Equations and charts for the evaluation of the hypersonic ...AEDC-TDR-64-25, " .r ABSTRACT The pressure distribution predicted by the modified Newtonian theory is used to develop equations

TECHNICAL DOCUMENTARY REPORT NO. AEDC-TDR-64-25

\

JUt 0) (I

'" rJ 1981

L PET CE TE

AFSC Program Element 65402034

March 1964

E. L. Clark and L. L. Trimmer

yon Karman Gas Dynamics FacilityARO', Inc.

By

(Prepar:ed under Contract No. AF 40(600}.1000 by ARO, Inc.,contract operator of AEDC, Arnold Air Force Station, Tenn.)

'AIR F RC 5 C

U ITED STATES AIR F RCE

o 64 25

EQUAT S CHARTSFOR THE EVALUATION OF THE

HYPERSONIC AERODYNAMIC CHARACTERISTIC.SOF LIFTING CONFIGURATIONSBY THE NE TONIANTHEORY

ARN LD E I EE I

·...,.-

AI' • AEDCArnold AP'S Tenn

A EOC- TO R-64-25

EQUATIONS AND CHARTS

FOR THE EVALUATION OF THE

HYPERSONIC AERODYNAMIC CHARACTERISTICS

OF LIFTING CONFIGURATIONS

BY THE NEWTONIAN THEORY

By

E. L. Clark and L. L. Trimmer, ,

von Karman Gas Dynamics Facility

ARO, Inc.

a subsidiary of Sverdrup and Parcel" Inc.

March 1964

ARO Project No. VT8002

AEDC-TDR-64-25

FOREWORD

The authors wish to express their appreciation toMrs. P. Trenchi for her help with the prepa!ration ofthis report.

AE DC- TDR-64-25

, " ­.r

ABSTRACT

The pressure distribution predicted by the modified Newtoniantheory is used to develop equations for the aerodynamic forces,moments, and stability derivatives for components of hypersonic lift­ing configurations. In conjunction with the equations, a set of chartsis presented to enable simple determination of the aerodynamic char­acteristics of swept cylinders, swept wedges, spherical segments, andcone frustums at zero sideslip and angles of attack from 0 to 180 deg.This method allows evaluation of most delta wing-body combinationswithout the need for numerical or graphical integration. As an exampleof the procedure" the theoretical characteristics of a blunt, 75-degswept delta wing are calculated and compared with experimental results.

I -

PUBLICATION REVIEW

This report has been reviewed and publication is approved.

/!-\ AJ /\,~ #J?1c:(/~

Jean A. Jack{/Colonel, USAFDeS/Test

Darreld K. CalkinsMajor, USAFAF Representative, VKFDCS/Test

v

AEDC-TDR-64-25

CONTENTS

1. 02. 0

ABSTRACT ...NOMENCLATURE ....INTRODUCTION. . • • .DEVELOPMENT OF EQUATIONS

2. 1 Delta-Wing Components . .2. 2 Body Components . . . . . . . . . . • .2. 3 Composite Configurations

REFERENCES . . . . . . . . . .APPENDIX - Application of Method to a Typical

Delta Wing . . . . . . . . . . . • . . . .

vix1

5223536

39

2. Aerodynamic Characteristics of Spherical-Wedge Nosesa. Normal Force. . . .b. Axial Force. • . . • . . . • • • .c. Side- Force Derivative, f3 = 0 • • • • • •

ILLUSTRATIONS

--11 • .-

Figure

1.

3.

4.

Axis and Coefficient Nomenclature

Aerodynamic Characteristics of Flat-ToppedSpherical-Wedge Noses

a. Normal Force. . . .b. Axial Force. . . . .c. Side-Force Derivative, f3 = 0

Aerodynamic Characteristics of Swept-Cylinder LeadingE dge s (ep' = 1T / 2 )

a. Normal Force. . . •b. Axial Force. . . . .c. Side-Force Derivative, f3 = 0 • • • • •••••

d. Rolling-Moment Derivative, f3 = 0 •

41

424344

454647

48495051

.. -~I-

5. Aerodynamic Characteristics of Flat-ToppedSwept-Cylinder Leading Edges (ep' = rr/2)

a. Normal Force. . . . .b. Axial Force. . . . . . . . .c. Side-Force Derivative, f3 = 0 • •

d. Rolling-Moment Derivative, f3 = 0 •

vii

52535455

A E DC- TO R-64-25

Figure

6. Aerodynamic Characteristics of Spherical Segmentsa. Normal Force" 0 = 0 to 45 deg. . . • .b. Normal Force" 0 = 45 to 70 deg . . • • • . .c. Axial Force . . • . . . . . . • . . . •d. Side-Force Derivative (f3 = 0), 0 = 0 to

45 deg. . . . . • . . . . • • . . • •e. Side-Force Derivative (f3 = 0), 0 = 45 to

70 deg. . . . . . . . . . . • • .

7. Aerodynamic Characteristics of Flat-ToppedSpherical Segments

a. Normal Force" 0 = 0 to 45 deg. • .b. Normal Force" 0 = 45 to 70 degc. Axial Force. .••...d. Side-Force Derivative (f3 = O), 0 = 0 to

45 deg. . • • . . • . . . . • .

Page

56 '-

5758

-'I.,

59

60

616263

64e. Side-Force Derivative (f3 = 0), 0'= 45 to

e.

8.

9.

10.

11.

70 deg. . • • • . . • . . • . .

Aerodynamic Char~cteristicsof Cone Frustumsa. Normal Force . . . . • • . .b. Axial Force" 0 = 5 to 25 deg. . . • .c. Axial Force" 0 = 25 to 40 degd. Side-Force Derivative, f3 = o.

Aerodynamic Characteristics of Flat-Topped ConeFrustums

a. Normal Force . • • . . . . .b. Axial Force . . • . . • . . •c. Side-Force Derivative" f3 = o. . . . . . .

Details of 75-deg Swept Delta Wjng. • • .

Aerodynamic Characteristics of a 75-deg Delta Winga. Normal Force . • . • . • . • • • •b. Axial Force . • • • . • . . • • • • • .c. Pitching Moment, Referenced to O. 6 Lnd. Lift... . . . . . . . .

Drag . • • • . • . • • • . •f. Lift-to-Drag Ratio . . • . • • • .g. Side - Force Derivative" f3 = o. •h. Yawing-Moment Derivative" f3 0,

Referenced to O. 6 Ln • • •i. Rolling-Moment Derivative, f3 = 0 • • • •

viii

65

66676869

707172

73

74757677787980

8182

AEDC-TDR-64-25

NOMENCLATURE

0,

0,

0,

Body surface area

Base area of swe'pt-wedge wing half

Planform area of swept-wedge wing half

Side area of swept-wedge wing half

Half-span of swept-wedge wing

Axial-force coefficient, FA /qoo S

Drag coefficient, drag/qoo S

Lift coefficient, lift/qoo S

Rolling-moment coefficient, Mx/qoo S,t

Rolling-moment coefficient derivative, aC,t / af3 at f3l/radian

Pitching-moment coefficient, My Iqoo S1,

Normal-force coefficient, FN /qoo S

Yawing-moment coefficient, Mz I qoo S 1,

Yawing-moment coefficient derivative, iJ Cn I af3 at f3l/radian

Pressure coefficient, (p - Poo) I qoo

Pressure coefficient at stagnation point

Pressure coefficient at nose of pointed body

Side-force coefficient, Fy Iqoo S

Side-force coefficient derivative, aCy I af3 at f3l/radian

b

A

CPnos e

Gy

:

.. -.,.

c

F

1, I, k

Chord of swept-wedge wing

Function defining body surface

Axial force

Normal force

Side force

Vertical displacement of swept-wedge wing half fromwing centerline

Unit vectors directed along the X-, y -, and Z-axes:lrespectively

ix

A EDC- T DR-64-25

K

L

Ln

LID

1.

Mx

My

Mz

(n,x),(n,y),

tn, z)

p

R

r

st

X,Y,Z

XT,ZT

x, y, Z

a

a

Proportionality constant used in the modified Newtoniantheory

Body length

. Delta-wing length~ measured from theoretical apex

Lift-to-drag ratio

Moment coefficient reference length

Rolling moment

Pitching moment

Yawing moment

Free-stream Mach number

Angles between unit normal vector ~ n, and the pos itiveX-, Y -, and Z-axes~ respectively

Inward directed unit vector normal to the body surface

Surface static pressure

Stagnation pressure behind normal shock

Free-stream static pressure

Free-stream dynamic pressure

Radius of curvature

Base radius of cone frustum

Nose radius of cone frustum

Local body radius on cone frustum

Reference area

Thicknes s of s wept -wedge wing half

Free -stream velocity

Orthogonal body axes

Moment transfer lengths

Coordinates along X-, Y-, and Z-axes

Angle of attack

Angle of attack where epo = - ep~ on swept-cylinder leadingedge

Angle between body X- axis and free -stream velocityvector

x

.-- -

SUBSCRIPTS

L

u

A EDC- T D R-64-25

Angle of sideslip'

Dihedral angle of swept -wedge wing

Ratio of specific heats and wedge angle normal to lead­ing edge of swept-wedge wing

Half-angle of cone frustum and base tangent angle ofspherical segment

Nose half-angle of pointed body

Centerline angle of swept-wedge wing measured inX, Z- plane

Angle between surface unit inner normal vector and free­stream velocity vector

Angular coordinate which defines cross-sectional planes

Angle defining base location of spherical segment

Sweepback angle and basp. angle of spherical wedge

Cone frustum bluntness ratio, Rn/Rb

Angle of body roll measured in Y, Z- plane

Angular coordinate which defin~s circumferential positionin a cross-sectional plane

Angle defining circumferential position where surfacebecomes shielded from the flow

Angle defining circumferential extent of swept-cylinderleading edge

Lower half of swept-wedge wing

Upper half of swept-wedge wing

xi

AEDC-TDR-64-25

1.0 INTRODUCTION

In the design and testing of lifting re-entry configurations, there isoften the need for a simple, approximate method of predicting the pres­sures and forces acting on the vehicle at hypersonic speeds. The New­tonian theory has proven very useful for this purpose. A number ofstudies have shown the accuracy of this simple theory in predicting thepressures and forces on such configurations as sharp and blunted cones(Refs. 1 through 5), circular cylinders (Refs. 6 and 7), hemispheres(Ref. 7), and delta wings (Ref. 8). Although the Newtonian theory iseasily applied to the calculation of pressure distribution, integration ofthe pressure over the body surface to obtain total forces and momentscan be difficult and time consuming. Hence, the theory is not alwaysused to full advantage. Design charts which simplify the evaluation ofbody loads have been developed for complete and partial bodies of revo­lution (Refs. 5 and 9 through 13), elliptic cones (Refs. 5, 14, and 15),delta-wing components (Ref. 13), and three--diInensional bodies (Ref. 16).The charts of Refs. 5, 9, 11, 13, 14, and 15 provide total loads andderivatives for selected bodies, while the methods of Refs. 10, 12, and16 apply to arbitrary bodies but require numerical or graphical integration.

The purpose of the present report is to extend the scope of the pre­vious design charts by providing additional aerodynamic characteristicsand an increased angle-of-attack range. To avoid a requirement ofnumerical integration, only selected configurations consisting of typicaldelta-wing and body components are considered. Equations are derivedfor the pressure distribution on each component, and this distribution isthen integrated over the surface area in closed form to obtain total forcesand moments. Equations and charts are given for the longitudinal sta­bility and performance coefficients (CN, CA, Cm): and the directional andlateral stability derivatives (CYf3' Cnf3 , Cl(3 ) for an angle-of-attack rangeof 0 to 180 deg at zero sideslip.

