Aerodynamics

AND THEIR VALIDATION IN

HYPERSONIC FLOWS

AND THEIR VALIDATION IN

HYPERSONIC FLOWS

AK SREEKANTH

Defence Research & Development Organisation Ministry of Defence

New Delhi – 110 011 2003

AERODYNAMIC PREDICTIVE METHODS AND THEIR VALIDATION IN HYPERSONIC FLOWS

AK SREEKANTH

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SREEKANTH, A.K.

Aerodynamic predictive methods and their validation in hypersonic flows.

DRDO monograph series. Includes index and bibliography. ISBN 81-86514-11-2

1. Aerodynamics 2. Hypersonic flows I. Title (Series) 629.132.306.072

© 2003, Defence Scientific Information & Documentation Centre (DESIDOC), Defence R&D Organisation, Delhi-110 054.

All rights reserved. Except as permitted under the Indian Copyright Act 1957, no part of this publication may be reproduced, distributed or transmitted, stored in a database or a retrieval system, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

The views expressed in the book are those of the author only. The editors or publisher do not assume responsibility for the statements/opinions expressed by the author.

Preface xi

Acknowledgement xiii

PART - I AERODYNAMIC PREDICTIVE METHODS IN

HYPERSONIC FLOWS 1

CHAPTER 1

AERODYNAMIC PREDICTIVE METHODS IN HYPERSONIC

FLOWS 3

CHAPTER 2

METHODS 5

2.1 Introduction 5

2.2 Newtonian Theory 6

2.3 Modified Newtonian Theory 7

2.4 Embedded Newtonian Flow 8

2.5 Newtonian & Prandtl-Meyer Mode1 10

2.6 Tangent Wedge & Tangent Cones 13

2.7 Tangent Wedge, Tangent Cone & Delta

Wing Empirical Method 14

2.8 OSU Blunt Body Method 17

2.9 Hankey Flat Surface Empirical Method 17

2.10 Dahlem-Buck Empirical Method 17

2.11 Blast Wave Pressure Increments 18

2.12 Shock Expansion Theory 18

2.12.1 First Order Theory 18

2.12.2 Second Order Shock Expansion Theory (SOSET) 19 2.13 Blunt Bodies of Revolution at Small Angles of Attack 21

2.14 Van Dyke Unified Theory 24

2.15 2-D Airfoil Theory in Hypersonic Flows 25

2.16 High Mach Number Base Pressure 30

CHAPTER 3

AERODYNAMIC CHARACTERISTICS OF VEHICLE

COMPONENTS 33

3.1 Introduction 33

3.2 Body-Alone Aerodynamics 34

3.2.1 Forces & Moments on the Body 34

3.2.2 Axial Force 37

3.2.3 CA_f – Skin Friction Coefficient 37

3.2.4 C_A

b – Base Pressure Coefficient 39

3.2.5 Determination of C_{A N}, the Axial Pressure

Coefficient of Nose Portion of the Body 40

3.2.5.1 Pointed Cone 40 3.2.5.2 Pointed Ogive 41 3.2.5.3 Hemispherical Nose 41 3.2.6 Normal Force 41 3.2.6.1 Pointed Cone 41 3.2.6.2 Pointed Ogive 42 3.2.6.3 Hemispherical Nose 42 3.2.6.4 Cylinder 42

3.3 Allen & Perkins Viscous Cross Flow Theory 43

3.4 Moments 43

3.4.1 Pointed Cone 44

3.4.2 Pointed Ogive 44

3.4.3 Hemisphere 45

3.4.4 Circular Cylinder 45

3.5 Wing Alone Aerodynamics 45

3.5.1 Hexagonal Shape Wing Section 45

3.5.1.1 Axial Force 47

3.5.1.2 Normal Force 50

3.5.1.3 Axial Component of the Rudder 51

3.5.1.4 Normal Component (Wing or Rudder) 51

3.5.1.5 Pitching Moment 52

3.5.2 Other Wing Sections 52

3.5.2.1 Airfoil Characteristics by 2-Dimensional

Hypersonic Airfoil Theory 52

CHAPTER 4

SKIN FRICTION FORCE CALCULATION 61

4.1 Introduction 61

4.2 Sommer & Short Method 62

4.3 Van Driest-II Method 63

4.4 Spalding & Chi Method 64

4.5 Empirical Equations 65

References 66

CHAPTER 5

AERODYNAMIC HEATING AT HYPERSONIC SPEEDS 67

5.1 Introduction 67

5.2 Heating Analysis 67

5.3 Stagnation Point Heat Transfer 69

5.3.1 Spherical Nose 69

5.3.2 Cylinder Normal to the Stream 72

5.3.3 Swept Wing Stagnation Line Heat Transfer 72

5.3.4 Perfect Gas 73

5.3.5 Real Gas 74

5.3.6 Heat Transfer Coefficient h 75

5.3.7 Heat Transfer on Flat Surfaces and Fuselage Panels 77 5.4 Heat Transfer Analysis by the Method of Quinn & Gong 80

5.4.1 Stagnation Point Heating Rate 80

5.4.2 Convective Heating Equation for Small or Zero

Pressure Gradient Surfaces 83

5.4.3 Boundary Layer Transition 86

5.5 High Speed Convective Heat Transfer Methodology of

Tauber 87

5.5.1 Stagnation Point Heat Transfer 88

5.5.2 Swept Infinite Cylinder 88

5.5.3 Cone & Flat Plate Heating Rate 89

5.5.3.1 Laminar Boundary Layer 89

5.6 Empirical Equation for Convective Heat Transfer 90

5.6.1 Stagnation Point 91

5.6.2 Flat Plate in Laminar Flow 91

5.6.3 Flat Plate in Turbulent Flow 91

PART - II VALIDATION OF PREDICTION METHODS 93 CHAPTER 6

VALIDATION OF PREDICTION METHODS 95

6.1 North American X-15 Research Aircraft 110

6.1.1 Walker & Wolowicz’s Work 115

6.1.2 Lift Characteristics 115

6.1.3 Wing 116

6.1.4 Horizontal Tail 118

6.1.5 Fuselage 118

6.1.6 Pitching-Moment Characteristics 119

6.1.7 Wing & Horizontal Tail 119

6.1.8 Fuselage 128

6.1.9 Maughmer et al. Analysis of X-15 128

6.2 Hypersonic Research Airplane 139

6.3 Space Shuttle Orbiter 156

6.4 Conclusions 158

References 169

PART - III AERODYNAMICS OF RAREFIED GASES 173 CHAPTER 7

AERODYNAMICS OF RAREFIED GASES 175

7.1 Introduction 175

7.2 Free Molecule Flow Analysis 177

7.2.1 Surface Interaction Parameters 177

7.2.2 Forces on an Surface Element in Free Molecule Flow 179 7.3 Aerodynamic Forces for Typical Bodies 187

7.3.1 Flat Plate 187

7.3.2 Infinite Right Circular Cylinder at an Angle of Attack,α 191

7.3.3 Sphere 194

7.3.4 Cone Frustrum 195

7.3.5 Spherical Segment 198

7.4 Aerodynamic Forces in Slip & Transitional Flows 200 7.5 Energy Transfer in Free Molecule Flow 203 7.5.1 Equilibrium Temperatures for Simple Shapes 207

7.5.2 Heat Transfer for Typical Bodies in Free Molecule Flow 208 7.5.3 Heat Transfer in Slip & Transitional Flow Regimes 212

References 213

Appendix 215

This monograph presents a summary of engineering methods most commonly employed for preliminary aerodynamic analysis of bodies travelling at hypersonic speeds. To the extent possible, an attempt has been made to make the present work self-sufficient. However, references are cited if one is interested in the source or more details.

