Aerodynamics
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Hypersonic Aerodynamics...
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AERODYNAMIC PREDICTIVE METHODS AND THEIR VALIDATION IN HYPERSONIC FLOWS AK SREEKANTH
AERODYNAMIC PREDICTIVE METHODS AND THEIR VALIDATION IN HYPERSONIC FLOWS AK SREEKANTH
Defence Research & Development Organisation Ministry of Defence New Delhi – 110 011 2003
DRDO MONOGRAPH SERIES AERODYNAMIC PREDICTIVE METHODS AND THEIR VALIDATION IN HYPERSONIC FLOWS AK SREEKANTH Series Editors Editor-in-Chief Dr Mohinder Singh
Editors Dr JP Singh, A Saravanan
Coordinator Ashok Kumar
Cover Design A Saravanan
Asst. Editor Ramesh Chander
Editorial Asst. AK Sen, Kumar Amar Nath
Production Printing JV Ramakrishna, SK Tyagi
Marketing RK Dua, Rajpal Singh
Cataloguing in Publication 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.
Printed and published by Director, DESIDOC, Metcalfe House, Delhi-110 054.
CONTENTS Preface
xi
Acknowledgement
PART - I
AERODYNAMIC PREDICTIVE METHODS IN HYPERSONIC FLOWS
xiii
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
2.5
Newtonian & Prandtl-Meyer Mode1
10
8
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
References
31
(vi)
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
37
3.2.4
C A – Base Pressure Coefficient b
39
3.2.5
Determination of CA 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
f
– Skin Friction Coefficient
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
References
59
(vii)
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
References
92
(viii)
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
(ix)
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
Index
225
PREFACE 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 Date: June 2003
AK Sreekanth
ACKNOWLEDGEMENT 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.
2.
3.
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. Dr. J.Agrell, Head of Experimental Aerodynamics Department, FFA, Sweden, for permission to include material in the monograph from the FFA Technical Note AU1661. Mr. Dan Pappas, Chief Librarian, NASA Ames Research Center, Moffett Field, CA. for allowing me to use the Ames Library freely.
PART - I AERODYNAMIC PREDICTIVE METHODS IN HYPERSONIC FLOWS
CHAPTER 1 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 timeconsuming 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 wellknown 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
4
Aerodynamic Predictive Methods In Hypersonic Flows
experimental data are discussed in Part II(Chapter 6) of this monograph, to illustrate the applicability and validity of the approximate methodology.
2.1
INTRODUCTION
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 Embedded Newtonian Modified Newtonian + Prandtl-Meyer Tangent wedge and Tangent cone Tangent wedge-Tangent cone (empirical)
Newtonian (Cp=O) Modified Newtonian + Prandtl-Meyer Prandtl-Meyer from free stream OSU blunt body empirical ' Van Dyke unified Contd...
6
Aerodynamic Predictive Methods and their Validation i n Hypersonic Flows Imwact Flow OSU blunt body empirical Van Dyke unified 2-Dim. Hypersonic airfoil theory
Shadow Flow High Mach number base pressure Shock expansion 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 mass of particles striking the surface area A in unit time is p, V , A sin S . The total normal momentum carried by these
Methods
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. It 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. MODIFIED NEWTONIAN THEORY In the Modified Newtonian theory (MNT), the pressure coefficient is written a s
2.3
Various values of K have been suggested depending on the Mach number, body shape, angle of attack and ratio of is the specific heats. The most common one is K = Cpswhere Cps stagnation pressure coefficient behind a normal shock. For this case we have
and mainly used for blunt bodies. For 1.839andfor Y = 1.0and M,
-+
Y
=
1.4 and M, +co, K+
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 L,= nose length and a = angle of attack.
=
body diameter,
7
8
Aerodynamic Predictive Methods and their Validationin Hypersonic Flows
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 K =1.95 +
0.3925
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 at low angles of attack requiring a different methodology. 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
2.4
Methods
9
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 an 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 ( ~ sin 1 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
P2
=
+
'P2 4,
Newtonian
10
Aerodynamic Predictive Methods and their Validationin Hypersonic Flows
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 at 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
I
L
At the stagnation point 6 = 90"
Substitutingfor
YM,~K
from Eqn. 2.8 in 2.7, we have
Methods
(2.9)
Po,
Po,
The slope of this pressure curve is d
PIP^,)
= 2 (1-P) sin 6 cos 6 dS according to Newtonian theory. For an isentropic expansion downstream of the stagnation point, on the blunt nose, we have
P Po,
;
[
2+(y-1)
kf2
The Prandtl-Meyer angle for expansion from sonic flow to supersonic Mach number M is v
=JzJw)d $ Z ] tan-
-
tan
(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 )
dM2-1
dv
( 2.15)
Equating the expressions for the pressure gradients (Eqns. 2.12 and 2.15), noting that dv = - d6
ykf: ( P 9 /Po,) =
2(1-P) sin 6cos 6
(M~z-I)~
where, subscript q refers to conditions at the point on the body
11
12
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
where the pressure distribution slopes are equal. The location of this point can be easily determined once the value of Mqis 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 PrandtlMeyer relation.
Methods
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
Figure 2.3. Tangent cone method
13
14
Aerodynamic Predictive Methods and their Validationin Hypersonic Flows
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 at 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 Mnsis
M,,
=
(0.87~ -~ 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 5% when the above expression is used for all Mach than numbers above 1.5.
*
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 2.7
sin 0, =
(1 - E )
sin 8, COS (0, - 6, )
Methods
and for a cone (thin shock layer approximation) sine,
where,
E =
(P
sin 6 , = ,
,/ 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 2
-sinti,
For a wedge:
sine,
For a cone:
sin 0 ,= ---- sin 6 ,
=
2 6 +I)
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 Mnsand M rsin F that satisfies the following requirements:
15
16
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
(a)
Shock detachment is neglected
(b)
A t M s i n g = O , MrlS=l
(c)
The solution asymptotically approaches the M,
(d)
Has the correct slope, d ( ~ ,
4%
)
+ cc
value
a t MI; sin& = 0
These conditions lead to a relationship of the form
M,,
K,,M,sind, for a wedge, and =
M,, = K , M , sins, for a cone, where
K,
=
y+l
and 3
-
t
e -(K,+w,,
-
e sin&)
KC=
sins,)
/2
2 (Y + 1)
(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
The static pressure jump across an oblique shock is given by
Methods
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.
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
2.8
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
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. For impact angles up to 22.5", the pressure coefficient is given by
2.10
17
18
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
C, =
[I
+ ~in(46)~/~]sin(6)~/~ (2.39)
[4 cos 6 cos (26)] 3'
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, d x
is is is and the values a s follows: Flow
2.12
the nose drag coefficient the nose diameter or thickness and the distance from the nose stagnation point of coefficients A, B and nose drag coefficients are CI
J
A
B
SHOCK EXPANSION THEORY
2.12.1 First Order Theory 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 2dimensional bodies or the conical shock relations for ,axisymmetric 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.
Methods ,LEADING TANGENT BODY COMPOSED OF CONICAL SEGMENTS
EDGE SHOCK
SHOCK WAVE
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).
The pressure gradient is approximated by the relation16
19
20
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows 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.
22
Aerodynamic Predictive Methods and their Validation in HypersonicFlows
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 are shown below.
4
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
24
Aerodynamic Predictive Methods and their Valzdationin Hypersonic Flows
AC,
=-
( 2 a )s i n ( 2 ~ ) s i n ( ~ ) + ( ~6c) oa's ~
+
[ ( 4 / 3 ) sin(26) sin
(o)] . . a'
(2.49)
where
AC P = -
(2a) sin (26) sin(@) 3
(2.50)
Eqn 2.49 is used for pointed body configurations as 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
Mci is replaced by the factor
f i 6 a s suggested by Van Dyke.
Methods
25
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 = applicability in both supersonic and hypersonic flow 2.15
m1 S for
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 the pressure coefficient C
2
,- Y M :
P-Pm m
j
Let the subscript 2 denote the conditions immediately behind a n oblique shock wave, there follows
From the oblique shock relations we have 2
1
Y + l
sin P = 4 CP2
+
-
and
(c~2;"")2
=
[(
1 - -'i2
where p is the shock angle and
) e
sin P tan
e2
I'
the flow deflection angle.
For thin airfoils in hypersonic flow, we can approximate a s follows:
Methods
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 becomes
'pW2
from Eqn. 2.57, the above equation
27
28
Aerodynamic Predictive Methods and their lralidation in Hypersonic Flows
The above equation leads to two cases viz., no shock and no expansion. Let us consider the no shock case. For this Ow,
=
0.
Hence, Mw?= 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.,
30
Aerodynamic Predictive Methods and their Validation i n Hypersonic Flows
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 at any point on the surface of an 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 at 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
Methods
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, 1981. TN-AU1661. Hankey, Jr., 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. 11 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- 1102.
31
32
Aerodynamic Predictive Methods and their Validatiott in Hypersonic Flows
Dahlem, V. & Buck, M.L. Experimental and analytical investigations vehicle designs for high lift-drag ratios in hypersonic flight. June 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, 1951. 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 12THFluid 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).
CHAPTER 3 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 + ∆C N B (W ) + ∆C N B (T ) + ∆C N W ( B ) + ∆C N T ( B ) + ∆C N T (W ) + ∆C N W (T )
The subscripts B, W, and T refer to the body, exposed wing and exposed tail, respectively. The terms ∆CNB (W ) , ∆CNW (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 CN and axial CA, force coefficients) or the wind oriented coordinate system (the lift (CL ) and drag (CD ) force coefficients). These can be converted from one system to the other by the following relations.
34
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows Z
CN CL
CD cm y
V∞
α
x CA
Figure 3.1. Connection between wind-oriented coefficients (CL, CD) and body-oriented coefficients (CN, CA).
CL = CN cosα – CA sinα CD = CA cos α + CN sinα CN = CL cos α + CD sinα CA = CD cos α – CL 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 flow 1 is given by C p = K (sin è cos α − sin α cos è sin â ) 2
(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.
Aerodynamic Characteristics of Vehicle Components Z ZN
L
SHIELDED AREA
θ V∞
Z
r
α
tan θ βu = sin − 1 tan α
β
y
Figure 3.2 Modified Newtonian theory
In the Newtonian theory, the pressure coefficient C p u in the shielded region of the body is zero. However, from gasdynamic considerations, the value of C p u lies 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 C p u appears 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 π/2 βu A = 2q∞ ds C p r sin θ dβ+ C p u r sin θ dβ surface − π / 2 βu
∫
∫
∫
Cp as given by the Eqn. 3.1, C p u = 0 and ds = dx/cos θ
(3.2)
35
36
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
∫
A = 2q ∞ K
βu
r tanèdx
∫(sinè cosα −sin α cosè sinβ)
2
dβ
A Axial force coefficient, CA = q S ∞ ref
2K CA = Sref
dC A dx
=
r tan θ dx
∫ (sinè cosα − sinα cos θ sin β) dβ 2
−π / 2
length
S ref
(3.4)
βu
∫
2 Kr
(3.3)
− π /2
length
tan θ sin 2 θ cos 2 α
βu π sin 2 β u + − 2 4 4
(3.5)
π β u + + cos 2 θ sin 2 α 2
+ 2 sin θ cos α cos θ sin α cos β u
(3.6)
Similarly, the normal force is given by
π/2 βu N = − 2q∞ dx C p r sinβ dβ + C p u r sin β dβ −π 2 βu length /
∫
∫
∫
(3.7)
with Cp given by the Eqn. 3.1 and C p u assumed zero,
dC N dx
=
Kr S ref +
π sin 2 α cos 2 θ β u + tan θ 2
(
)
cos β u cot α tan 2 θ + 2 tanα 3
1
(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
Aerodynamic Characteristics of Vehicle Components
M = − Kq ∞ {(L N −x ) − r tan θ }dx length
∫
βu C p r sinβ dβ + −π / 2
∫
dC m dx
=
Kr S ref L
C p u r sin β dβ βu
π/2
(3.9)
∫
sin 2α cos 2 θ [(L N − x ) − r tan θ ]
π 1 2 β u + tan θ + cos β u cot α tan θ + 2 tan α 2 3
(
)
(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 f +C A b +C A N
(3.11)
where CAf
= Coefficient of axial force on the body due to skin friction
C Ab
= Coefficient of axial force due to the pressure on the base area, and = Coefficient of axial force on the body excluding friction. (wave drag)
C AN
3.2.3 C A 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
37
38
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
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 C A f = C f
S B wet
(3.12) S ref The mean value of Cf 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 6 Re < Re cr = 10 (Reynolds number based on the length of the body). For laminar flow Cf =
1.328 C f Re C f i
lam
(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 Cf Cf i
= 1 − 0 .028 M ∞ lam
(3.14)
An alternate expression suggested for Cf in laminar flow is C
f
=
1 Re
[1.328 − 0.0236 M
+ 0.000349 M
3 ∞
− 0.00335 M
∞
− 8.54 M
4 ∞
]
2 ∞
(3.15)
For Re > Re cr = 10 6 the boundary layer is turbulent with a laminar part in front of it. For this case the value of Cf is given by
Re − Recr C f = C fi Re
lam
(3.16)
C f i = 0.427 (ln Re − 0.407 ) − 2 . 64 − 3300 Re −1
(3.17)
Cf Cf i
Recr + Re turb
Cf Cf i
where
C f = 1 − 0.0689 M ∞ − 0.0343 M ∞2 + 0.0061 M ∞3 − 0.000278 M ∞4 C f i turb (3.18)
Aerodynamic Characteristics of Vehicle Components
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 d 1 + 1.5 L
3 2
d + 7 L
3
(3.19)
3.2.4 C A 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 D
base
=
−C P
− C p base =
Cp
C Ab
cyl .
S base
S ref 1 M
2 ∞
1. 4 2 = 2 γ M ∞ γ +1
2
base
base
−
0 . 57 M ∞− 4
1 M ∞
2 .8
for M ∞ ≥ 1
(3.203)
2 γ M 2 − ( γ −1 ) ∞ −1 + 1 ( ) γ
(3.214)
= (1/cos α )
(0.004714 M ∞2 − 0.06307 M ∞ + 0.2455 ) 4πSd
2
(3.225)
ref
C Aboattail = (0.0071 M ∞ + 0.782 ) C Ab
cyl
(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.
39
40
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
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 = θv, 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, βu = π / 2 and all surfaces of the cone are exposed to the flow. For θ v ≤ α ≤ π – θ v, certain portions of the cone are shielded from the flow and for this case βu = sin –1 (tan θ v / tan α ). For π – θ v ≤ α ≤ π, none of the surfaces of the cone are exposed to the flow and βu = – π/2. For the above range of angles of attack, the axial pressure coefficients of the cone are
C Acone =
Kπd 2 sin 2 α cos 2 è 2 2 cos α sin è + 4S ref 2
; 0 ≤ α ≤ èv (3.24)
where, d is the base diameter of the cone. If the axial force coefficient is based on the base area, then Sref = ( π d2/4). The axial force coefficient is
CA
cone
= K cos 2 α sin 2 è +
sin 2 α cos 2 è 2
; 0 ≤ α ≤ èv
(3.25)
For the case, θv ≤α ≤π −θ , the upper limit of the integration βu = sin − 1 (tan θ/tan α ) .
CA
= cone
K ð 2 2 cos α sin è â u + 2 π
+
sin 2 α cos 2 è ð â u + 2 2
sin 2 α sin 2è cos â u + 8 3
and C Acone = 0
for π − θv ≤ α ≤ π
(3.26)
Aerodynamic Characteristics of Vehicle Components
3.2.5.2 Pointed Ogive
C Aogive
= K 2 1 + F 2
(
[1 + 0.22 F
2
)
1+ F2 1 − F 2 ln F2
(
sin 2 α M 2 − 1
− 1
)] 4πSd
2
(3.27)
ref
where, F = Ln /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
CA
= hemisphere
Kπ R2
(1 + cos α )2
4 S ref
2
(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:
C N cone = K
C N cone =
K 2π
cos 2 èv sin 2 α π d 2 2
4 S ref
; 0 ≤ α ≤ èv
(3.29)
cos 2 è v sin 2 α
2 π 1 πd â u + 2 + 3 cos â u (cot α tan è v + 2 cot è v tan α ) 4 S ref
: èv ≤ α ≤ π − èv CN
cone
=0
;
π − èv ≤ α
(3.30)
≤ π
(3.31)
41
42
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
3.2.6.2 Pointed Ogive
CN
ogive
π d2 1+ F2 = K (sin α cos α ) sin α F 2 2 1 + F 2 ln − 2 F2 4 S ref
(
)
(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 C N versus α for the two nose shapes cross, so that α = θv , 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
C N sphere =
πR 2 K 4S ref
sin α (cos α + 1 )
; 0 ≤ α ≤ π;
(3.33)
Calculations for other nose shapes are more involved since r, θ and βu are 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
Aerodynamic Characteristics of Vehicle Components
CN
= cylinder
1.5 d (L B − L N S ref
) sin 2 α
; 0 ≤ α ≤ π (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
(∆ CN )NL
= η C dc
Ap Aref
sin 2 α cos α
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). C d c is the average cross flow drag coefficient, A p is the planform area of the body in the cross flow plane and A re f the reference area used in lift force coefficient term. C d 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. Cm = C
m
+ Cm N
cylind er
43
44
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
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
K
C m cone =
Cm
cone
6
=
(1 − 2 tan 2 è v )cos 2 è v sin 2 α
(
)
;
0 ≤ α ≤ èv
K 2 2 2 1 −2 tan θv cos θv sin α 6π π 1 βu + 2 + 3 cos βu cotα tan θv + 2 cotθv tan α θv ≤ α ≤ π − θv
C m cone = 0 ;
(
(3.35)
);
(3.36)
π −θ v ≤ α ≤ π
(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
x cp LN
=
2 1 3 cos 2 èv
(3.38)
3.4.2 Pointed Ogive CN ogive C m ogive = (Lmr - l N ) + 3K (sin α + cos α )sin α F 2 1 + F 2 L ref
(
1 1 − F arctan F
2 d lN 1 π d − 3 4 S ref L ref L ref
) (3.39)
where F Lmr L re f
= = =
( L n /d) the fineness ratio the moment arm length measured from the nose 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
Aerodynamic Characteristics of Vehicle Components
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, C m hemisphere = 0 ;
0≤α≤π
(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
Cm 3.5
= cylinder
CN
cylinder
L ref
[L mr − (l N + l cylinder )]
(3.41)
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) (b) (c)
Hexagonal Biconvex, and 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 PrandtlMeyer 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.
45
46
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Compression Flow The pressure coefficient behind an attached oblique shock is given by the relation,
C pc =
(
4 M ∞2 sin ε − 1
(γ + 1) M ∞2
)
(3.42)
where, ε is the shock angle obtained from oblique shock tables or from the solution of the cubic shock equation in sin2 ε
sin 6 ε + A sin 4 ε + B sin 2 ε + C = 0 where A=−
B=
2M
M
2 ∞
2 ∞
+1
M
C =−
+2
2 ∞
M
4 ∞
− γ sin 2 è
(γ + 1 ) 2 γ − 1 sin 2 è + + 2 4 M ∞
cos 2 è 4
where, θ is the flow deflection angle and it is assumed that for the given free stream Mach number its value is such that it always gives rise to an attached shock at the leading edge. Expansion The pressure coefficient behind an expansion fan is
C pe =
2 γM ∞2
pe 2 − 1 = γ M ∞2 p∞
pe p o∞
po∞ p∞
− 1
(3.43)
Where Pe is the pressure behind expansion obtained either from P-M expansion flow tables or from the following relations poe γ −1 2 M∞ 1 + p∞ 2
γ γ −1
from the isentropic relation (3.44)
Aerodynamic Characteristics of Vehicle Components
and
pe poe
1 = 1 + cos γ +1
2
γ −1 υ e + arctan γ +1
M
−1
2 e
γ −1 γ
(3.45) where, pe is the pressure and Me the Mach number downstream of the expansion. υe is the corresponding Prandtl Meyer angle which is obtained from the Prandtl-Meyer function
υe =
γ +1 γ −1
arctan
γ −1 γ +1
(M
2 e
)
−1 − arctan M
2 e
−1
(3.46)
similarly
υ∞ =
γ +1 γ −1
arctan
γ −1 γ +1
(M
2 ∞
)
−1 − arctan
M
2 ∞
−1
(3.47) and, νe = ν∞ + θ From the known value of νe the Mach number Me is obtained by an iterative solution of the Eqn. 3.46, then from pe Eqn. 3.45 and C p e from Eqn. 3.43. 3.5.1.1 Axial Force The axial force on a wing surface consists of two parts, one due to friction and the other due to inviscid flow over it. The friction force is determined similar to the case of a body in hypersonic flow (section 3.22). CAf = C
S wet f
S ref
(3.48)
In case the vehicle has horizontal and vertical wing surfaces (cruciform arrangement as in a missile) the axial force is the sum of these two. The wing surfaces might also act as rudders by wing deflection (see Fig. 3.3, 1& 2). The pressure forces on the leading and trailing wedge surfaces are calculated by the
47
48
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
application of appropriate oblique shock or Prandtl-Meyer theories. The inviscid flow over the straight portion does not contribute to the axial force. HORIZONTAL WING OR RUDDER WHEN φ = 0
CN
CA X,X'
Z' Z
φ = ROLL ANGLE
Y' Y
φ δ RUDDER DEFLECTION
φ α'
M∞
VERTICAL WING OR RUDDER WHEN φ = 0
Cruciform wing arrangement
M∞
ξ1′ ξ 2′
EXPANSION
M∞
ε
ν
SHOCK
Figure 3.3 Shock and expansion method applied to hexagonal crosssectional wing.
Aerodynamic Characteristics of Vehicle Components
ξ11 INNER WING
t
Λi ,1
SECTION-AA D
Λ0 ,1
OUTER WING
A
Λi ,2 A
B
Λ0 ,2
ξ21
B SECTION-BB bi bo
Figure 3.4. Planform showing inner and outer wing panels
The axial pressure coefficient on a horizontal wing is given by C AwH =
t 2 S ref
{ (b i − d )
2
2
∑ ∑ (− 1)
m +1
m =1 n =1
+ (btot − bi
)
2
2
∑ ∑ (− 1)
m =1 n =1
m +1
(
C p σ im , n , M
(
C p σ om , n , M
σ′i m , n i m ,n
)σ
σ′o m , n o m ,n
)σ
}
(3.49)
where t is the wing thickness, m = 1 for LE and m = 2 for TE , n = 1 for upper and n = 2 for lower surfaces, σ = | σ ' | is the deflection angle. If σ ' > 0 (shock) and Cp is determined from Eqn. 3.42 and σ ' < 0 (expansion) from Eqn. 3.43. Let the semi-wedge angles normal to the leading and trailing edges be denoted by ξ1' and ξ2' respectively. Subscript i refers to inner wing, and o to the outer wing. The angle σi′ for the inner horizontal wing is given by
′ cos Λ cos α' σ ′i m,n = ( − 1)m + arcsin sin ξ m im 1
′ sin Λ + sin ξ m i
m
1 ′ ] sin α' sin ϕ + (− 1 )n ( − 1)m + sin α' cos ϕ cos ξ m
(3.50)
49
50
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
where, m = 1 for LE and m = 2 for TE, n =1 for upper and n = 2 for lower wedge surfaces, φ = roll angle, α' = the non-rolled angle of attack and subscript i refers to the inner wing. ' Similarly, σo m ,n for the outer wing is as given above with Λo m replacing Λim
The axial force for the vertical wing C A wV is also obtained from the Eqn. 3.49. However, the deflection angle σ ' is obtained by substituting φ with (90º - φ ) in Eqn. 3.50. The wing total axial force is given by C
A
w
=C A
f
+C A
wH
+C A
wV
(3.51)
In case there is deflection of the rudder by an angle δ then one has to consider the contribution to the axial force by this deflection. The change in σ ' due to rudder deflection angle δ and hence to C A R is usually neglected. This is because the axial component of the force normal to the rudder, C A R , is based on the total wing planform including the areas on the wedge shaped LE and TE. The total axial force on the rudder is given by C A = C A + CA R
f
RH
+ CA
RV
+ C AR δ
(3.52)
where, C A f is obtained from Eqn. 3.48 and C A R H and C A R V are obtained from the Eqn. 3.49 with correct values of σ ' . C A R δ from the normal force as outlined below. 3.5.1.2 Normal Force An assumption is made that the normal force acts through the area centroid of the wing planform. As mentioned earlier, for normal force calculation, the wing is considered as a flat plate with zero thickness. The actual wing or rudder angle of attack ϑ ' expressed in non-rolled angle of attack α ' , roll angle ϕ and rudder deflection angle δ is
Aerodynamic Characteristics of Vehicle Components
[
ϑ′j ,k = arc sin sin δ j ,k cos α′
{(
+ cos ( j − 1) 90 o − (− 1 ) j ϕ
) }cos δ
j ,k
sin α ′
]
(3.53)
where, j = 1 for the horizontal wing and j = 2 for the vertical wing. k =1 for the right and upper wing halves and k = 2 for left and lower wing halves at roll angle ϕ = 0. When the above angle is known the pressure coefficients are determined by use of Eqns. 3.42 and 3.43. The above gives the forces acting normal to the wing or rudder chord. This has to be converted to get the axial component of the force and the component of the force normal to non-rolled xy plane ( i.e., negative z axis direction). 3.5.1.3 Axial Component of the Rudder
C AR
δ
=
SR
2
2
∑ ∑
4 S ref
j =1 k =1
C (ϑ , M ) − C (ϑ , M ) p j ,k j ,k pe c sin δ j , k
}
(3.54)
In the above equation SR is the planform area of either horizontal wings (right and left combined) or vertical wings (top and bottom combined). 3.5.1.4 Normal Component (Wing or Rudder)
CNw =
Sw 4S
ref
2
∑
j =1
2 ∑ C p (ϑ j ,k , M ∞ ) − C p (ϑ j ,k , M ∞ ) c e k =1
ϑ' j ,k cos ( j − 1) 90 o − ( − 1) j ϕ cos δ j , k ϑ j ,k
(3.55)
' with ϑj ,k = | ϑj ,k |, j = 1 for horizontal wing and j = 2 for vertical wing, k = 1 for right and upper wing halves and k = 2 for left and lower wing halves at zero roll angle viewed from nose towards tail.
It is suggested that as a means for conversion from 2-dimensional to 3-dimensional lift at hypersonic speeds, the following approximation for wing tip effects, based on supersonic linear theory, may be applied.
51
52
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
C Lα
CN =
Cn
4 M
2
(3.56)
−1
where, C N is the normal force coefficient, C L α is the lift-curve slope from linear theory for the 3-dimensional planform, as given by Harmon & Jeffreys7, and Cn is the 2-dimensional normal force coefficient calculated using 2-dimensional shock and expansion theory. 3.5.1.5 Pitching Moment The pitching moment is given by
Cm
= W
CN
w
X ref
(X
mr
− X cp
)
(3.57)
3.5.2 Other Wing Sections For wings having a blunt leading edge, the commonly used method to analyze the pressure on the wing surface is by the Newtonian + Prandtl-Meyer method as described in section 2.4. For wings having a sharp leading edge giving rise to an attached oblique shock and having curved surfaces like the biconvex airfoil, etc., the tangent wedge method is applicable. For thin sharp edged wing sections at low angles of attack, the approximate 2-dimensional airfoil method described in section 2.14 can be applied to determine the pressure distribution over the airfoil surfaces. From this, the axial force coefficient, the normal force coefficient and the moment coefficient can be determined. Based on methods described by Kaufman et. al 8 and Dorrance9, the results of calculations done on some common airfoil sections are given. The same is reproduced in the accompanying table.
Aerodynamic Characteristics of Vehicle Components
3.5.2.1 Airfoil Characteristics by 2-dimensional Hypersonic Airfoil Theory All equations are based on a dimensionless coordinate system( ξ , η ) with origin at midchord.
ξ=
2x ; c
η=
2z ; c
c = chord length, ;
α = angle of attack, K = M ∞ ε ;
ε = thickness ratio
Moment Cm taken about the origin; A1
=
2 , K
A2 =
γ + 1 , 2
t = thickness;
A3 =
(γ
α t ; β = ; ε c
Validity 3 ≤ M ∞ ≤12
+ 1) K 6
53
54
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Profile
Airfoil Characteristics
CN = 2 ( A1 + A3 ) α2 Cm =0 α2
Flat Plate
CD = 2 ( A1 + A3 ) α3
{
}
CN = 2 â A1 + A3( 3 + â 2 ) ; å2
Symmetrical Double Wedge
Cm =A2 â å2
[
]
[
]
CD = 2 A1 â2 + 1 + 2 A 3 â 4 + 6 â2 + 1 3 ε
ξ1
Double Wedge
CN = 2A1 â + 2A3 â â 2 + 3 2 ε2 − ξ 1 1
Cm 1 + î1 1 − î 1 = A2 â − 3 A3 â − 4 ε2 1 − î1 1 + ξ1
CD 1 1 = 2A1 â 2 + 1 2 + A2 − 2 2 3 å (1 − î1) 1 − î1 (1 + î1) 1 1 + A3 + + 2 â 4 + 6 â 2 1 3 3 1+î + − ( ) ( ) î î 1 1 1 1
− 1
1 1 − î1
Aerodynamic Characteristics of Vehicle Components
Profile
Airfoil Characteristics
CN 3 1 = 2 A 1â + 2 A 3 â â 2 + 2 − ( 2 1 î å
ξ1
ξ2
Modified Double Wedge
1)
+
CD 1 1 = A1 2 â 2 + + 3 1 − î1 1 − î 2 å 1 1 + A2 − 2 2 (1 − î 2 ) (1 − î1 )
1 1 + 6 â 2 + 1 − î1 1 − î 2
(
CN = 2 A1 â − 4 A2 + 2 A3 â 3 + â 2 å2
)
Cm 1 3 = − A 1 + A 2 â − A3 2 + â 2 2 2 å2
(
)
CD = 2 A 1 2 + â 2 − 12 A 2 â å3 + 2 A 3 â 4 + 12 â 2 + 8
(
)
CN 3 = 2 â (A 1 + A 2 ) + A 3 â + 2 â 2 2 å 2
Blunt T.E. Double Wedge
3 1 − ( 2 1 î
2 )
1 + î1 1 + î 2 Cm 1 3 = A2 â ( î 1 + î 2 + 2 ) − A3 â + 2 2 4 å î1 − 1 1 − î 2
1 1 + A3 2 â 4 + + 3 (1 − î1 ) (1 − î 2 )3
Single Wedge
55
CM =0 å2 CD 1 1 = A 1 + 2 â2 + A 2 + 3 â2 å3 2 4 1 + A 3 + 3 â2 + 2 â 4 8
56
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Profile
Airfoil Characteristics
t1
CN 8 = 2 A 1 â + A 2 (ç 1 + ç 2 ) 2 3 å
(
+ 2 A 3 â 3 + 4 A 3 â ç 12 + ç 22 t2
η1 = 2t 1 / t η2 = − 2t 2 / t t = t1 + t 2
)
Cm 4 1 = A 2 â − A 1 (ç 1 + ç 2 ) 3 3 å2 4 − A 3 ç 13 + ç 32 − A 3 â 2 (ç 1 + ç 2 ) 5
(
)
Double Circular Arc
(
)
CD 2 = 2 A 1 â 2 + ç 12 + ç22 3 3 å + 8 A 2 â (ç 1 + ç2 )
(
)
(
)
8 ç 14 + ç 24 + 2 A 3 â 4 + 4 â 2 ç 12 + ç 22 + 5
ξ1
ξ1
Modified Circular Arc
( (
) )
CN 1 − î 13 = 2 A1 â + 2 A3 â â 2 + 4 2 2 å 1 − î 12
Cm å2
CD å3
=
4 A2 3
(1 − î ) â (1 − î ) 3 1 2 1
= 2 A1 â + 2
4 3
+ 2A 3
â
4
(1 − î ) (1 − î ) (1 − î ) +8â (1 − î ) 3 1
2 2 1
2
3 1
2 2 1
+
16 5
(1 − î ) (1 − î ) 3 1
2 4 1
Aerodynamic Characteristics of Vehicle Components Profile
Airfoil Characteristics
CN å2 Cm å2
CD Symmetrical Biconvex
ε3
= 2 A1 â + 2 A
3
8 â â2 + 4 + å2 5
2 2 = 2 A2 â + ε2 3 15
8 1 = 2 A1 + â 2 + å2 15 3 16 16 + 2 A2 8 â 2 + + â4 + å2 5 5
CN å
2
Cm å Double Parabolic Arc
CD
CN å
[
2
(
= 2 â A1 + A 3 4 + â 2 4
=
2
ε3
Cm
ε
Blunt T.E. Double Parabolic Arc
3
CD å
3
12 â2 + 7
)]
A2 â
3 1 = 2 A1 + â 2 + 2 A 3 3 = 2 A 1 â + 8î 1 β + 2A 3
ξ1
57
=
=
(1 + î 1 ) 2
(
4
4A 2
(1 + î 1 )
)
2
8
)
+ 8î 1 A 3 â
4 2 + â (1 + î 1 3
2A 1
+
A2
4 1 + 3î 12 2 â â + (1 + î 1 ) 4
4 1 A2 â 3 (1 + î 1
(1 + î 1 )
16 â4 + 8 â2 + 5
)4
1
(1 + î 1 ) 4 + 4î 12
[ 3î 1 â 2 ( 1 + î 1 ) 4
+ 4î 1 ( 1 + î 12 )] + 2
A3
(1 + î 1 )
8
[ â 4 (1 + î 1
)8
Contd ...
58
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
1 4 + 8 â 2 1 + 3 î 12 (1 + ξ1 ) + 16 î 14 + 2 î 12 + ] 5
(
CN ξ1
å
2
Cm Single Parabolic Arc
å3
CN ξ1
= 2A 1 â −
=−
å2 CD
å2
)
2
A1 +
3
(
16 A2 + 2A3 â 8 + â 2 3
)
32 A2 â − A3 2 â 2 + 3 5 4
8 = 2 A1 + â 2 − 16 A 2 â 3 128 + 2 A 3 â 4 + 16 â 2 + 5 2î 1 = 2 A1 â − 2 (1 + î 1 )
+ Blunt T.E. Single Parabolic Arc
8 A2
(1 + î 1 )2
1 2 + î 12 î 1 â − 3 2 (1 + î 1 )
6î 1 â 2 3 + 2 A3 â − (1 + î 1 )2 +
Cm å2
=−
(
8 â 1 + 3 î 12
(1 + î 1 )4 2
A1
2 32î 1 (1 + î 1 ) − 8 (1 + î 1 )
−
4
A2
3 (1 + î 1 ) 2 A3 2 4 î 1 − â (1 + î 1 ) − (1 + î 1 )6 3
(1 + î 1 )2
)
[
1 16 + î 12 5
4
]
− 8 ξ â (1 + î )2 + â 2 (1 + î )4 1 1 1
Aerodynamic Characteristics of Vehicle Components
CD å
3
1 + î 12 8 3 = 2 A1 â 2 + (1 + î 1 ) 4
+
8 A2
(1 + î 1 )
2
−
4î1 â
(1 + î 1 ) 2
(
)
(
)
3 2 â 1 + 3î 12 8î 1 1 + î 12 + ξ1 â 2 − (1 + î 1 )2 (1 + î 1 )4 2
128 1 + 2î12 + î14 4 5 + 2A3 â + ( + ξ1)6 1 −
(
128î 1 â 1 + î 12
(1 + î 1 )6
) + 16 â (1 + 3î ) + 2
(1 + î 1 )4
2 1
8î 1 â 3 (1 + î 1 )2
REFERENCES 1.
2. 3.
4. 5.
6.
7.
Grinminger, G.; Williams, E.P. & Young, G.B.W. Lift on inclined bodies of revolution in hypersonic flow. JAS. 1950, 17(11). Hoerner, S.F. Fluid dynamic drag. published by the author. Bonner, E.; Clever, W. & Dunn, K. Aerodynamic preliminary analysis II, Part-I. Theory, NASA, April 1991. Contractor Report, 182076. Gabeaud, A. Base pressure at supersonic velocities. J. of Aero. Sci. 1950, 17(8), 525-26. Weibust, E. Status report on the FFA version of the missile aerodynamics program LARV for calculation of static aerodynamic properties and longitudinal aerodynamic damping derivatives FFA. The Aeronautical Research Institute of Sweden, Stockholm, 1981. Technical Note AU-1661. Allen, H.J. & Perkins, E.W. Characteristics of flow over inclined bodies of revolution. NACA, March 1951. RM A50L07. Harmon, S.M., & Jeffreys, I. Theoretical lift and damping in roll of thin wings with sweep and taper. NACA, 1950, TN-2114.
59
60
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
8.
9.
Kaufman-II, L.G. & Scheuing, R.A. An introduction to hypersonics. Grumman Aircraft Engineering Corporation August 1960. Research Report RE-82. Dorrance, W.H. Two-dimensional airfoils at moderate hypersonic velocities. JAS, 1952, 19(9).
CHAPTER 4 SKIN FRICTION FORCE CALCULATION 4.1
INTRODUCTION
In the previous chapter (section 3.2.2), expressions are given for the skin friction forces which are simple modifications of incompressible flat plate values. Being empirical relations they are only approximate. It may be desirable to have a much better evaluation of the skin friction forces experienced by a body in high speed flow. Towards this, an engineering approach is adopted and no attempt is made to calculate the detailed skin friction distribution on the exact shape of the body. Instead, the vehicle surface is divided into a large number of flat surface panels in a manner that adequately approximates the true shape. Leading edge surfaces and the curvature are omitted. In the skin friction analysis, the number of panels chosen to represent the body is usually much less than the panels required in the pressure calculation analysis. For each surface element, its normal end coordinates of the area centroid are determined. The shear force on each surface element is assumed to act through its centroid on the surface in a direction parallel to the plane containing the surface normal and the free stream velocity vector. Approximate laminar or turbulent skin friction relations are used to calculate the skin friction force on the element. The net shear force on the body is obtained by summing up over the vehicle. In this type of approach the problem of determining the viscous force on the two- or three-dimensional body is reduced to one of solving the skin friction force on a number of constant property flows over flat plate panels. For the skin friction calculation, the local flow properties such as pressure, temperature, velocity and density over each of the element under consideration are required and these are determined from the approximate inviscid aerodynamic analysis programs such as tangent wedge, tangent cone, modified Newtonian, Shock Expansion, Newtonian plus P.M. expansion, etc.
62
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Compressible turbulent flow over a flat plate at an arbitrary Mach number and plate temperature has been the subject of investigation by many workers over a number of years. Numerous theoretical and finite-difference computational methods have been suggested to calculate the skin friction coefficient. Amongst these, the commonly used engineering methods are that due to Sommer and Short1, Van Driest2 and Spalding and Chi3. All three methods are suitable modification of the incompressible turbulent skin friction coefficient based on Reynolds number suitably transformed to take into account the compressibility and temperature effects. Brief summaries of these methods are given below. 4.2
SOMMER & SHORT METHOD
This method is based on finding a temperature T * at which the density and viscosity for compressible flow have to be evaluated if incompressible flow relations for zero heat transfer are to apply for any Mach number, Me , at any wall temperature ratio (Tw/Te ), for a given Reynolds number, Rexe. The subscript e refers to conditions at the outer edge of the boundary layer over the surface element under consideration and x, the distance of the centroid of the element under consideration from the nose or the leading edge. The calculation procedure is as follows: Evaluate T
*
from the equation
T* = 1 + 0.035 M Te
2 e
T w + 0.45 Te
− 1
(4.1)
* from the relation Evaluate Reynolds number R e xe
Re *xe Re xe
ñ* = ñ
ìe * ì
(4.2)
Since the pressure is constant across the boundary layer, ñ* ñe
=
Te T
*
Aerodynamic Characteristics of Vehicle Components
Therefore
ì e * ì
T e = T *
Re *xe Re xe ì e The ratio * ì
µe * µ
(4.3)
is determined from the Sutherland relation, viz.,
Te * T
1.5
* T + 115 T e + 115
( temperatur e in o K )
(4.4)
Evaluate from the Karman-Schoenherr equation given by
0.242 C F*
= log 10 C F* Re *xe
(4.5)
or alternately from the Prandtl-Schlichting equation, viz.,
C F* =
0.46 * log 10 Re xe
2.6
(4.6)
Evaluate CF the compressible turbulent skin friction coefficient from the relationship
CF C F*
4.3
=
ñ* ñe
=
Te T
*
(4.7)
VAN DRIEST-II METHOD
The following is the formula for calculation of local skin friction coefficient due to turbulent compressible flow over a flat plate due to Van Driest2.
63
64
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
ìe ≈ 4.15 log 10 Re x C F ìw −1
sin −1 A + sin −1 B T aw CF Te
+ 1. 7
(4.8) where A =
2a
−b
2
2 b + 4a
2
0.5
and
B=
b 2 b + 4 a 2
0.5
The parameters a and b are given by the following relations γ − 1 a = M 2
2 e
T T e aw − 1 and b = Tw Tw
The adiabatic wall temperature is given by the relation
Taw Te
=1 + ℜ
γ −1 2
M e2
where, ℜ = the recovery factor = (Prandtl No.)1 / 3 4.4
SPALDING & CHI METHOD
Similar to the above two methods in the Spalding-Chi method, the compressible skin friction is given by the incompressible form with appropriate correction factors to account for compressibility and viscosity effects.
Aerodynamic Characteristics of Vehicle Components
CF =
1 Re F CF xe Re x inc Fc
(4.9)
where
Fc =
T /T − 1 aw e sin −1 A + sin −1 B
and
F Re x F c =
ìe ìw
(4.10)
where expressions for A and B are as given in Eqn. 4.8. One first computes Fc and then FRex. The equivalent incompressible Reynolds number is given by Rexe FRex. Using this equivalent Reynolds number, the corresponding incompressible turbulent skin friction coefficient is determined using any one of the well-known formulae such as Karman and Schoenherr, (Eqn.4.5), Prandtl-Schlichting, (Eqn.4.6) or the Sivells & Payne relation, viz. CF
inc
=
0.088 log R xe − 1.5
2
Dividing the incompressible flow skin friction coefficient by the factor Fc one obtains the desired compressible flow skin friction coefficient. For most flows, a portion of the flow over the body is laminar. For these regions the mean friction coefficient is determined from the approximate laminar flow value, viz.,
C F inc =
1.328 Re x
where the Reynolds number is based on the length from the leading edge to the point of transition. 4.5
EMPIRICAL EQUATIONS
Over a wide range of angle of attack, velocity and altitude conditions, Schmidt4,5 has presented data on skin friction coefficients of hypersonic flow over flat plates both for laminar and turbulent flows in terms of free stream parameters rather than the flow parameters at the edge of the boundary layer. To this data
65
66
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
empirical curves have been fitted6 to obtain simpler relations particularly useful in preliminary design. The suggested equations are: Laminar flow
C f lam
Re ∞ = 0.45 cos α + 4.65
V∞ 3050
sin α cos 2.2 α
where, α is the angle of attack and the V the velocity in m/s. The above equation deviates no more than 20 per cent from the data presented by Schmidt for low altitude, high angle of attack flight and is closer to ± 10 per cent for the rest of the altitude and angle of attack ranges: 15 ≤ α ≤ 45 ; 3050 ≤ V ∞ ≤ 8000 ; 45700
≤ h
≤ 91400
Turbulent flow
C
f
turb
Re ∞
0.2
V∞ = 0.048 sin 4.5 α + 0.70 cos 2.25 α sin 1.5 α 3050
(In the ranges of 5 ≤ α ≤ 50 ; 3050 ≤ V ∞ ≤ 8000 ; 30500 ≤ h ≤ 91500).
REFERENCES 1.
2. 3.
4.
5.
6.
Sommer, S.C. & Short, J.B. Free-flight measurements of turbulent boundary layer skin friction in the presence of severe aerodynamic heating at mach numbers from 2.8 to 7.0. NACA, March 1955. TN 3391. Van Driest, E.R. Turbulent boundary layers In compressible fluids. JAS, 1951, 18(3). Spalding, D.B.M. & Chi, S.W. The drag of compressible boundary layer on a smooth flat plate with and without heat transfer. J. Fl. Mech., 1964, 18(1), Part 1. Schmidt, J.F. Laminar skin friction and heat transfer parameters for a flat plate at hypersonic speeds in terms of free stream flow properties. NASA, September 1959. TN D-8. Schmidt, J.F. Turbulent skin friction and heat transfer parameters for a flat plate at hypersonic speeds in terms of free stream flow properties. NASA, May 1961. TN D-869. Hankey(Jr), W. L, & Alexander, G.L. Prediction of hypersonic aerodynamic characteristics for lifting vehicles. September 1963. ASD-TDR-63-668.
CHAPTER 5 AERODYNAMIC HEATING AT HYPERSONIC SPEEDS 5.1
INTRODUCTION
The rate of heat transfer to a vehicle due to aerodynamic heating is a function of its geometry, orientation in space and its flight trajectory. For flights at high supersonic and hypersonic speeds, the problem of aerodynamic heating becomes important and needs to be analysed particularly for Mach numbers greater than about 2.0. The induced thermal stresses due to aerodynamic heating can seriously affect the structural integrity of the vehicle and may result in a component failure. Consequently, the determination of the heating rates and the skin temperature are needed to complement the pressure field calculation. Several methods have been developed by different authors to account for aerodynamic heating effects at high speeds. Some of these are rigorous in which the full governing equations are numerically solved on a computer for a given body geometry, incidence and flight condition. These types of analyses are usually reserved for complicated cases such as shock-boundary layer interactions, flow separation, etc., or in final detailed design. However, approximate analytical, semi-analytical and empirical methods have been proposed for easy and fairly accurate and acceptable calculations of heat transfer at high speeds that are particularly useful in the preliminary analysis stage. No attempt is being made here to collate all such methods and critically review them. Instead some simple methods familiar to the author and having good correlation with experiments or more exact analysis are presented in this section. 5.2
HEATING ANALYSIS
The basic equation used to calculate the time rate of change of heat stored in a thin skin element is given by
68
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
q& = (ρ w C p ,w τ w
)
d Tw dt
= h (H st − H w ) − σ s ε F T w4
(5.1a)
for 3-dimensional stagnation points or unswept leading edges and
q& = (ρ w C p ,w τ w
)
dT w dt
= h (H R − H w ) − σ s ε F T w4
(5.1b)
for swept leading edges, body, wing and control surfaces. where q
=
heating rate, J/m2-sec
ρw Cp,w τw Tw t h
=
density of surface material, kg/m3
=
specific heat of surface material, J/kg- °K
=
wall or skin thickness, m
=
wall or skin temperature, °K
=
time, sec
=
local heat transfer coefficient based on enthalpy, kg/m2 sec
HR =
boundary layer recovery enthalpy, J/kg
Hw =
enthalpy at wall temperature condition, J/kg
ε
=
emissivity of wall surface
σs
=
Stefan-Boltzman constant, 5.67 × 10-8 W/m2-°K - 4
F
=
radiation geometry factor normally taken as 1.0
Tw =
wall temperature, °K
The above equation neglects the heat absorbed by the wall from solar radiation and heat lost from internal radiation and conduction. Normally, the radiation effects are compensating. For cases in which the internal conduction is large, it could be accounted for by incorporating an appropriate conduction term in the above heat balance equation. It is to be noted that for high supersonic and hypersonic flows, the heat transfer rate is expressed in terms of the enthalpy whereas for temperatures below approximately 800°K the heat balance equation is normally formulated in terms of temperatures. As mentioned1 and to quote "to obtain good surface temperatures and to a lesser extent good surface heating rates, proper engineering judgement must be exercised to determine the heat capacity in the above equation. Since the values of specific heat of the surface material and the density of the surface material are
Aerodynamic Heating
thermal properties of the material, the only way to vary the heat capacity significantly is to change the value of the material thickness. For metallic surfaces, the thickness of the skin will give satisfactory results. For surfaces that are insulated with low conductivity insulation (such as space shuttle), a material thickness should be used that will result in a heat capacity of approximately 2044 J/m2 - ° K". The heat transfer rate varies from a maximum value at Tw = 0 (cold wall) to zero at Tw = Taw (adiabatic wall). Knowing the heat transfer coefficient, the Mach number, angle of attack and altitude combination, it is possible to compute the actual heating rate for any wall temperature. From this, the time rate of change of skin temperature can be determined if one knows the properties of the wall material. By carrying out the heat transfer calculations at a series of selected points along the vehicle trajectory, the time history of the surface temperature of the body can be obtained. The skin equilibrium temperature for each skin element under consideration can be calculated from the above equation when the time rate of change of temperature goes to zero. 4 h H R − H eq = σ s ε F T eq
The temperature and heat flux for each skin element at time J are calculated from
h (H R − H j − 1 ) − σ s ε F T j4−1 T j = T j −1 + ∆ t j ñw c p , w ô w and
q& j = h (H R − H j ) − σ s ε F T j4 respectively. In practice, computational intervals are spaced closely during the early times and farther apart at later times so that precision is maintained during periods of rapid temperature rise but excessive computational time will not be required after the period of initial temperature rise. 5.3
STAGNATION POINT HEAT TRANSFER
5.3.1 Spherical Nose The nose tip of a vehicle is usually blunt and spherical in shape. Consequently a normal detached shock is present in front of
69
70
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
the stagnation point. From the known free stream conditions the downstream properties of the shock are determined using the normal shock relations for the case of perfect gas. If however, the temperature is such that one has to consider the gas as real, then the downstream properties of the shock are determined for an equilibrium flow either by the use of real gas flow charts if one is available or by numerical iterative solution as outlined in Andersons book2. From the known properties downstream of the normal shock, the stagnation enthalpy, recovery enthalpy and stagnation pressure are determined as follows: V 22 (for both perfect and real 2 gases). Recovery enthalpy, H R = H 2 + ℜ( H 0 − H 2 ) where, ℜ is the recovery factor (In the above relations subscript 2 refers to conditions downstream of the normal shock)
Stagnation enthalpy, H 0 = H 2 +
ℜ= Pr = for laminar flow, and
ℜ= (Pr )
1
3
for turbulent flow;
Pr = Prandtl Number. The flow downstream of a normal shock at high speeds is low subsonic. Hence, one can use the incompressible Bernoulli’s equation without much of an error to calculate the stagnation pressure both for perfect and real gas flows.
po
= p2 +
ρ 2 V 22 2
For a perfect gas the above value could be checked if necessary with the exact calculation of stagnation pressure using the isentropic relation as the downstream Mach number is known. For real gases the condition of constant entropy can be used to calculate the stagnation pressure by trial and error. However, these are not really necessary as the downstream Mach number will be very low subsonic at high Mach numbers and the incompressible Bernoulli’s equation is more than adequate. The commonly used expression to calculate the heat transfer coefficient for a spherical nose stagnation point as given in the book by White3 is
Aerodynamic Heating
h = 0.763 Pr
− 0. 6
(ρ o µ o )
0.5
ρw µw ρo µo
0. 1
du e dx
s
(5.2)
The above expression is derived from the work of Fay and Riddell4 but restricting to the case of non-reacting gases. It is equally applicable to both perfect and real gases. (due/dx)s is the velocity gradient at the stagnation point. Subscript o refers to stagnation conditions downstream of the detached normal shock in front of the body and w to the wall conditions at the stagnation point. A relation identical to Eqn. 5.2 but without the term 0.1
ρ µ w w is also mentioned in the book of Anderson2. This term ρo µo is of the order of unity and alters the heat transfer rate by about 10 per cent depending on the wall temperature conditions. The stagnation point streamwise velocity gradient is given by the Newtonian impact theory, viz.,
du e dx
=
1
2( p o −p ∞ )
ρo
rN
where, rN is the spherical nose tip radius. Supposedly a more accurate relation than that of the Newtonian for the velocity gradient as given by Adams and associates and mentioned by Dejarnette, et al 5 is du e dx
V∞ = r N s
1.85
ρ∞ ρo
In the case of a perfect gas, a constant value of 0.7 is taken for the Prandtl number and the viscosity is based on the Sutherland law corresponding to the stagnation temperature. For a real gas the stagnation temperature is determined from the known values of stagnation pressure and stagnation enthalpy by curve fits6,7. For real gases, the Prandtl number and viscosity are functions of stagnation temperature and stagnation density, viz.,
71
72
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Pr = Pr (To, ρo) and µo= µo (To, ρo) Their values are obtained by curve fits6,7. As the flow is laminar near the stagnation region, the recovery factor dependence on the Prandtl number is taken to be that corresponding to the laminar flow, viz., Pr when calculating the heating rate, viz., q . 5.3.2 Cylinder Normal to the Stream The stagnation point heat transfer coefficient for a 2dimensional circular cylinder normal to the incoming flow is given by the expression3 viz.,
h = 0.57 Pr
− 0.6
ρ
o
µ
o
0.5
ñ µ w w ρo µo
0.1
du e dx
s
(5.3)
The method of determining the relevant parameters in the above expression is exactly identical to the axisymmetric case described above. Comparison of Eqn. 5.3 above with the heat transfer coefficient expression for the case of axisymmetric stagnation point case, Eqn. 5.2 shows that both the expressions are the same except for the leading numerical coefficient. All other terms being the same, it is apparent that the stagnation point heating to a sphere is larger than to a 2-dimensional cylinder. 5.3.3 Swept Wing Stagnation Line Heat Transfer The wing or control surface leading edge is considered to be cylindrical and flow past it is analogous to a flow past a yawed inclined circular cylinder of infinite span. The detached shock will be parallel to the leading edge whereas it is oblique to the incoming free stream. Let ‘ Λ ’ be the sweep angle. The incoming velocity is now decomposed in to components, V ∞ n = V ∞ C os Λ, normal to the leading edge and V ∞ t =V ∞ Sin Λ , tangential to the leading edge. The normal component of the incoming velocity is now made to pass through the detached shock which is normal to it. The tangential component of the velocity remains unchanged as it passes through the shock. Let the resulting conditions downstream of the shock be as
Aerodynamic Heating
p2
= pressure downstream of the normal shock
T2
= temperature downstream
ρ2
= density downstream
V2 n = normal component of the velocity downstream V2 t = tangential component of the velocity downstream equal to V ∞ S in Λ S2
= downstream entropy or change in entropy
H2
= downstream enthalpy
All of the above quantities can be obtained from normal shock tables for perfect gases and by an iterative numerical solution of the governing equations across the shock for real gases2. After passing through the shock, as the flow approaches the leading edge, the normal component of the velocity viz., V2 n goes to zero on the surface edge, leaving the tangential component V2 t unchanged, giving rise to a non-zero velocity along the attachment line called the stagnation line analogous to the stagnation point on a cylinder normal to the free stream. The loss of the normal component of the velocity on the surface results in increase of pressure and enthalpy as follows: Pressure on the surface can be calculated by the Bernoulli’s equation as V2n is low subsonic pe = p
2
+
1 ρ 2
2
V 22n
This value holds good both for perfect and real gases as V2n is low subsonic. 5.3.4 Perfect Gas For a perfect gas, the temperature on the surface is given by
γ −1 2 Te = T 2 1 + M 2n 2 where
M
2n
=
V2n
γ RT 2
73
74
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
and the density 1
ñe = ñ 2
γ −1
+
2
M 22n
1 γ −1
The Mach number of the flow with a velocity V2t along the stagnation line is M
e
=
V 2t
γ RT e
The stagnation pressure, temperature, density and enthalpy are
po = pe
γ −1 M 1+ 2
γ −1 To = Te 1+ M 2
ρo = ρe
1+
H o = C p Te +
γ −1 2
2 e
2 e
M
γ γ −1
2 e
1
γ −1
2 V 2t
2
respectively. 5.3.5 Real Gas The enthalpy of the flow along the edge is given by
He = H2 +
V 22n 2
and the stagnation enthalpy is
Aerodynamic Heating
Ho = He +
2 V 2t
2
The stagnation pressure has to be obtained to know the stagnation temperature. Since the entropy or the change in entropy is known downstream of the shock, this value has to be constant as the flow is decelerated isentropically from downstream of the shock to the stagnation conditions. The stagnation pressure can be determined from the Mollier diagram for equilibrium air (if one is available) corresponding to the stagnation enthalpy and entropy. Otherwise a trial and error iterative technique is adopted in which the invariance of entropy is maintained. Once the stagnation pressure is thus determined with the known value of stagnation enthalpy, the other thermodynamic variables like temperature, viscosity, etc., can be obtained by curve fits6,7. 5.3.6 Heat Transfer Coefficient h Beckwith and Gallagher8 have suggested that the leading edge heat transfer rate for a yawed cylinder is the same as that of a normal cylinder but modified by a cosine factor involving the sweep angle Λ . Accordingly, the laminar flow stagnation line heat transfer coefficient is
(
h = 0.57 Pr − 0.6 ρ o µ o
×
du e dx
s
)
0.5
ρw µw ρo µ o
0.1
(cos Λ )1.1
(5.4)
and the convective heat transfer rate is
q
l . e ,lam
= h ( H aw − H w
)
At high Reynolds numbers and angles of sweep from 40° to 60°, it was found that the boundary layer on a swept cylinder was completely turbulent even at the stagnation line. For turbulent flow the corresponding equation is
q i .e , turb = 1.04 Pr
− 0.6
(ρ* µ* )− 0 . 8 ( V ( µo ) 0. 6
∞
sin Λ 2
)
0.6
du e dx
0.2
(5.5)
75
76
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
The superscript* denotes evaluation at the Eckert reference enthalpy9 which is given by H * = 0.5 ( H w + H e ) + 0.22 ( H aw − H e
)
where He = enthalpy at the outer edge of the boundary layer For a perfect gas a reference temperature is used as follows which is multiplied by Cp to get the reference enthalpy.
T
*
= 0 .5 ( T w + T e
The streamwise streamline is given by
1 = r le sl
du e dx
) + 0.22 ( Taw − Te ) velocity
gradient
at
the
stagnation
2 p e − p ∞
ρe
where, rle is the leading edge radius. The nature of the flow, whether it is laminar or turbulent is based on the free stream value of the Reynolds number based on the leading edge diameter, viz., Re D =
ρ ∞V ∞ D µ∞
If the Reynolds number is below a specified lower limit value it is laminar and, if it is above a specified upper limit it is turbulent. In between these lower and upper limits the flow is considered transitional. The heating rate in the transitional region is given by
q& = q&
lam
Re − Re low D + Re upp − Re low
q& lam
Aerodynamic Heating
5.3.7 Heat Transfer on Flat Surfaces & Fuselage Panels The approximate engineering method of determining the heat transfer rate consists of treating the surface element under consideration as a part of a flat plate. It has been shown by many investigators that for a wide range of Mach numbers and temperatures, a close approximation to the actual compressible skin friction coefficient is obtained when the incompressible value of the skin friction coefficient is evaluated at a temperature corresponding to a reference enthalpy (or a reference temperature for a perfect gas). Application of suitable Reynolds analogy factor relating the skin friction coefficient to Stanton number is used to get the appropriate heat transfer coefficient. The basic expression for the heat transfer coefficient is the same as given by Eqn. 5.1, viz.,
h =
ρ w C p ,w τ w
dT w
+ σ s ε F T w4 dt ( HR − Hw )
The recovery enthalpy is computed from
HR =H∞ +
V ∞2 − V e2 2
+ ℜ
V e2 2
where, ℜ is the recovery factor. For heat transfer determination it is necessary to know the properties of the local flow parameters over the surface element under consideration. These are obtained by any one of the aerodynamic predictive methodologies such as shock expansion, second order shock expansion, tangent cone, tangent wedge, etc. For a real gas, knowledge of local pressure and entropy or enthalpy by any of the approximate analysis methods is able to give all other flow parameters. The heat transfer coefficient h is given by
h =
* f
Pr * 2
C
−0.667
ρ *Ve
77
78
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
where, Cf* is the skin friction coefficient and Pr* is the Prandtl number C
* f
=
0.332 Re * N lam
1 2
for a laminar flow, and C *f =
0.185 Re * log 10 N tur
2.584
for a turbulent flow The * conditions are evaluated at the reference enthalpy H * for a real gas given by H * = He + 0.5 (Hw – He ) + 0.22 ( HR – He )
(5.6)
H * = He + 0.5 (HR – He )
(5.7)
or
known as adiabatic reference enthalpy For a perfect gas for * conditions the reference temperature T * is used given by T * = 0.5 (Tw + Te ) + 0.22 (TR – Te )
(5.8)
The subscripts e refer to the local conditions in the inviscid shear layer or at the outer edge of the boundary layer, w the wall conditions and, R adiabatic wall or boundary layer recovery conditions respectively. The Reynolds number is based on boundary layer running length s and is given by Re * =
ρ *Ve s µ
*
and the Prandtl number
Aerodynamic Heating
Pr * =
µ*C k
* p
*
ρ * = f (H *, pe ), for a real gas6,7
ρ * = pe /R T* for a perfect gas. T*
= f ( H *, pe ) for a real gas6,7
T*
= reference temperature as given by Eqn 5.8 above for a perfect gas.
The viscosity is evaluated by using the Sutherland relation at a temperature corresponding to T * and the thermal conductivity corresponding to temperature T * and pe. Substituting the above values of the skin friction coefficient in the heat transfer coefficient relations we get h H = 0.332 ( Pr * ) − 0.667
ρ *V e Re * N lam
(5.9)
for laminar flow, and
h H = 0.185 ( Pr * )
− 0.667
ρ* V e Re * log10 N tur
2.584
(5.10)
for turbulent flow. In the above expressions, Nlam and Ntur are the laminar and turbulent Mangler transformation factors.
shown
Nlam
=
3 for a body of revolution and 1 for a planar body (wing)
Ntur
=
2 for a body of revolution and 1 for a planar body (wing)
Some experiments conducted on an X-15 aircraft10, have that the correlation between the heat transfer
79
80
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
measurements and above prediction is better when the * conditions were evaluated at the adiabatic reference enthalpy as defined in Eqn. 5.7 above, instead of the reference enthalpy as proposed by Eckert, Eqn 5.6. 5.4
HEAT TRANSFER ANALYSIS BY THE METHOD OF QUINN & GONG
A real time aerodynamic heating programme for hypersonic flight simulation has been developed by Quinn and Gong1. This algorithm is capable of calculating 2- and 3-dimensional stagnation point heating rates and surface temperatures. The leading edge sweep is also accounted for in the program. In addition, upper and lower surface heating rates and surface temperatures on flat plates, wedges and cones can be calculated both for laminar and turbulent flows together with boundary layer transition which is made a function of free-stream Reynolds number and free-stream Mach number. The results of this method of analysis when compared with more exact values obtained by the use of a NASA in-house aeroheating program showed that the heating rates and surface temperatures as predicted by the real time heating analysis were well within the required accuracy to evaluate heating trajectories. One unique feature of this methodology is the use of free stream conditions instead of the local flow quantities to calculate the heat transfer rate. Because of its simplicity and good prediction capabilities this method is described in detail. 5.4.1 Stagnation Point Heating Rate The basic equation used to compute the surface temperatures and heating rates (J/m2-s) for stagnation point calculation is
q& = ρ w C
p ,w
τ
dT w dt
= h H st − H
w
−σ εFT 4 w
(5.11)
for 3-dimensional stagnation point or unswept leading edges. For swept leading edges and surfaces the corresponding expressions is
q& = ρ w C
p ,w
τ
dT w dt
= h H R − H
w
− σ εFT 4 w
(5.12)
The only difference between the above two equations is the replacement of stagnation enthalpy by the recovery enthalpy for swept leading edges and surfaces.
Aerodynamic Heating
To solve the above two equations we have to know the heat transfer coefficient h. For this, a modified version of the solution given by Fay and Riddell4 is used. For 3-dimensional flow
h = 0.94 K 1 (ρ st µ st
) 0.4 (ρ w µ w )0.1 (dU / dx )x = 0
(5.13)
and for 2-dimensional flow
h = 0.706 K 2 (ρ st µ st
)0.4 (ρ w µ w )0.1 (dU / dx )x = 0
(5.14)
The subscripts st and w are stagnation and wall conditions. K1 and K2 are 3-dimensional and 2-dimensional stagnation factors respectively. Quinn and Gong1 have used U.S. Customary Units in their work. However, in the present work, S.I. Units are employed and so whereever it is necessary, the relations taken from the work of Quinn and Gong have been modified. To solve the Eqns. 5.11- 5.14 suitably one has to know the local flow conditions behind a normal shock wave. However, to minimize the computational time, a method has been developed in which the free stream values are used along with a table of values for K1 and K2 maintaining the accuracy of the results. The Eqns. 5.11- 5.14 are solved as The stagnation enthalpy H (J/kg ) is given by
U
H st = H ∞ +
2 ∞
cos 2 Λ (5.15)
2
for 3-dimensional calculations, and
HR =H∞+
U
2 ∞
2
+ 0.85
U
2 ∞
sin 2 Λ 2
(5.16)
81
82
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
for 2-dimensional calculations.
dU 1 = dx R x=0
7 M
2 ∞
cos 2 Λ −1 M
2 ∞
M
cos 2
cos 2 Λ + 5 P∞ ρ∞ Λ 2 ∞
(5.17) where R = body nose radius
ρ st µ st = ρ ∞ µ ∞
T ∞ o K
= 2.43 × 10 − 7 T w o K
µ ∞ = 2.43 × 10
µw
6 M ∞2 cos 2 Λ M ∞2 cos 2 Λ + 5
−7
T st T∞
0.75
(5.18)
0.75
(5.19)
0.75
(5.20)
P w ≈ P st ≈ P ∞
ρw =
Pw 287 T w
7M
2 ∞
cos 6
2
Λ −1
(5.21)
(5.22)
For 3-dimensional stagnation point calculations, the sweep angle Λ in the above equations will be zero. Values of H ∞ , Hw , TR
Aerodynamic Heating
and Tst are obtained by the curve fits given by Gupta et. al.6 and, Srinivasn7. Values of K1 and K2 are obtained by linear interpolation from the following table. Table 5.1 Stagnation point heating factors Mach No.
K1
K2
1
1.00
1.00
5
1.16
1.20
10
1.14
1.18
15
1.16
1.16
20
1.23
1.16
25
1.40
1.25
30
1.45
1.26
5.4.2 Convective Heating Equation for Small or Zero Pressure Gradient Surfaces The basic equation is the same as Eqn. 5.12 viz.,
dT q& = (ρw C p ,w τ) w = h (H R − H w ) − βTw4 dt To determine the heat transfer coefficient h one has to know the local flow conditions. However, for simplicity and to save time the free stream values are utilized similar to the case of stagnation point heat transfer analysis. The heat transfer coefficient is written as
h= C 5
h + o
h α
(5.23)
where, ho is the heat transfer coefficient for a flat plate at zero angle of attack and h α is that portion of the heat transfer coefficient caused by angle of attack and wedge or cone angles. For laminar flow the suggested value of C5 is 1.73 and for turbulent flow C5 is 1.15. For turbulent flow, the equations to calculate the heat transfer coefficient are as follows:
83
84
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Lower Surface
ho
C 1 ( 0.0375 ) ρ ∞ U ∞ = 0.2 x
hα
0.8 ρ U ∞ ∞ = A1 x 0.2
0.8
µ
∞
0.2
T ∞ * T
0.65
δ + α
(5.24)
(5.25)
and for Upper surface
hα
0.8 ρ U ∞ ∞ = A1 x 0.2
δ −α
(5.26)
The constant C1 used is an empirical value used to account for the approximation involved in developing the Eqn. 5.24. The suggested best values for C1 are 1.0 for the upper surface and 0.90 for the lower surface. The difference between the heat transfer coefficient calculated using the free stream flow conditions and the local flow conditions are represented by the Eqns. 5.25 and 5.26. Similarly for laminar flow, the relevant equations are Lower Surface
ho
ρ U µ ∞ ∞ ∞ = 0.4 x
hα = A2
ρ U ∞ x
∞
0.5
T ∞ * T
0.125
(5.27)
0. 5
δ +α
(5.28)
Aerodynamic Heating
Upper Surface
ho
ρ U µ ∞ ∞ ∞ = 0.421 x
ρ U ∞ x
hα = A2
∞
0.5
T ∞ * T
0.125
(5.29)
0. 5
δ − α
(5.30)
In the above expressions, δ is the cone or wedge half angle in degrees and α the angle of attack in degrees. Values for A1 and A2 are as given in the tables to follow. Table 5.2 Turbulent flow correction factors, A 1 × 10 −4
M∞
Wedge or cone angle ± angle of attack, deg. -1 0 +1 +5 +10
-10
-5
+20
+40
2 3 5
0.844 1.50 1.30
0.920 1.62 1.79
1.04 1.80 2.38
1.04 1.62 1.84
1.30 1.80 2.07
1.41 2.08 2.34
1.62 2.08 2.61
1.73 2.37 2.87
1.84 2.49 2.90
10
1.19
1.66
2.49
2.16
2.61
3.32
4.06
4.80
4.88
15
1.01
1.57
2.60
2.16
2.77
3.97
5.18
6.58
6.77
20 25
0.747 0.582
1.41 0.989
2.30 2.03
2.16 1.95
2.87 2.87
4.47 4.54
6.08 6.49
9.94 9.20
9.49 8.66
30
0.472
0.826
1.88
1.84
2.87
4.87
6.49
6.49
6.49
Table 5.3 Laminar flow correction factors, A 2 × 10 −4 M∞
-10
-5
Wedge or cone angle ± angle of attack, deg. -1 0 +1 +5 +10
+20
+40
2
0.366
0.378
0.390
0.354
0.366
0.378
0.403
0.415
0.427
3
0.488
0.507
0.533
0.488
0.528
0.547
0.561
0.558
0.561
5
0.732
0.799
0.871
0.763
0.455
0.950
1.01
1.04
0.819
10
0.997
1.18
1.51
1.46
1.62
1.85
2.03
2.07
1.87
15
1.13
1.53
2.20
1.83
2.29
2.68
2.94
2.96
2.51
20
1.17
1.77
2.48
2.51
2.90
3.54
3.82
3.75
3.20
25
1.12
1.74
2.75
2.48
3.33
4.15
4.27
3.66
3.11
30
1.03
1.71
2.95
2.30
3.66
4.27
4.27
2.44
2.44
85
86
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
It is to be noted that in Eqns. 5.25 and 5.28 the value for wedge angle plus the angle of attack ( δ + α ) is limited to 40°, and in Eqns. 5.26 and 5.30 the wedge angle minus the angle of attack ( δ - α ) is limited to –10°. The values of the other parameters in the heat transfer coefficient equations are
Re ∞ =
ρ∞ U ∞ x µ∞
µ ∞ = 2.43 × 10
T ∞
−7
H R = H ∞ + 0.85
U
0.75
2 ∞
2
For laminar flow,
U
H R = H ∞ + 0.89
∞
2
2
and for turbulent flow, H * = H ∞ + 0 .5
Hw = f H
∞
T
*
( H w − H ∞ ) + 0 .22 ( H R − H ∞ )
( Tw , P∞ )
= f T ∞ , P ∞ = f H
*
, P ∞
In the above the values of Hw , H ∞ and T * are obtained from curve fittings given6-7 and the values of A1 and A2 from the tables above. 5.4.3 Boundary Layer Transition The methods outlined above, utilizes only the free stream flow properties to calculate the local heat transfer rates. In keeping
Aerodynamic Heating
up with the same methodology only the free stream Reynolds number, free stream Mach number are utilized as criteria to determine the boundary layer transition by Quinn and Gong as follows:
[log Re
If log Re ∞ ≤
T
[log Re
If log Re ∞ >
T
( ∞ )] the flow is assumed laminar
+Cm M
( ∞ )] the flow is assumed turbulent
+Cm M
Values of log R e T and Cm depend on type of flow, angle of attack, sweep angle and leading edge or nose bluntness. The authors suggest the following table of values, which according to them produce satisfactory transition results. Table 5.4 Transition Reynolds numbers and transition Mach number coefficients.
(a) Conical flow α, deg
Cm Sharp leading edge
log ReT
0-7 7 - 20 20 - 40
5.3 5.3 5.3
Blunt leading edge
0.25 0.20 0.15
0.20 0.18 0.12
(b) 2-Dimensional flow Cm Λ,deg
Log ReT
Sharp leading edge α ≤ 7° 7°< α < 20°
α ≥ 20°
Blunt leading edge α ≤7°
7° < α < 20°
α ≥ 20°
0 - 45
5.3
0.23
0.20
0.18
0.20
0.18
0.15
45 - 60
5.3
0.20
0.18
0.15
0.18
0.15
0.12
60 - 75
5.3
0.17
0.17
0.13
0.15
0.13
0.11
5.5
HIGH SPEED CONVECTIVE METHODOLOGY OF TAUBER
HEAT
TRANSFER
An engineering type formulation to calculate the convective heat transfer rates at high speeds has been made by Tauber11. The final pertinent relations taken from that reference are as follows:
87
88
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
5.5.1 Stagnation Point Heat Transfer The expression for the stagnation point heat transfer for a sphere is
q& ws = 1.83 10 − 4
ρ ∞ rn
0.5
Hw V ∞3 1 − Hs
watts/m2
(5.31)
where the nose radius rn is in metre, the free-stream density ρ∞ in kg/m3, and the flight velocity V ∞ , in m/sec. Hw is the wall enthalpy 2 and Hs the stagnation enthalpy = H ∞ + V ∞ in J/kg. 2 5.5.2 Swept Infinite Cylinder The analogous expression for the swept infinite cylinder is
(
q& w ,cyl = 1.29 10
−4
)
ρ∞ r cyl
Hw × V ∞3 1 − H aw
0.5
(1 − 0.18 sin 2 Λ eff )
cos Λ eff
(5.32)
where
(
H aw = H ∞ + 0.5 V ∞2 1 − 0.18 sin 2 Λ eff
)
sineff = sin Λ cos α ,
Λ = the sweepback angle In deriving the above equations the same Newtonian value for velocity gradient at the stagnation point was used both for the sphere and cylinder, viz.,
du e dx
s
2 p st − p ∞ 1 = rn ρ st
0.5
Aerodynamic Heating
As mentioned by Tauber in his report, the velocity gradient calculated as above is correct for the case of a cylinder but a bit low for the sphere. It has been suggested by him that if a correction is made by using the appropriate value, the constant in Eqn. 5.31 changes from 1.83 to 1.90. 5.5.3 Cone and Flat Plate Heating Rates 5.5.3.1 Laminar boundary layer The expressions for laminar boundary layer heating of bodies without pressure gradients such as sharp cones, wedges and flat plates at angle of attack is given by
ñ ∞ cos δ c q& w , Cone = 4.03 10 − 5 x Hw ×V ∞3.2 sin δ c 1 − H aw
(
)
ρ ∞ cos δ FP q& w ,FP = 2.42 10 − 5 x Hw ×V ∞3.2 sin δ FP 1 − H aw
(
)
0.5
W / m 2
(5.34)
0.5
W / m 2
(5.35)
where Hw
= enthalpy of the wall
Haw = adiabatic wall enthalpy = H ∞ + 0.40V ∞2
δc
= cone half angle
δFP
= wedge angle or angle of attack of the flat plate
5.5.3.2 Turbulent Flow For the case of flat plate and for V ∞ > 3960 m/sec
89
90
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
q& w ,FP ≈
(
2. 45 10
−5
)( ρ ∞ sin 2 δ cos 2 . 62 δ )0 . 8 ( x − x bt )0.2
Hw × V ∞3.7 0.9 − He
for 3960 > V ∞ > 1560 m /sec
qW FP =
(
) ( ρ sin (x − x ) (T
3.72 10 − 4
∞
2
δ cos 2.2 δ
0 .2
bt
W /555
)
)
0 .8
0 .25
W /m 2
and for a cone
q&
w cone
= 1.15 q& w FP
where Hw =
static enthalpy at the wall
He =
stagnation enthalpy at the outer edge of boundary layer
x
distance from the leading edge for flat plate and from nose for cone
=
xbt =
5.6
distance at which boundary layer transition begins
EMPIRICAL EQUATION FOR CONVECTIVE HEAT TRANSFER
Anderson2 in his book, reference has given a very simple method to estimate aerodynamic heating at hypersonic speeds. The heat transfer rate is expressed in a generalized form, viz.,
q& w = ρ ∞N V ∞M C The units for q w, ρ∞ and V∞ are W/m2, kg/m3 and m/sec respectively. The values of M, N and C are as follows:
Aerodynamic Heating
5.6.1 Stagnation Point M = 3, N = 0.5 H C = 1.83 × 10 − 8 R − 1/2 1 − w Ho
where, R is the nose radius in metres, and H w and Ho are wall and total enthalpies respectively. 5.6.2 Flat Plate in Laminar Flow M = 3.2 and N = 0.5 C = 2.53 × 10
−9
cos α
0.5
sinα x
− 0.5
Hw 1− Ho
where, α is the local body angle with respect to the free stream and x is the distance measured along the body surface in metres. 5.6.3 Flat Plate in Turbulent Flow For V∞ ≤ 3962 m/s, N = 0.8, M = 3.37 and
C = 3.89 × 10 − 8 (cos α )1.78 (sin α )1.6 (x T Hw × 1 − 1.11 Ho For V∞ > 3962 m/s,
Tw 556
)−0.2
−0.25
N = 0.8, M = 3.7 and
C = 2.2 × 10 − 9 (cos α )2.08 (sin α )1.6 (x T
)− 0.2 1 − 1.11
Hw H o
where, xT is the distance measured in metres along the body surface in the turbulent boundary layer. As mentioned by Anderson “the above are useful for preliminary analysis and are not recommended for more detailed work ”.
91
92
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
REFERENCES 1.
2. 3. 4. 5.
6.
7.
8.
9.
10.
11.
Quinn, R.D. & Gong, L. Real-time aerodynamic heating and surface temperature calculations for hypersonic flight simulation. NASA, August, 1990. 4222. Anderson(Jr), J.D. Hypersonic and high temperature gas dynamics. McGraw-Hill Book Co. 1989. White, F.M. Viscous fluid flow. McGraw Hill Book Co. 1991. Fay, J.A. & Riddell, F.R. Theory of stagnation point heat transfer in dissociated air. J. of Aero. Sci., 1958, 25(2), 73-85. DeJarnette, F.R.; Hamilton, H.H.; Weilmuenster, K.J. & Cheatwood, F.M. A review of some approximate methods in aerodynamic heating analysis. J. Thermophysics, 1987, 1(1). Gupta, R.N.; Lee, K.P.; Thompson, R.A. & Yos, J.M. Calculations and curve fits of thermodynamic and transport properties for equilibrium air to 30,000 °K. NASA, October 1991. 1260. Srinivasan, S.; Tannehill, J. & Weilmuenster, K.J. Simplified curve fits for the thermodynamic properties of equilibrium air. NASA , 1987. 1181. Beckwith, I.E. & Gallagher, J.J. Local heat transfer and recovery temperatures on a yawed cylinder at mach numbers of 4.15 and high reynolds numbers. NASA, 1961. R-104. Eckert, E.R.G. Survey of boundary layer heat transfer at high velocities and high temperatures. Wright Air Development Center, April 1960. 59-624 p. Quinn, R.D. & Palitz, M. Comparison of measured and calculated turbulent heat transfer on the X-15 airplane at angles of attack up to 19.0°. NASA, TM-X-1291. Tauber, M.E. A review of high speed, convective, heat transfer computational methods. NASA , July 1989. 2914.
PART - II VALIDATION OF PREDICTION METHODS
CHAPTER 6 VALIDATION OF PREDICTION METHODS Approximate methods are normally used in preliminary design stages to predict flight control forces and moments experienced by a flying vehicle in the entire speed range covering the flight envelope. To ascertain the validity and the range of applicability of these methods, particularly for hypersonic vehicles, some detailed studies have been reported in the literature both in the past and fairly recently. In general, the approach is to examine several vehicle designs, such as the wing-body, the blended body, the cone body, etc., which cover a broad range of proposed hypersonic vehicle configurations and compare the predicted values with the available experimental data and then draw conclusions. In the subsonic and supersonic speed ranges, many methods are available to predict the aerodynamic characteristics. Some of these have been used for the analysis of a few hypersonic vehicle configurations. The vortex lattice method for subsonic flow analysis described by Lamar and Gloss1, which includes the leading edge suction effects based on theory, described by Polhamus2 has been applied to three different hypersonic vehicle configurations and compared with the wind tunnel results3 at a Mach Number of 0.2. The theoretically predicted lift, drag due to lift and the pitching moment, correlated well with the experimental results. Similar results using the same type of analysis were also reported4. Since the method described by Lamar and Gloss1 did not have the capability to include vertical surfaces, it was not possible to predict the lateral-directional characteristics of the configurations analysed. For this reason, the vortex lattice method used was not of much use in the preliminary design analysis. The most commonly used prediction method in subsonic and supersonic flows is the Panel method. Panel method has been applied to some specific hypersonic vehicle configurations such as X-15, Space Shuttle and Hypersonic Research Airplane, in the subsonic and supersonic
96
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
speed ranges. The results of these investigations are reviewed in the later part of this work. In supersonic speed ranges, the linear supersonic theory and the hypersonic impact methods were applied to three non-slender Hypersonic Airplane concepts at Mach numbers from 1.1 to 2.865. The configurations used were similar to those of Penland, et al 3. Lift, pitching moment, and drag due-to-lift values were calculated by the planar method6. The mean camber surface of the body and the wing were inputs to the linear theory program, but the vertical tail surfaces that could contribute to the pitching moment were ignored. A drag buildup analysis was carried out in which the drag due to skin friction, the wave drag at zero angle of attack due to volume and the camber drag at zero lift, were independently evaluated and summed up. The hypersonic impact methods were also used to check the limit of applicability of hypersonic methodology in the supersonic speed range. Tangent-cone empirical method was used on the body and tangent wedge method on the wing and tail surfaces. Prandtl-Meyer flow was assumed on expansion surfaces, where the minimum expansion pressure coefficient was limited to −1/M ∞2 . Also evaluated was the tangent-cone empirical method on all the configuration components. According to this work, the linear theory gave good lift prediction and fair to good pitching-moment prediction over the Mach number ranges tested, except in the transonic region. This good agreement between data and the linear theory predictions of CL and Cm showed that the linear theory with its thin-airfoil and planar assumptions was valid for the class of low fineness ratio, blunt-base high volume vehicle configuration. The linear theory drag prediction was generally poor with good agreement only below M ≤ 1.2. The inaccuracy of the zero lift drag prediction using the linear theory was attributed to the violation of slender body assumption, as the vehicle configurations had low-fineness ratio bodies. The tangent cone theory predictions were good for lift, and fair to good for the pitching moment for M ≥ 2.0. The combined tangent cone/tangent wedge theory (tangent cone for the fuselage and tangent wedge for the wing and tail) gave the least accurate prediction of lift and pitching moment. For all theories, the zero lift drag was overestimated especially for M < 2. The tangent cone method predicted the zero lift drag most accurately for M ≥ 2.0. The level of agreement depended on the configuration being studied, very good for one configuration but fair for others. No definite conclusions were reached regarding the lower Mach number limit of applicability of the hypersonic impact methodology.
Methods
An experimental investigation was conducted to provide a systematic set of longitudinal characteristics and lateraldirectional stability data for a simplified wing-body model with a series of vertical-tail arrangements7. The range of Mach numbers covered were 1.60 to 2.86 at nominal angles of attack from – 8º to 12º. Comparisons were made of the experimental data at zero angle of attack with three theoretical methods, a Second-order Shock-expansion and Panel method (MISLIFT)8, a slender body and first order panel method (APAS)22 and PAN AIR9, a higher order panel method. Since the results from MISLIFT and APAS were invariant with angle of attack, only the results from PAN AIR were used for comparisons of the stability parameters at angles of attack. Overall, the results were quite good except for a couple of vertical tail configurations. The differences were attributed probably to the inability of the program to account for the presence of vortices, etc. The conclusion reached were: (a)
(b)
(c)
PAN AIR generally provided accurate predictions at moderate angles of attack for complete configurations with either single or twin vertical tails. APAS provided fairly accurate predictions at zero angle of attack for complete configurations with either single or twin vertical tails. MISLIFT provided estimates only for the simplest bodyvertical-tail configurations at zero angle of attack.
Experimental and theoretical aerodynamic characteristics of two hypersonic cruise aircraft concepts at Mach numbers of 2.96, 3.96 and 4.63 were studied10. The test models consisted of a fuselage with lenticular cross-section, two geometrically similar wings, one set of horizontal tails sized for each wing, a wedge-centre vertical tail, a set of fuselage mounted twin vertical tails, a flowthrough body-mounted nacelle, and a set of flow-through wingmounted nacelles formed other parts of the configurations. The large wing airfoil had a circular arc upper surface and a flat lower surface, whereas the smaller wing airfoil and the horizontal tail airfoils were symmetric circular arcs. The airfoil of the twin vertical tails was flat on the outboard surface and circular arc on the inboard surface. Estimates from first order supersonic linear theory and hypersonic impact theory were compared with the experimental data. Two types of calculations were made under the hypersonic impact method, viz.,
97
98
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
(a)
(b)
application of the tangent cone empirical calculations to all the components in the impact region and Prandtl-Meyer relations in the expansion region, and the application of tangent cone theory to the fuselage and tangent wedge theory to the wing, horizontal tail and vertical tail with Prandtl-Meyer theory in the expansion regions.
In both the calculations, the minimum expansion was limited to 0.7 vacuum pressure coefficient. To account for viscous effects, Spalding-Chi method was utilised. For the linear theory analysis, an integrated program11, which included an empirical skin friction calculation based on the work of Sommer and Short12, was used. The conclusions reached were: (a)
(b) (c)
The results of tangent-cone empirical theory to the fuselage and tangent-wedge theory to the wing, horizontal and vertical tails gave very good overall agreement with the experimental data but the estimates of the aerodynamics of the individual components were significantly different from the data. The tangent-cone theory applied to all the configurations generally showed poor estimates. The predictions of first order supersonic theory were also bad except for lift and drag data at low angles of attack.
A simple and rapid method for the computation of the aerodynamic coefficients in the high Mach number range for small angles of attack and tail deflection and arbitrary roll angles was suggested13. The method was limited to the following configurations: cylindrical bodies of circular cross-section with different nose shapes, sharp cones, sharp ogives and hemispheres, arbitrary position of the wing of any planform, consisting of flat plates with sharp leading and trailing edges. The prediction methodology used was the modified Newtonian, exact relations for oblique shocks, and Prandtl-Meyer theory for expansion regions. Test results at M = 3.08 and M = 4.63 were compared with predictions and it was found that the agreement was very satisfactory. The aerodynamic characteristics of three hypersonic configurations at a Mach number of 6 were studied3. Tangent cone theory on the body, tangent wedge on the wing and vertical tail surfaces and Prandtl-Meyer expansion for all expansion regions, were utilised to compare the predicted data with experimental values. The general trend of lift, pitching moment and drag were observed but the comparison between the predicted drag and pitching moment with experimental values were not too good.
7.67 C.G.OF MODEL
80°
47.55 53.52 48.92
HORIZONTAL CONTROL PIVOT
(a) BHVCB
31.60
b
12.52 19.25
Figure 6.1. HYFAC model (all dimensions in centimetres)
CANARD PIVOT
.810 0 STATION
6.5°
MOMENT CENTRE
l
48.11
Methods 99
1.18 76.5°
77.8°
65°
4.44
3.38
CANARDS
0.0238 RAD
0.665
3.46
4° 36'
1° 36'
HORIZONTAL CONTROL SURFACE
6.05
0.0254 RAD
0.292
8.95 3.485
3.92
70°
2.10
(b) Components
65°
6.3
Figure 6.1. HYFAC model (all dimensions in centimetres) (Concluded)
VERTICAL TAIL
1.382
1.49
0.47
8° 35'
100 Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Methods
In the hypersonic speed range, the most commonly used prediction method is the Gentry Hypersonic Arbitrary-Body Aerodynamic Computer Program (HABP)14. Although this program is more than thirty years old, it is still being used by many, with some additions and modifications. An experimental program was conducted at Mach 6 to determine the aerodynamic characteristics of an all-body, delta planform, hypersonic research aircraft (HYFAC Configuration)15. The sketch of the model tested is given in Fig. 6.1. The body had a delta planform with a 6.5º half-angle conical nose faired to an 80º swept leading edge afterbody. Aft of the conical nose the fuselage had modified rhombic cross-sections. Computed theoretical values were compared with experimental data. Predictions from various methods like tangent cone, tangent wedge, shock-expansion, Dahlem-Buck empirical, etc., were evaluated and compared with experimental data. For skin friction effects, the reference temperature method for laminar region and Spalding-Chi method for turbulent layers were used. It was found that the tangent cone for all the components gave the best correlation. For small control deflections the agreement between predicted and experimental values was good. However, for large control deflections, particularly for negative deflections, none of the methods adequately predicted the longitudinal characteristics. This was attributed to the use of free stream dynamic pressure values over the control surfaces rather than the local dynamic pressure. Some representative comparisons between the predictions and wind tunnel measurement taken from Clark15 is shown in Figs. 6.2, 6.3 & 6.4. In these figures, the symbols, B, H, and V represent the body, horizontal tail and vertical tail respectively. In Fig. 6.4, the side force parameter δCY/δβ, (CY = side force coefficient), the effectivedihedral parameter δCl /δβ, (Cl = the rolling moment coefficient), and the directional stability parameter, δCn/δβ, (Cn = yawing moment coefficient) are respectively indicated by the symbols C Y β , C 1 β and C n β . β is the angle of side slip.
In order to determine the effect of wing leading-edge sweep and wing translation on the aerodynamic characteristics of a wing body configuration, a series of experiments were conducted at a Mach Number of 616. Seven wings with leading edge sweep angles from –30º to 60º were tested on a common body over an angle of attack ranging from –12º to 10º and side slip angles of 0° and 2º. All wings had a
101
102
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Tangent cone on body and vertical tails andTangent cone on horizontal controls Modified Newtonian (K = 2.4) on horizontal controls Shock expansion on horizontal controls Tangent wedge on horizontal controls Wind-tunnel data δh'deg 0 10 -30
0.04 0.03 0.02 0.01
δh'deg
Cm
-30
0 -0.01
0 10
-0.02 -0.03 Tangent cone Modified Newtonian (K = 2.4) Shock expansion, and Tangent wedge
4 3 2 1 L/D 0 -1
δh'deg - 30
-2 -3 -4 -0.15
-0.10
-0.05
0
0.05
0.10
0.15
0.20
0.25
CL
Figure 6.2. Comparison of several hypersonic theories with wind× 106 tunnel data for HYFAC configuration (BHV) at R∞, = 10.5× ∞,l (Continued).
Methods
Tangent cone on body and vertical tails andTangent cone on horizontal controls Modified Newtonian (K = 2.4) on horizontal controls Shock expansion on horizontal controls Tangent wedge on horizontal controls
.11 .10
δh'deg -30 0
Wind-tunnel data
.09
δh'deg
.08 .07
0 10 -30
CD .06 .05 .04 .03 .02 .01 0
-30
20
δh'deg 0
10
15
α, deg
10
5
0
-5 -.15
-.10
-.05
0
.05
.10
.15
20
.25
CL
Figure 6.2. Comparison of several hypersonic theories with windtunnel data for HYFAC configuration (BHV) at R∞,l= 10.5 × 106 (Concluded).
103
104
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
.11
.10 .09 .08
δh'deg -30
.07 CD
.06 .05
-20
.04 .03
-10
.02 .01 0 Tangent cone on body, horizontal controls and vertical tails Wind-tunnel data
20
δh'deg 10 0 -10 -20 -30
15
10 α,deg
5
0 δh'deg -30 -20 -10 0 10
-5 -.15
-.10
-.05
0
.05
.10
.15
.20
.25
CL
Figure 6.3. Comparison of tangent-cone theory with wind-tunnel data for longitudinal aerodynamic characteristics of HYFAC configuration BHV at R∞,l = 10.5 × 106 (Contd...).
Methods
.04 .03
δh'deg -30
.02 -20
Cm
.01 -5 0 5 10
0 -.10 -.02
Tangent cone on body, horizontal controls and vertical tails Wind-tunnel data
δh'deg 10 5 0 -5 -10 -20 -30
4 3 2 1 L/D
0 -1 -2
δh'deg -30
-3 -4 -.15
-20 -10
0
-.10
-.05
0
.05
.10
.15
.20
.25
CL
Figure 6.3.
Comparison of tangent-cone theory with wind-tunnel data for longitudinal aerodynamic characteristics of HYFAC configuration BHV at R∞,l = 10.5 × 106 (Concluded).
105
106
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
0
CY
-.005 β
-.010 -.015 Wind-tunnel data (M = 6)
.001
Cl
0 β
-.001 -.002 Tangent cone on body, and horizontal controls
and -
Tangent cone on vertical tails Tangent wedge on vertical tails Modified Newtonian (K = 2.4) on vertical tails
.003 .002
Cn
β
.001 0 -.001 -4
0
4
8
12
16
20
α, deg
Figure 6.4.
Comparison of several hypersonic theories with windtunnel data for lateral-directional stability characteristics of HYFAC configuration (BHV) δh= 0°; R∞= 10.5 × 106.
Methods
common span, aspect ratio, taper ratio, planform area and thickness ratio. The wings were translated longitudinally on the body to make tests possible with the total and exposed mean aerodynamic chords located at a fixed body station. The theoretical estimates were based on Gentry’s Program14. Tangent cone pressure distribution was applied on the body and tangent wedge on the wings, (method 1). In the expansion region a limiting expansion pressure coefficient of 70 per cent of vacuum conditions, (i.e., Cp, limit= ( −1/M ∞2 ) was utilised for all calculations. The base pressure on the body base was assumed to be equal to the freestream static pressure. Spalding-Chi method was used for viscous effects, assuming 100 per cent turbulent boundary layer. The drag contribution from the body nose bluntness and the wing leading and trailing edges were not taken into account as they were estimated to be very low. An alternate analysis in which tangent cone only on the fuselage fore body and tangent wedge on the wing and body aft of the wing was also done (method 2). Comparison of the wind tunnel data with the theoretical predictions lead to the following conclusions: (a)
(b)
(c) (d)
Good to excellent predictions of longitudinal forces (normal, axial, lift and drag) and lift to drag ratio were obtained throughout the angle of attack range on all wing body configurations tested by using the tangent cone theory on the body and tangent wing theory on the wings and by limiting the expansion pressure to 70 per cent vacuum conditions. Unsatisfactory predictions of the magnitude of the pitchingmoment coefficients were obtained on all the wing-body combinations tested with the tangent cone/tangent wedge analysis. This was attributed to the wing body interference, wing tip losses and the change in dynamic pressure over part of the wing due to bow shock, etc. The Gentry program was found unsatisfactory for estimating the lateral-directional stability. It was also suggested that the mean aerodynamic chord of the exposed wing should be used as the design reference. The independence of the longitudinal and lateral forces and pitching moments from wing leading edge variations when the exposed wing mean aerodynamic chords were located at a fixed body station verified the Hypersonic Isolation Principle.
A wing cone configuration was identified as one of the potential designs for a trans-atmospheric vehicle and a study was conducted on a generic model17. The base line wing-cone model consisted of a 5° half angle cone body, a cylindrical mid body, and a 9° truncated cone afterbody. The fuselage was fitted with a
107
108
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
4 per cent thick diamond airfoil delta wing of aspect ratio 1 that could be located at three longitudinal positions while maintaining a smooth wing body juncture. The model components included two nose geometries which varied in bluntness, two canards which differed in planform and three vertical tail arrangements. The test Mach Number ranged from 2.5 to 4.5. The model was designed to allow the wing to be positioned at five incidence angles (– 5°, – 2.5°, 0°, 2.5°, and 5°). Angle of attack was varied from – 4° to 28° and angle of slide slip from – 8° to 8°. Theoretical analysis was performed using three prediction methods: The Gentry Hypersonic Arbitrary Body Program14, Linear Theory18, and Supersonic Implicit Marching Program (SIMP)19. The HABP method employed the tangent cone theory for the body compression pressure and the tangent wedge methodology for the wing, canard and tail compression pressures. The Prandtl-Meyer expansion was used for lee-side pressures. Spalding-Chi method was used for viscous forces. The linear theory method was based on linearized supersonic potential theory and slender body estimates for inviscid lift, far field wave drag using supersonic area rule for inviscid zero lift drag and Sommer-Short skin friction estimate12. The SIMP method solved the full potential equation to provide inviscid characteristics and the skin friction estimate was done using the Sommer-Short estimates. Comparisons of the experimental data with several analysis methods to predict the longitudinal aerodynamic characteristics of the wing-body showed the following:
•
The HABP predicted lift and L/D values were overpredicted, and the drag slightly underpredicted.
•
The stability level was reasonably well predicted, although the non-linear aspects of the pitching moment curve were not predicted.
•
Linear theory underpredicted lift and L/D, and overpredicted drag and stability level.
•
The SIMP predictions agreed well with the experimental lift, drag and pitching moment results although the absolute pitching moment values differed. HABP
results
for
the
wing-body
and
two
canard
configurations showed minimal changes in C L α , CD,o and (L/D)max due to the addition of the canard, consistent with experiments but failed to predict the change in the centre-of-pressure location. For the configuration with canard, Linear theory predictions showed a
Methods
small increase in CL, no change in CD,o and an increase in (L/D)max compared to the baseline wing body combination. These results were in contrast to the experimental data which showed no change in CL, CD,o and (L/D)max. These discrepancies between linear theory and experiment were attributed, perhaps to the inability of the linear theory to accurately model the downwash of the canard and its influence on the wing. Three hypersonic vehicles, viz., Shuttle Orbiter, the FDl-7, and the X-24C-10D were chosen20, to establish a rationale for choosing the best combination of hypersonic analysis methods. In this work, three Mach number ranges were defined: a low hypersonic Mach number range from 3.0 – 6.5, a high hypersonic Mach number range above 8.5, and a transitional region in between. In the high hypersonic case, Modified-Newtonian was suggested for all the components and the Prandtl-Meyer expansion for the shadow regions. In the low hypersonic range, the methods suggested were: Dahlem-Buck empirical for a round nose, tangentwedge for flat nose, inclined cone for a round body; tangent cone for a strake: and tangent wedge for lifting surfaces. Different methods for calculating viscous effects had very little impact on the overall results and the Reference Temperature/Spalding-Chi method was recommended. In the transitional hypersonic region, it was felt that either one of the above low or high hypersonic approaches would work well. Gentry’s program (HABP)14, has been in wide use since the early 1970’s to predict the hypersonic aerodynamic characteristics of several vehicle configurations and their comparison with experimental data as discussed above. However, it was felt by Maughmer, et al 21, that there was a need for a comprehensive and systematic study to explore the ability of the simple local surface inclination methods to predict control forces and moments generated by aerodynamic flight controls for a variety of configurations in hypersonic flows. For their work they made use of the industry developed single analysis program for subsonic, supersonic and hypersonic flow regimes called the Aerodynamic Preliminary Analysis II (APAS II)22. The APAS II, is an aerodynamic analysis program based on potential theory at subsonic/supersonic speeds and impact type finite element solutions at hypersonic speeds. It was developed at Rockwell International by integrating their version of the Woodward subsonic/supersonic panel method, called the Unified Distributed Panel (UDP) with an enhanced version of Gentry’s program. The
109
110
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
first order panel method is capable of analyzing a complete aircraft configuration with relatively short computation time. The method is based on potential flow theory but includes surface leading edge and side forces and semi empirical techniques for the calculation of skin friction drag. In addition to exploring the validity of hypersonic capability of APAS methods, Maughmer et al, made use of the subsonic and supersonic panel methods of that program, including wetted area drag prediction and compared the results with experimental data in all the speed ranges. In their work three representative hypersonic vehicle configurations were considered, viz., (a) the X-15 Research Aircraft, (b) the Hypersonic Research Airplane, (HRA), and (c) the Space Shuttle Orbiter. For these vehicles experimental results were available in the entire speed range from subsonic to hypersonic, so that comparisons could be made between the predicted values and experimental data. Being of a comprehensive nature and being fairly recent, their work is considered in some detail. 6.1
NORTH AMERICAN X-15 RESEARCH AIRCRAFT
The North American X-15 was developed in the late 1950's and test flown in 1960's. It was designed to reach flight velocities of about 2000 m/s and an altitude of about 76,000 m. Wind tunnel tests were done on the X-15 model at subsonic, transonic, supersonic and hypersonic Mach numbers. The fuselage was basically cylindrical in shape with fairings along both sides. The wing had an aspect ratio of 2.5, a quarter chord sweep angle of 25°, and was equipped with conventional trailing edge flaps for use during landing. The horizontal tail had a quarter chord sweep angle of 45° and a dihedral angle of -15°. This all-movable tail was deflected asymmetrically for roll control. During the development of the North American X-15, the original upper and lower vertical tails were found to be insufficient in producing the required stability. These original surfaces consisted of a large upper vertical and small lower vertical, each having a diamond shaped airfoil. Both the upper and lower verticals were all movable and the rear portion of each surface could be deflected to form speed brakes. The final vertical tails had wedge shaped airfoils and the lower vertical was only slightly smaller than the upper vertical. On both the upper and lower surfaces, the inner portion was fixed and contained speed brakes while the outer portion was all movable. The lower movable portion was jettisoned
Methods Table 6.1
X-15 Research characteristics)
airplane
(airplane
geometric
Wing (extended to body centre line) Area, (sq ft)
200
Aspect ratio
2.50
Taper ratio
0.20
Mean aerodynamic chord, (in.)
123.23
Sweep of leading edge, (deg)
36.75
Span, (ft)
22.36
Root chord, (in.)
178.89
Tip chord, (in.)
35.78
Dihedral angle, (deg)
0
Incidence angle, (deg)
0
Twist, (deg)
0
Airfoil section
NACA 66005(modified)
Fuselage station for 20 per cent mean aerodynamics chord, (in.)
339.19
Wing station for 20 per cent mean aerodynamic chord, (in.)
52.17
Flap area, (sq ft)
15.48
Flap travel, (deg)
40 Wing (exposed)
Area, (sq ft)
105
Aspect ratio
2.15
Taper ratio
0.27
Root chord, (in.)
131.95
Tip chord, (in.)
35.78 Horizontal tail (exposed)
Area, (sq ft)
51.76
Aspect ratio
2.81
Taper ratio
0.21
Mean aerodynamic chord, (in.)
60.07
Sweep of quarter-chord line, (deg)
45
Span, overall, (ft)
17.64
Root chord, (in.)
84.27
Tip chord, (in.)
25.28
Dihedral angle, (deg)
-15
111
112
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows Airfoil section
NACCA66005(modified)
Fuselage: station for 50 per cent horizontal-tail mean aerodynamic chord, (in.) Span station for 50 per cent horizontal-tail mean aerodynamic chord, from fuselage, (in.) Tail arm, 20 per cent wing mean aerodynamic chord to 50 per cent horizontal-tail mean aerodynamic chord, (in.) Incidence range, normal to plane of symmetry, (deg) — Pitch control Roll control
537.52 26.96 1 98.33 35 (down), 15 (up) +7.5
Vertical tail (upper, exposed) Area, (sq ft)
40.8
Aspect ratio
1.03
Taper ratio
0.74
Mean aerodynamic chord, (in.)
1 07.5
Sweep of leading edge, (deg)
30
Span, (exposed), (in.)
55
Root chord, (in.)
1 22.5
Tip chord, (in.)
90.75
Airfoil section
10° wedge
Fuseulage station for 50 per cent vertical-tail mean aerodynamics chord, (in.) Span station for 50 per cent vertical-tail mean aerodynamic chord, from fuselage, (in.) Tail arm, 20 per cent wing mean aerodynamic chord to 50 per cent vertical-tail mean aerodynamic chord, (in.)
5 20.25 26.15 1 81.06
Movable outboard panel area, (sq ft)
26.5
Angular travel of movable area, (deg)
+ 7.5
Vertical tail (lower, exposed) Area, (sq ft)
34.2
Aspect ratio
0.785
Taper ratio
0.79
Mean aerodynamic chord, (in.)
1 09.2
Sweep of leading edge, (deg)
30
Span, (exposed), (in.)
44
Root chord, (in.)
1 21.4
Tip chord, (in.)
96
Airfoil section
10° wedge
Fuseulage station for 50 per cent vertical-tail mean aerodynamics chord, (in.)
519.4
Methods Span station for 50 per cent vertical-tail mean aerodynamic chord, from fuselage, (in.) Tail arm, 20 per cent wing mean aerodynamic chord to 50 per cent vertical-tail mean aerodynamic chord, (in.)
21.15 1 80.21
Movable (jettisonable) area, (sq ft)
19.9
Angular travel of movable area, (deg)
+ 7.5 Fuselage
Length, high-speed nose, (ft)
49.17
Length, low-speed nose, less boom, (ft)
50.16
Width, including side fairings, station 346 to station 411, (in.)
88.0
Height, station 186 to station 530, (in.)
56.0
Maximum cross-sectional area, (sq ft)
21.4
Fineness ratio, average
9.4
Nose apex angle, (deg)
31.0 Speed Brakes (Upper and Lower)
Location hinge line, fuselage station, (in.) Side area, each, (sq ft) Angular travel, fuselage centre line, (deg)
534 4.88 41
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Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
50.16
17.64
22.36
Figure 6.5.
Three view drawing of the X-15 airplane (All dimensions in feet).
Methods
for landing. The original tail configuration results were used21, for comparisons in the subsonic and transonic speed ranges and the final configuration results in the supersonic and hypersonic speeds. The X-15 aircraft geometric characteristics are given in Table 6.1. The three view drawing of the aircraft is given in Fig. 6.5. Since, this aircraft configuration was extensively studied in 1960's during its development, it is interesting to study the comparisons between theoretical predictions and measurements made then 23 (period in which the finite element panel methods for subsonic and supersonic flows had not yet been developed), and now fairly recently 21, wherein, there has been an extensive use of combined subsonic/supersonic computer codes available for analysis. A summary of these studies is as follows: 6.1.1
Walker & Wolowicz’s Work
The stability and control derivatives for the X-15 research airplane in power-off flight at supersonic and hypersonic Mach numbers have been presented in this work, both as derived from the then existing theoretical methods and as measured in various wind tunnel facilities. Calculations were made for Mach numbers within and beyond the estimated flight envelope and for angles of attack from 0° to 25°. The results were compared with the experimental data in the Mach number range from 2 to approximately 7 and, for the static derivatives, with the limiting values given by the Newtonian theory. Because the report23 is old, originally classified and not easily available, some details of the prediction methodology used in their work is presented below. 6.1.2
Lift Characteristics
The lift for the complete airplane is calculated by the method24, in which the total lift is considered initially to be the sum of the individual lifts of the exposed wing and horizontal-tail surfaces and of the fuselage, each treated as an isolated body. Incremental lifts are then added which represent corrections for the interference that arises when the components are placed adjacent to one another in the overall configuration. The interference is reciprocal, consisting of reflection-plane and upwash effects on the wing due to the presence of the fuselage, and of the carryover lift on the fuselage due to the exposed wing and tail panels. Both these effects were treated as wing contributions in accordance with the method described by Pitts, et al 24. The forces on the horizontal-tail
115
116
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
surfaces at zero incidence (controls fixed) were similarly treated. According to the procedure24, the X-15 lift coefficient can be expressed as
CL =
SW C L′W (K W B + K B W ) S S cos ΓT C L′T (K T B + K B T +Q T S
dε ) 1 − + C LB dα
(6.1)
where the K terms represent the interference factors which account for the lift of the wing and the horizontal tail in the presence of the body, KWB and KTB, and for the lift of the body in the presence of the wing and the horizontal tail, KBW and KBT respectively. In the above equation, the aerodynamic coefficients when used without a superscript, were based on the dimensions of the wing with leading and trailing edges extended to the plane of symmetry of the airplane, and when primed, on the dimensions of the isolated surface or body. Sw and ST are the exposed wing and tail areas respectively, ΓT is the dihedral angle of the horizontal tail measured from the x-y plane, positive when rotated upwards, ε is the downwash angle at the tail in degrees.
Q =
q 1 (C L α )1 q ∞ (C L α )∞
where, q∞ is the free stream dynamic pressure and q1 is the dynamic pressure of downstream flow where the local stream pressure behind the bow shock wave is equal to the free stream static pressure; (C L α )1 is the lift curve slope in the downstream based on the local Mach number and (C L α ) ∞ is the lift curve slope in freestream flow. 6.1.3
Wing
Assuming that both the wing and the tail surfaces can be considered as flat plates for the lift analysis, Van Dyke’s unified small disturbance theory was used for calculating the lift coefficient. This method is suitable to both the supersonic and hypersonic speed regimes. For a 2-dimensional flat plate wing at an angle of attack α,
Methods
c n = (C p )low er − (C p )u p p er
cn = α 2
γ +1 + 2
γ +1 2
2
+
4 H
2
−
(6.2)
2
γH
2
2γ 1 − γ −1 H γ −1 −1 2
where, H is the similarity parameter M
2
(6.3)
− 1 α .
The lift as calculated above was converted from 2-dimensional to 3-dimensional lift at hypersonic speeds by the following approximation based on the linear theory. C L′ α C N' =
4 M
2
cn
(6.4)
−1
In the above expression + L′ α is the lift curve slope from linear theory for the 3-dimensional planform as given in ref 25, and cn as given by the Eqn. 6.3. Neglecting the wave drag and skin friction drag, the lift coefficient for the isolated wing becomes
C L′ W = C N′ cos α
(6.5)
while that for the wing in the presence of the body (based on reference area S ), C L′ α C L W = (K W B + K B W )
4 M
2
− 1
cn
SW S
cos α
(6.6)
Approximate values for the interference terms, KWB and KBW taken from Pitts, et al 24, were substituted in the above equation to get the value of CLW. The Newtonian theory limit was also evaluated with KWB = 1 and KBW = 0.
117
118
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
6.1.4
Horizontal Tail
The lift characteristics of the horizontal tail at zero incidence were calculated in a similar manner to that of the wing with dihedral angle taken into account. The fuselage-induced upwash at the tail plane was considered negligible and also the term KTB corresponding to KWB in Eqn. 6.1 was taken as unity. The wing downwash parameter dε/dα was estimated from the charts of Haefeli, et al 26. This downwash was found to be negligible for Mach number values greater than 4.0. 6.1.5
Fuselage
The lift from the fuselage was derived both from the inviscid flow and the viscous crossflow. Second-order Shock-Expansion theory was used to calculate the inviscid lift. Since the fuselage cross-section of the X-15 aircraft was noncircular, the second-order shock-expansion expressions, as given in Syvertson’s report27, was multiplied by a factor equal to the ratio of the actual planform area to that of an equivalent body of revolution having the same local cross-sectional area as the X-15 configuration. This approximation lead to the relationship for the inviscid flow as
(C ) LB
In which
in viscid
(C ) ' Nα
B
= Rα
( )
SB C N′ α S
B
α cos α
(6.7)
was obtained from Syvertson & Dennis27
(Appendix C), and Rα =
T otal fuselage plan form area Planform area of equ ivalen t body of revolu tion
The lift due to viscous crossflow was calculated using the Allen and Perkins theory28
(C ) LB
viscou s
= ηc d c
Aα 2 α cos α S
(6.8)
The term Aα, is the planform area of the fuselage, consisting of forebody area only (vertex to the leading edge approximately, since the wing and tail in effect block the crossflow over the remaining sections). A value of c d = 1.2 was taken in the overall c Mach Number and angle of attack ranges. Newtonian theory was applied approximately assuming that the X-15 fuselage may be represented from the vertex to a station
Methods
immediately rearward of the canopy by a circular cone and over the remaining length by a cylinder of constant diamond-shaped crosssection similar to that of the combined fuselage and side fairings. 6.1.6
Pitching-Moment Characteristics
From the calculated values of the lift coefficients for various components as determined above and from the centre-ofpressure charts given in Pitts, et al24, the buildup of the moments about a centre of gravity location of 20 per cent of the mean aerodynamic chord (based on area S, the reference area equal to area of wing with leading and trailing edges extended to the plane of symmetry) was calculated as follows. 6.1.7
Wing & Horizontal Tail
The moment arm for the lift of the wing in the presence of the body differs in general from that for the lift induced by the wing on the body, with the difference depending primarily on Mach number and fuselage diameter. The moments from the two sources therefore must be determined separately. For consistency with the lift calculations described earlier, both effects are charged to the wing. The characteristics for the horizontal tail at zero incidence are also determined likewise, although the moment arms for the various interference effects, due to the absence of the fuselage afterbody, are essentially equal. The pitching moment of the combined wing and tail in the presence of the fuselage is given by
C m W +C m T =
SW x x C LW K W B W B + K B W B W S c c S Tcos Γ T x dε + Q C L T 1 − (K TB + K B T ) T dα S c
(6.9)
where, x is the longitudinal distance from centre of gravity to centre of pressure of component lift measured in direction of fuselage centre line. The subscripts WB, BW, TB, and BT refer to wing in the presence of the fuselage, fuselage in the presence of the wing, horizontal tail in the presence of the fuselage, and fuselage in the presence of horizontal tail respectively. c is mean aerodynamic chord. Other symbols are defined in the earlier lift section.
119
.2
.4
.6
0 4 8 α,deg
12
Horizontal tail off
16
Horizontal tail on (iT=0°)
.1
Airplane trimmed
(a) M = 2.01
20
Wing
Horizontal tail on, wind tunnel (ref.1) Calculated
0 -.1
Horizontal tail off
-.2 Cm
-.3
Horizontal tail on
-.4
Wing
-.5
Figure 6.6. Comparison of calculated and experimental lift and pitching-moment characteristics for the X-15 airplane at various Mach numbers.
CL
.8
1.0
1.2
1.4
120 Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
CL
0
.2
.4
.6
.8
1.0
1.2
1.4
4
Horizontal tail on
8
12 α,deg
16
Wing
} wind tunnel (ref.3)
Horizontal tail off
Horizontal tail on, Horizontal tail off Calculated
(b) M = 2.29
.1
0
Figure 6.6. Continued
20
Airplane trimmed
-.1
Horizontal tail off
-.2
Cm
-.3
Wing
Horizontal tail on
-.4
-.5
Methods 121
CL
0
.2
.4
.6
.8
1.0
1.2
1.4
4
8
12
.1
(c) M = 2.98
20
Airplane trimmed
Figure 6.6. Continued
16
Wing
α, deg
Horizontal tail off
Horizontal tail on
}
Horizontal tail on, Horizontal tail off wind tunnel .3) Calculated
0 -.1
Horizontal tail off
-.2 Cm
-.3
Wing
Horizontal tail on
-.4
-.5
122 Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
0
.2
.4
CL .6
.8
1.0
1.2
4
8
12 (d) M = 4.65
20
.1
Airplane trimmed
Figure 6.6. Continued
16
α,deg
Wing
Horizontal tail on
Horizontal tail off
}
Horizontal tail on, wind tunnel (ref.3) Horizontal tail off Calculated
0
-.1
Horizontal tail off
-.2 Cm
Wing
-.3
Horizontal tail on
-.4
Methods 123
CL
0
.2
.4
.6
.8
1.0
4
8
Horizontal tail off
12
Horizontal tail on
}
20
24
Figure 6.6. Concluded
(e) M = 6.86
α,deg
Wing
16
Horizontal tail on, Horizontal tail off wind tunnel (ref.1) Calculated
.1
Airplane trimmed
0
-.1
Cm
-.2
Wing
Horizontal tail on
Horizontal tail off
-.3
124 Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Methods
22
Calculated (iT, deg (refs. 3,4)
20
– 20
15
18 16 14
α, deg
12
10 8 6 4 2
iT = 15°
iT = 15°
iT =–20°
iT =–20°
0 (a) M = 2.29
(b) M = 2.98
20 18 16 14 12
α, deg 10 8 6 4
iT = 15°
iT = –20°
iT = –20°
iT = 15°
2 0 -.4
-.2
0 ∆Cm
.2
(c) M = 4.65
.4 -.4
-.2
0 ∆Cm
.2
.4
(d) M = 6.86
Figure 6.7. Comparison of the calculated and experimental stabilizer effectiveness of the X-15 airplane for incidence settings of 150 (leading edge up) and -200 (leading edge down) at various Mach numbers.
125
126
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
20
M=2.98
M=2.29
M=4.65
M=6.86
16
α, deg
12 8 4 0
-.008 -.004 0 C n ∂ , per deg R
20
-.008
Cn∂
-.004 0 , per deg R
-.008
-.004 0 , per deg
Cn∂
R
-.008
Cn∂
-.004 0 , per deg R
Calculated Data (refs. 3,4)
16 12 α, deg 8 4 0
M=2.98
M=2.29
-.001
C l∂
0
.001
, per deg R
-.001
C l∂
0
.001
, per deg R
M=6.86
M=4.65
-.001
0
.001
C l ∂ , per deg R
-.001
C l∂
0
.001
, per deg R
Figure 6.8. Comparison of the calculated and experimental directionalcontrol derivatives for the X-15 airplane at several Mach numbers.
Methods
20
M=2.98
M=2.29
M=6.86
16
M=4.65
12 α, deg 8 4
Calculated Data (refs. 3,4) 0
C l1'
T
.001 .002 , per deg
0
C l1'
-.001 .002 , per deg
T
20
0
C l1'
.001 .002 , per deg
-.001
C l1'
T
M=4.65
M=2.98
M=2.29
0
T
002 003 , per deg
M=6.86
16 12 α, deg 8 4 0
.001 .002
C n1
'T '
, per deg
0
.001 .002
C n1
'T '
, per deg
0
.001 .002
C n 1'
, per deg
T'
0
.001 .002 .003
C n 1'
, per deg
T'
Figure 6.9. Comparison of the calculated and experimental lateralcontrol derivatives for the X-15 airplane at several Mach numbers.
127
128
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
6.1.8
Fuselage
The centre of pressure for the lift due to the inviscid flow over the fuselage was calculated by the Second-order ShockExpansion method27, (Appendix C) and that due to viscous cross flow by the procedure described by Perkins & Jorgensen29. The former was found to vary slightly with the Mach number and the latter to be essentially constant. The moment coefficient for the fuselage was expressed as x xB + C L C m B = C L B B c in viscid B c viscous
(6.10)
Figures 6.6 to 6.9 show some of the comparisons of the experimental data with the theoretical predictions in the Mach number ranges of 2.01 to 6.86. The figures are reproduced from Walker and Wolowicz 23. Walker also deals with the study of longitudinal control characteristics such as lift variations due to incidence as well as angle of attack for wing body combinations, damping in pitch and lateral directional derivatives by the methods outlined in Pitts et al 24. The main conclusions reached were that the calculated longitudinal characteristics for the most part were in close accord with the results from the wind tunnel data. The lateral and directional characteristics agreed well with the wind tunnel results in the lower angle-of-attack range. However, due to the interference of the bow shock wave on the tail surfaces and other effects not accounted for in the theory, some disagreement was found at higher angles for the stabilizer effectiveness and several of the lateral directional characteristics at high angles of attack. Both stability and controllability were maintained well beyond the estimated speed limit. Further, it was found that the limiting values predicted by the Newtonian theory for the static derivatives were found, in general, to be lower than the trends shown by the unified supersonic-hypersonic small disturbance theory used for wing and tail lift calculations and the second-order shock-expansion theory used for flow over the fuselage. 6.1.9
Maughmer et al. – Analysis of X-15
As mentioned earlier, the APAS code was used for the analysis. Panel method was the basis for subsonic and supersonic
Methods
flows. For the hypersonic analysis, the impact methods chosen for the various components were: tangent cone empirical for the bodies; tangent wedge empirical for the surfaces; and modified Newtonian with the factor K =C p m ax
for the leading edges and blunt trailing
edges and blunt end of the fuselage. Prandtl-Meyer empirical was used for all shadow regions except for blunt ends or trailing edges where the high Mach number base pressure method was used. Viscous corrections were calculated assuming 100 per cent turbulent boundary layer and the use of reference temperature/ Spalding-Chi method. The HABP had a choice of whether or not to ignore the aerodynamic contributions of components shielded(or shadowed) in the wake of upstream parts of the vehicle. For the X-15 aircraft it was found that the shielding had very little impact on the results and as such the shielding option was not utilised. Predicted results were compared with experimental values at Mach numbers of 0.56, 0.8, 1.03, 1.18, 2.96, 4.65 and 6.83, covering the entire speed range. Some representative comparison figures taken from the above work are reproduced in Figs. 6.10 to 6.17. Summary of their findings are as follows: At very low speeds, the lift and drag were well predicted up to angle of attack of 25º, both for zero flap deflection and 40º flap deflection. The change in lift and drag due to elevator control deflection were all reasonably well predicted although the absolute values of the coefficients were somewhat in error. However, this was not the case with the pitching moment versus angle of attack and versus lift coefficient. The general trend of the changes in pitching moment coefficient with angle of attack and CL with elevator deflections were captured but not the values. Regarding the lateral/directional coefficients, viz., the side force, yawing moment and rolling moment, due to aileron and vertical tail deflections, the predictions were good enough for conceptual design purposes. The inclusion of edge forces did not make much difference in the results. In the transonic speed regime, the longitudinal characteristics were compared with the panel method. The lift and drag predictions at angles of attack up to 20º were good with and without control deflections. Similar to the low speed case, the pitching moment values were unsatisfactory apart from the correct trend in the changes with horizontal tail deflection.
129
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
0.3
PITCHING MOMENT COEFFICIENT
NASA RM L57D09 APAS II
0.2
δf 40° 0°
0.1
0.0
-0.1
-0.2
0
10
40
30
20 ANGLE of ATTACK (deg)
0.3
0.2 PITCHING MOMENT COEFFICIENT
130
0.1
0.0
-0.1
-0.2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
LIFT COEFFICIENT
Figure 6.10. Comparison of experimental and theoretical pitching moment coefficients for the North American X-15 at Mach 0.056 for flap deflections of 00 and 400.
Methods
0.4 NASA RM TM X-24 APAS II
PITCHING MOMENT COEFFICIENT
0.2
δh 0° -3° -6°
δa 0° 3° 0°
0.0
– 0.2
– 0.4 – 0.5
0.0
0.5
1.0
1.5
LIFT COEFFICIENT
Figure 6.11. Comparison of experimental and theoretical pitching moment coefficients for the North American X-15 at Mach 1.18 for various horizontal tail deflections.
131
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
2.0 NASA RM TM X-820 APAS II
DRAG COEFFICIENT
1.5
δh 0° -20° -35° -45°
1.0
0.5
0.0
-10
0
10
30
20
40
50
ANGLE OF ATTACK (deg)
2.0
1.5 DRAG COEFFICIENT
132
1.0
0.5
0.0
-0.5
0.0
0.5
1.0
1.5
2.0
LIFT COEFFICIENT
Figure 6.12. Comparison of experimental and theoretical drag coefficients for the North American X-15 at Mach 2.96 for various horizontal tail deflections.
Methods
PITCHING MOMENT COEFFICIENT
0.5
0.0
– 0.5
– 1.0 – 0.5
NASA TM X-820
0.0
APAS II
δh 0° -20° -35° -45°
0.5
1.0
1.5
2.0
LIFT COEFFICIENT
Figure 6.13. Comparison of experimental and theoretical pitching moment coefficients for the North American X-15 at Mach 2.96 for various horizontal tail deflections.
133
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
1.00
NASA TM X-236 APAS II APAS II with Shielding
0.75
LIFT COEFFICIENT
0.50
0.25
0.00
– 0.25 –5
0
5
15
10
20
25
ANGLE OF ATTACK (deg)
0.4
0.3 DRAG COEFFICIENT
134
0.2
0.1
0.0 – 0.2
0.0
0.2
0.4
0.6
0.8
LIFT COEFFICIENT
Figure 6.14. Comparison of experimental and theoretical longitudinal aerodynamic coefficients for the North American X-15 at Mach 6.83 showing the effect of including the hypersonic shielding option.
Methods
NACA RM L57D09 APAS II Potential + LE +Tip
CY
δV
0.000
– 0.002
– 0.004
0
10
20
30
40
ANGLE OF ATTACK (deg)
Cn
δV
0.002
0.000
– 0.002 0
10
20
30
40
30
40
ANGLE OF ATTACK (deg)
Cl
δV
0.004
0.002
0.000
0
10
20 ANGLE OF ATTACK (deg)
Figure 6.15. Comparison of experimental and theoretical vertical tail effectiveness for the North American X-15 at Mach 0.056.
135
136
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
NASA TM X-820 APAS II Potential APAS II Potential + LE APAS II Potential + LE + Tip
0.000
CY
δV
– 0.005
– 0.010 – 10
0
10 20 ANGLE OF ATTACK (deg)
30
40
0.004
Cn
δV
0.002
0.000 – 10
0 10 20 ANGLE OF ATTACK (deg)
30
40
0.001
Cl
δV
0.000
– 0.001 – 10
0
10 20 ANGLE OF ATTACK (deg)
30
40
Figure 6.16. Comparison of experimental and theoretical vertical tail effectiveness for the North American X-15 at Mach 2.96.
Methods
NASA TM X-236 APAS II
0.050
C Yδ
V
0.025
– 0.000
–5
0
10 5 ANGLE OF ATTACK (deg)
15
20
15
20
15
20
0.000
Cnδ
V
– 0.002
– 0.004 –5
0
5
10
ANGLE OF ATTACK (deg) 0.001
C lδ
V
0.000
– 0.001 –5
0
5
10
ANGLE OF ATTACK (deg)
Figure 6.17. Comparison of experimental and theoretical vertical tail effectiveness for the North American X-15 at Mach 6.83.
137
138
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
At supersonic speeds, the lift and drag predictions were good. The prediction of the slope of moment coefficient against lift curve was much better when compared with the results at low speed. The prediction of lateral/directional coefficients missed the nonlinear behavior observed in the experimental results but was found acceptable for preliminary design work, provided the angle of attack was not too great. The comparison of theoretical and experimental results of the longitudinal data for the hypersonic Mach numbers showed that the lift, drag and the pitching moment coefficients were good. Regarding the lateral/directional coefficients, at low angles of attack, the comparison between theoretical values and experimental data was reasonably good for the case of aileron deflection. However, at higher angles of attack, only the sign of the force and moment were predicted correctly. The predictions of side force, yawing moment, and rolling moment due to vertical tail deflection were relatively good and acceptable for conceptual design studies. Comparison of the work of Walker and Wolowicz 23 done in 1960 with that of Maugmer, et al.21, in 1991, on the same X-15 aircraft reveal the following: both at supersonic and low hypersonic speeds the theoretical results obtained by using the second-order shock-expansion method for the fuselage, inclusion of viscous crossflow contribution as proposed by Allen and Perkins, and the use of the small disturbance pressure coefficients for hypersonic flows with the unified similarity parameter as suggested by Van Dyke for lifting surfaces and utilizing the data presented in the report by Pitts, et al.24 for the interference effects between various components of the vehicle and their method of determining the lift and centre of pressure of wing-body tail combinations, etc., gave very good correlation with the experimental data. However, in this work, the drag was not computed. Prediction of the longitudinal characteristics and the directional derivatives due to vertical tail deflections were as good and somewhat better than that presented in Maughmer, et al.21, where the panel method was utilised for analysis in the supersonic case. In the hypersonic regime, the predictions from both the Gentry’s program contained in Bonner, et al.22 and the prediction methods used in Walker and Wolowicz 23, agreed equally well with the experimental data.
Methods
6.2
HYPERSONIC RESEARCH AIRPLANE
A wing-body concept for a Hypersonic Research Airplane (HRA) was developed at NASA Langley Research Center in the mid 1970's to serve as a hypersonic flight technology demonstrator. The wind tunnel results on this configuration are reported3, 30, 31, 32 in the Mach number ranges from subsonic to low hypersonic. The configuration of the vehicle consisted of a body, a cropped delta wing having a 2.1º negative incidence and 10º dihedral and a centre vertical tail. The airfoil was a modified circular arc with a leading edge radius (normal to L.E.) followed by a wedge section. It was equipped with full span elevons that could be deflected symmetrically for pitch control and asymmetrically for roll control. The centre vertical tail had a dual hinge line that allowed for a diamond shaped airfoil at subsonic and supersonic speeds. At hypersonic speeds, the rear portion deflected to form a wedge shaped airfoil. They could also be deflected to form speed brakes at high speeds. The dual hinged rudder could also be deflected in the same direction to provide yaw control. Dillon and Pittman32 deal with an experimental investigation of the static aerodynamic characteristics of scale model of the above research airplane at a Mach number 6 and comparison of the measurements with theoretical predictions using Gentry’s method. The geometrical characteristics of the model tested are given in Table 6.2, the model dimensions in Fig. 6.18. Comparisons between some of the predicted values and wind tunnel measurements, taken from the above referenced report are reproduced in Figs. 6.19 to 6.21. In these figures, the symbols B, BW, and BWVCH refer to the body or fuselage, the body and wing, and the combination of body, wing and centre vertical tail (wedge airfoil used in hypersonic testing) respectively. The following options were used in the Gentry’s program for theoretical predictions: for compression regions, tangent cone on the fuselage and tangent wedge on the wing and vertical tail; for expansion regions, Prandtl-Meyer expansion; and for skin friction, Spalding-Chi method (with 100 per cent turbulent boundary layer). The conclusions reached were that, in general, the Gentry’s Hypersonic Arbitrary Body Aerodynamics computer program gave reasonable predictions of longitudinal aerodynamic characteristics. At lower angles of attack the lift was underpredicted and drag and pitching moment overpredicted, whereas, at the higher angles of attack the lift was overpredicted, and the drag and pitching moment were underpredicted. However, lateral-directional stability parameters were not well predicted except for the isolated body.
139
140
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows Table 6.2
Geometric characteristics of model (Hypersonic Research Airplane) Wing
Area (includes fuselage intercept), m2 (in2)
0.060 (92.63)
Area, exposed, m2 (in2)
0.030 (47.00)
Area, wetted, m2 (in2)
0.064 (98.98)
Span, m (in.)
0.244 (9.62)
Aspect ratio
0.999
Root chord (at fuselage center line), m (in.)
0.371 (14.59)
Tip Chord, m (in.)
0.119 (4.7)
Taper ratio
0.322
Mean aerodynamic chord (includes fuselage intercept), m (in.)
0.294 (11.57)
Sweepback angles: Leading edge, deg
67.5
25-perent chord line,deg
61.1
Trailing edge, deg Dihedral
0
angle,deg
Incidence
10
angle,deg
-2.1
Airfoil thickness ratio: Exposed root
0.051
Tip
0.078
Leading-edge radius (normal to leading edge),cm (in.)
0.064 (0.025)
Trailing-edge thickness, cm (in.)
0.064 (0.025)
Elevons: Tip chord, percent wing tip
36.6
Span, percent total span
59.8
Area, both, m2 (in2)
0.0064 (9.89)
Vertical Tail Area, exposed, m2 (in2)
0.007 (10.93)
Span, exposed m (in.)
0.077(3.06)
Aspect ratio of exposed area
0.857
Root chord at fuselage surface line, m (in.)
0.101 (3.99)
Tip chord, m (in.)
0.057 (2.256)
Taper ratio
0.565
Mean aerodynamic chord exposed area, m (in.)
0.097(3.804)
Sweepback angles: Leading edge, deg
49.9
Trailong edge, deg
18.5
Methods
Vertical Tail Hinge line location, per cent chord
68.7
Arudder/Atotal
0.295
Leading-edge, radius, cm (in.)
0.064 (0.025)
Fuselage Length, m (in.)
0.584 (23.0)
Nose radius, cm (in.)
0.159 (0.063)
Maximum height, m (in.)
0.076 (2.98)
Maximum width, m (in.)
0.097 (3.83)
Fineness ratio of equivalent round body
6.86
Planform area, m2 (in2)
0.042(65.12)
Wetted area: Without components of base, m2 (in2)
0.122 (188.6)
With wing on, m2 (in)
0.116 (179.4)
Ab, m2 (in2)
0.0023 (3.54)
Complete model: Planform area, m2 (in2)
0.072 (112.12)
Aspect ratio of planform
0.825
141
10°
.045 .132
x-STATION 0.0 .045
.018
.132 .408
.531 .561
7.6°
.075
7.6°
.075
(a) Model details
MODEL REFERENCE LINE
.365
1.350
.297 .212 .182 .254
.026
air foil section at exposed wing root
.506
.204
10° .016 .238
.057
.687
67.5°
.65
c.g.
AIRFOIL SECTION AT WING TIP
.751
.830 .870 HL
1.088
y-STATION .209 1.000 1.075
HL
Y-STATION .209
Figure 6. 18. Model general dimensions. All dimensions have been normalized by body length (l = 58.4 cm)
.254 .212 .182
.531 .408 .297
.561
.751
.830
.870
X-STATION 1.000
° BASE PRESSURE MEASUREMENT LOCATION
142 Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Methods 7.75°
12°
.0312
12° .0226
12°
.0022 .0161 .0673 .0981
HL .1326
49.9° 1.8°
22.8° .0216 .1735 .1485
12° SUBSONIC AIRFOIL
12° .0384
HYPERSONIC AIRFOIL
7.75 12° .0454
SPEED BRAKES
(b) Vertical tail variations Figure 6.18. Model general dimensions. All dimensions have been normalized by body length (l = 58.4 cm) (Concluded).
143
144
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
.025
.020
.015
.010 Cm
.005
0 - .005
-.010
20
15 10 α, deg
5 Exp.
Theory
.15
.20
B
0
BW BWVCH
-5 -.10
-.05
0
.05
.10
.25
CL
Figure 6.19(a). Comparison of experiment with theory for body buildup
.30
Methods
4
2
0 L/D
-2
-4
.12
.10
Exp.
Theory
B BW
.08
BWVCH
.06 CD
.04
.02
0 -.10
-.05
0
.05
.15
.10
.2.0
.25
.30
CL
Figure 6.19(b). Comparison of experiment with theory for body buildup (Concluded).
145
146
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows Exp.
Theory
B BW BWVCH
0
CY
β
-.02
.002
Cl
β
0
-.002
0
C n -.002 β
-.004 -5
0
10
5 α, deg
15
.20
Figure 6.20. Comparison of experiment with theory for effect of body buildup on static lateral-directional stability characteristics ∆β = −2° −2°.
Methods
.015 .010 .005 Cm
0 -.005 -.010 -.015 -.020
20 15 10 α, deg
-5 δe,deg Exp. Theory
10 5 0 -5 -10 -15
0 0 -5 -.10
-.05
0
.05
.10
.15
.20
.25
.30
.35
CL
Figure 6.21(a). Comparison of experimental with theory for effects of elevon deflection on BWV CH configuration.
147
148
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
.18 δe, deg Exp. Theory
10 5 0 -5 -10 -15
.16 .14 .12
.10 CD
.08
.06 .04
.02 0 0
0 0 0 0 -.10
-.05
0
.05
.10
.15
.20
.25
.30
.35
CL
Figure 6.21(b). Comparison of experimental with theory for effects of elevon deflection on BWV CH configuration (Continued).
Methods
4 2 0 0 0 L/D
0
0 δe, deg Exp. Theory
10 5 0 -5 -10 -15
0 -2 -4 -.10
-.05
0
.05
.10
.15
.20
.25
.30
.35
CL
Figure 6.21. Comparison of experimental with theory for effects of elevon deflection on BWV CH configuration (Concluded).
The same aircraft design was analysed by Maughmer et. al.21 using the APAS code at Mach numbers of 0.80, 0.98, 1.20 and 6.00. The main results of their work were as follows: a)
The agreement between the panel method predictions with experimental data both at the low and high speeds for the longitudinal aerodynamic characteristics were good, as in the case of X-15. The lift coefficient was slightly underpredicted possibly due to the fact that the vortex lift was not fully taken into account. The change in lift coefficient with symmetric elevon deflection was predicted well enough for conceptual design. The pitching moment coefficient predictions were much better for HRA than for X–15.
b)
In the case of lateral/directional coefficients, the change in side force, yawing moment, and rolling moment with
149
150
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
asynchronous elevon deflection as predicted by the potential flow plus the leading edge suction force agreed well the experimental data. However, the changes in the coefficients due to vertical tail deflection were not at all predicted well and were typically in error by 50 per cent or more. Further, for these cases the trends in general were also not captured. c)
In the hypersonic flow, the predictions were computed using HABP, both with and without shielding. For longitudinal cases, the shielded values compared better with the experimental data. For lateral coefficients with aileron deflections, the predictions without shielding agreed somewhat better with experimental data than those with shielding. The lateral derivatives due to vertical tail deflection at zero angle of attack agreed well at M = 6.0 only, unlike the low and supersonic Mach number cases. Figs. 6.22 to 6.26 illustrate some of the typical comparisons between the predictions based on APAS code and wind tunnel experiments as reported21.
Methods
PITCHING MOMENT COEFFICIENT
0.10
0.05
0.00
-0.05
-0.10 -5
0
5
10
15
25
20
ANGLE OF ATTACK (deg)
0.10
PITCHING MOMENT COEFFICIENT
NASA TP-1189
APAS II
δe
5°
0° -5° -10° -15°
0.05
0.00
-0.05
-0.10 -0.4
-0.2
0.0
0.2
0.4
0.6
0.8
LIFT COEFFICIENT
Figure 6.22. Comparison of experimental and theoretical pitching moment coefficients for the hypersonic research airplane at Mach 0.20 for various elevon deflections.
151
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
PITCHING MOMENT COEFFICIENT
0.02
0.01
0.00
NASA TP-1249 APAS II
-0.01
δe,
10° 5°
0° -5° -10° -15°
-0.02
-0.03 -5
0
5
10
15
20
ANGLE OF ATTACK (deg)
0.02 PITCHING MOMENT COEFFICIENT
152
0.01
0.00
-0.01
-0.02
-0.03 -0.1
0.0
0.1
0.2
0.3
0.4
LIFT COEFFICIENT
Figure 6.23. Comparison of experimental and theoretical pitching moment coefficients for the hypersonic research airplane at Mach 6.0 for various elevon deflections. The hypersonic shielding option was not included in the theoretical calculations.
Methods
NASA TP-1189 APAS II POTENTIAL APAS II POTENTAIL + LE APAS II POTENTAIL + LE + TIP
C
Y
δr
0.004
0.002
0.000 -5
0
5
10
15
20
15
20
15
20
ANGLE OF ATTACK (deg)
Cn
δr
0.000
-0.002
-0.004 -5
0
5
10
ANGLE OF ATTACK (deg)
Cl
δr
0.001
0.000
-0.001 -5
0
5
10
ANGLE OF ATTACK (deg)
Figure 6.24. Comparison of experimental and theoretical rudder effectiveness for the hypersonic research airplane at Mach 0.20.
153
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
NASA TP-1249 APAS II APAS II WITH SHIELDING
CY
δr
0.001
0.000
-0.001 -5
0
5
10
15
20
15
20
15
20
ANGLE OF ATTACK (deg)
Cn
δr
0.000
-0.001
-0.002 -5
0
5
10
ANGLE OF ATTACK (deg)
δr
0.002
Cl
154
0.001
0.000 -5
0
5
10
ANGLE OF ATTACK (deg)
Figure 6.25. Comparison of experimental and theoretical rudder effectiveness for the hypersonic research airplane at Mach 6.0 showing the effect of including the hypersonic shielding option in the theoretical calculations.
Methods
EXPER
APAS II
CY
δr
0.004
α 0°
0.002
0.000 0
2
4
6
4
6
4
6
MACH NUMBER
Cn
δr
0.000
-0.002
-0.004 0
2 MACH NUMBER
Cl
δr
0.001
0.000
-0.001 0
2 MACH NUMBER
Figure 6.26. Comparison of experimental and theoretical rudder effectiveness as a function of Mach number for the Hypersonic research airplane.
155
156
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
6.3
SPACE SHUTTLE ORBITER
The Space Shuttle Orbiter configuration consists of a body, a double delta wing, a body flap and a centre vertical tail. The double delta wing is equipped with full span elevons broken into two panels in each side. These can be deflected symmetrically as an elevator for longitudinal control or asymmetrically as ailerons for roll control. The body flap, located on the bottom side at the rear of the Orbiter is used as the primary trim device. Body flap deflections range from +22.5º (trailing edge down) to –11.7º. The vertical tail consists of a split rudder that can be deflected together for yaw control and separated to act as a speed brake. The Space Shuttle Orbiter physical geometry and hypersonic analysis model figure used in the APAS analysis is given in Fig. 6.27. Results of the comparison of the predicted values by the application of APAS code with the experimental data for three different Mach number values as reported by Maughmer, et. al. are as follows: *
Longitudinal aerodynamic coefficients as predicted by the subsonic panel method of APAS code were compared with the experimental data at a Mach number of 0.25. Included in the comparisons were the results of full span elevon deflections from - 20º to +10º . * The predicted lift coefficients for the most part were within 10 per cent of the experimental data although the lift curve slopes were not well predicted. However, the flap effectiveness of the elevon CL δ was predicted reasonably well except at low angles of attack and large negative deflection angles. * Drag predictions were also good for preliminary design analysis purposes. * The pitching moment coefficients as a function of angle of attack did not agree well. * The slopes of the predicted and experimental curves were sometimes of opposite sign. * Regarding the lateral control derivatives, the potential flow modified with the leading edge suction analogy only yielded the best agreement with experiments. * Similar were the cases for directional control derivatives. The same kind of results were also observed with the tests at Mach 0.8. Comparison of the predictions using the HABP without employing the shadowing of downstream components at the hypersonic Mach number of 5.0 showed excellent agreement of
Figure 6. 27. Space shuttle orbiter physical geometry and hypersonic analysis model
Methods 157
158
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
the lift coefficient upto an angle of attack of 20º and remained reasonably good till even upto 40º. The predicted CD vs α and CD vs CL curves agreed well with the experiments. The flap effectiveness for symmetrical elevon deflection was also well predicted. The characteristic trends of the pitching moment coefficient variation with angle of attack and lift were captured. However, the control effectiveness was significantly overpredicted for large negative elevon deflections. The changes in the values of CD, CL and Cm due to the Shuttle body flap deflections as observed in the experiments were more or less captured by the HABP. The lift and drag data were not affected or influenced much whether the shielding due to upstream influences were taken into account or not. However, for the case of the pitching moment the shielded results were better at low angles of attack while the unshielded results were better at higher angles of attack. Comparisons between HABP predictions without shielding for the control derivatives showed that both the trend and magnitudes were captured quite well. But this was not the case for the directional control derivatives. Similar comparisons were also exhibited for Mach 20 case. It was noted that HABP predictions improve as the Mach number increases. Some representative figures taken from Maughmer’s work showing comparisons between theoretical predictions and experiments are reproduced in Figs. 6.28 to 6.36. CONCLUSIONS From a critical study of the comparisons between various prediction methods and the experimental data on a wide variety of hypersonic vehicles, some of which have been described above, the following general conclusions are drawn. At subsonic speeds, the vortex lattice and the panel methods give reasonably accurate values for the lift and drag in the linear angle of attack range. But the pitching moment was not well predicted. For the case of lateral and directional coefficients the potential flow panel method with leading edge suction analogy yields the best agreement with the experiments, and the results are satisfactory for preliminary design analysis. The vortex lattice method that was used to analyze some hypersonic configurations in subsonic flow did not have the ability to include vertical surfaces, and for this reason it could not be used to determine lateral-directional control characteristics.
Methods
PITCHING MOMENT COEFFICIENT
0.4
0.2
0.0
-0.2 -5
0
10
5
20
15
ANGLE OF ATTACK (deg)
0.4
PITCHING MOMENT COEFFICIENT
ADDB
APAS II
δe,
10° 0°
-10° -20°
0.2
0.0
-0.2 -1.0
-0.5
0.0
0.5
1.0
1.5
LIFT COEFFICIENT
Figure 6.28. Comparison of experimental and theoretical pitching moment coefficients for the space shuttle orbiter at Mach 0.25 for various elevon deflections.
159
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
1.25 ADDB
DRAG COEFFICIENT
1.00
APAS II
δe
10° 0°
-10° -20°
0.75
0.50
0.25
0.00 -10
20
10
0
30
40
ANGLE OF ATTACK (deg)
1.25
1.00 DRAG COEFFICIENT
160
0.75
0.50
0.25
0.00 -0.25
0.00
0.25
0.50
0.75
1.00
1.25
LIFT COEFFICIENT
Figure 6.29. Comparison of experimental and theoretical drag coefficients for the space shuttle orbitter at Mach 5.0 for various elevon deflections. The hypersonic shielding option was not included in the theoretical calculations.
Methods
PITCHING MOMENT COEFFICIENT
0.05
0.00
-0.05 ADDB
APAS II
δe
10° 0°
-0.10
-10° -20°
-0.15
-10
0
10
20
30
40
ANGLE OF ATTACK (deg)
PITCHING MOMENT COEFFICIENT
0.05
0.00
-0.05
-0.10
-0.15 -0.25
0.00
0.25
0.50
0.75
1.00
1.25
LIFT COEFFICIENT
Figure 6.30. Comparison of experimental and theoretical pitching moment coefficients for the space shuttle orbiter at Mach 5.0 for various elevon deflections. The hypersonic shielding option was not included in the theoretical calculations.
161
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
1.5
LIFT COEFFICIENT
ADDB APAS II APAS II WITH SHIELDING
1.0
0.5
0.0
-0.5 0
-10
20
10
30
40
ANGLE OF ATTACK (deg)
1.00
DRAG COEFFICIENT
162
0.75
0.50
0.25
0.00 -0.25
0.00
0.25
0.50
0.75
1.00
1.25
LIFT COEFFICIENT
Figure 6.31. Comparison of experimental and theoretical longitudinal aerodynamic coefficients for the space shuttle orbiter at Mach 5.0 showing the effect of including the hypersonic shielding option.
Methods
PITCHING MOMENT COEFFICIENT
1.00 ADDB
APAS II
δe
10° 0°
0.75
-10° -20°
0.50
0.25
0.00 -10
10
0
30
20
40
ANGLE OF ATTACK (deg)
PITCHING MOMENT COEFFICIENT
1.00
0.75
0.50
0.25
0.00 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
LIFT COEFFICIENT
Figure 6.32. Comparison of experimental and theoretical drag coefficients for the space shuttle orbiter at Mach 20.0 for various elevon deflections. The hypersonic shielding option was included in the theoretical calculations.
163
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
0.04
ADDB
APAS II
δbf
-11.7° 10.0° 22.5°
∆c L
0.02
0.00
-0.02 -10
0
10
20
30
40
30
40
ANGLE OF ATTACK (deg)
0.06
0.04
0.02
∆c D
164
0.00
-0.02 -10
0
10
20
ANGLE OF ATTACK (deg)
Figure 6.33. Comparison of experimental and theoretical increments in longitudinal aerodynamic coefficients for the space shuttle orbiter at Mach 20.0 for various body flap deflections.
Methods
ADDB SHUTTLE FLIGHT TEST APAS II
CY
δa
0.001
0.000
-0.001 -10
0
10
20
30
40
30
40
30
40
ANGLE OF ATTACK (deg)
Cn
δa
0.001
0.000
-0.001 -10
0
10
20
ANGLE OF ATTACK (deg)
Cl
δa
.002
.001
.000 -10
0
10
20
ANGLE OF ATTACK (deg)
Figure 6.34. Comparison of experimental and theoretical increments in lateral force and moment coefficients due to differential elevon deflections for the space shuttle orbiter at Mach 5.0.
165
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
ADDB
0.002
APAS II
α 0° 10°
SHUTTLE FLIGHT TEST
CY
δa
0.000
-0.002
-0.004 0
5
10
15
20
15
20
15
20
MACH NUMBER
X10-3
0.5
Cn
δa
1.0
0.0 -0.5 0
5
10 MACH NUMBER
δa
.0050
Cl
166
.0025
.0000 0
5
10 MACH NUMBER
Figure 6.35. Comparison of experimental and theoretical increments in lateral force and moment coefficients due to differential elevon deflections as a function of Mach Number for the space shuttle orbiter.
Methods
ADDB APAS II APAS II WITH SHIELDING
CY
δr
0.001
0.000
-0.001 -10
0
10
20
30
40
30
40
30
40
ANGLE OF ATTACK (deg)
0.000
Cn
δr
0.001
-0.001 -10
0
10
20
ANGLE OF ATTACK (deg)
Cl
δr
0.001
0.000
-0.001 -10
0
10
20
ANGLE OF ATTACK (deg)
Figure 6.36. Comparison of experimental and theoretical rudder effectiveness for the space shuttle orbitter at Mach 20.0 showing the effect of including the hypersonic shielding option in the theoretical calculations.
167
168
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
At low and moderate supersonic speeds, the PAN AIR method predicts the longitudinal aerodynamic characteristics very well. Due to the separated flow, some results at supersonic speeds for the lateral/directional control derivatives are found to be unacceptable. The rolling moment derivative due to aileron deflection and the yawing moment derivative due to rudder deflection are predicted well enough for conceptual design purposes. The methods based on the linear supersonic flow theory and slender body approximations give reasonably good predictions only in the low supersonic speed ranges. Utilisation of the second-order shock-expansion theory for the body, Van Dyke’s unified hypersonic-supersonic similarity relations and the approximate methods of the Pitts, Neilsen, and Kaattari24 for the interference and downwash effects gave very good predictions in the supersonic and as well as low hypersonic speed ranges, both for longitudinal and lateral control aerodynamic characteristics. In the hypersonic flow regimes there are no intermediate theoretical models between the simple flow inclination methods and the nonlinear CFD methods (Euler or Navier Stokes). Setting up of computational grids as well as its execution requirements, make CFD method unsuitable for preliminary design analysis. For this reason, the original HABP as developed by Gentry or with slight improvements or additions to the basic program as contained in APAS is being universally used as a prediction methodology at hypersonic speeds. To account for viscous effects, the most commonly adopted method is Spalding-Chi. The longitudinal aerodynamic characteristics particularly the lift and drag are well predicted by the HABP when the tangent cone theory is applied on the body for compressive forces and the tangent wedge on the lifting surfaces and the Prandtl-Meyer expansion for the lee-side surfaces with limiting value of a pressure coefficient of 70 per cent of vacuum conditions. The magnitudes of the pitching moment coefficients are not always well predicted in almost all of the cases tested due to errors in the locations of the centres of pressures. However, the trends in the changes in the pitching moment as a function of angle of attack and flap or elevon deflections, are generally captured. Lateral control has been well predicted when the shielding is not taken into account. The prediction methodology using the surface inclination method improves as the Mach number increases. Some conventional aircraft configurations like X-15 may be influenced by body interference effects at low hypersonic Mach numbers, which are not accounted for in the HABP. For cases
Methods
like this it is better to use in addition, an alternate approach like the one suggested by Walker and Wolowicz23 that utilizes the procedures outlined in the work of Pitts, et. al 24 to account for interference and downwash effects between various components of the vehicle. REFERENCES 1.
2.
3.
4.
5.
6.
7.
8.
9.
Lamar, J.E. & Gloss, B.B. Subsonic aerodynamic characteristics of interacting lifting surfaces with separated flow around sharp edges predicted by a vortex lattice method, NASA, 1975. TND-7921. Polhamus, E.C. Concept of the vortex lift of sharp-edge delta wings based on a leading-edge suction analogy, NASA, 1966. TND-3767. Penland, J.A.; Dillon, J.L. & Pittman, J.L. An aerodynamic analysis of several hypersonic research airplane concepts from M=0.2 to 6.0. 16th Aerospace Sciences Meeting, AIAA. January 1978. pp. 78-150. Pittman, J.L. & Dillon, J.L. Vortex lattice prediction of subsonic aerodynamics of hypersonic vehicle concepts, J. of Air., 1977, October. 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, 1979. TP-1539. Middleton, W.D.; & Carlson, H.W. Numerical method of estimating and optimizing supersonic aerodynamic characteristics of arbitrary planform wings. J. of Air., 1965, 2(4). Lamb, M.; Sawyer, W.C. & Thomas, L. Experimental and theoretical supersonic lateral-directional stability characteristics of a simplified wing-body configuration with a series of tail arrangements, NASA, August 1981. TP-1878. Jackson(Jr), C.M.; & Sawyer, W.C. A method for calculating the aerodynamic loading on wing-body combinations at small angles of attack in supersonic flow. NASA, 1971. TN D-6441. Ehlers, F.E.; Epton, M.A.; Johnson, F.T.; Magnus, A.E. & Rubbert, P.E. An improved higher order panel method for linearized supersonic flow. AIAA 16th Aerospace Sciences Meeting. AIAA. January 1978. pp. 78-15.
169
170
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Pittman, J.L.; & Riebe, G.D. Experimental and theoretical aerodynamic characteristics of two hypersonic cruise aircraft concepts at Mach numbers of 2.96, 3.96, and 4.63. NASA, TP-1767. Middleton, W.D. & Lundry, J.L. A computational system for aerodynamic design and analysis of supersonic aircraft, part 1-general description and theoretical development, NASA, 1976. CR-2715. Sommer, S.C. & Short, B.J. Free-flight measurements of turbulent boundary layer skin friction, in the presence of severe aerodynamic heating at Mach numbers from 2.8 to 7.0. NACA, 1955. TN-3391. Stock, H.W. Ein Verfahren zur Berechnung der aerodynamischen Beiwerte von Flugkorpen im hohen Uberschall fur massige Anstell-und Ruderausschlagwinkel und bei beliebigen Rollagen, Z.Flugwiss, 1976, 24(Heft 4). Gentry, A.E. & Smyth, D.N. Hypersonic arbitrary-body aerodynamic computer program (Mark III version), Vol. II. McDonnel Douglas Corp, April 1968. DAC61552. Clark, L.E. Hypersonic aerodynamic characteristics of an all-body research aircraft configuration. NASA, December 1973. TN D-7358. Penland, J.A. & Pittman, J.L. Aerodynamic characteristics of a distinct wing-body configuration at Mach 6. NASA, 1985. TP-2467. Covell, P.F.; Wood, R.M. & Bauer, S.X. Configuration trade and code validation study on a conical hypersonic vehicle. AIAA/AHS/ASEE Aircraft Design, Systems and Operations Conference. September 1988. Middleton, W.D.; Lundry, J.L. & Coleman, R.G. A system for aerodynamic design and analysis of supersonic aircraft, Part-1. NASA, 1980. CR–3352. Shankar,V.; Szema, K. & Bonner, E. Full potential methods for analysis of complex aerospace configurations. NASA, 1987. CR-3982. Moore, M.E.; & Williams, J.E. Aerodynamic prediction rationale for analyses of hypersonic configurations. AIAA 27th Aerospace Sciences Meeting. AIAA, January 1989. pp. 89-525, Maughmer,M.; Ozoroski, L.; Straussfogel, D. & Long, L. Validation of engineering methods for predicting hypersonic
Methods
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
vehicle control forces and moments, J. of Guid., Co., & Dyna., 1993, 16(4). Bonner, E.; Clever, W. & Dunn, K. Aerodynamic preliminary analysis system II, Part I-Theory. NASA, April 1981. CR165627. Walker, W.J. & Wolowicz, C.H. Theoretical stability derivatives for the X-15 research airplane at supersonic and hypersonic speeds including a comparison with windtunnel results. NASA, August 1960. TM X-287. Pitts, W.C.; Nielsen, J.N. & Kaattari, G.E. Lift and center of pressure of wing-body-tail combinations at subsonic, transonic and supersonic speeds. NACA, 1957. 1307. Harmon, S.M. & Jeffreys, I. Theoretical lift and damping in roll of thin wings with arbitrary sweep and taper at supersonic speeds. Supersonic leading and trailing edges. NACA, 1950. TN-2114. Haefeli, R.C.; Mirels, H. & Cummings, J.L. Charts for estimating downwash behind rectangular, trapezoidal, and triangular wings at supersonic speeds, 1950. NACA, TN-2141. Syvertson, C.A. & Dennis, D.H. A second order shockexpansion method applicable to bodies of revolution near zero lift, NACA, 1957. 1328. Allen, H.J. Estimation of the forces and moments acting on inclined bodies of revolution of high fineness ratio. NACA, 1949. RMA9126. Perkins, E.W. & Jorgensen, L.H. Comparison of experimental and theoretical normal force distributions (including reynolds number effects) on an ogive–cylinder body at mach number 1.98. NACA, 1956. TN-3716. Dillon, J.L. & Creel(Jr.), T.R. Aerodynamic characteristics at Mach number 0.2 of a wing-body concept for a hypersonic research airplane. NASA, 1978. TP-1189. Dillon, J.L. & Pittman, J.L. Aerodynamic characteristics at mach numbers from 0.33 to 1.20 of a wing-body design concept for a hypersonic research airplane. NASA, 1997. TP-1044. Dillon, J.L. & Pittman, J.L. Aerodynamic characteristics at Mach 6 of a wing-body concept for a hypersonic research airplane. NASA, 1978. TP-1249.
171
PART III AERODYNAMICS OF RAREFIED GASES
CHAPTER 7 GAS DYNAMICS OF RAREFIED FLOWS 7.1
INTRODUCTION
Tsien1 was the earliest to study the aerodynamics of rarefied gas flows. The earliest detailed theoretical study of forces and heat transfer on bodies in free molecule flow and their comparison with experiments were done by Stalder and his associates2-4. Then followed review articles by Schaaf and Chambre5 and Schaaf and Talbot6. These works still serve as basic introduction to mechanics of rarefied gases. The characteristics of the flow over a body flying at very high altitudes depend on the degree of rarefaction of the atmosphere. As the aerodynamic and heat transfer effects depend on the type of flow, it is necessary to identify this. Towards this, a dimensionless parameter called the Knudsen number is introduced. Knudsen number is defined as the ratio of the molecular mean free path λ, of the gas, to the characteristic dimension of the body d Kn =
λ
(7.1.1) d
An alternative is also suggested in which the vehicle length L and width D are combined as a function of effective angle of attack α to determine d, the characteristic length. For this case, the relation for Knudsen number becomes Kn =
λ
(L sin α + D cos α )
(7.1.2)
176
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Values of λ as a function of altitude may be obtained from standard atmosphere tables. From the kinetic theory of gases, the mean free path is given by λ=
16
µ RT
(7.1.3)
5 2π p
where µ is the viscosity of the gas, p, the gas pressure, T, the gas temperature and R, the gas constant. From the above equation, one can obtain a relation for the Knudsen number in terms of the Mach and Reynolds number as Kn = 1.27
γ M Re
(7.1.4)
Based on the Knudsen number, the flow regimes are arbitrarily defined as Slip flow Transition Flow Free Molecule flow
0.01 ≤ Kn ≤ 0.1 0.1≤ Kn ≤ 10 Kn ≥ 10
Slip flow is still governed by the continuum theory (Navier– Stokes equations), except for the introduction of modified boundary conditions for the finite velocity, known as the slip velocity, and a temperature jump at the body surface. In free molecule flow, molecules reemitted from the body surface after collision, do not collide with the incoming stream until very far away from the body. This implies that the free stream molecular velocity distribution is unaffected by the presence of the body. Flow phenomena are entirely governed by the moleculesurface interaction, and the kinetic theory of gases is utilised to analyse the flow. In transitional flow from free molecular to continuum, inter-molecule collisions near the body and molecule-surface collisions are of equal importance, and the theoretical analyses are quite formidable, and no results of direct aerodynamic interest are available. Except for the highly computer oriented Direct Simulation Monte Carlo Technique to analyse the flow in this regime to get accurate values, approximate, empirical bridging formulae between free molecular and continuum flows are invariably made use of.
Gas Dynamics of Rarefied Flows
7.2
FREE MOLECULE FLOW ANALYSIS
Inherent in the definition that the flow is free molecular is that there are no collisions between the incident molecules and the surface reflected ones near the body. This allows us to consider the effects of incident and reflected molecules on the body independently. Further, we assume that the reflected molecules do not collide again with any other part of the body, implying that the body surface is concave everywhere. In free molecule flow, the molecule-surface interaction determines the amount of energy and momentum transfer. Towards this, interaction parameters, one for the energy and the other for momentum transfer are defined. 7.2.1 Surface Interaction Parameters The degree of energy equilibrium attained between the surface element and the molecules before they are reemitted is expressed in terms of a parameter known as the thermal accommodation coefficient defined as
α=
dE i − dE r dE i − dE w
(7.2.1)
where dEi = dEr = dEw =
energy flux incident on the surface of unit area in unit time. energy flux reemitted from the surface of unit area in unit time. energy flux of the reemitted molecules if all the incident molecules were reemitted in Maxwellian equilibrium with the surface.
It has been suggested that, if necessary, separate thermal accommodation coefficients for each degree of freedom, viz., translational, rotational and vibrational can also be similarly defined, as all of them are involved in energy transfer. Experiments have indicated that vibrational degrees of freedom are not affected by collision with a surface, while only the translational and rotational energies are involved. Similarly, two momentum reflection coefficients are defined, one for the tangential momentum and the other for normal momentum transfer.
177
178
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
The tangential momentum reflection coefficient αt is defined by
σt =
τ i − τr τi − τw
=
τ i − τr τi
(7.2.2)
where τi
=
τr
=
τw
=
tangential momentum carried to unit area of surface by incident molecules tangential momentum carried away from unit area by reflected molecules tangential momentum carried away from unit area by diffusely reflected molecules in thermal equilibrium with the surface. By definition τw = 0.
Normal momentum reflection coefficient σn is defined by
σn =
pi − pr pi − pw
(7.2.3)
where pi
=
pr
=
pw
=
flux of normal momentum (pressure) incident on the surface flux of normal momentum (pressure) carried away by reflected molecules flux of normal momentum reemitted if all incident molecules were reemitted in Maxwellian equilibrium with the surface.
The case for which α = σt = σn = 1 is called diffuse reflection. It corresponds to an interaction in which the incident molecules come to a complete thermodynamic equilibrium with the surface. The case in α = σt = σn = 0 is called specular reflection. In this type of reflection, there is no energy and tangential momentum transfer to the surface. It has been observed that the materials used in space vehicles are of such a type that the accommodation and reflection coefficients are close to one.
Gas Dynamics of Rarefied Flows
7.2.2 Forces on an Surface Element in Free Molecule Flow For a gas in equilibrium, kinetic theory states that the molecular velocities are specified by the Maxwellian equilibrium velocity distribution function f. The distribution function f is the probability that the velocity of a randomly selected molecule will have velocity components lying in an element of velocity space (dU, dV, dW ), centered at (U,V,W ). It is given by the expression
f (U,V,W ) dU dV dW =
(
1 π cm
−
)
3
e
1
( c m )2
(U 2 + V 2 + W 2 ) dU dV dW
(7.2.4) Here, cm = most probable molecular speed = 2 R T , where R, is the gas constant and T, the temperature of the gas. U, V and W are the velocity components of the random or thermal motion of the molecules. To calculate the forces experienced by a body, it is first necessary to obtain the basic momentum transfer to a differential elemental area dA and then integrate over the body. The motion of a body with a velocity - q may be transformed into an equivalent dynamical problem by considering the body to be at rest and the gas moving towards it with a velocity q. The distribution of velocities for the incident molecules relative to an observer moving with the steady gas velocity q is Maxwellian. Consider an orthogonal coordinate system such that the axes x and z lie on the plane of the surface element and the y axis is the inward directed normal. Let u , v and w be the components of the mass velocity along x, y and z axes respectively. The molecules approaching the element will have the thermal velocity components superimposed on the mass velocity components, viz., u = u + U , v = v + V and w = w + W
The number of molecules in the velocity range u to u + du, v to v + dv and w to w + dw per unit volume is ni f du dv dw where, ni is the number density of incident molecules.
(7.2.5)
179
180
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
z
y
q x
Figure 7.1. Coordinate system used for a surface element
The number of molecules with velocity components in the range u to u + du, v to v + dv and w to w + dw that strike the surface element of area dA in unit time must lie in a cylinder of height v and base area dA and is ni v f du dv dw dA
(7.2.6)
Each molecule that strikes the surface has a momentum component mu in the x direction where m is the mass of a molecule. The number of molecules in the velocity range u to u + du, v to v + dv and w to w + dw which strike the front surface will impart to it a force in the x direction of magnitude. m n i uv f du dv dw dA
(7.2 .7)
The total x direction force on the front surface of the element dA due to incident molecules in all velocity ranges is +∞ +∞ +∞
m ni
∫ ∫ ∫ u v f du dv dw dA
−∞
o
(7.2.8)
−∞
Similarly, the y component of the force is +∞ +∞ +∞
m ni
∫ ∫ ∫v
−∞
o
2
f du dv dw dA
(7.2.9)
−∞
and the z component of the force is ∞
m ni
+∞ +∞
∫ ∫ ∫ v w f du dv dw dA
−∞
o
−∞
(7.2.10)
Gas Dynamics of Rarefied Flows
In Eqns. 7.2.8 to 7.2.10, the limits of integration for v are from 0 to ∞ because only the molecules with a velocity component in the positive y direction will hit the surface. The vector sum of Eqns. 7.2.8, 7.2.9 and 7.2.10 gives the total force on the element due to molecules in all the velocity ranges. We are interested in calculating the components of these forces in a particular direction. Let ax, ay and az be the direction cosines between the direction in which the force is required and the x, y and z axes respectively. Then the component in a particular direction of the total force on the element due to incident molecules is
dF = m ni dA incident
+∞ +∞ +∞
∫ ∫ ∫ (a
−∞
o
x
u + a y v + a z w )v f du dv dw
−∞
(7.2.11) In a similar manner we can compute the force due to reflected molecules. It is assumed that the molecules are reemitted from the surface randomly with a Maxwellian velocity distribution at a temperature of Tr . The reflected molecules are thought of as a fictitious gas issuing from the rear side of the surface at a temperature Tr. The number of molecules reemitting from the elemental surface of area dA in unit time in the velocity range u to u + du, v to v + dv and w to w + dw is n r (− v ) f du dv dw dA
(7.2.12)
where, (-v) is used because only the molecules having a component of velocity component in the negative y direction will leave the surface. nr is the number density of the reflecting molecules. The force on the surface element due to reflected molecules must be equal to the change in momentum between the initial and final conditions. Force =
(initial momentum – final momentum) of reflecting molecules.
The initial momentum is zero as the gas is at rest on the surface. Therefore, the force in the x direction on the surface due to reemitted molecules in all the velocity ranges is
− −m nr
+∞
0
u v f du dv dw dA −∞
+∞
∫ ∫ ∫
−∞ −∞
(7.2.13)
181
182
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Similarly, the force in the y direction due to reflected molecules is
− −m nr
+∞ 0
+∞
∫ ∫ ∫
− ∞ − ∞ −∞
v 2 f du dv dw dA
(7.2.14)
while the force in the z direction is +∞ 0 +∞ v w f du dv dw dA − − m n r (7.2.15) − ∞ − ∞ −∞ The limits of integration for v are from – ∞ to 0 as only the molecules having a velocity component in the negative y direction will leave the surface.
∫ ∫ ∫
The sum of the components of the above in the desired direction having the direction cosines ax, ay and az is dF dA
= m nr reflected
+∞
+∞
0
∫ ∫ ∫ (a
x
)
u + a y v + a z w v f du dv dw
− ∞ − ∞ −∞
(7.2.16) The sum of Eqns. 7.2.11 and 7.2.16 give the total force on the surface element in a specified direction due to both the incident and reemitted molecules and is given by dF dA
= m n i
+∞ +∞ +∞
∫ ∫ ∫ (a
−∞
+ m nr
u + a y v + a z w )v f du dv dw
−∞
0
+∞
x
0
+∞
∫ ∫ ∫ (a
x
u + a y v + a z w ) v f du dv dw
(7.2.17)
− ∞ − ∞ −∞
The reflecting molecules are having no mass velocity, only the thermal velocity components. Performing the integration results in dF = m n 1 (a u + a v + a w ) i x y z dA 2 π v π (1 + erf v β) + 1 e − β2 v 2 β ay a + (1 + erf v β) + m nr y2 4 β2 4 βr
(7.2.18)
Gas Dynamics of Rarefied Flows
1
In the above equation β =
βr =
1
, ni and nr are 2 RTi 2 RTr the number densities of incident and reflected molecules and x
2
∫e π
erf x =
,
−y2
dy
0
Since the number of molecules impinging on the surface in unit time must be equal to the number of molecules reflected in unit time, a relation exists between ni and nr. The number of molecules in the velocity range u to u + du, v to v + dv and w to w + dw that strike the unit area of the surface in unit time is given by ni v f du dv dw
(7.2.5)
The total number of molecules incident on unit area of surface in unit time having all possible velocity ranges is given by +∞ + ∞ + ∞
N i = ni
∫ ∫ ∫ vf du dv dw
−∞
0
(7.2.19)
−∞
Substituting the expression for f, the Maxwellian distribution function 3
1 N i = ni 2 πR T i
2
+ ