 8

IF (N.GjT.1) GO TO 7CNA= .033 + #00,6*ALFýNA-- e03*(AM-3*) GC TO 147 CNA= .043 + .0cJ4*ALL.NA - 94(M3)GOTO1

6 IP (ALtiNAeG1S.60) GO TO 9CNA= *C!.,6 + *001*ALLNA - 203*(AM-3.) GO TO 14

9 CNA=*049 GO TU 14410 IF (ALENAo.UT.2.5) GO TO 12

IF- (N.GTr.1) GO TO 11CNA= .0.33 + *010*ALENA - *03*(AM-2e) GO TO 14.

.11 CNA= -e0Q4.5 *0.O5*ALIENA' - e04*(AM-i-2e) GO TO 1412 IF (ALffNA&.j~e..) GO'TO 13

CNA= .053 + e602*ALENA - 08*(AM-2o) GO TO 1413 CNA=,C4514 CNA=CNA*l1:0&/3*14I59c RETURN END

TABLE XII. DCMDA

TO COMPUTE DCMDA, IF NOT SUPPLIED

F-UNCTION CMA (DCNDA,ALENM,AM)ARM=( .5-.O'33*AM)*ALENMCMA=ZCCNDA*ARM RFTURN END

28

13. OUTPUT

The output format is shown in Tables XIII and XIVo. The valuesprinted at zero angle, of attack are to remind the reader that the proceduresused are all based on simple bodies of revolution which create only drag(no nortnal force or moment) when the angle of attack is zero. The dragcoefficient printed is that of the last increment of integration on themudel afterbody, without corrections for transient effect and finite lehigtheffect.,

29

i

T1BLE XIII. OUTPUT (LEFT HALF)

RUN OF NUSE CODE I MODEL, WITH A 2.83 CALIBER NOSEiAND AN OVERALL LENGTH OF d.8.3 CALIBERS. MACH NUMBER IS '6.84

AND RE(PER CAL) OVt:k MACH NO. IS 22700o

ALPHA LSUBN CSUBM FSTTMN

DCNDA IS 2.46, AND DCMDA IS 8.43

0.00 0.00 0.00

5.00 ,0.24 Lij. 81 0a2

1 0,00 Oo85 3, 11 0,042

lso 5,I0143 5'922 0962

2 0' 0 0 2,,14 7*84 0.7§

25.00 2.97 10.93 0,94

3 0.00 3.92 14.44 1.07

35.000 4.,92 18,17 I.16

40.00 5.95 22.00 1.21

45s00 b.97 25.82 1.23

50.00 7.95 29.51 1.21,

5b. 00 •87 32.96 i.16

60.00 9.69 36.07 1.07

65.00 10.39 38.73 0994

70.00 10.95 40.88 0979

75900 11*35 42.44 0.62

80.00 11.58 43.37 0.42

85.00 11.63 43.63 0.21

90.00 11.50 43.23 0.00

30

TABLE XIV. OUTPUT (RIGHT HALF)

SECTMN FSTTMM SECTMM CD

0902 0.73 0008 1.59

0.43 1044 1.67 1.59

0.81 2.11 3.11, 1.32

1.35 2o71 5o13 1*.4

2.03 3.23 7.70 1.22

2.85 3.65 10o79 1.22

3.76 3.96 14.21 1.22

4.74 40.15 17.85 1.22

5."74 4.21 2-1.61 1.-e22

6.74 •.15 25.36 1.22

7.71 3.96 29.00 1.22

r862 3.65 32.42 1.22

9i45 3.23 35*50 1.22

10.16 2.71 38.17 1*22

1:0.73 2.11 40.33 1.22

1.16 1.44 41.92 1.22

11.42 0.73 42.90 1o22

11050 0000 43.23 1.22

31

SECTION VRESULTS

The computer program was executed more than 40 times, with over 25runs producing useable data. Many paired runs were made to observe theeffect of changing a single parameter'or constant. Near the final stagesof the program development, a separate regression program was U... 4 to fitone set of experimental data. A function of the form (constant, 5si'n acos a + (constant) sin3a/cos a was used for this regression program,since it represents the alpha-dependent quantities of the theory. Thehigh degree, of conformity indicated the method was of the proper form.

Three sources were used for comparisons with the output of the program.Almost no data for the desired angle-of-attack range are available.Additional data are required for angle of attack up to 90 degrees, parti-cularly at transonic speeds.

The moment coefficient will not be discussed in detail, since thenormal force is calculated for incremental areas, and if it is accurateand no couples exist, the moment must also be accurate. This can beobserved from comparisons with the moment data in References 15 and 16,where deviations in normal force coefficient are reflected almost directlyin the moment coefficient.

1. SUBSONIC

The subsonic runs were compared with Hauer and Kelly's data in Reference,15 which reports primarily on the Magnus effect of spinning projectilemodels, but also presents data for a zero spin rate. As shown in Figure 11,the computed values of the coefficients generally take the same shape asthe experimental data up to about 25 degrees. From there to about 55degrees , the data show, a marked divergence. Hauer and Kelly describe aregion of flow unsteadiness at 55 degrees which is attributed to a rapidtransition between an axial and a transverse flow pattern. Above thisregion of unsteady flow, the comparison shows another divergence, but ata different degree of error.

Much of the error in the comparison seems to come from the values ofthe cross-flow drag coefficient used since the computations gave excellentcomparisons to 55 degrees and then severe overpredi'c('•ons to90 degrees, ifa laminar value were used. The difficulty lies in the uncertainty involvedin predicting the transition from laminar to turbulent flow, and the rateof recovery of the drag coefficient above the transition point. Reference11 presents several different cases which could apply, but the selectionof the proper one would"probably be based on prior knowledge of theexperimental data. Transition for a cylinder in cross-flow usually occursat a diameter-based Reynolds number Pf 500,000, but this may vary by asmuch as 200,000. If the test case lies within this region of Reynoldsnumbers, the actual transition Reynolds number would have to be known topermit selection of the proper cross-flow drag coefficient.

32

* Experimental 3 Computed

• ; o, ... . . .t- - ----- --- ----

4 _•L ... Lt L•,_

2,

j r

F 10

15

10

Cone: .00 Nse, 700,Modl,.Mah.0.2

Fgr m1 C to E i Df

.j I

0~0

0 20 460 80 300

3 ,33

*Experimental 13 Computed

2

---L-- ---

4

2

aOgive: 4.00 Nose, 7.00Model, Mach = 0.26

Figure 11. (Concluded)

34

2. SUPERSONIC

The supersonic runs (Figure-12) were made with an input value for,DCNDA. The slope used for input either was, taken from the-curve inReference 14, or was computed from' the experimental data between zero andfive degrees angle of attack. This computatioh can be used because thevarious runs show that potential effects account for 95 percent or moreof ,the total normal force coefficient in thi'sregion.

The runs show excellent agreemeft-with. the data presented by Penland'in Reference 13. The agreement in, the hypersonic region is excellent,since the flow approaches the-conditiOns that satisfy the Newtonifn theory.The agreement is slightly better than that produced by the Modified,Newtonian theory. The data used were in thel form of lift--and drag coeffi-cients referenced to planform area, but a short program on desk-tqp,computer converted the data to normal force coefficient referenced to basearea. The error introduced by these computations is not significant forthe purpbse 6f comparison.

The comparisons at Mach 2.37, 2.98, and 3.90 require additional con-sideration. In an effort to determine nj the 90 degree angle-of-attackruns with 6ylindrical noses were used to-ascertain the effects of,.Machnumber. When the experimental data were converted 60 useable -Form, thedrag coefficient exceeded the value for an infinite cylinder. Such acondition is contrary to all previous knowledge of this ýtype of flow, andan error is indicated. Since the test runs Were-made with a wall-mountedsting, the corrections for the mounting condition'may contain some error.However, the data are useful for plotting ,the shape of the curve and wereadequate for the original rcuort. The wall mounting limited the datastart to 30 degrees and prevented a check on the test apparatus, at zeroangle of attack. Therefore, the source-of the apparent error-cannot beevaluated.

3. INDICATED IMPROVEMENTS

As the program developed, several areas were indicated where'anincrease in complexity might improve the results; however, the return, from,the additional effort would be low. Also, the lack of experimental data,for cporrelation with the theoretical results precluded:any real value inan adempt to improve the accuracy'of the method. If valid data wereavailable and if greater accuracy were required, some;improvement in theresults would be possible.

The best candidate area for improving accuracy would, be the dragcoefficient in the region of transition Reynolds numbers. A method ofreading into the computer the transition Reynolds number, basefl on surfaceroughness and other pertinent factors, could be used to adjust thetransition induced dip in the Cd versus Reynolds number curve. Certainly,

c

35

* Experimental o Computed

, L

1710

-- - -- -- -I

5

-- -- -30 20 4. 60 8010

Ii .

1- --- .----

20.

-- ---

10 "- -F i

20 40 60 80 100

Ia

Ogive: 5.00 Nose, 7.00 Model, Mac-h =2.37

Figu re 12. Comparisons to Experimental Data - Supersonic (Reference 16)36

* Experimental o Computed,

0

0 20 0 60 0 10

30

20

0 -

0 20 4060810

Ogive: 5.00 Nose, 7.00 Model, Mach =2.98

Figure 12. (Continued)

37

*Expearimental 0 Computed

15

10 il 1 I i

0 2 8010aO

gve 5.0 Noe 7.0MdA ah=39

Figure~ 12 Cotnud

if38

*Experimental 0 Computed

+,Modified Newtonian Theory

10

CN

5

0

60 4 0 0

I ------ --

40

Cone: 2.83 Nose, 8.83 Model, Mach =6.83

Figure 12. (Continued) (Reference 17)

39

* Experimental o Computed

* Modified Newtonian Theor

10

JI-

CN

5

-Wj-

0 -0 20 40 60 80 100

a

60 L L L

- ----- -----. E .

404

--- - -- - - --- ----- ------

-------- -------7u -J -------

20 40 60 80 100

Cone: 1.87 Nose, 7.87 Model, Mach =6.83

Figure 12. (Concluded)

40

additional experimental data 'in the subsonic range would be required fora significant improvement in this function. More attention to the changefrom axial to' transverse flow dominance could lead to a factor to defineits manner and causes. Since both the axial and the transverse flowsare involved in transition, and gince transition is, at best, not a verypredictable phenomenon, a reliable prediction may not be possible.

There-are other improvement areas which would give lesser benefits inthe improvement of accuracy, and they are mentioned in the order in whichthey appear in the program.

The volume in the subsonic subroutine assumes an inscribed cone fornoses which are not true cones. The radius function could be expanded,or an additional function could be added, to provide a value of volumethat would be more accurate for non-conical noses. In particular, thismodification would yield better results for the blunter class of powerseries and parabolic noses.

Since the DCNDA function is approximate, it could easily be improved

with additional experimental data. The insertion of experimental dataon the input data cards would be the easiest method of correcting thissource of error.

The theta subroutine is an averaging process, and it is necessary thattheta be averaged for the apparent mass factor. However, when using thenose angle to correct n, the local value of theta would be more accuratefor noses• which are not true cones.

All the functions that are approximated with line segments could befitted with higher order curves that would be more accurate.

Any of the factors mentioned might produce some refinement in theaccuracy of the predicted aerodynamic characteristics; however, the addedcomplexity is not justified at this time, since experimental data foractual verification are not available. In addition, some of the recommend-ed techniques require experimental data for their implementation. Further,it is a primary goal of this study to retain a minimum of complexity.Additional study should be performed to improve the technique, but withoutsignificantly increasing the level of complexity.

41

SECTION VI

FUTURE DEVELOPMENT

The most significant improvement that might be made to the programwould be to include more complex models. The first additions might be:nose blunting and boattailing of the afterbody, since each of these couldbe quite readily incorporated in the present program with only minormodifications. The addition of fins, wings, control surfaces, modelirregularities, and changes in shape to include other than bodies ofrevolution are some of the extensions that would be desirable. Theseadditions require extensive effort for theoretical development, as wellas large amount of experimental data; however, the results would be mostvaluable to future designers and analysts.

The requirements of the Air Force, the intended use of the program,and the success of early advances in complexity would determine how muchextension should be undertaken. At this time,, the program developmentconfirms the basic theories involved and verifies the value of the com-puter as • basic research tool in this area.

42

REFERENCES

1. Munk, Max M., The Aerodynamic Forces on Airship Hulls, NACA RepNo. 184, 19i24.

2. Kelly, Howard R., The Estimation of Normal Force and Pitching MomentCoefficients for-BluntTBased-Bodies of Revolution at'Large Angle ofAttack, NOTS TM-998, May 27, 1953

3. Kelly, HowardR., The Estimation of Normal Forces and Pitching, MomentCoefficients for Blunt-Based Bodies of Revolution atLarge Angles ofAttack, Journal of the Aeronautical Sciences, Vol. 1, No. 1,A-gust 1954, pp. 549z555, 565.

4. Allen, H. Julian, and Edward W. Perkins, A Study of Effects ofViscosity on Flow Over Slender Inclined Bodies of Revolution,NACA Rep. No. 1048, 1951.

5. Ward, G. N., Supersonic Flow Past Slender Pointed Bodies, QuarterlyJournal of Mechanics and Applied Mathematics, Vol. 2, Part I,•pp. 75-97,, March 1949.

L6. Goldstein, S., Modern Developments in Fluid Dynamics, Oxford, 1938.

7. Schwabe, M., Pressure Distribution in Nonuniforrm Two-DimensionalFlow, NACA-TM-1039, 1943.

8. Hill, J. A. F., Forces on Slender Bodies at Angles of Attack, NavalSupersonic Laboratory, Massachusetts Institute of Technology,R-a 100-59, May 9, 1950.

9. Upson, Ralph H., and W. A. Klikoff, Application of PracticalHydrodynamics to Airship Design, NACA Rep. No. 405, 1931.

10. Van Dyke, Milton D., First- and Second-Order Theory of SupersonicFlow Past Bodies of Revolution, Journal of the Aeronautical Sciences,Vol. 18, No. 3, p. 161-178, 216, March 1961.

11. Hoerner, Sighard F., Fluid Dynamic Drag, Published by the author, 1958

12. Gowen, Forrest E. and Edward W. Perki~ns, Drag of Circular Cylindersfor a Wide Range of Reynolds Numbers and Mach Numbers, NACA TN-2960,

_953.

13. Penland, Jim A., Aerodynamic Characteristics of a Circular Cylinderat Mach Number 6.86 and Angles of Attack up to 900, NACA TN-3861,1957. (Supercedes NACA RM L54AI4.)

43

REFERENCES

1. Munk, Max M., The Aerodynamic Forces on Airship Hull's, NACA RepNo. 184', 1924-.

2. Kelly, Howard R., The Estimation of Normal Force and Pitching MomentCoefficients for Blunt-Based Bodie. of Revolution at Large Angle ofAttack, NOTS TM-998, May 27, 1953

3. Kelly, Howard R.,,The Estimation of Normal Forces and PitchingMomentCoefficients for Blkjnt-ýBased Bodies of Revolution at Large Angles ofAttack, Journal of the Aeronautical Sciences, Vol. 1, No. 1,Aug!ist 1954, pp. 549-555, 565.

4. Allen, H. Julianr. and Edward W. Perkins,.A Study of Effects ofViscosity on Flow Over Slender InclIhed Bodies of"Revolution,NACA Rep. No. 1048, 1951.

5. Ward. G. N., Supersonic Flow Past Slender Pointed Bodies, QuarterlyJournal' of Mechanics and Applied Mathematics, Vol, 2, Part I,pp. 75-97,, March 1949.

6. Goldstein, S., Modern Developments in Fluid'Dynamics, Oxford, 1938.

7. Schwabe, M., Pressure Distribution in Nonuniform Two-DimensionalFlow, NACA-TM-1039, 1943.

8. Hill, J.A. F., Forces on Slender Bodies at Angles of Attack, NavalSupersonic Laborýatory, Massachusetts Institute of Technology,

,R-a 100-59, May 9, 1950.

9. Upson, Ralph H., and W. A. Klikoff, Application of PracticalHydrodynamics t6 Airship Design, NACA Rep. No. 405, 1931.

lu. Van Dyke, Milton D., First- and Fecond-Order-.Theory of SupersonicFlow ,Past Bodies of Revolution, Journal of the Aeronautical Sciences,Vol. 18,'No. 3, p. 161-178, 216, March 1961.

11. Hoerner, Sighard F., Fluid Dynamic Drag, Published by the author, 1958

12. Gowen, Forrest E. and Edward W. Perkins, Drag of Circular Cylindersfor a ,Wide Range of Reynolds Numbers and Mach Numbers, NACA TN-2960,1953.

13. Penland, Jim A., Aerodynamic Characteristics of a Circular Cylinderat Mach Number 6.86 and Angles of Attack up to 900, NACA TN-3861.1957. (Supercedes NACA RM L54A]4.)

43

.4. Grimminger, G., 'E. P. Williams, and G. B. W. Young, Lift on InclinedBodies-oftRevolution inHypersohic "Flow, Journal of AeronauticalSciences, Vol-, 17, No. Ui, pp. 675-690, November 1950.

15, Hauer, Herbert .. , and-,oward R. Kelly, The Subsonic AerodynamicCharacteristics o'f Spinning Cone-Cylinders--.', Ogive-CYlinders atLarue ngles of .Attack, NAVORD Report 3529, 7T..

16. Smith, Fred M., A Wind Tunfhel Investigation of the AerodynamicCharoacteristics of Bodies of Revolution at Mach Numbers .of 2_37,22ý98 6nd 3.90 at Angles of Attack to 90°, NASA TM X-311, 1960.

17. Penland, Jim A., Aerodynamic Force Characteristics of aSeries dfLifting Cone and Cone-C -linder Configurations at a Mach Number of6.83 and Angies.of Attack up-to 130°, NASA TN D-840, 1961.

44

APPENDIX

The following 'block diagrams are intended to $hbw ,the various +relation-ships of the different parts of the programtoa e`ach other and'to t06 MainProgram. Only those, subprograms which uti:l'ze another subprogram areincluded. These diagramns are generalized by the variouS'+parts of the sub-program being considered and should not be construed to be flow charts.The equations necessary to write the flow charts have alrieady been :presented.

Main Program

Sub o Supson

Theta,-lRadius Jecti CNA CM

Radius Theta CDRM CDCI DCORR

o Data Input O

Run Control W.-Stop

Write Headings

Subson W Mach Number Do- Supson

Figure I-1. Main Program

45

-- Evaluate constants-.* wTheta

Evaluate first terms Radius

Evaluate second terms - Seci.t

Add terms

Write resultz

Alpha control

Return

Figure 1-2. Subson Subprogram

46

CNA- qI.-- Evaluate constants--o Theta

CMA *SWrite DCNDA, DCMDA Radius

Evaluate first terms

Evaluate second terms-a--- sSecti

Add terms

Write results

Alpha control

Return

Figure 1-3. Supson Subprogram

47

Initialize variables

IniJement X

Radius-No valuate Jari ab es '~CD,

CDC Evlaeincrement, crSum increments

Nose length control

Reset X and Delx

CDC -r EValuate riabes -owCD

Evaluate increment Dc6rr

Sum increments

Model length control

Return

Figure 1-4. Secti Subprogram

48