An example of the use of the charts is given in the appendix, wherethe aerodynamic characteristics of a blunt, 75-deg swept delta wing arecomputed and compared with experimental results.

2.0 DEVELOPMENT OF EQUATIONS

The Newtonian theory has been discussed in a number of references,and only a brief summary will be given here. A thorough analysis of thistheoretical method is given by Hayes and Probstein in Ref. 17.

Manuscript received January 1964.

1

A E DC- T D R-64-25

Newton calculated the force on a body by assuming that the impactof fluid particles was completely inelastic for the normal component ofmomentum and was frictionless. Thus, the normal component of mo­mentum is converted to a pressure force on the body while the tangen­tial component remains unchanged. The analysis based on theseassumptions gives the surface pressure coefficient, Cp , as

At high Mach numbers the disturbed region in front of a body be­comes very limited in extent. The bow shock wave has approximatelythe same inclination as the body and is separated from the body surfaceby a very thin, practically inviscid, shock layer. With this flow geom­etry, the normal momentum of impinging molecules is lost inelasticallyand the tangential component of momentum is conserved. Hence,Newton I s analysis is realistic for this type of flow, and the validity ofthe analysis increases as the shock-layer thickness decreases. .Forthe shock wave to approach the inclination of the body, the gas dynamicequations show that the ratio of the density ahead of the shock to thatbehind the shock must approach zero. The equations further show thatfor the density ratio to approach zero, the Mach number must approachinfinity and the ratio of specific heats must approach unity. If theseNewtonian conditions (Moo -? 00, Y -? 1) are satisfied, the Newtonian pres­sure coefficient, Eq. (1), is identical to that given by the oblique shockrelations for the pressure immediately behind the shock wave.

2

Cp = 2 cos TJ

where TJ is the angle between the free-stream velocity vector and theinward directed unit vector normal to the surface.

(1)

In Newton I s analysis, the impinging molecules leave the body surfacealong an unaccelerated path. However, in the case of a curved body witha thin shock layer, the particles are constrained in the shock layer andmust follow an accelerated path. Therefore, for a correct analysisEq. (1) must be modified to allow for the pressure gradient resulting fromthe centrifugal forces acting on the particles. This correction was firstobtained by Busemann (Ref. 18), and the rational th~ory including thecorrection has been called the Newton-Busemann theory in Ref. 17. How­ever, despite the theoretical correctness of the Newton-Busemann rela­tion, . the simple Newtonian theory has been found to agree much betterwith experimental data (e. g., Ref. 7), and the equations given in the pres­ent paper have not been corrected for centrifugal effects.

Equation (1) has been modified by a number of investigators to pro­vide a better correlation with experimental data for several classes ofbodies. The modified forms of the equation have the general relation,

Cp = K cos 2 7J ( 2)

2

A EDC- TDR-64-25

where K is a multiplicative factor which is ,used to match certain limit­ing conditions. For a flat plate with attached shock (i. e., at low anglesof attack), Love (Ref. 19) suggested that K = y + 1 provides betteragreement with the exact oblique shock solution. For a slender pointedbody with attached shock, best agreement with exact theory is obtainedby using either the simple Newt'onian value of K = 2 or the valuesuggested in Ref. 19 of K = . C2Paose ~ where Doose is the surface angle

SIO nose

at the nose and Cpnose is the exact value of pressure coefficient for thisangle. LJees (Ref. 20) suggested that for a blunt body with detachedshock wave the Newtonian theory could be modified to match conditions

p - pat the stagnation point by letting K = Cp = t

200 , which is closely

max qoo

approximated by K = y + 3/y + 1 for large Mach numbers. In the present

derivations, the modified form of the Newtonian approximation as givenin Eq. (2) will be used with an arbitrary value of K.

where (n, x), (n, y), and (n, z) are the angles between n and the positiveX-, Y-, and Z-axes, respectively, and their cosines are given by

The angle, 11, between the velocity vector V00 and the surface unitinner normal vector n is determined by the scalar product of the twovectors. The velocity vector (Fig. 1) is

(3)

(4a)

(4b)

-I cos (n, x) + j cos (n, y) + k cos (n, z )o

-V00 = - V00 (1 cos a cos f3 + 1 sin f3 + k SIn a cos (3)

cos (n, x)

cos (n, y)

cos (n, z)

where 1, 1, and r are unit vectors directed along the X-, Y-, andZ-axes. The body su'rface may be described by the equationF (x, y, z) o. Then the inward directed unit vector normal to the bodysurface is

Thus,

cos 11 (5)

= - [cos a cos f3 cos (n, x) + sin f3 cos (n, y) + sin a cos f3 cos (n, z )J

3

A E DC- T D R-64-25

The Newtonian theory predicts pressures only on surfaces which facethe flow. For surfaces which are shielded from the flow, it isassumed that the surface pressure is equal to the free-stream staticpressure and Cp = o. Therefore, Eqs. (1) and (2) are applicable onlyfor cos TJ :::. o.

Force and moment coefficient nomenclature utilized in the deriva­tion is shown in Fig. 1. The coefficients are non-dimensionalized byan arbitrary reference area, S, and, in the case of moment coefficients,by an arbitrary reference length, P... The moment reference point ofeach component is given in the corresponding figure. The coefficientsare obtained by integrating the Newtonian pressure distribution over thebody surface area, A, as indicated in the following general equationswhere the moment reference point is at the origin of the axes:

CN = ~ K II 2cos (n, z) dA (6)- cos TJ

qlXl S S A

CA ~K II 2

cos (n, x) dA (7)- - cos TJqlXl S S A

CmMy K [II 2

cos (n, z ) dA - II 2cos (n, x) dAJ (8)-- x cos TJ z cos TJ

qlXl S t St A A

Cy Fy K II 2cos (n, y) dA (9)- cos TJ

qlXl S S A

CnMz K [II 2

cos (n, y ) dA - II 2cos (n, x) dA J (1 0 )- x cos TJ Y cos TJqlXl S t St A A

CtMX K [I I 2

cos (n, z) dA - If 2cos (n, y) dA J (11)- y cos TJ z cos TJ

qlXl S t St A A

In the integration over the surface area, it is assumed that Cp = 0

on all surfaces shielded from the flow and on all flat surfaces (exceptin the case of the swept wedge) because these surfaces are usually con­cealed by other body components. Equations and charts are given forcomponents having vertical symmetry and for the corresponding flat­topped components. For flat-bottomed components, the loads may bedetermined by taking the difference between the loads acting on the com­plete component and those acting on the flat-topped component and addingthe pressure load of the flat lower surface to the normal force andpitching moment.

4

-. -

AEDC-TDR-64-25

2.1 DELTA-WING COMPONENTS

The basic components of a delta wing are the nose" leading edge"and wing. The loads on these components are computed in the fol­lowing sections. The method of combining the components to give acomplete wing is described in Section 2.3.

2.1.1 Spherical-Wedge Nose

The nose of a delta wing is usually a spherical wedge. If the wingcenterline angle f. (see Fig. F) is not zero" the nose is not exactlyspherical" but the error in force coefficients will be negligible when f.

is small. The nomenclature used in the derivation is shown in Fig. A.y

x

x z

Section A-A

Moment Reference Point

Fig. A Spherical-Wedge Nose

The direction cosines of the inward directed unit normal vector" n, asobtained from Eq. (4b) or by analytic geometry are

cos (n, x)

cos (n, y)

cos (n, z)

- cos ¢ cos e

cos ¢ sin e

- sin ¢

5

(12)

AEDC-TDR-64-25

The angle TJ between the free-stream velocity and the normal vector isobtained from Eq. (5) which gives

cos TJ = cos f3 (cos a cos ¢ cos 0 + sin a sin ¢) - sin f3 cos ¢ sin 0

The pressure distribution over the nose is given by Eq. (2) as

2Cp = K cos TJ

(13)

The value of ¢ at which the surface becomes shielded from the flow isdesignated as ¢o and is defined by Cp = 0 or cos TJ = 0, thus

sin f3 sin 0 - cos f3 cos a cos 0 (14)tan ¢o =

cos f3 sin a

and at f3 = 0

,/.. -1 (cosO)'P = - tan --o tan a

The elemental surface area is given by

dA = R2

cos ¢ d ¢ d 0

(14a)

(15)

As was mentioned previbusly, it is assumed that Cp = 0 on the flatbase surfaces even at angles of attack where these surfaces are notshielded from the-flow. All coefficients and derivatives are evalu­ated at f3 = o.

2.1.1.1 Normal-Force Coefficient

The normal-force coefficient is given by Eq. (6):

CN = - -.!L f f cos2

TJ cos (n, z) dAS A

Since the body has lateral symmetry, this equation may be integratedover the left side and the results multiplied by 2. Then, for 0 S. a S. TT,

A TTh

fo

f¢o cos2

TJ si.n ¢ cos ¢ d ¢ d 0(16)

Substituting the value for cos TJ at f3 = 0 from Eq. (13) and performingthe indicated integrations gives the normal-force coefficient as afunction of a and A. Note that the equation must be evaluated by inte­grating first between ¢ = ¢o and ¢ = TT /2 and then integrating the r.esult­ing function between 0 = 0 and 0 = A since ¢a is a function of O.Then,

sin a

2 [COS a sin A (JL + tan -1 ~-.-A) + tan -1 (sin a tan A)J

2 tan a

6

(1 7)

Equation (1 7) was evaluated for a = 0 to 180 deg and Aand the results are presented in Fig. 2a.

2.1.1.2 Axial-Force Coefficient

The axial-force coefficient is given by Eq. (7):

AEDC-TDR-64-25

60 to 90 deg

JJA

COS 2 TJ cos (n, x) dA

Then, for 0 S a S TT,

Integration of Eq. (18) gives

(18)

sKJt2 -} [(sin

2a sin A + 3 cos

2a sin A - cos

2a sin

3A) (~ + tan-

1

+ 2 cos a tan-1

(sin a tan A) + sin a cos a sin A cos A J

cos A )tan a

(19)

"--

The numerical evaluation of Eq. (19) is presented in Fig. 2b.

2.1.1.3 Pitching-Moment Coefficient

Since the force on any element of surface is directed toward thecenter of curvature, the resultant force acts through the center, andthe moment about the reference point is zero.

2.1.1.4 Side-Force Coefficient Derivative

where the upper sign corresponds to the left side of the nose and thelower sign corresponds to the right side. The derivative of the side­force coefficient with respect to (3 is

The side-force coefficient is given by Eq. (9):

Cy = .x Jf cos2

TJ cos (n, y ) dAS A

For either side of the nose, at 0 S a S TT,

Cy = ±±A TT/2

K R2 2 2J J cos TJ cos e:p sin e d e:p d (jS 0 e:po <f3 )

(20)

± K R2

S

±AJo

sin eTT/2

f cos2

TJ cos2

e:p d ¢ d (je:po <(3)

7

(21)

AEDC-TDR·64·25

Since the lower limit, 9 0 , is a function of f3, the Liebnitz rule is usedto obtain the derivative

2

a cos 7J cos2 cP d cPaf3

By definition, cos2

7J (cPo) = 0, and acPo/ af3 is finite, so the first term iszero. Thus

-i..

±A

Jo

7T/2

JcPo <f3>

a COS~ 7J

af3 cos2 cP sin e d 9 d e (22)

Since 9 and e are independent of f3, the derivative at f3 0 is

J±A J 7T/2 (a cos2

7J) . cos2 ¢ sin e d ¢d e (23)o ¢o (f3 = 0) \" af3 f3 = 0

From Eq. (13)

( aacf30s2

7J)f3 = 0 -- - 2 cos cP sin e (cos a cos cP cos e + sin a sin cP) (24)

Substituting Eq. (24) in Eq. (23) shows that CYf3 ifi the same for bothsides of the nose. Then, multiplying Eq. (23) by 2 and performing theintegration gives

-;- [ cos a sin3

A (; + tan-1 cos A)

tan a (25 )

+ tan-1

(sin a tan A) - sin a sin A cos A ]

The numerical evaluation of Eq. (25) is presented in Fig. 2c.

2.1.1.5 Yawing-Moment and Roll ing-Moment Coefficient Derivatives

As 'was the case with pitching moment, the yawing-moment androlling-moment coefficients and their derivatives are zero about, thereference point.

2.1.2 Flat-Topped Spherical- Wedge Nose

The geometry of the flat-topped spherical wedge is shown in Fig. B.

The direction cosines, pressure coefficient, and elemental areaare the same as for the complete spherical wedge. For 0 S; a S; 7T/2 no

8

A EDC- T D R-64-25

x

x

/

"'xzSection A-A

Moment Reference Point

Fig. B Flat-Topped Spherical-Wedge Nose

part of the curved surface is shielded from the flow, so the limits ofintegration on ¢ become 0 to "/2. Then, Eqs. (16), (18), and (23) givethe following results:

CA S -l [ 32" cos

2a (sin A sin: A) 3!...-. . 2

sin A-K R2 = + SIn a4 2

+ 2 cos a sin a (A + sin A cos A)]

CY{:3s _-1

[ " cos a sin3

A + 2 sin a (A - sin A cos A)JI(R2 4

(26)

(27)

('28)

-; [" sin a cos a sin A + cos2

a sin A cos A + A (1 + sin2

a)]S--2

KR

where A is in radians. The moment coefficients and their derivativesare zero for the indicated reference point. For a ~ 17 /2 the equationsfor the complete spherical wedge also apply to the flat-topped sphericalwedge.

9

AEDC-TDR-64-25

The characteristics of the flat-topped spherical wedge are pre­sented in Fig. 3.

2.1.3 Swept-Cylinder Leading Edge

Section A-A

..........__-- ---JE::-- M_o_m_e_n_t_R_efe7

X

x

The delta-wing leading edge which is analyzed in this section con­sists of two symmetrically swept circular cylinders. The nomencla­ture used in the derivation is shown in Fig. C .

z

Fig. C Swept.Cylinder Leading Edge

The two sides are treated as a unit in the analysis, and the sweepbackangle A is taken as positive in the equations for the leading-edge coef­ficients. Where the two sides must be considered separately, as inEqs. (29) and (30), the sign convention is A > 0 for the right side andA < 0 for the left side.

....

10

,,',

AEDC-TDR-64-25

The direction cosines of the inward directed unit normal vector are

cos (n, x) - cos ¢ cos A

cos (n, y) - cos ¢ sin A(29)

cos (n, z) = - sin ¢

Then, from Eq. (5)

cos 1] = cos f3 (cos a cos ¢ cos A + sin a sin ¢) + sin f3 cos ¢ sin A (30)

At f3 = Q the surface becomes shielded from the flow along a line de­fined by ¢ = ¢o' where

_ tan -1 (cos A)tan a

(31)

The elemental surface area is

dA = 2LR d ¢ (32)

A

The integration of the pressure distribution over the surface has asits limit the geometric angle ¢', which is always positive. For a lead­ing edge which is tangent to the wing surface, ¢' is determined by thewing sweepback and dihedral angles A and r, as shown in Fig. D.

A

~ --l~A /

-.

Section A-A

Fig. 0 Leading Edge and Wing Geometry

11

AEDC-TDR-64-25

Thus,

<P' ~ - y 2 (33a)

and -1

(~) (33b) y tan 8m A

Therefore -1 (~) (33c) <P' tan

tan ['

For a leading edge which is not tangent to the wing surface, Eq. (33) is not valid and <P' must be determined from the leading- edge geometry. It is assumed that Cp = 0 on all flat surfaces, and the coefficients and their derivatives are evaluated at (3 = o.

2.1.3.1 Normal-Force Coefficient

The normal-force coefficient is given by

<P' CN 2 K L R f COS27J sin <P d <P

S '<Po (or-<p')

If <P' < \ <Po \, the lower limit of integration is <P = -<p'. The limit changes to <P = <Po at <P'o = - <P' or at a = ao , where

-1 ao tan (~) ,tan A

Evaluating Eq. (34) for 0 ::; a ::; ao with the lower limit - <p',

CN KSL R + sin a cos a cos A sin3

<P'

and forao ::; a ::; (7T - ao) with the lower limit <Po '

CN K~R + [(si~2 a- cos2

a cos2

A) (cos3 <P' - cos

3 <Po)

(34)

(35)

(36)

(37)

- 3 sin2

a (cos <p' - cos <Po) + 2 sin a cos a cos A (sin3 <p' - sin

3 <po)J

where <Po and <p' are given by Eqs. (31) and (33c), respectively. If the wing dihedral, I', is small, the leading edge will be closely approxi­mated by a complete hemicylinder (<p' = 7T / 2). Then, from Eqs. (31) and (37) for 0 ::; a ::; 7T

CN KSL R t sin a (cos a cos A + Y 1 - sin2

A cos2

a )

Equation (38) was evaluated for a = 0 to 180 deg and A and the results are presented in Fig. 4a.

12

'-=--==--=---.--::-:~=--=-=-=------~~-

(38)

60 to 90 deg,

AEDC-TDR-64-25

2.1.3.2 Axial-Force Coefficient

The axial-force coefficient is given by

¢' 2 K L R f cos

2 TJ cos ¢ cos A d ¢

S ¢o(or-¢') (39)

Then, for 0 ::; a ::; a o

CA _s - = 4 cos A [(sin2 a _ cos

2 a cos 2 A) sin

3 ¢' + 3 cos2

a cos2 A sin cp'J (40)

KLR 3

C _S __ A KL R -

2 cos A 3

For the hemicylinder (¢' = 7T /2) ,

The numerical evaluation of Eq. (42) is presented in Fig. 4b.

2.1.3.3 Pitching-Moment Coefficient

(41)

The resultant force acts through the center of curvature at a point midway between the ends. Thus,

L sin A 2 j,

2.1.3.4 Side-Force Coeffic ient Derivative

(43)

The side-force coefficient for either side (with proper sign con­vention on A) is

Cy KLR ¢' 2 --- f cos TJ cos ¢ sin A d ¢

S ¢o(or-¢') (44)

Using the same procedure as was used with the spherical-wedge nose,

¢' 2 ) K L-1L f ( a cos TJ cos cp sin A d ¢

S ¢o(or-¢') a(3 (3=0 (45)

13

AEDC·TDR·64·25

where

(acos 2 TJ)'

= 2 cos 1> sin A (COS a cos 1> cos A + SID a sin ep)a(3 fJ=O

(46)-It •

Substituting Eq. (46) in Eq. (45) shows that CY(3 is the same for both .sides. Then, multiplying Eq. (45) by 2 and performing the integrationgives for 0 ~ a ::; a o

(47)3

• 3 "/"')sIn 't'Cyf3 K {R = - 8 cos a cos A sin' A( sin 1" ­

and for ao ~ a ~ (11 - a o )

'5' 4 2 [ 3 3Cy -- = - - sin A cos a cos A (3 sin ¢' - 3 sin epa - sin ¢' + sin ¢o)(3 KLR 3 (48)

- s in a (c os3¢' - cos

3¢ a )]

For the hemicylinder (¢' = 11 / 2) ,

CY(3 _5_ = - --.!.. sin2

A [2 cos a cos A +KLR 3

• 2 2 2 Jsm a + 2 cos a cos A4/ 2 2Y I - 8 in A c as a

(49)

The numerical evaluation of Eq. (49) is presented in Fig. 4c.

2.1.3.5 Yawing-Moment Coefficient Derivative

The yawing-moment coefficient is given by Eq. (10):

Cn = SKt [f fAx cos 2 TJ cos (n, y) d A - f fA y cos2

TJ cos (n, x) d AJFrom Fig. C, x = ± .~ sin A and y = ± ~ cos A, where the upper sign

applies to the right side and the lower sign applies to the left side.Since x and yare not functions of ¢, the yawing-moment coefficient foreither side is

Cn = ± _L_ (Cy sin A + CA cos A)2 t

Comparing Eq. (39) and Eq. (44) gives

-Cy

tan A

Then,

Cn = ± Cy 2~ ( 28~n2 A - I)smA

The total yawing-moment coefficient for both sides is

Cn = (CYright + CYleft) _L_ (2 8 i?2A - I)

2 t 8Ill A

14

AE DC- T D R-64-25

and the derivative is

2.1.3.6 Rolling-Moment Coefficient Derivative

Cnf3 = (CY Q + Cy~ ). fJrigh t fJle ft

(50)

(2 sin~ A - I )

sID AL

2 1-

( 2 s~n2 A - I )SID A

_ C Y f32

_L_2 t

Cyf31eft

But, CYf3 righ t

Therefore

JO""

The rolling-moment coefficient is given by Eq. (11):

Ct = S~ [ffA

Y cos2

." cos(n,z)dA -:- ffA

Z cos2

." cos(n,y)dA]

From Fig. C, y = ± ~ cos A and Z = 0, where the upper sign appliesto the right side and the lower sign applies to the left side. Since y isnot a function of ¢, the rolling-moment coefficient for either side is

The total rolling-moment coefficient for both sides is

\ -

and the derivative is

C1f3 = L cos A (CN - CN )2 1, f31eft f3right

For either side, Eq. (34) gives

CNf3 = -.K L R_ f ¢' ( a cos2

." ) sin ¢ d ¢S ¢o(or-¢') af3 f3=o

Integrating gives

.... -

2KLR3S

¢'sin A [sin a sin

3 ¢- cos a cos A cos3 ¢J

¢o (or-¢')· (51 )

-~ . Thus,

CN = - CNf31eft f3righ t

and

-2KL2

R cos A sin A3 SJ,

¢'[ sin a sin

3¢ - C os a cos A cos

3¢ ] ' (5 2 )

¢o (or-¢ )

15

AEDC-TDR-64-25

For 0 ~ a ~ a o

s .e 43

sm a cos A sin A sin3¢' (53)

...... -

and for a o ~ a ~ (7T - a o )

S .t - ..1- cos A sin A3 [

. (.3,+.' .3,+.)l?ln a SIn '/"' - SIn '/"'0

- COS a COS A (cos3¢' - cos

3¢o) ]

(54)

For the hemicylinder (ep' = 7T /2)

st = - -t cos A sin A sin a (1 + cos a cos A )Y 1 - sin 2 A cos 2 a

(55)

The numerical evaluation of Eq. (55) is presented in Fig. 4d.

2.1.4 Flat-Topped Swept-Cylinder Leading Edge

The geometry of the flat-topped swept-cylinder leading edge isshown in Fig. E.

.-

Section A-AI

\f

_________M_o_m_e_n_t_R_e_

fere7x

~

x

Fig. E Flat-Topped Swept-Cylinder Leading Edge

16

..

AE DC- T D R-64-25

The directional cosines, pressure coefficient, and elemental~rea are the same as for the complete cylindrical leading edge. Foro ~ a ~ TT /2, the limits of integration on ¢ are from 0 to ¢'. ThenEqs. (34), (39), (45), and (52) give

CN__5_ 2 [ (sin

2a - cos2 a cos

2 A) (c os3 ¢' - 1)- 3 . 2

(cos ¢' - 1)SIn aKLR 3

+ 2 sin a cos a cos A . 3 ¢'] (56)SIn

CA _5_ 2 COS A [(sin2

a -2

cos2

A) . 3 ¢' 3 2cos

2A sin ¢'cos a sm + cos a

KLR 3 (57)- 2 sin a cos a cos A ( cos

3 ¢' - 1) ]

CYf3_5_ =_--±- sin

2A [ c os a c os A (3 . ¢' . 3 ¢') . ( cos

3¢' - 1)J(5 8)sm - SIn - sm aKLR 3

- -}- c os A Sin A [s in a sin3¢' - c OS a c OS A ( cos3 ¢' - 1)] (59)

For the hemicylinder with ¢' = TT /2 ,

C _5_N KLR

4 (. 2 A cos2

~ cos2

A )3 sm a + sm a cos a cos + (60)

C _5_A KL R

4(

. 2

COS A SIll a3 2

+ sin a cos a cos A+ cos2

a cos2 A) (61 )

4 sin2

A (s in a + 2 c os a c os A)3

(62)

Cif3 Ki/' R = - + cos A sin A (sin a + cos a cos A) (63)

11-.. -

The pitching-moment coefficient and yawing-moment coefficient de­rivative are given by Eqs. (43) and (50), respectively, with eN andCYf3 determined from Eqs. (56) and (58) or (60) and (62). For a ~ TT/2,

the equations for the complete leading edge also apply to the flat-toppedleading edge.

The characteristics of the flat-topped swept-cylinder with ¢' = TT /2are given in Fig. 5.

17

AEDC-TDR-64-25

2.1.5 Swept-Wedge Wing

A planar wing having sweepback and dihedral is analyzed in thissection. The nomenclature used in the derivation is shown in Fig. F.

Leading Edge Centerline

x

x

Fig. F Swept-Wedge Wing

The sweepback angle, A, is taken as positive in the equations for thecoefficients of the entire wing. Where the sides must be consideredseparately, as in Eqs. '(65) and (66), the sign convention is A > 0 forthe right side and A < 0 for the left side. The centerline angle, £, isalways positive and is related to the dihedral angle, r, and the sweep­back angle, A, by

( = tan .... 1 (tan r )tan A

(64)

The direction cosines of the inward directed unit normal vector are

cos (n, x)

cos (n, y)

cos (n, z)

- sin (

'II 1 + tan 2 A sin 2 £

- tan A sin £

+ cos (v)+ tan 2 A sin 2 £

18

(65)- ...

AEDC·TDR·64·25

where the upper sign applies to the upper surface and the lower signapplies to the lower surface. Then, from Eq. (5)

where «( + a) is used for the lower surface and «( - a) is used for theupper surface. At f3 = 0, the upper surface becomes shielded from theflow at a = (, and the lower surface is shielded at a = 7T - (. Equa­tion (66) is valid only within these limits.

Since the pressure coefficient is constant over each of the wedgesurfaces it is not necessary to integrate to obtain total loads. Theforce and moment coefficients are most easily calculated by using theprojected planform, base, and side areas. The total planform area,Ap , is

_t .. "'_

'III>cos 1/ =

cos f3 sin «( ± a) + sin f3 tan A sin (

V I + lan 2 A sin 2 (

(66)

In the following derivations it is assumed that Cp = 0 on the base ofthe wing and that the wing is at f3 = o. Separate equations are given forthe lower and upper halves, and the total wing loads may be obtained bycombining the two halves. No graphical results are presented becauseof the simplicity of the equations.

Ap =hc

The base area, Ab, for either half of wing is

Ab = ht

The side area, As, for either half of wing is

(67)

(68)

(69)

2.1.5.1 Norma I-Force Coefficient

For the lower half of the wing at 0 ~ a ~ (1T - d the pressure coef­ficient is

(70)

and the normal-force coefficient is...-- -

(71 )

at (7T - () ::; a ::; 7T, eN L = 0

For the upper half of the wing at 0 ::; a ::; (, the pressure coefficient is

K sin 2 «( - a)I + tan 2 A sin 2 (

19

(72)

A EDC- TD R-64-25

and the normal-force coefficient is

(73)

At ( ~ a ~ 17, CNU = O.

2.1.5.2 Axial-Force Coefficient

For the lower half of the wing at 0 ~ a ~ (17 - d

For the upper half at 0 ~ a ~ (

(74)

(75)

where CPL and Cpu are given in Eqs. (70) and (72)

2.1.5.3 Pitching-Moment Coefficient

c3£

For the lower half of the wing at 0 ~ a ~ (17 - ()

; (++h)or f~om Eqs. (71) and (74)

(76)

For the upper half at 0 .:s; a ~ (

c:u [T - tan ( (-+- + h)J (77)

If the leading edge is tangent to the wing surface, h is given by

h = R cos y (78)

where R is the leading edge radius and y is given by Eq (33b). If theleading edge is not tangent to the wing, Eq. (78) is not valid and h mustbe determined from the leading edge geometry.

2.1.5.4 Side-Force Coefficient Derivative

For the lower half of the wing at 0 ~ a ~ (17 - ()

(79)

20

AEDC-TDR-64-25

where, by definition

(80)

and

Then,

(81)

(82)tan A sin ( sin «( + a)

I + tan 2 A sin 2 (

- 4 K cos {3 sin {3 tan A sin ( sin «(+ a)

l+tan 2 A sin 2(

(C) 4 K AsYf3 L = - S

and from Eg. (66)

- ,,-

For the upper half at 0 ~ a ~ (

4K As

S

tan A sin ( sin «(- a)

1 + tan 2 A sin 2 ( (83)

2_1.5.5 Yawing-Moment Coefficient Derivative-

.1- -For the lower half of the wing at 0 ~ a ~ (77 - ()

(84)

From Egs. (74), (79), and (80)

( CA - CA ) = - Cyright left L L

and (85)

For the upper half at 0 ~ a S (

. (86)

2.1.5.6 Rolling-Moment Coeffic ient Derivative

-'." For the lower half of the wing at 0 ~ a S (77 - ()

(C')L= +[-CYL(-3t +h)+ _b_(CN -eN) ] (87)AI k 3 left right L

21

AEDC-TDR-64-25

From Eqs. (71), (79), and (80)

( CN - CN ) =left right L

and

For the upper half at 0 ~ a S. (

(88)

(89)

-

2.2 BODY COMPONENTS

..•.

Body components typical of lifting re- entry configurations are the,spherical segment, cone frustum, and circular cylinder. The loadson these components are computed in the following sections. Themethod of combining components to give a complete configuration isdescribed in Section 2. 3.

2.2.1 Spherica I Segment

The spherical segment is a basic nose for bodies of revolution. Thenomenclature used in the derivation is shown in Fig. G.

x

x R~

Reference Point

y

R sin :-z

Section A-A

....

Fig. G Spherica I Segment

22

A E DC- T D R-64-25

The direction cosines of the inward directed unit normal vector are

cos TJ = cos f3 (cos a cos e + sm a sin e sin e:P) + sin f3 sin e cos ¢ (91)

Then, from Eq. (5)-- .J-

cos (n, x)

cos (n, y)

cos (n, z)

- cos e

sin e cos ¢

sin e sin ¢

(90)

At f3 = 0 the surface becomes shielded from the flow along a curve de­fined by ¢ = ¢o' where

• -1 ( I )= - SIn¢o tan a tan e (92)

This equation is valid only for e 2: (7T/2 - a), since no shielding occursfor e ~ (7T /2 - a). The elemental surface area is'

dA = R2

sin e de d ¢ (93)

It is assumed that Cp = 0 on the base at all angles of attack, and thecoefficients and their derivatives are evaluated at f3 = o.

2.2.1.1 Norma I-F oree Coeffie ient

The normal-force coefficient is given by2

CN= KsR ffA

cos2

'T]sin2esin¢d¢de (94)

Because of the limitations on the shielding equation, the evaluation ofEq. (94) must be treated as three separate cases. In each case, theintegration is taken over the right side of the body and the result ismultiplied by 2.

(I ) 0 ~ a ~ (7T/2 - eb)

eh 7T/2

f f cos2

'T] sin2 e sin ¢ d ¢ d e

o -7T/2

which gives.

CN S = -!L cos a sma sin4

ebJCilT 2

(95)

(96)

(II) {7T/2

-- CN 2 K R2 [/h -a f 11h 2

sin2 e sin ¢ d ¢ des cos 'T]

o -7T/2

eh 7T/2

sin ¢ d¢ dO]+ ff¢o

2sin

2 e (97)cos 'T]7T/2-a

23

AEDC-TDR-64·25

then

This equation reduces to Eq. (98).

(99)

+ sin- 1 ( __l )Jtan a tan 8b

(98)

cos' 0b }sin12 a ) - 5 ] V sin 2 a -+

sin a { -1 (COS 8b ) • "-2- cos sin a + cos a sm 8b

To make the spherical segment compatible with the cone frustum,let 8b = TT /2 - o. Then from Eqs. (96) and (98), for 0 :S,a :S 0,

C S TT • "se-N -- = - cos a SIn a COS uKR 2 2

and for 0 :S a :S (TT - 0)

(100)

cos" 0 [; + sin -1 (~:: ~ ) J(101)

.\ ) - 5J Vsin2

a - sin2 o}

sIn a

(s~n 0) + COS aSIn a

sin 03

+

{

-1COSsin a

2

For (TT - 0) :S a :S TT, CN = 0,

Equations (100) and (101) were evaluated for a = 0 to 180 deg ando = 0 to 70 deg, and the results are presented in Figs. 6a and b.

2.2.1.2 Axial-Force Coefficient

The axial-force coefficient is given by2

CA = ---lUL I I cos2

TJ cos 8 sin 8 dep d8, S A (102)

Evaluating Eq. (102) for the three cases:

(I) . o :S a :S (TT/2 - 8b)

2 K R2 8b TT/2

CA Io I2

COS 8 sin 8 d ¢ d 8S -TT/2

cos TJ

so

S• 2 . " 8bCA

TT ( sIn a sIn 2" 2 )KRT= - - COS a COS 8b + COS a

2 2

24

(103)

(104)"

AE DC- TDR-64·25

2 K R2 [f

rrh-

afrrl2 2

SCOS ." COS () sin () d ¢ d ()

.0 -7T/2

-- ..... ()h rr/2 ]+ f f../,.o cos

2." cos () sin () d¢ d ()

rrh-a 'jJ(105)

then

+{cos a-1

cos

- cos2 acos4 8b + cos

2 a) [..!!.- + sin -1 ( 1 ())]2 tan a tan h

+cos a cos 8h

2(l - 3 cos' Ob) Vsin' a - cos' Ob } (106)

The equation for this case reduces to Eq. (106). Letting 8b

Eqs. (104) and (106) give for 0 ::; a ~ o.rr/2 - 0,

; ( sin2

a cos4

02

(107)

and for 0 ::; a ~ (rr - 0)

1 { -12 cos a cos (~)sIn a

+ ( sin2

a 2COS

4O. _ cos

2 a sih4 0 + cos

2a) [~ + sin-

1

+ cos a sin 0 ( 2 _/. 2 2 }-'----..;;;.'-2-"'-=~ 1 - 3 sin 0) V sin a - sin 0

(~)Jtan a

(108)

......

For (1T - 0) ~ a ~ rr, CA = 0 •

The numerical evaluation of Eqs. (107) and (108) is presented in Fig. 6c.

2.2.1.3 Pitching-Moment Coeffic ient

The resultant force acts through the center of curvature, and thepitching moment about the reference point is zero.

2.2.1.4 Side-Force Coefficient Derivative

A general relation for the side-force coefficient derivative for allbodies of revolution can be obtained. Consider an axisymmetric body

25

AEDC-TDR-64-25

which is pitched through an angle a' relative to the free- stream veloc­ity and is then rolled through an angle <I> about the X- axis as shown inFig. H.

y .....

v00

-Cy

z

Fig. H Force Coefficients on Body of Revolution atCombined Angles of Attack and Sideslip

The force acting normal to the X-axis in the X, V00 -plane is defined, incoefficient form, as (CN ){3=o and remains constant as the body isrolled. Then

cy = - (C N ){3 = 0 s in <I> ( 109)

By resolving the velocity vector V00 along the body X-, Y-, and Z- axes, itcan be shown that

or

Then,

tan <I>

sin <I>

tan {3sin a

tan {3Vtan 2 {3 + 'sin 2 a

(110)

(110a)

)tan {3

Cy = - (CN Q=ofJ Vtan 2 {3 + s in 2 a

Differentiating with respect to {3 and then letting {3 o gives

(111)

-eNsin a

(112)

26

AEDC-TDR-64-25

Thus, CYf3 may be obtained from Egs. (100) and (101). At a = 0, CY{3is determined by substituting Eg. (100) into Eg. (112) and then lettinga = o. Thus,

_ 17 K R2cos

4 825

(113)

- ...... -

The numerical evaluation of Egs. (112) and (113) is presented inFigs. 6d and e.

2.2.1.5 Yawing- and Rolling-Moment Coefficient Derivatives

As was the case with pitching moment, the yawing- and rolling­moment coefficients and their derivatives are zero at the momentreference point.

2.2.2 Flat-Topped Spherical Segment

The geometry of the flat-topped spherical segment is shown inFig. 1.

x

x-fA

Reference Point

--r--"~'" y

R sin e

Section A-A

Fig. I F lat-Topped Spherica I Segment

27

AEDC-TDR-64-25

The direction cosines, pressure coefficient, and elemental areaare the san1e as for the complete spherical segment. For 0 ~ a ~ 1T /2

the limits of integration on ¢ are from 0 to 1T /2 and the limits on e arefrom 0 to eb. Then, from Eqs. (94) and (102), with eb = (1T/2 - 0) ,

C 5N K"Jl2

+ sin 02 cOS 0 (2 cos2

8 - 1 - sin2

a - 13° sin2

a cos2

o)J

( 114)

+[; 2cos a (1 - sin 4 8) + -!L sm a cos"'o

4(115)

+ cos a sin a (; - 8 - sin 0 cos 0 + 2 sin 0 cos3

0) J

where 0 is in radians.

Equations (112) and (113) apply only to complete bodies of revolu­tion, and CY,Bfor the flat-toppe~ spherical segment must be obtainedfrom Eq. (9) which gives

KR 2 f 2.2Cy = - -5- fA cos 71 SIll e cos ¢ d ¢ de

For the right side of the segment

C;Yf3 ~ feb f1Th (a cos2

71) sin2 e cos ¢ d ¢ de5 0 0 af3 f3=o

where

(116)

(117)

Substituting Eq. (118) in Eq. (117) shows that CYf3 is the same for bothsides of the segment. Then, multiplying Eq. (117) by 2 and performingthe integrations, with eb = (1T/2 - 8),

CYfJ K i, ~ - t [ ; cos a cos' li + sin a ( ; - li - sin li cos li - t sin li cos' li)] (119)

The moment coefficients and their derivatives are zero for the indicatedreference point. For a 2: 1T /2 the equations for the complete sphericalsegment also apply to the flat-topped spherical segment. The charac­teristics of the flat-topped spherical segment are presented in Fig. 7.

2.2.3 Hemisp here

Although the hemisphere is a limiting case of either the sphericalwedge (A = 90 deg) or the spherical segment (0 = 0 deg), the body is of

28

A E DC- T D R-64-25

enough interest to warrant a presentation of the equations. Letting(; = 0 in Eqs. (101), (~08), and (112) gives, for 0 ~ a ~ ",

CN S JL SIn a (1 + cos a)-. - - J("RT 4

S ...!L2

CA I(R"2 (1 + cos a)-- ..... 8

CY/3s - JL (1 + cos a)J("RT 4

(120)

(121)

(122)

The moment coefficients and their derivatives are zero for a momentreference point at the center of curvature. The characteristics of thehemisphere are presented in Figs. 2 and 6.

2.2.4 Flat-Topped Hemisphere

For 0 ~ a ~ "/2 and (; = 0, Eqs. (114), (11.5), and (119) give

CN S JL (1 + 2 cos a sin a +• 2 a)KR 2 8

SIn

CA S JL (1 2 cos a sin a2

a)KR 2 + + cos

8

CY/3s - JL (cos a + sin a)J("RT 4

(123)

(124)

(125)

The moment coefficients and their derivatives are zero for a momentreference point at the center of curvature. For a 2:: "/2 the equationsfor the full hemisphere apply. The characteristics of the flat-toppedhemisphere are presented in Figs. 3 and 7.

2.2.5 Cone Frustum

The cone frustum is frequently used:as a nose or flare section oflifting bodies. The nomenclature used in the derivation is shown inFig- .T

-'

-.

Moment Reference Point

-~------..

y

LFig. J Cone Frustum

29

AEDC-TDR-64-25

The direction cosin~s of the inward directed unit normal vector are

cos (n, x) = - sin 0

cos (n, y) - cos 0 cos cP (126)

cos (n, z) - cos 0 sin cP

Then, from Eq. (5 )

cos 1] = cos f3 (cos a sin 0 + sm a cos 0 sin ep) + sin f3 cos 0 cos cP (127)

At f3 = 0 the surface becomes shielded from the flow along a line de­fined byep = epo' where

,I.. • -1 (tan ,0 )'fJ = -sIn --o tan a (128)

This equation is valid only for a ~ 0, since there is no shielding of thesurface for a :s; o. The elemental surface area is

'\

dA =r dcp dr

sin 0(129)

It is assumed that Cp = 0 on the flat surfaces, and the coefficients andtheir derivatives are evaluated at f3 = o.

2.2.5.1 Norma I-F oree Coeffie ient

Since CPo is not a function of r, the first integral may be evaluated togive

where ( = Rn:lRb .

Be~ause of the limitations on the shielding equation, the integration ofEq. (131) must be treated as two separate cases. In each case, theintegration is taken over the right side of the body. and the result is multi­plied by 2.

(I) o:s; a ::; 0

The normal-force coefficient is giyen by

(130)

(131)

(132)

(133)

I cos2

1] sin ep d epep.

Rb

Iep IRn

r cos2

1] sin ep dr d cpKStan 0

K L Rh (I + ()2 S

K L Rh (I + () ITT /2 2cos 1] sin ep d cp

S -TT/2

30

TT Cos a s in a sin 0 cos 0

GN

which gives

CN K L RbS

( I + ()

AEDC-TDR-64-25

(II) 0 ~ a ~ (TT - 0)

K L Rb ( 1 + ()S

TT /2

I¢o cos2"lsin¢d¢ (134)

O. Equations (133) and (135) were evaluated foro to 40 deg, and the results are presented in

(135)

(136)

(137)

(~)Jtan a

d¢

• -1+ sm

2 2 2)COS 0 + sin 0 cos a V· 2 • 2 03 sin a cos 0 sm a - sm

• 2SIn a

cos a sin a sin 0 cos 0 [ ;

K L Rb (1 + () s:- I 2---=-----'::......- tan U cos "l

2 S ¢

The axial-force coefficient is given by

s

which gives

At (IT - 0) ~ a ~ TT, CN

a = 0 to 180 deg and 0

Fig. 8a.

2.2.5.2. Axia I-Force Coeffic ient

Rb

CA = ~ I¢ fRn

r COS2"l dr d¢

Integrating over r ,

Evaluating Eq. (137) for the two cases:

(I) 0 ~ a ~ 0

K L Rb (1 + t) 7ThCA tan 0 f

2d¢ (138)cos "l

S -TTh

which gives

CA S TT tan 8 (2 2sin

20 +

. 2cos

20)

K L Rb (1 + ()cos a SIn a (139)

2

(II ) o ~ a ~ (TT - 8)

K L Rb (1 + t) 11/2(140)

CA tan 0I¢o

2d¢

ScOS'TJ

- - which gives

CA S tan 0{ (2

2sin

20 +

. 2cos

20) [; +

• -1

(~)Jcos a SIn a smKLRb (1+() 2 tan a

+ 3 cos a sin 8 V·' .'oJSIn a - sm (141)

31

At (TT - 8) ~ a ~ TT, CA = O. The numerical evaluation of Eqs. (139) and(141) are presented in Figs. 8b and c.

2.2.5.3 Pitch ing-Moment Coeffic ient

(143)

(142)

(144)

(145)

(146)

sin if> dr dif>J

2

2 2r cos ."

2 3 3 JRb(r 2Rb r) r J 2- -3- - -3- ¢ cos .". sin ¢ d ¢R n

The pitching-moment coefficient is given by

~ [J JR b r ( Rb - r) 2C 0 S

2

." sin ¢ d r d ¢S 1- ¢ R

ntan [)

(c ) S = - TT sin 0 cos 0YfJ a= 0 K L Rh (1 + ~)

The side-force coefficient derivative is given by Eq. (112),

CY~ = _ ~NfJ SIn a

C __K_ [. 1m - St -t-an--:2=--.-=-0-

lntegrating over r,

2.2.5.4 Side-Force Coefficient Derivative

Substituting Eq. (131) into Eq. (143) gives

Since the limits of integration on ¢ do not enter into this derivation,Eq. (144) is valid for 0 ~ a ~ 17.

and CYfJ may be obtained from Eqs. (133) and (135). For a 0,

substituting Eq. (133) into Eq. (145) gives

The numerical evaluation of Eqs. (145) and (146) is presented in Fig. 8d.

2.2.5.5 Yawing-Moment Coeffic ient Derivative

It is obvious from Eq. (145) that a general relation for all bodies ofrevolution is

sin a (147)

Substituting Eqs. (144) and (145) in Eq. (147) gives

32

AEDC.TDR·64·25

CY{32 ( 1 - ~3 ) ]

( 1 - (;2 ) (148)

Like Eq. (144), this equation is valid for 0 ~ a ~ 71 •

- --"'-

2.2.5.6 Rolling-Moment Coefficient Derivative

The resultant force acts through the center of the cone, and thereis no rolling moment about the indicated reference point. There­fore, Cf,{3 = o.

2.2.6 Flat-Topped Cone Frustum

The geometry of the flat-topped cone frustum is shown in Fig. K.

Moment Reference Point

The direction cosines, pressure coefficient, and elemental areaare the same as for the complete cone frustum. For> 0 ~ a ~ 71/2, thelimits of integration on ¢ are from 0 to 71/2. Then, from Eq. (131) and(137)

- t_ ....X Rn T y

f Rb

--l-. L ~I

lz~ Rn/RbFig. K Flat-Topped Cone Frustum

C S 71 cos a sin a sin [) cos [) + cos2

a sin2

[) + ~ sin2

a cos2

[) (149)N K L Rh (I + ~) = 2" 3

and

Equations (145) and (146) apply only to the complete cone frustum, andCy{3 for the flat-toppe"d cone frustum must be obtained from Eq. (9).

(150)tan [) [2 cos a sin a sin [) "cos [)

33

A E DC-T D R-64-25

For the right side of the cone frustum,

Cy = -K

5 tan 02

r cos TJ cos ¢ d r d ¢ (151 )

and

where

K L Rh ( 1 + l:) 7Th (a 2)CYf3 2 s 'f

O. ~o; 'I f3~O cos ¢ d¢ (152)

tel ~ol '1)f3=O ~ 2 cos Ii cos ¢ (cos a sin Ii + sin a cos Ii sin ¢) (153)

Since CY(3 is the same for both sides of the cone frustum" Eq. (152) ismultiplied by 2 and integrated to give

(154)Gy(3 K L Rh (1 + e) = - ; cos a sin 0 cos 0 - + sin a cos2

0

The pitching-moment coefficient and yawing-moment coefficientderivative are given by Eqs. (144) and (148)" respectively" with CNand CY(3 determined from Eqs. (149) and (154). The rolling-momentcoefficient derivative is zero. For a ~ 1T / 2 the equations for the com­plete cone frustum apply to the flat-topped cone frustum. The charac­teristics of the flat-topped cone frustum are presented in Fig. 9.

2.2.7 Circular Cylinder

The circular cylinder is a special case of the cylindrical leadingedge (A = 90 deg) and the cone frustum (0 = 0 and e = 1). LettingA = 90 deg in Eqs. (38), (42)" (43)" (49)" (50)" and (55) gives

CN _5_ -!.. • 2 (155)SID aKLR 3

CA 0 (156)

Cm CN _L_ (157)21-

CY(3 _5_ 4 . (158)KLR 3 sma

Cn(3 CY(3 _L_ (159)21-

C1-(3 o . (160)- .-

where the moment coefficients are referenced to the base of the cyl­inder. The characteristics of the circular cylinder are presented inFigs. 4 and 8. The Eqs. (155) through (160) also apply to the flat­topped circular cylinder.

34

AEDC·TDR-64-25

2.3 COMPOSITE CONFIGURATIONS

The Newtonian analysis is based on the local flow deflection angle and assumes that the only interference between components is that due to shielding. Therefore, the configuration being analyzed can be broken into independent elements corresponding to the components described in the previous sections. From the equations and charts, the aerodynamic characteristics of each component can be determined based on the refer­ence area and length of the complete configuration. The contributions of the individual components are added together to give the total coefficients of the configuration. When this method is used to obtain the character­istics of a composite configuration, one component may be in a position where it shields another component from the flow, and the effects of this shielding must be considered.

The summation of moment coefficients and their derivatives re­quires a moment transfer from each component reference point to a common reference point. Let XT and ZT be the moment transfer distances, positive as shown in Fig. L.

Component Reference Point

T'- Reference Point of Complete Configuration

Fig. L Moment Transfer Lengths

Then,

em f' ' con 19 ura tlon (161)

Cn(3 .c onfigura tion

C n (3 - CY(3 component

(162)

Cg = Cg(3 + Cy ~ (3configuration component (3 p,

(163)

The characteristics of a typical composite configuration are evalu­ated in the Appendix.

35

A EDC- TD R-64-25

REFERENCES

Neal, Luther, Jr. "Aerodynamic Characteristics at a MachNumber of 6. 77 of a 9° Cone Configuration, with and withoutSpherical Mterbodies, at Angles of Attack up to 180° withVarious Degrees of Nose Blunting." NASA TN D-1606,March 1963.

Fohrman, Melvin J. "Static Aerodynamic Characteristics of aShort Blunt 10° Semivertex Angle Cone at a Mach Number of15 in Helium." NASA TN D-1648, February 1963.

Wells, William R. and Armstrong, William O. "Tables ofAerodynamic Coefficients Obtained from Developed NewtonianExpressions for Complete and Partial Conic and SphericBodies at Combined Angles of Attack and Sideslip with SomeComparisons with Hypersonic Experimental Data.. "NASA TR R-127, 1962.

Penland, Jim A. "Aerodynamic Characteristics of a CircularCylinder at Mach Number 6. 86 and Angles of Attack up to90°." NACA TN 3861, January 1957.

Julius, Jerome D. "Experimental Pressure Distributions overBlunt Two- and Three- Dimensional Bodies Having SimilarCross Sections at a Mach Number of 4.95." NASA TN D-157,September 1959.

Bertram, Mitchel H. and Henderson, Arthur, Jr. "RecentHypersonic Studies of Wings and Bodies." ARS Journal,Vol. 31, No.8, August 1961, pp. 1129-1139.

Fisher, Lewis R. "Equations and Charts for Determining theHypersonic Stability Derivatives of Combinations of ConeFrustums Computed by the Newtonian Impact Theory. "NASA TN D-149, November 1959.

Rainey, Robert w. "Working Charts for Rapid Prediction of Forceand Pressure Coefficients on Arbitrary Bodies of Revolutionby Use of Newtonian Concepts." NASA TN D-176,December 1959.

1. Penland, Jim A. "Aerodynamic Force Characteristics of a Seriesof Lifting Cone and Cone - Cylinder Configurations at a MachNumber of 6.83 and Angles of Attack up to 130°. "NASA TN D-840, June 1961.

Ladson, Charles L. and Blackstock, Thomas A. "Air-HeliumSimulation of the Aerodynamic Force Coefficients of Cones atHypersonic Speeds." NASA TN D-1473, October 1962.

2.

3.

4.

5.

7.

9.

10.

· 8.

36

16.

15.

AEDC-TDR-64-25

11.- Gray, J. Don. "Drag and Stability Derivatives of l\1issileComponents According to the Modified Newtonian Theory. "AEDC-TN-60-191, November 1960.

12. Margolis, Kenneth. "Theoretical Evaluation of the Pressures,Forces, and Moments at Hypersonic Speeds Acting onArbitrary Bodies of Revolution Undergoing Separate andCombined Angle-of-Attack and Pitching Motions. "NASA TN D-652, June 1961.

13. Malvestuto, Frank S., Jr., Sullivan, Phillip J., Marcy,William L., et al. "Study to Determine AerodynamicCharacteristics on Hypers onic Re - Entry Configurations:Analytical Phase, Design Charts." WADD-TR-61-56,Part II, Vol. 2, August 1962 .

. 14. McDevitt, John B. and Rakich, John V. ",The AerodynamicCharacteristics of Several Thick Delta:'Wings at'MachNumbers to 6 and Angles of Attack to 50°." (TitleUnclassified) NASA TM X-162, March 1960. (Confidential)

Seaman, D. J. and Dore, F. J .. "Force and Pressure Coeffi­cients of Elliptic Cones and Cylinders in Newtonian Flow. "Consolidated Vultee Aircraft Corporation, San Diego,Califorpia, ZA-7-004, May 16, 1952.

Jackson, Charlie M., Jr. "A Semigraphical Method of ApplyingImpact Theory to an Arbitrary Body to Obtain the HypersonicAerodynamic Characteristics at Angle of Attack and Side­slip." NASA TN D-795, May 1961.

17. Hayes, Wallace D. and Probstein, Ronald F. Hypersonic FlowTheory. Academic Press, N:ew York, 1959.

18. Busemann, A. "Fliissigkeits-und-Gasbewegung."Handworterbuch der Naturwissenschaften, Vol. IV,2nd Edition, pp. 276-277, Gustau Fischer, Jena, 1933.

19. Love, Eugene S., Henderson, Arthur, Jr., and Bertram,Mitchel H. "Some Aspects of Air-Helium Simulation andHypersonic Approxima.'tions." NA.SA TN D-49, October 1959.

20. Lees, L·ester. "Hypersonic Flow." Fifth International Aero-nautical Conferenc~ (Los Angeles, California, June 20-23,1955), Institute of the Aeronautical Sciences, pp. 241-276.

37

A EDC- TD R-64-25

APPENDIX

APPLICATION OF METHOD TO A TYPICAL DELTA WING

As an example of the us e of the equations and charts given' in thisreport, the aerodynamic characteristics of a typical delta wing werecomputed. The configuration which was analyzed is shown in Fig. 10,and the lengths, angles, and areas used in the calculations are givenbelow:

The coefficients were based on the planform area (5 = 0.267 Ln2), with

the mean aerodynamic chord as reference length (1, = 0.667 Ln) for thepitching-moment coefficient and the span as reference length (1, = 0.536 Lj))

for the yawing- and rolling-moment coefficients. For all angles of attackit was assumed that the base of the delta wing contributed no axial force,i. e., CPh = o. The modified form of the pressure -coefficient given byEq. (2) w~s used with K = 2 .

39

A EDC- T D R-64-25

The characteristics of the nose component were determined fromFig. 2 and the moment transfer equations, Eqs. (161), (162), and (163).Since ¢' was not equal to 90 deg, it was necessary to use the equationsof Section 2. 1. 3 and the moment-transfer equations to evaluate theleading-edge component. The wing component was evaluated with theequations of Section 2. 1.5 and the moment-transfer equations.

The aerodynamic coefficients of each component and of the completedelta wing are presented in Fig. 11. To provide a comparison of thetheory with hypersonic experimental results for this delta wing, dataobtained in the 50-in. Mach 8 Gas Dynamic Wind Tunnel, Hypersonic (B)of the von K~rm~n Gas Dynamics Facility, Arnold Engineering Develop­ment Center, are also given. The data were obtained at a Mach numberof 8. 1 and free-stream Reynolds numbers of 1. 3 to 5. 2 x 106 based onmodel length. The theory predicts the experimental results with goodaccuracy at angles of attack up to about 57 deg, where the shock wavebecomes detached. Above this angle of attack, the theory will givebetter agreement with the experimental data if K = Cp = 1.83 is used.

max

40

AE OC~TO R·64·25

Fig. 1 Axis and C foe ficient Nomenc lature

41

. \I... f I...... , \, (

I iI,

I, \ .. : t

I ~ I I t

A, degr::r.

90 (Hemisphere)

}>­

mo()

~o;;0

I

0­J:>.

~lJ1

180

.r

rrm

+

160

"

140

A1(3

~ WI/~,

120100a, deg

80

80

75

70

65

60

60

1.0

0.8

-.C\l

f20.6"lZl

'-"

~ Z!'V u

0.4

~-+t1

0.2

20 40

a. Normal Force

Fig.2 Aerodynamic Characteristics of Spherical-Wedge Noses

R 64-25AEDC-TD .

-cC1I C1IU :l0 C

-.;:LL c0

D UOx« N

..D mLL

o

~$ +I

V'J( 'H)I/S)~

43

I r., ~.... ' ., , \

• I=,1 I' • ,;

" I I • \-

»moQ-4o:;:0

I

0-

~IV<.n

~ -1.2C'j

'M"0C'j

~

~Q)

0.

~-0.8

-..C"l

~

p::~

~,UJ~

co...>c -0.4

C)

A, deg

90 (Hemisphere)Em L1±UnUllJ

85

807570

65

60

1\tG-' CD (DO

00 20 40 60 80 100 120 140 160 180

0" deg

c. Side-Force Derivative, (3 = 0

Fig.2 Concluded

... ~ \ f ~ .1. •. ... : "

, \ " :• ~ t I • ! ...

A, deg -#fJ 1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIumum /\

1.0 ~iiiiiiiii 100.8 ~llllllllllllllllrlll[filKaJRIU¥l'1't1IIII!II!jlllllln'fit70

65 f60

~~ CD.-C'il

ea"- 0.6rn.....,

Z~

C,)

CJ1

0.4

0.2

Fig.3 Aerodynamic Characteristics of Flat-Topped Spherical-Wedge Noses

oo 20 40 60 80 100

a, deg

a. Norma I Force

120 140 160 180

>mon.-lo:::0,0­l:>.

IV\.f,

1.0

0.8

-..(\1

~ 0.6""-U,l--

~ u<OJ

0.4

0.2

A, degI "'"''"IIIIII!''''''',,''''!'''

UL gO (Flat-Topped Hemisphere)illLiIttLLWII11JTI

7560

AtGL~\3J \D\

»mon~

o;:0

0...::..t-.)

111

oo

'iI '"-...

20 40 60 80

0:, deg

b. Axial Force

Fig. 3 Continued

100 120 140 160

, ""

180

1\,

... '. \t

'.., I"",.

IIf ,I'

11 I •I- j •

-1.2

A, deg~

90 (Flat-Topped Hemisphere)~ A-1 . 0 mttllllllllllllllHfllllllllllllllttti I1I111I111I1I111111111111111 niillllllllllllllllllittllllllllllllllTlllfl rmTfTI 1(;)

80

~ -0.8 75 (ro'M

CD'0

y\S]roH 70-1

H\Q)

0465

~-0.6

--N

f2 rtltttitttttllllllllllllllllllllllllllllllllllllllllllllllll160~ "-J CZl-....,;

co..:>t -0.4C)

c. Side-F orce Der ivative, f3 := 0

oo 20 40 60 80 100

a, deg

Fig. 3 Concluded

120 140 160 180

»li1on,-loAl,0-

f"tv111

--... L ----.....JA, deg

( ~ 60

( --0 65

~ 0.470

\ 75-~ 80

;;; 0.290-z

I C,)

00 1 20a, deg

A, deg60

65

70

75

80

90 (Circular Cylinder) ~

1.6

.- 1.2s:t:

~ ~00 "-en-

ZC,) 0.8

0.4

oo .20 40 60 80

a, degI

100 120 140 160 180

»mon-lo:::0I

0­l>o.I

"'"<.n

...

a. Normal Force

Fig. 4 Aerodynamic Characteristics of Swept-Cylinder Leading Edges (¢" = 7T /2)

I'" t! .

: \.. t

~ I 1 ' '".

0.5

,1 \ r I ~

'I l.. " ,!,... I t l ~

~

co

0.4

.-~

~;n 0.3.....,

u<

0.2

0.1

~ARA, deg -60

~L65

~{ ( - (f~

70 ,75

80

oo 20 40 60 80 100 120 140 160 180

a, deg

b. Axial Force

Fig. 4 Continued

»mon.-lo;u

I

0-

t"""VI

>m0n

I

-l0

-2.0

~;;0I

A, deg 0-

~....IV

60 tTl

65

-1.6 70

l::~ 75oM

"t:l~

S-t -1.2S-tQ) 1tttttt1111111111111111111800..

",..."

CJ1:s~ -0.80 ........(/)'-'

co. 1111111111111111111111111190 (Circular Cylinder)l>lC)

-0.4

oo 20 40 60 80 100 120 140 160 180

a, deg

e. Side-Foree Derivative, (3 = 0

Fig. 4 Continued

...

, \ . ,' .... I" ", 't l

;.

I

\ \ rl". .

-0.4. ii""lilli"'ii"'Ii'Ii' I:A

I:l-l-l-

CIS.r-! -0.3 Em

~

'0

~CIS

A, deg ttl

f.lf±

enm

+lCI

$.4

60

Q,)~

~

f:t

65if (

L'

'" -0.2

=+1

~

CJ1

~

ttJ 1-l..H,

tt

C'I

m +tt:tm= ~~

~

~

H-t I+R

~

IttttlflTI1I1I I ffi 170

- (I

~

,III, 6I411i*f111111 tJmllllllllllll·~II ..I-I·I·I·liiitw:f11111 rll [II [II~75

I:t •

rt.l-- <0..~

U

oo 20 40 60 80

0, deg

100 120 140 160 180

d. Rolling-Moment Derivative, f3

Fig. 1 Concluded

o >moQ-lo;:0

0­~

""01

CJ1t\J

1.2

........p::;g~ 0.8.........

zu

0.4

A, degrn- 60

65

70

75

80

90

,~

~,--- 11

»mon

I

-IoAlI

0­A

~VI

oo 20 40 60 80

a, deg

a. Normal Force

100 120 140 160 180

Fig. 5 Ae·rodynamic Characteristics of Flat-Topped Swept-Cylinder Leading Edges (¢' =: 7T /2 )

AE DC- TDR-64-25

000r-t

7~

--0tor-t

~

....:l

<: UI 0~r-t

00r-t

btl "'tJCIt CIt

Q) U ~

't3 0 cLL ~

cd 0

.~ u0 x00 4 in

mbtl ..0

LLQ)'t3 0 ll) 0 ll) 0

to to l:"- I:"- 00..~

0to

ll)

oM

oC'iI

o

53

o

oo

~

_1.21!CIS

- -160

.~

6-5

"0

CJ1CIS

' 70

t+::>oM

Mttti"mttH75

Q)~

-0.8... __ 801-;

........

e1~

"-en-- en. -0.4t>4

C) 1-14-+-l~ 90

I~\

~

\ -------:11

1-1 i±'l-­;"ttttt+kt

-1-',T

r

>mon.~

o;;0.0-

t'1'..>t.1I

00 20 40 60 80 100 120 140 160 180

a, d~g

c. Side-Force Derivative, f3 == 0

Fig. 5 Continued

, T ~.J \ 1

;"

l \ .. I ••~ ,~. ·"'1

I

" t"',, I \1 ..\

'.

-0.4

I::-0.3~'·· ....................................

~•.-4

......... A, deg

'0~

J.4

J.4

60 1Q)04

65

CJ1 "-0.2

~c== \I

~CJ1 C'l

t2

\

"""~{f)-.-

co..~

u

oo 20 40 60 80 100 120 140 160 180

a, deg

d. Rolling.M~ment Derivative, f3 = 0

Fig. 5 Concluded

»mI:'n.~

I:';:0

I

0-

~t-J111

1.0

0.8

-.N

~"- 0.6rn--

CJ1 Zen

C,,)

0.4

0.2

o20 40 60

0, deg fffttffil II II II 111111 II III - Q~011.'(j~~'~i'~';h~~~ \

s1fI:l:tl)tlllllllllllllllllllrJ II--;-

b(]+ 8101#11 tltllllliNd 11I1I111111111 ,UUUUNrml1 NIIIIIIN 1III1I1115

I20

II25$30tfH35"1"T

40:r.:m:45---

i80 100 120 140 160 180

»mon.-lo;:0.0-

~

""t.n

0, deg

a. Normal Force, 0 "'" 0 to 45 deg

Fig. 6 Aerodynamic Characteristics of Spherical Segments

~ , J __

., ~

C'Icu

-c0r...

0..b.O -cQ)

I.t') cu'0

-.::t :::»

U .:~

..d GO

c0

cu"U

u ~

0.~U-U-

cE0

Z

..c

AEDC-TDR-64-25

oC\l

o

+

:±

R:R=l±

~oo

00oo

~ I-I-i-H- m-+

++

o

Hf+

=ffFffF ++-H+ 1::l±H+l+ \+

::l c:t

l-l-l-l-

+f+

~ ttl±t+ t=t +1=H- :+ ~

~I-t uI+-

+++ l-l-I +11+

H-f+ :t:I:q::1:tl

-+~ 1-1-+ m

1+ m:: t:P1+:++

:+ l+

+ +

~

1+ ~a H-

y-~.J-

~ ~+ f-+ H-

1+

+

++ -+f+

o

f+±

tt-+

u.

I

:H

I-t+

Ff

l± H­

!±l±Irt++±H:!::

o

oC\1

--~. ....

57

1.6

1.2

-.(\1

ea......... 0.8f/.l........

CJl u<co

0.4

0, dego (Hemisphere)

1020

Q*25303540 \45

~50

y(]+ 860

70

»mo()

-lo:;0.0­.t:..I

tvV1

oo

., '

20 40 60 80

a, deg

c. Axial Force

Fig.6 Continued

100 120 140 160

!loS 11 } ,.

180

, i,. " .I

• t l :1 I r.,".. t:

.' t

-1.6

0, deg LilliLilli liIHIHIH ilUlllliillllllllllllllllllllll.lllllllllllll

-1.2

o (Hemisphere) crs:=

5

~

'M

10

'tj~

f..I

15 ~f..I -0.8Q)

20

0.

CJ1 - H

25 L(] 8co C':I

~t:E

~ a + +~ 30:: 01.1+

I+"W- ++*+"Id-+N,.+ ~.~."'- t±±:I FH

N,

rJ.l -0.4

T.TJ::q w. l'oI. ..L ~

..L

......i 35

-t- =--

>ten..'t'I"t'ti' ........

t)

40

45

o 20 40 60 80

a, deg

100 120 140 160 180

d. Side-F oree Derivative ({3 :.: 0), 0 = 0 to 45 deg

Fig.6 Continued

»moQ-Io;0.0­~

tvt1I

0, deg

R. 45r

Q* -,j. +

tT,

l"; ri- 0

++ ~.

50++ 0 I.'

~

b(]+ 8 ~ ~ 0

- 0

-- 0

..- - -. ~ - - -

-,

._'

r~ .,\-,

.,~ -1-o.

.j.

- , 60

- ~ -.Il• '-ff

- --

- ::t - - II

-- ·r+

70",.-

eno

-0.4

-0.3

~~

.r-!"0~

~

~ -0.2~

........C\1

r;a"e -0.1

:>teo..C)

oo 20 40 60 80 100

a, deg

120 140 160 180

»mo<;1-lo;;0.0­~.I'Vtn

e. Side-Force Derivative (f3 == 0),8 = 45 to 70deg

Fig. 6 Cone luded

..., .\'

1.2

0, degJlttLITilnBHTTFFFR=FR=FFrlTH1TI+r1Tlllllll ! 1III1I111 1tttttt11 I I I I I I I I ! ! II I !

1.0lJ'+lifff£fIIIIIIIIIlIIIIlIIIIIIIIIIlIIIlIIIIIIIlJ>HttI1 0 (Flat-Topped Hemisphere)! Q*5 E11I11I11111 rflllllllllllllllllllllllllllllU IIII \

10 "i-0.8

15lHH1ffiAfln""i"ilN"'"II\1IIIIIUWWW b\J +- 0-N 20 ,~

~

......... 0.6en......,

m z 25...... C)

0.430

35

m40

0.2 45

a. Normal Force, 0 = 0 to 45 deg

Fig. 7 Aerodynamic Characteristics of Flat-Topped Spherical Segments

oo 20 40 60 80

a., deg

100 120 140 160 180

" ..

}>mon,~

o;0

I

0­.c:...,'"tn

'1 \

\

~

.... 'l

"I,,' ,

:>mon

0.24

0.20

0.16

---.C'1

~

O':l"- 0.12Cf.l

t\:) '-'

Zu

0.08

0.04

0, deg

45

50

60

70

(}~~

I~\J+ 0

o:;u

I

0-

~"->01

oo 20 40 60 80

a., deg100 120 140 160 180

b. Norma I Force, 8 = 45 to 70 deg

Fig. 7 Continued

.I

, "

'..,J \ l

1.0

0, dego

;tQ!Q25

30

, \

Flat-Topped Hemisppere

"it,l

0.'8

-1J 0.6"tf}......,

<u-mw

0.4

0.2

oo 20

35

40

45

50

60

70

40 60 80

a, deg

t. Axial Forte

Fig. 7 Continued

100 120 140 160 180

»mon.-Io:;0.0­:...:.,til

oo 20 40 60 80

U, deg

100 120 140 160 180

d•. Side-Force Derivative (f3 = 0), 0

Fig.7 Continued

o to 45 deg

I, ,;y.

I ,

J '\ '-,, , ,t ,

I

~.I

-0.24

-0.20

d

-0.16 Q*l::t ti

~

:{:l:

H+I=I++

.r-l+l

f:l:±a FR

++1-

't:l~

l=t

M

I:t tH

H it=! ~MQ) -0.12'

~ ~,~O

0-

EE~

m

:ttt: :t K:tt:!:H

L

CJ1""",,"'""""" 4- ,,~~ 111111 "'" II II II II II ttl ," II " "'" 111111 ~ ~ ~I!t <I: Itt!!

~\J+ 0OJ N"'""""'""'"""""'"~ <0. - 0 . 08 . j '" II , II II , II II II II II II

>cu

_0.04-0

70

e. Side-Force Derivative (f3 =: 0), 0

Fig.7 Concluded

o 20 40 60 80

a, deg

100 120 140

45 to 70 deg

160 180 >mon

I

-lo;0I

0­J:".I

tvIn

}>­

mo()

~oAlI

0­~

~<.n

20

:-~

15

10

o

0, deg20

~ffI'~'4-,.-'-~p.:;::t.;;' ,, I

'-t , +-;:!::j 5

=7=FF~T~

10a, deg

25.,

++,§l40

3530~~-

I-t++t-H

oo

~ 0.2+M..........

.0P::

~"'" 0.1it.:l

'JZ

u

~.

fr-r-:_

,....;.::"

H++t+--Jj

O~(Circular Cylinder"~ ""' 1)

~t-Hi-~.f,-'~-,

.-~-tfh

0,_ deg

403530

25

20

15

10 ' -i-+-', ,-

5

M-'+h-y,I-t-

~:t~.--'-+.

-;:~

~J!t.f"-

+/;-¥-H·F·

~--:l:'

~

~

R_l ~RbnT---'

~=R /R l-n b I L I

1~8\ J

\..J..+-l.

~t"7

l-l-

.LJ.++H.,J-Y.-+

~

-H-

0.41+i--.-i~

~

r;:::'i.u.J'+M..........

.0P::

£2L--J

"'"{f)-..-.z

u

OJOJ

oo 20 40 60 80 100 120 140 160

-4+-~

180a, deg

a. Normal Force

Fig. 8 Aerodynamic Characteristics of Cone Frustums

"

, \.. .,1 \ •.'t., 1-,

"~ r'

OJ-J

~

,....--,­,uJ'

+r-4-,c~

~.........('J)

----u<

0.4

0.3

0.2

0.1

0, deg

25

20

15

10

5

R ~ FG-.-1n- Rb

~-R /R Tr -- fn b Ll . --"1

~~ 8+

oo 20 40 60 80 100 120 140 160 180

a, deg

b. Axial Force, 0 = 5 to 25 deg

Fig. 8 Continued

>moQ....o::0.0­./:>.

I

t-..>In

I>mo':1-lo;:0.0­~I

'"t1'l

ISO

'rn

~h- --t-L~t- +i--

.w

,~

1.1

-

160

8

140120

+

Rnci r:::=:::Jf1~ b

r L-1

t

~

llTTITrmmrIolIII

llllllL11 tmnm

~=Rn/Rb

EEl-

SO 100

~

f+!-

60

+l

l:±H±m

++mum,...

.J...l"j".I.I R=!+

c:

I=l=l+t+m

~

tt

++-I

I-H

f+

40

f:i±l:l±E:l:

mH=R=R=

f=j:

1:0:

:g25ff:§ffi

•

l:l:l

20

111111111111111

o

III1111111111111111111111111111

",eg

40

o

.6

0.4

0,211

1.0

o.slIIlI35

--'-~

-uJl+,-t-.c~

0~

L-...J..........

O':l rnco --...-

C)<

a, deg

c. Axial Force, 0 '"" 25 to 40 deg

Fig. 8 Continued

, . :,\ (

-1.6- 0, deg

= 40Cl:S

'M 35-0Cl:S -1.2 ITn

1-4 301-4Q,)

25p.

--"- 20,...--,.- -0.8oU1' 111111111!11111111111111111111115+

0") ~

co .......c ILll11UIIIIII t II! IIII! III t1j;j:110~

r2 1111111111111111I1111,IIIIIII!U 5L..-J.......... -0.4

...l!3- 0 (Circular Cylinde~, ~=1)

<Il~t)

R ~ F3-~n- RT -- b

~==Rn/Rb L= - TI L I

~~ 8of-,

~

o 20 40 60 80 100 120 140 160 180a, deg

d. Side-Force Derivative, f3 ... 0

Fig.8 Concluded

»mCJn.-ICJ:::0I

0­J:>,.I

'"<..n

, \. \

...,;, \ 1

, . tI I l; t I ~'

~

iI : '

1. O.

»m0()

0.8

-l0:;:0.0-.I:>-

~<.n

0.6

0

~

~-uJ'

+-J

M

0-.....;

rf~

"""

5

lZl~

0

uZ

'imr-,+ . +-Tr, ;-+-

oo 20 40 60 80 100 120 140 160 180

a, deg

a. Normal Force

Fig.9 Aerodynamic Characteristics of Flat-Topped Cone Frustums

"~I

~, -', /'It;-'I .. ....1

., : t ~

1.0

0, deg l

0.8 El33ffilHlfIfI:I:I:IN+~11111111111111111111111 ~~llllImJ 111111111111111111111111111111111111111111 R~8-i-~=Rn/~ ~ t --.i

35 ,..-.-...f~

~-uJ' 0.6 0+

~r-i'-'

..c 30 \~

~'"fI) o . 4EUil111111111 !IIIII bur 1IIIIII±I:1J:I:1±tJHIliAd2 5-J ~

I-l

(3<

20

0.2 15l:H:\:10tttt

5tt+tt l II mTlllllllll ! ! , III f II ~IIIIIIIIIIIIII

00 20 40 60 80 100 120 140 160 180

a, deg }>m

b. Axial Force '='n

Fig. 9 Continued -I'=';;0.00-~

t..JIn

s:: -0.8cd

..-i"0cdM

MQ)~ -0.6~~~

+r-i--.c-J t:z:: -0.4C\:J

~"-rJ.)~

~C,)

-0.2

0, deg

403530

25

20

15

10

5

o

RLF3-~n _ Rb

~=Rn/Rb T, -- ,( I L-1~c==±1, ~ o

»mo()

-Io:::0,0­.t>-,tvU'1

60oo

.

20 40 80 100

0, deg

c. 'Side.Force Derivative, f3 I: 0

Fig. 9 Concluded

120 140 160 180

II ,

.:.'"• 1

• c

.;.. \ JI· •

LD

\ n ~~ ".., . r1

Fig. 10 Details of 75.deg Swept Delta Wing

-JW

O.60LD

><:(+

L12LnSpherical Radius

,O.0212Ln RadiUS~I->

¢'=74.5°

+ "€=4.11o

I

O.536LD

----,r=150

O.088Ln-l .-Moment Reference Point

»mon

I

-Io:;0I

0­::..

I

NV1

-.:J~ CN

1,6-<-

1.2

0.8

0.4

Theory, Complete Delta Wing

Wing

Leading Edge

Nose

M'00

8.1 Data

0.10

0.08CN

0.06

0.04

0.02

-0o 2 4 6

a, deg-8 10

m',+h,t,+:).:;

,rrn:

-aB,

»mo()I

-to:::0.0-

t'tvtn

Ouo 20 40 60 80 100 120 140 160 180

.'

a, deg

a. Normal Force

Fig. 11 Aerodynamic Characteristics of a 75-deg Delta Wing

., • f

I'•. I,t

:

0.12

.~ k 1

~~Theory, Complete Delta Wing

0.022

0.020

0.016

0.014

... ,

a, deg

b. Axial Force

Fig. 11 Continued

-.JCJl

CA

0.08

0.04

\.

oo 20 40

Wing

Leading EdgeIIIII

Nose

60 80

M00

8.1 Data

100 120

2

140

4 6a, deg

160

8 10

180>rno().~

o;0.0­,I:>..

t..:,.."

-Jm

-0.2

CmO' 6LD

-0.1

[I'heory, Complete Delta Wing

Wing

M00

8.1 Data

-0.015

0.6LD-0.005

'0o 2 4 6

a, deg'8 10

»mon.:..o;u.0­A.

I'Vtn

o

0.1o 20 40 60

Nose

Leading Edge

80 100 120 140 160 180

a, deg

c. Pitching Moment, Referenced to 0.6 Lo

Fig. 11 Continued

'"..i(,

..I I

, ~ .... • I ',I.. .

0.12

0.10

<~0.08

LD~0.06

CL0.8iiiii ••• i•••••• ~ ~ ~

0.04

0.02

0.40

0 2 4 6 8 10CL a, deg-J Theory, Complete-J

Delta Wing0

M 8.1 Data

-0.4

a, deg

dl. Lift

Fig. 11 Continued

-0.8o 20 40 60 80 100 120 140 160 180

»mo·n,-lo:;0,0­~

;..,LfI

-JD:l

1.6

1.2

CD

0.8

0.4

Theory, Complete Delta Wing

~~---.

75°

LD~

M = 8.1 Data

0.03

CD

0.02

0.01

oo 2 4 6

a., deg-8 10

>mo(),~

o;u

0­.t:>-

I

t-..)

111

oo 20 40 60 80

a., deg

100 120 140 160 180

e. Drag

Fig. 11 Continued

-Jco

, \

2

1

LID

o

-1

M = 8.1 Data00

Theory, CompleteDelta Wing

I >,\'r .,r.. t ,_. .

-0"

Fig. 11 Continued

f. Lift-to-Drag Ratio

40-2

o 20 60 80 100

U, deg

120 140 160 180»mo(),-io;uc,.,l:>.,,"",(,\

-0.4

tl

~ -0.3"t:l

c:l1$-I

$-IQ)0.

co ...o en. -0.2

~C)

-0.1

Theory, CompleteDelta Wing

Wing

Leading Edge

,Nose

~~

»mo().-to:AJI0-

~....,0'1

00 20 40 60 80 100 120 140 160 180

a., deg

g. Side-Force Derivative, f3 = 0

Fig. 11 Continued

• I

'""

• f " .... .t "t

0.04

_' t~ '1

Wing

I A,.J

~ •• f ""'fI- III ....

O:l~

c:~

.r-!"0~~

~Q)0.

(llc:

u

0.02

o

Theory, Complete Delta Wing

No'se

Leading Edge

+J

-0.02o 20 40 60 80 100 120 140 160 180

Ct. deg

h. Yawing-Moment Derivative, f3 = 0, Referenced to 0.6 Lo

Fig. 11 Continued

>mon.-lo;uI

0­J:>..

tVVI

CX)[\j

-0.16

d~~ -0.12-0~M

M(1)04

"'en. -0.08~

C)

-0.04

Theory, Complete Delta Wing

Wing

Leading E9ge

Nose

:-.......

}>

mon.-Io::::0.0-

t-"->tn

oo 20 40 60 80

a, deg

100 120 140 l60 180

- ""

,..

i. Rolling-Moment Derivative, f3 = 0

Fig. 11 Concluded

...