The work is in three parts. Part 1 deals with Predictive Methodology, Part 2 covers Validation of Prediction Methods and Part 3 the Aerodynamics of Rarefied Gases.

Secunderabad AK Sreekanth

The writing of this monograph has been made possible by the financial assistance received from the Defence Scientific Information and Documentation Centre (DESIDOC), Ministry of Defence, Government of India, New Delhi.

The author would like to place on record his sincere thanks and appreciation to the following persons.

1. Prof. M. Maughmer, Department of Aerospace Engineering, The Pennsylvania State University, University Park, PA. U.S.A. for permission to freely use the figures and material from the thesis of his student L.P.Ozoroski and from the NASP Contractor Report 1104.

2. Dr. J.Agrell, Head of Experimental Aerodynamics Department, FFA, Sweden, for permission to include material in the monograph from the FFA Technical Note AU-1661.

3. Mr. Dan Pappas, Chief Librarian, NASA Ames Research Center, Moffett Field, CA. for allowing me to use the Ames Library freely.

AERODYNAMIC PREDICTIVE METHODS

IN

AERODYNAMIC PREDICTIVE METHODS IN

HYPERSONIC FLOWS

1.1 INTRODUCTION

The conceptual design of an efficient hypersonic cruise vehicle or a missile requires a detailed knowledge of how various geometrical configuration parameters affect the aerodynamic performance of such a vehicle. Besides, it is desirable to have the ability to compare one configuration’s performance with another in a relatively short amount of time. During the preliminary design phase involved in arriving at feasible configurations for a specified mission, simple engineering-type empirical and semi-empirical methods are invariably employed. The expensive and time-consuming wind tunnel tests and sophisticated computational techniques are reserved for possible designs evolved from the preliminary analysis.

A variety of engineering methods applicable to flows at hypersonic Mach numbers have been reported over the years in open literature. Each of these methods works well on very specific types of components. Therefore, it is necessary to choose a combination of these methods to analyse the complete vehicle made up of various components, such as body, lifting, and control surfaces.

The present work is a compilation of some of the well-known prediction methods, their applicability and limitations. Examples of the application of a few of these methods to calculate aerodynamic parameters of some specific components of vehicle configurations have been made and the results presented. Some published work on the aerodynamic characteristics of a few of hypersonic configurations, their predictions and comparison with

experimental data are discussed in Part II(Chapter 6) of this monograph, to illustrate the applicability and validity of the approximate methodology.

A majority of the methods for calculating the pressure forces in hypersonic flow are based on non-interfering constant pressure finite element analysis. The geometry of the configuration is represented by a system of quadrilateral panels. The only parameter required to calculate the pressure is the impact angle of the free stream flow with the panel or the change in impact angle from one panel to another. The surface elements that see the oncoming flow directly, are said to be in the impact region and the others, either shielded by the front portion or other surrounding elements, are in the shadow region. Depending upon whether the element is in the impact or shadow region, the appropriate method is chosen for the analysis. Body components are typically broken into separate analysis regions. The forward most body component may have a nose and body region. The rear most body consists of a body region and probably a blunt base. In each of these separate regions, an analysis method must be chosen for the impact flow region and a suitable one for the shadow region. Similar division is also done for the lifting and control surfaces. viz., a leading edge region, a surface (mostly flat) region and a blunt region if the trailing edge is blunt. Most commonly used methods are listed in Table 1.1.

Table 1.1. List of most commonly used methods

Imwact Flow Shadow Flow

Newtonian and Modified Newtonian Newtonian (Cp=O)

Embedded Newtonian Modified Newtonian + Prandtl-Meyer Modified Newtonian + Prandtl-Meyer Prandtl-Meyer from free stream Tangent wedge and Tangent cone OSU blunt body empirical '

Tangent wedge-Tangent cone Van Dyke unified

Imwact Flow Shadow Flow

OSU blunt body empirical High Mach number base pressure

Van Dyke unified Shock expansion

2-Dim. Hypersonic airfoil theory Rarefied gas flow Shock-expansion

Input pressure coefficient Hankey flat surface empirical Delta wing empirical

Dahlem-Buck empirical Blast wave

Rarefied gas flow

Brief reviews of some of the above listed methods are discussed in following pages.

2.2 NEWTONIAN THEORY

Newtonian theory is a local surface inclination method. In this, the pressure coefficient depends only on the local surface deflection angle and not on any other aspect of the surrounding flow field.

Newton originally assumed that the medium around a body was composed of identical non-interfering independent particles. When these particles collide with the surface they lose their normal component of the momentum resulting in a pressure force on it. After collision, the particles move along the surface with their tangential component of the momentum unchanged. The regions of the body that do not see the oncoming particles directly are said to be in the shadow region and the pressure coefficient in these regions are normally set equal to zero.

Figure 2.1. Newtonian theory

The normal component of the velocity is

V,

sin

6 . The m a s s of particles striking the surface area A in unit time is

particles is pm

v,2

sin2 6 . This momentum is transferred to the

surface element and acts a s the normal pressure force.

If F / A is interpreted a s the difference in pressure above the free stream, we have

For blunt bodies in a high Mach number flow, the surface pressure is fairly well predicted by the above Newtonian theory. I t is to be observed that according to the Newtonian theory the pressure coefficient is independent of the Mach number, so long a s the flow is hypersonic.

2.3 MODIFIED

NEWTONIAN THEORY

In the Modified Newtonian theory (MNT), the pressure coefficient is written a s

Various values of

K

have been suggested depending on the Mach number, body shape, angle of attack and ratio of specific heats. The most common one is

K

=

Cps

where

Cps

is the stagnation pressure coefficient behind a normal shock. For this case we have

and mainly used for blunt bodies. For

Y

= 1.4 and M, +co,

K+

1.839andfor

Y =

1.0and

M,

-+

co, K j 2 . 0 .

For pointed cones and ogives, the suggested1 value for

K

is

where, k = 1 for cones and k = 0 for ogives, d = body diameter, L,= nose length and a = angle of attack.

For a hemisphere1,

For real gas flows,

for

Y =

1.4, K-2.083,andfor

Y

= 1 , K = 2 . 0 .

Although the Newtonian and Modified Newtonian theories are mainly applicable for blunt body flows, attempts have been made to see whether the same form of modified equation could be made applicable to flat surfaces such a s wings. One such suggested relation by Hankeyqs

0.3925 K =1.95

+

M : . ~

tan

6

where, 6 is the local flow inclination angle. The above expression is applicable in the Mach number range of 2 to 22 and angles of attack from 10" to 90" for surfaces with highly swept leading edges.

For surface inclinations below 10 degrees, particularly for wings, Newtonian theory is not applicable. Interaction and induced pressure effects also become dominant a t low angles of attack requiring a different methodology.

2.4 EMBEDDED NEWTONIAN FLOW

For bodies having compression corners on the surfaces such a s flares or flap controls, the Newtonian theory may not correctly predict the pressure on these surfaces a s there may be a oblique shock in front of the ramp if the local flow is supersonic a s shown in the Fig. 2.2.

Newtonian theory assumes that the bow shock wgve in front of the body wraps around the body very closely thereby not giving rise to secondary shock that might be normally present in local supersonic regions. For configurations like this, a method

9

EMBEDDED

-

SHOCK FLARE (a)

-

EMBEDDED SHOCK FLAP (b)

Figure 2.2. Embedded Flows

has been suggested by Sieffj in which the flow over the ramp is viewed a s a n embedded Newtonian impact flow if the flow is not extensively separated and the ramp sh0c.k wave is thin, with conditions along the surface of the secondary shock wave a s initial conditions. According to this postulation, the pressure on the ramp surface is given by the following expression:

P2

-

PI

= PI ( ~ 1 sin

dl2

where, the subscript 1 refers to conditions along the front surface of the ramp shock wave, which are the initial conditions for the application of Newtonian theory on the ramp surface. The above relation can be expressed in the form of pressure coefficient based on free stream static and dynamic pressure conditions,

viz.,

41

= +

-

' P 2 Newtonian

P2

where, Cp,,,O,i, is the pressure coefficient given by the usual Newtonian Impact theory and q the dynamic pressure.

For the application of the above formulation one needs to know the properties of the stream that is incident on the ramp surface. Towards this, one can utilize the methods presented in Sieff,

et

~ l . , ~ and Maslen,

et

~ l . , ~ or any other known procedures.

2.5 NEWTONIAN 88 PRANDTL-MEYER MODEL

In this flow model applicable to blunt bodies with a detached shock, it is assumed that the flow expands around the body surface to supersonic conditions isentropically from the stagnation point. Modified Newtonian theory is combined with Prandtl-Meyer Expansion theory. The technique involves matching the Modified Newtonian and Prandtl-Meyer Expansion methods a t the point where the pressure gradients calculated using each method are equal. Downstream of this point the Prandtl-Meyer Expansion theory is used6. The calculation procedure is a s follows:

Calculate the ratio of the free stream static to stagnation pressure behind a normal shock.

According to Newtonian theory

L

I

At the stagnation point 6 = 90"

Y M , ~ K

Po, Po,

The slope of this pressure curve is

d

PIP^,)

dS = 2 (1-P) sin 6 cos 6

(2.9)

according to Newtonian theory.

For an isentropic expansion downstream of the stagnation point, on the blunt nose, we have

P -;

[

Po, 2 + ( y - 1 ) k f 2

The Prandtl-Meyer angle for expansion from sonic flow to supersonic Mach number M is

v

=Jz

tan-

Jw)

- tan

d $ Z ]

_(2.14)

The rate of change of pressure w.r.t .the Prandtl-Meyer angle v is

d ( P / P o , )

-

-

-7 M 2 ( P / P O 2 )

( 2.15)

dv

dM2-1

Equating the expressions for the pressure gradients (Eqns. 2.12 and 2.15), noting that dv =

-

d6

y k f : ( P 9 / P o , )

= 2(1-P) sin 6cos 6

( M ~ z - I ) ~

where the pressure distribution slopes are equal. The location of this point can be easily determined once the value of

Mq

is known.

Eliminating 6 in Eqn. 2.16 by using Eqn. 2.1 1, we have

where

For a given free stream conditions, the value of P is known and it is necessary to solve the Eqn. 2.17 by an iterative process. This is done a s follows:

Assume a starting value for the matching Mach number

Mq,

say 1.30. For this value calculate Q using Eqn.2.18. Calculate P from Eqn. 2.17. Assume a new matching Mach number say 1.70 and repeat the above steps to get a new value of P. A linear interpolation between these two calculated values is made to get a new matching Mach number corresponding to the actual value of P given by the free stream conditions (Eqn. 2.6). This process is repeated until the solution converges. The location of the matching point is easily determined once

Mq

is known.

From Eqn. 2.11

The pressure at the matching point, in terms of the free stream static pressure and the ratio P is simply

Starting from the matching point, the pressure on the body surface downstream is calculated by the Prandtl-Meyer theoretical relationship. It is found that the use of Prandtl-Meyer relations from the matching point onwards gives a betfer correlation with experimental data and exact theories than by using the sonic point a s the starting point for use of the Prandtl- Meyer relation.

2.6 TANGENT WEDGE 8s TANGENT CONES

Although not based on any theoretical grounds, it is found that the tangent wedge and tangent cone methods give reasonably accurate results at hypersonic speeds. Tangent wedge theory determines the pressure at each point by calculating the pressure on a wedge of the same half angle a s the local inclination angle at the point. The pressure on the equivalent wedge is found by using the oblique shock theoretical relations at the free stream Mach number of

Ma.

In a similar manner, tangent cone method uses an equivalent cone at each point to calculate the pressure on axisymmetric bodies.

It is found that the tangent wedge method works well for airfoils with sharp leading edges and the tangent cone method for bodies with pointed noses.

TANGENT WEDGE

I

Figure 2.2. Tangent wedge method

When applying the tangent cone method to bodies of revolution or equivalent bodies of revolution, sometimes it is convenient to use an empirical equation for the pressure coefficient rather than the use of Sims conical flow tables. The suggested equation7 for the pressure coefficient is

which is a function of the Newtonian impact angle

8

and a so called effective Mach number normal to the shock, Mns. The angle 6 is defined a s the smallest angle between the free stream direction and the tangent to the vehicle surface a t the point of interest. The above equation is a physical representation of Cp for %-dimensional oblique theory when the actual Mach number normal to the shock is employed with

Y

= 1.4. The suggested effective Mach number Mns is

M,, = ( 0 . 8 7 ~ ~

-

0.554) sin6

+

0.53 (2.22)

which is only a function of free-stream Mach number and Newtonian impact angle. For impact angles up to 30°, the deviation from the Sims tabulated conical flow table values is less than

*

5% when the above expression is used for all Mach numbers above 1.5.

2.7 TANGENT WEDGE, TANGENT CONE 8a DELTA

WING EMPIRICAL METHOD

Experiments on large surfaces of blunt, highly swept delta wings in hypersonic flows have shown the following trend. For angles of attack between 5" and 15" the tangent wedge theory appears representative of the mean data. For angles of attack above 15", the flow appears to change in nature such that the tangent cone approximation appears valid up to 40" angle of attack. Probably based on this, an approximate method has been reported by Gentry, et. aL8, the details of which are a s follows: For a wedge, the shock angle is given by

sin 8,

sin 0, =

and

for a cone (thin shock layer approximation) sin 6 ,

sine, = ,

where, E = ( P

,/

P ,) the density ratio across the shock given by

In the limit

as

M ,

-+

co

The limiting values of the shock angles are:

y + 1

For a wedge: sine, =

-

sinti, 2

2

6

+ I )

For

a

cone: sin 0

,

= ---- sin 6

,

Y

+ 3

From Eqns. 2.23 and 2.24

Far a wedge

and,

for a cone

The parameter (8

-6

) is approximately constant and independent of the Mach number indicating that

Mm

is a function of

M sin

F

only. For calculation purposes we need a relationship between Mns and

M r

sin

F

that satisfies the following requirements:

(a) Shock detachment is neglected (b) A t M s i n g = O , M r l S = l

(c) The solution asymptotically approaches the

M,

+

cc value

4%

)

a t M I ;

sin&

= 0 (d) Has the correct slope, d ( ~ ,

These conditions lead to a relationship of the form

-(K,+w,, sins,)

M,,

= K,,M,sind, t

e

_{/ 2}

for a wedge, and

sin&)

M,, = K , M , sins,

-

e for a cone, where

y + l

K,

=

-

and

K C

= 2 (Y + 1)

3

-

(Y

+ 3)

The pressure coefficient may now be obtained by the following relationships for a wedge and a cone respectively,

As mentioned earlier, a t low angles of attack, the centre line pressure distribution on a delta wing agrees well with the 2-Dimensional theory (wedge flow) and a t higher angles of attack with the conical flow theory. A s such, the relationships for

Mns a s given above for a wedge and a cone can be combined to yield a relation

It has been observed that the value of Cp a s given by the above expression with the value of Mns given by Eqn. 2.33 correlates well with the experimental data of pressure distribution on the centre line of delta wings.

2.8 OSU BLUNT BODY METHOD

The Ohio State University (OSU) Blunt Body Empirical equationg predicts the pressure distribution around circular cylinders in supersonic flows. The suggested expression is

where p, is the surface pressure, P o , is the stagnation pressure through the normal shock and 0 is the peripheral angle on a

cylinder (= 0 at the stagnation point). The pressure coefficient is given by

where p is the free stream static pressure and ( P 0 , / P

,

) from

the normal shock relations.

2.9 HANKEY FLAT SURFACE EMPIRICAL METHOD

This method is mainly used for estimating the lower surface pressure on blunt flat plateslO. It approximates tangent wedge a t low impact angles and approaches Newtonian method at high impact angles.

The pressure coefficient is given by

2.10 DAHLEM-BUCK EMPIRICAL METHOD

The suggested Dahlem-Buck E;mpirical metgod" approximates the tangent cone at low angles of attack and Newtonian at high angles.

[ I

+

~ i n ( 4 6 ) ~ / ~ ] s i n ( 6 ) ~ / ~

C, =

[4 cos 6 cos (26)] 3'

(2.39)

Above 22.5", the pressure coefficient is given by Cp = 2 sin2 6

2.11

BLAST WAVE PRESSURE INCREMENTS

Bluntness of the body and the lifting surfaces gives rise to a n over pressure. This additional amount must be added on to the pressure calculated by the various methods like tangent wedge, tangent cone, Newtonian, etc. According to the blast wave solution given by Lukasiewicz and quoted by Gentry,

et.

aL8, the pressure distribution downstream of the nose a s a function of x is given by

where

C, is the nose drag coefficient

d is the nose diameter or thickness and

x is the distance from the nose stagnation point

and the values of coefficients A, B and nose drag coefficients are a s follows:

Flow CI J A B

2.12

SHOCK EXPANSION THEORY

2.12.1

F i r s t Order T h e o r y

For bodies flying a t high supersonic speeds, Eggers,

et

all2-l4 and, Savin15 proposed a theory known as the Generalized Shock Expansion theory. I n this, the flow parameters immediately downstream of sharp nosed bodies are calculated using either the oblique shock relations for 2- dimensional bodies or the conical shock relations for ,axi- symmetric bodies. Downstream of the leading edge, the body surface is replaced by a tangent body composed of conical segments a s shown in the Fig. 2.4.

,LEADING EDGE SHOCK

TANGENT BODY

COMPOSED OF SHOCK WAVE

CONICAL SEGMENTS

Figure 2.4. First order theory

The change in the local surface slope in going from one tangent segment of the body or the airfoil to another tangent segment is determined and for this change, the Prandtl-Meyer relations are used to calculate the flow properties. It is assumed that the pressure is constant along each segment. Inherent in this theory is that the expansion waves created at each change of slope are absorbed by the shock and are not reflected back. Since the theory assumes that the pressure is constant along each conical tangent elements of the surface, the body should be slender or else one has to consider a large number of elements to obtain accurate pressure prediction.

2.12.2 Second Order Shock Expansion Theory (SOSET)

The previously outlined first order theory was extended by Syvertson, et all6 by defining the pressure along a conical frustum by a relation

instead of constant pressure along each segment. In the above expression, pc is the pressure on the conical segment, a s given by the conical flow over a cone of half angle equal to the slope of the conical segment with respect to the body axis of symmetry. p, is

the pressure just aft of the conical segment a s calculated from the Prandtl-Meyer relation from known values in region 1 (Fig. 2,5).

MACH LINES

t X

Figure 2.5. Flow about a frustum element

where, r is the radial coordinate in the (x,r, p ) cylindrical

coordinate system with origin at the nose, and

For negative angles such a s would occur on a boat tailed body, pc is replaced by p , . If q becomes negative, the Second Order theory is replaced by the First Order theory. This is because the Eqn 2.41 will not give the correct asymptotic cone solution for negative values of q

.

It has been observed that the Second Order Shock Expansion theory predicts fairly accurately the pressure distribution on the surface of the body at low to moderate angles of attack and the Mach number greater than 2.0.

The above relations give the coordinates of the streamlines in the wind axis system and these coordinates are used to generate bodies of revolution for each radial angle

4

of interest.

Y

Figure 2.6. The axis system

Bodies of revolution thus generated for several angles of

4

are shown below.

EQUIVALENT BODIES

Figure 2.7. Typical equivalent body shapes

The zero angle of attack method (Newtonian plus second order shock expansion) is applied to the equivalent bodies thus generated from the transformed streamlines to get the pressure distribution. There are two limitations in this method. First, the angle of attack should be low enough so that the stagnation point remains on the spherical surface and secondly, the angle of attack

AC, = - ( 2 a ) s i n ( 2 ~ ) s i n ( ~ ) + ( ~ c o s ~ 6 )

a'

+

[ ( 4 / 3 ) sin(26) sin

(o)]

. .

a'

(2.49)

where

(2a)

sin

(26)

sin(@)

AC _P = -

3 (2.50)

Eqn 2.49 is used for pointed body configurations a s well a s for blunt body configurations in the windward plane area, 60" c

4

5 180". For the leeward plane area on blunt bodies Eqn 2.49 is replaced by Eqn 2.50. In the above equations p = (M2

-

1)l! ; is the local surface

slope of the body with respect to body axis and

4

is the position on the body surface with

4

=

+

90' being the vertical plane and

4

= - 90" corresponds to leeward plane.

2.14 VAN DYKE UNIFIED

THEORY

A method based on the hypersonic small disturbance theory applicable to both the supersonic and hypersonic flow regimes was proposed by Van Dyke and is known a s the Unified Supersonic Hypersonic Flow theorylg. The det.ails of the derivation based on the similarity conditions are given by Shapiro20. The next section also gives the derivation of the pertinent equations which are further simplified. For small deflection angles a t high Mach numbers, the pressure coefficient on a compression surface is given by

where, H i s the hypersonic similarity parameter given by H = ' M6,

and 6 is the thickness ratio. The above relation can also be applied to supersonic flows if the hypersonic similarity parameter

A similar analysis has been applied to the high Mach number flow on a surface in expansion flow with no leading edge shock wave a s in the case of a flat plate at an angle of attack. The resulting expression20 is

A s before, the similarity parameter H =

m1

S for applicability in both supersonic and hypersonic flow

2.15

2-D

AIRFOIL THEORY IN HYPERSONIC FLOWS

It is possible to derive a much simpler expression for the surface pressure coefficient than those given by Eqns 2.5 1 and 2.52 above, based on additional assumptions, justifiable in hypersonic

flows over thin 2-dimensional bodies at small angles of a t t a ~ k ~ l - ~ ~ . Details of the analysis are

2 P - P m

the pressure coefficient _C

,

_-

m

j

Y M :

Let the subscript 2 denote the conditions immediately behind a n oblique shock wave, there follows

From the oblique shock relations we have

2 Y + l 1

sin

P

=

-

_{4 CP2}

+

-

and

(c~2;"")2 =

[

(

1 - - 'i2

)

sin

P

tan

e2

I'

where p is the shock angle and

e

the flow deflection angle. For thin airfoils in hypersonic flow, we can approximate a s follows:

The Prandtl-Meyer expansion is isentropic, hence we have the relation,

Combining the Eqns. 2.58 and 2.59 2nd neglecting higher order terms we have

The above is equivalent to

The surface pressure coefficient is

Making use of the Eqns. 2.53 and 2.61, the above can be

expressed as

Substituting for 'pW2 from Eqn. 2.57, the above equation

The above equation leads to two cases viz., no shock and

no expansion.

Let u s consider the no shock case. For this Ow, = 0.

Hence, M w ? = bfw

The above equation can be used to calculate the pressure coefficient on the upper surface of a flat plate at an

angle of attack of rr=8,. The case of no expansion after the shock implies that Q = 0. Hence,

The

above corresponds to the lower surface of a flat plate at an angle of attack

ow,

= a .

The lift coefficient of an airfoil of chord c is given by

For the flat plate case, at an angle of attack of

a ,

the lift coefficient can be obtained by the use of Eqns. 2.65 and 2.66, viz.,

A binomial expansion of the square root term of Eqn 2.69 results in

(2.72) The above Eqns. 2.71 and 2.72 are identical up to the first two terms and differs from each other in the third term by 10 per cent. It is reasonable therefore to assume that the Eqn. 2.7 1 is applicable to both the shock and expansion processes. This assumption is equivalent to neglecting entropy change across the shock. Both compression and expansion are considered a s isentropic. The pressure coefficient a t any point on the surface of a n airfoil is given by

Where, E is the thickness ratio of the airfoil and

K E

M,

E the well known hypersonic similarity parameter.

The above equation can be integrated for most of the thin airfoil shapes to give closed form solutions for the aerodynamic coefficients for various airfoils. Results of these calculations for the determination of aerodynamic characteristics of various types of airfoil shapes commonly encountered are g i ~ e n ~ ' . ~ ~ and they are reproduced in section 3.2.5

2.16 HIGH MACH NUMBER BASE PRESSURE

The base region of a hypersonic vehicle will invariably be in the shadow region. Further a t hypersonic Mach numbers the expansion of the flow from the body surface to the base region will be such that the base portion will be in a vacuum environment. For this condition the pressure coefficient at the base is given by:

However, due to viscosity and real gas effects some pressure is felt in the base region and according to some experiments for air this value is approximately 70 per cent

vacuum. Based on this, the base pressure coefficient can be taken as

REFERENCES

Weibust, Erling. Status report on the FFA version of the missile aerodynamics program LAIZV, for calculation of

static aerodynamic properties and longitudinal

aerodynamic damping derivatives FFA. The Aeronautical Research Institute of Sweden, Stockholm, 198 1. TN-AU-

1661.

Hankey, J r . , W.L. & Alexander, G.L. Prediction of hypersonic aerodynamic characteristics for lifting vehicles. WPAFB, Ohio, September 1963. ASD-TDR-63-668.

Sieff, A. Secondary floml fields embedded in hypersonic shock layers. NASA, May 1962. TN-D-1304.

Sieff, A., & Whitting, W.E. Calculation of flow fields from bow-wave profiles for the downstream region of blunt-nosed circular cylinders in axial hypersonic flight. NASA, 1961. TN-D- 1147.

Maslen, S.H., & Moeckel, W.E. Inviscid hypersonic flow past blunt bodies. J. of Aero.

Sci.,

1957, 24(9), 683-89.

Kaufman-11,

L.G.

Pressure estimation techniques for hypersonic flows over blunt bodies. J. of Aero. Sci., 1963, lO(2).

Pittman, J.L. Application of supersonic linear theory and hypersonic impact methods to three nonslender hypersonic airplane concepts at mach numbers from 1.10 to 2.86. NASA, December 1979. TP- 1539.

Gentry, A.E.; Smyth, D.N. & Oliver, W.R. The Mark IV supersonic-hypersonic arbitrary-body program. WPAFB, Ohio, November 1973. 1 1 p. AFFDL-TR-73- 159.

Gregorek, G.M., Nark, T.C. & Lee, J.D. An experimental investigation of the surface pressure and the laminar boundary layer on a blunt flat plate in hypersonic flow, Vol 1. March 1963. ASD-TDR-62-792.

Hankey

,

J r

.

,

W .L. Optimization of lifting re-entry Gehicles. March 1963. ASD-TDR-62- 1 102.

Dahlem, V. & Buck,

M.L.

Experimental and analytical investigations vehicle designs for high lift-drag ratios in hypersonic flight. J u n e 1967. AFFDL-TR-67- 138.

Eggers, A.J.; Sjvertson, C.A.; & Kraus, S.A. A study of inviscid flow about airfoils a t high supersonic speeds. NACA Report, 1953. TN-1123.

Eggers, A.J. & Savin, R.C. A unified tw-o-dimensional

approach to the calculation of three-dimensional

hypersonic flows with applications to bodies of revolution.

NACA Report, 1955. TN- 1249.

Eggers, A . J . & Savin, R.C. Approximate methods for calculating the flow about nonlifting bodies of revolution a t high supersonic airspeeds. NACA, 195 1. TN-2579.

Savin, R.C. Application of the generalized shock expansion method to the inclined bodies of revolution travelling at high supersonic airspeeds. NACA, 1955. TN-3349.

Sqvertson, C.A. & Dennis, D.H. Second order shock expansion method applicable to bodies of revolution near zero lift. NACA, 1957. TR-1323.

Jackson, C.M.; Sawyer,

W.C.

& Smith, R.S. A method for determining surface pressures on blunt bodies of revolution at small angles of attack in supersonic flows. NASA, 1968. TN P-4865.

Dejarnette, F.R.; Ford, C.P. & Young, D.E. A new method for calculating surface presures on bodies at an angle of attack in supersonic flow. Paper presented a t AIAA 12TH Fluid and Plasma Dynamics Conference, July 1974, Williamsburgh,

Va., AIAA Paper No. 79- 1522.

Van Dyke,

M.D.

A study of hypersonic small-disturbance theory. NACA, 1954. Report No. 1194.

Shapiro, A.H. The dynamics and thermodynamics of compressible fluid flow. The Ronald Press, Vol. 2, 1953, pp. 753-754.

Kaufman-11, L.G. & Scheuing, R.A. An introduction to hypersonics. Grumman Aircraft Engineering Corporation, August 1960. Research Report RE-82.

Linnel, R.D. Two-dimensional airfoils in hypersonic flows

JAS, 1949, 16(1).

Dorrance, W.H. Two-dimensional airfoils at moderate hypersonic velocities. JAS, 1952, 19(9).

AERODYNAMIC CHARACTERISTICS OF

VEHICLE COMPONENTS

3.1 INTRODUCTION

Having identified the method or methods to calculate the local pressure on an elemental area of a vehicle component based on its geometry, the forces and moments experienced by that component can be obtained by integrating the pressure and moment over the entire surface. To determine the aerodynamic characteristics of the complete vehicle, it is the usual practice to divide the general configuration into simple basic components such as nose, body, lifting surface, control surface, etc. By summing the aerodynamic characteristics of the isolated individual components and the effect due to the interference of one component on the other, the complete vehicle characteristics are determined. For example, the normal force coefficient of a vehicle can be expressed as

C_N = C_{N B} + C_{N W} + C_{N T} + ∆CNB W( ) + ∆CNB T( ) + ∆CNW B( ) + ∆CNT B( ) +

∆C_N

T W( ) + ∆CNW T( )

The subscripts B, W, and T refer to the body, exposed wing and exposed tail, respectively. The terms ∆CN_{B (W )} , ∆CN_{W (B)} , etc. represent the interference effects of the exposed wing on the body, of the body on the exposed wing, etc.

The forces and moments are normally expressed in coefficient forms referred to either in the body-fixed coordinate system (the normal C_N and axial C_A, force coefficients) or the wind oriented coordinate system (the lift (C_L) and drag (C_D) force coefficients). These can be converted from one system to the other by the following relations.

C_L = C_N cosα – C_A sinα C_D = C_A cosα + C_N sinα C_N = C_L cosα + C_D sinα C_A = C_D cosα – C_L sinα

3.2 BODY-ALONE AERODYNAMICS

3.2.1 Forces & Moments on the Body

The bodies of hypersonic vehicles usually have a blunt nose followed by a conical frustum and a cylindrical after body. Some of the vehicles may have an ogival nose followed by a cylindrical body. Forces on body shapes of these types can be analysed using Modified Newtonian theory (MNT) or the Second-order Shock Expansion theory, coupled with cross flow drag analysis on cylindrical portion. Some illustrative examples using MNT are worked out.

According to the MNT, the coefficient of pressure on surfaces exposed directly to the flow1 _{is given by}

(

_{sinècos sin cosèsin â}

)

2

K

C_p = α− α (3.1)

where

θ = Angle made by surface of body with body axis

α = Angle of attack of the body axis

β = Polar angle of any point on body surface, measured from positive xy plane and positive for counterclockwise direction when viewed from rear.

Figure 3.1. Connection between wind-oriented coefficients (C_L, C_D) and body-oriented coefficients (CN, CA).

Z C_L C_N y α C_A C_D x c_m V_∞

In the Newtonian theory, the pressure coefficient Cp_uin the shielded region of the body is zero. However, from gas-dynamic considerations, the value of C_p

ulies between zero and

( )

−2/ γM_∞2 depending on the free stream Mach number, shape of the body, and angle of attack. Experiments have indicated that the largest negative value of Cp_uappears to be of the order of

−1/M_∞2 for γ = 1.4. Therefore, for high Mach number flows, it is reasonable to assume that the shielded regions do not contribute to the forces and moments.

For any portion of the body shown in Fig. 3.2, the axial force is given by         β θ + β θ =

∫

∫

∫

β π − π β ∞ u u u / / p p surface d sin r C d sin r C ds q A 2 2 2 _(3.2)

C_p as given by the Eqn. 3.1, Cp_u = 0 and ds = dx/cos θ Figure 3.2 Modified Newtonian theory

Z_N Z SHIELDED AREA Z β θ α u=       − sin tan tan 1 θ α β L r y V_∞

∫

∫

β π − ∞ α− α β β = u / length d sin è cos sin cos è sin dx è tan r K q A 2 2 ) ( 2 _(3.3)

Axial force coefficient, C_A =

ref S q A ∞ (3.4)

(

α− α θ β

)

β θ =

∫

∫

β π − d sin cos sin cos è sin dx tan r S K C length / ref A u 2 2 2 (3.5)     β α θ α θ +         β − π + β α θ +        π + β α θ θ = u u u u ref A cos sin cos cos sin sin sin cos cos sin tan S r K dx dC 2 4 4 2 2 2 2 2 2 2 2 (3.6)

Similarly, the normal force is given by

∫

∫

∫

        β β + β β − = β π − π β ∞ length / / p p u u ursin d C d sin r C dx q N 2 2 2 (3.7)

with C_p given by the Eqn. 3.1 and C_p

uassumed zero,

(

)

   α + θ α β +    _θ     π + β θ α = tan tan cot cos tan cos sin S r K dx dC u u ref N 2 3 1 2 2 2 2 (3.8)

Likewise, the pitching moment, taken about the centroid of the area of the base of the nose section, for convenience, is given by

(

)

{

}

            β β + β β     θ − − − =

∫

∫

∫

β π − π β ∞ u u u / / p p length N d sin r C d sin r C dx tan r x L q K M 2 2 (3.9)

(

)

[

]

(

)

      α + θ α β + θ     π + β θ − − θ α = 2 3 1 2 2 2 2 tan tan cot cos tan u tan r x L cos sin L S r K dx dC u N ref m (3.10) where, L is the reference length. Eqns. 3.6, 3.8 and 3.10 can be integrated analytically or numerically to give axial force, normal force, and pitching moment coefficients, respectively for any arbitrary shaped body of revolution.

3.2.2 Axial Force

The axial force coefficient on a body can be considered to be made up of three parts,

C_A C_A C_A C_A

f b N

= + + _(3.11)

where

CAf = Coefficient of axial force on the body due to skin friction CAb = Coefficient of axial force due to the pressure on the base

area, and C_A

N = Coefficient of axial force on the body excluding friction. (wave drag)

3.2.3 CA_f – Skin Friction Coefficient

Several methods have been presented in the literature for the prediction of skin friction on bodies and flat plates in supersonic and hypersonic flows. Majority of these are complex, laborious and require a detailed knowledge of the flow field over the body for the evaluation of the skin friction coefficient. However, in preliminary design analysis, it is sufficient to go for simple

methods which give fairly acceptable values. Towards this, the incompressible flow results are utilised with an empirical correction for compressibility or Mach number effects.

Axial skin friction coefficient

ref B f A S S C C f wet = (3.12)

The mean value of C_f depends on whether the flow is laminar or turbulent. It is an usual practice to assume that the boundary layer over the body is laminar if

Re<Recr =10

6 _{(Reynolds number based on the length of the}

body). For laminar flow

lam f f f i C C Re . C _       =1328 _(3.13)

The above is the well known Blasius relation for the flow over a flat plate in incompressible flow multiplied by the compressibility factor. The suggested value of the compressibility factor is ∞ − =         M . C C lam f f i 028 0 1 (3.14) An alternate expression suggested for C_f in laminar flow

is

[

]

4 3 2 54 . 8 000349 . 0 00335 . 0 0236 . 0 328 . 1 Re 1 ∞ ∞ ∞ ∞ − + − − = M M M M C_f (3.15)

For Re>Re_cr =106 the boundary layer is turbulent with a laminar part in front of it. For this case the value of C_f is given by

                +         − = lam f f cr turb f f cr f f i i i _C C Re Re C C Re Re Re C C _(3.16) where

(

)

2 64 1 3300 407 . 0 427 . 0 − − − − = ln Re Re C_f . i (3.17) C C M M M M f f tu rb i    _ = − 1 0 0689. ∞ − 0 0343. ∞2 + 0 0061. ∞3 − 0 000278. ∞4 (3.18)

The above compressibility factors are for the case of an adiabatic wall. The skin friction coefficients (given above) both for laminar and the turbulent flows are for a flat plate at zero angle of attack. As the actual vehicle has a finite thickness, the pressure gradient and the boundary layer displacement effects influence the skin friction. To account for this, Hoerner2 _{has suggested that}

the skin friction coefficient values as given above be multiplied by the factor 3 2 3 7 5 . 1 1     +         + L d L d (3.19)

3.2.4 CA_b – Base Pressure Coefficient

The base drag of a hypersonic vehicle may be a substantial part of the total zero lift drag. Based on experimental data, various empirical formulae have been suggested. A few of these formulae are listed below. It is to be noted that some of the formulae are expressed in the form of axial force coefficient and some in the form of base pressure coefficient.

∆C C S S D P b ase ref b ase b ase =− 4 2 57 0 1 − ∞ ∞ − = − M . M C base p for M_∞ 1≥ (3.203₎ C M M M p_{b ase} = ₊          _  ₊− − −              ∞ ∞ ∞ 2 2 1 1 2 1 1 1 2 1. 4 2 8 2 γ γ γ γ γ . ( ) ( ) (3.214)

(

)

(

)

ref A S d . M . M . cos / C . cyl b 4 2455 0 06307 0 004714 0 1 2 2_{− +} π α = ∞ ∞ (3.22 5₎

(

)

cyl b boattail A A . M . C C = 00071 _∞ + 0782 _(3.235₎

It has been experimentally observed that for M ≥ 5.5 there is very little effect of angle of attack on the base pressure. It is well known that as the Mach number becomes very high, the base drag coefficient approaches zero.

3.2.5 Determination of C_AN , the Axial Pressure Coefficient of Nose Portion of the Body

3.2.5.1 Pointed Cone

Calculations for a conical nose are simple. Since θ = θ_v, the semi-vertex angle of the cone is constant and r is a linear function of x, (dr = dx tanθ). For positive angles of attack less than θ_v,β_u = π/ 2 and all surfaces of the cone are exposed to the flow. For _θv ≤ α

≤π– θ_v, certain portions of the cone are shielded from the flow and for this case β_u = sin –1 _{(tan θ}

v / tan α ). For π – θv≤α≤π, none

of the surfaces of the cone are exposed to the flow and β_u = – π/2. For the above range of angles of attack, the axial pressure coefficients of the cone are

v ref A è è cos sin è sin cos S d K C cone _{4 2} ;0 2 2 2 2 2 ≤ α ≤         α + α π = (3.24) where, d is the base diameter of the cone. If the axial force coefficient is based on the base area, then S_ref = (πd2_{/4). The axial}

force coefficient is v è è cos sin è sin cos K C cone A ₂ ;0 2 2 2 2 _{α +} α _≤α≤ =           (3.25)

For the case, θ α π θv≤ ≤ − , the upper limit of the integration

βu= sin−1(tanθ/tanα).

   α +       ₊ α +    _{α +} π =         u u u A â cos è sin sin ð â è cos sin ð â è sin cos K C cone 2 2 8 3 2 2 2 2 2 2 2 (3.26) and C_Acone =0 for π θ α π− ≤ ≤_v

3.2.5.2 Pointed Ogive

(

)

(

)

[

]

ref A S d M sin F . F F ln F F K C ogive 4 1 22 0 1 1 1 1 1 2 2 2 2 2 2 2 2 2 π − α +         −         + − + = (3.27)

where, F = L_n /D is the fineness ratio.

3.2.5.3 Hemispherical Nose

For the body having a hemispherical nose, the axial force coefficient is given by

(

)

2 1 4 2 2 _{+ α} π = cos S R K A C ref hemisphere (3.28) 3.2.6 Normal Force 3.2.6.1 Pointed Cone

When the Eqn. 3.8 is integrated for the case of a right circular cone having a semi-vertex angle of θv , and base diameter

d, the following expressions result for the normal force coefficients:

v ref v N è S d sin è cos K C _cone ;0 4 2 2 2 2 ≤ α ≤ π α = (3.29)

(

)

ref v v u u v N S d tan è cot è tan cot â cos â sin è cos K C cone 4 2 3 1 2 2 2 2 2 π       π _{+ α + α} + α π = :è_v ≤α≤π−è_v (3.30) π ≤ α ≤ − π = _v N ; è C cone 0 (3.31)

3.2.6.2 Pointed Ogive

(

)

(

)

ref N _S d F F ln F F sin cos sin K C ogive 4 2 1 1 2 2 2 2 2 2 π         − + + α α α = (3.32) It has been found that for angles of attack near zero for a circular arc ogival nose, the normal force can be adequately obtained from the simple equation for that of the cone of equal fineness ratio. For fineness ratios of unity or larger, calculations show that the difference in C_N for the cone and ogive are negligible at angles of attack up to θv for the cone. At angles of attack somewhat less than θv of the cone, the curves of CN versus α for the two nose shapes cross, so that α= θ_v, C_N for the ogive exceeds that for the cone. No explicit expression for the location of the centre of pressure can be given for the ogive as in the case of the cone; computations have shown, however, that for small angles of attack the centre of pressure of the ogive is nearer the vetex than that of the cone of equal fineness ratio and moves rearward with increasing angle of attack.

3.2.6.3 Hemispherical Nose

Integration of the Eqn. 3.8 for the case of hemispherical nose results in

(

1

)

;0 ; 4 2 π ≤ α ≤ + α α π = sin cos S K R C ref N_sphere (3.33)

Calculations for other nose shapes are more involved since r, _θ and βuare all functions of the lengthwise variable x. In

general, closed form solutions cannot be obtained and one has to go for numerical integration.

3.2.6.4 Cylinder

The normal force contribution from the circular cylindrical portion of the fuselage can also be obtained from Eqn. 3.8. If the normal force coefficient is based on the base area of the nose which is the same as the base area of the cylinder, the expression for the normal force coefficient is

(

)

sin α ;0 α π 5 . 1 ₋ 2 _{≤ ≤} = B N ref N _S d L L C cylinder (3.34)

In the above expression d is the diameter of the cylinder and a value of 2 is substituted for K, the factor in the modified Newtonian expression for the pressure coefficient. The normal force coefficient as given by the above agrees very well with the experimental data at small to moderate angles of attack and overestimates the force by only 5 per cent near α = 90°.

3.3 ALLEN & PERKINS VISCOUS CROSS FLOW THEORY

In cases in which the force on a body such as a fuselage is determined by some other method other than the Newtonian method, such as Second Order Shock Expansion method, Hybrid theory of Van Dyke, etc., then it is necessary to include the contribution to the lift by viscous cross flow. The widely used method to calculate the lift due to viscous cross flow is that proposed by Allen and Perkins6_{. According to this theory which is}

fairly simple yet quite powerful, the inviscid and viscous effects of the flow are assumed to be separate, with the inviscid form of the solution applying to the axial flow while the viscous part is confined to the cross flow giving rise to a nonlinear lift force coefficient. The viscous cross flow contribution to the lift coefficient is given by

(

∆

)

=η _sin2α_cosα A A C C ref p d NL N c

Here,

η

is the drag proportionality factor or cross flow drag of a cylinder or flat plate of finite length to one of infinite length, (for high Mach number flows, a value of

η

=1 is normally used). Cdcis the average cross flow drag coefficient, Apis the planform area of

the body in the cross flow plane and Are f the reference area used in

lift force coefficient term. Cd_c is taken from the experimental section drag coefficient. For simplicity, in the absence of available experimental data, a value of about 1.2 is normally assumed.

3.4 MOMENTS

The pitching movement is caused by the normal forces acting on the body nose and the body cylindrical part.

C_m C C

m m

N cy lin d er

3.4.1 Pointed Cone

From the integration of the Eqn. 3.10, where the moment is taken about the centroid of the base of the nose and taking the length of nose as the reference length, the pitching moment coefficient, for the case of a cone is given by

(

v

)

v v

m tan è cos è sin è

K C cone ₆ 1 2 ; 0 2 2 2 _{α ≤ α≤} − = (3.35)

(

)

(

)

C K ; m v v u u v v v v con e = − + + +     ≤ ≤ − tan cos

cos cot tan cot tan

6 1 2 2 1 3 2 2 2 2 π θ θ α β π β α θ θ α θ α π θ sin (3.36) π α θ π ; 0 − ≤ ≤ = v mcone C (3.37)

It is seen from the above results that the location of the centre of pressure of the conical nose is independent of the angle of attack. Its distance from the vertex is given by

v N cp è cos L x 2 1 3 2 = (3.38) 3.4.2 Pointed Ogive

(

)

(

)

(

)

ref N ref ref N mr ref N m L l L d S d F arctan F F F sin cos sin K l L L C C ogive ogive 4 3 1 1 1 1 3 -2 2 2 π    −       ₋     + α α + α + = (3.39) where

F = (L_n/d) the fineness ratio

L_{m r} = the moment arm length measured from the nose L_{re f} = the reference length in the moment coefficient

expression

No explicit expression for the location of the centre of pressure can be given for the ogive as in the case of the cone; computations have shown, however, that for small angles of attack

the centre of pressure of the ogive is nearer the vertex than that of the cone of equal fineness ratio and moves rearward with increase in angle of attack.

3.4.3 Hemisphere

When the moment is taken about the base of the sphere,

π ≤ α ≤ =0 ; 0 hemisphere m C _(3.40) 3.4.4 Circular Cylinder

The assumption is made that resultant normal force acts through the mid-point of the cylindrical body, thus giving

(

)

[

mr N cylinder

]

ref N m _L L l l C C cylinder cylinder + − = (3.41)

3.5 WING ALONE AERODYNAMICS

At hypersonic Mach numbers it is assumed that the different parts of the vehicle such as body, wing and rudder have no influence on each other. The wing or rudder is considered in isolation. Most of the engineering methods available to analyse the pressure over a lifting surface are based on 2-dimensional theory. In some cases it is possible that wing tips might have an appreciable effect on the predicted lift. In such cases 2-dimensional lift is corrected for 3-dimensional flow by the use of supersonic linear theory.

The common types of wing sections utilized in hypersonic vehicles are:

(a) Hexagonal (b) Biconvex, and

(c) Blunt nose leading edge

3.5.1 Hexagonal Shape Wing Section

The analysis is based on the formulations given by Weibust5_{. In this case the wing geometry has wedge sections at}

the leading and trailing edges with straight portion in between and treated as having six surfaces. The oblique shock and Prandtl-Meyer theories are applied to the leading and trailing edges depending on whether the surface under consideration gives rise to compression flow or expansion flow. For forces normal to the chord the wing is treated as a flat plate with zero thickness.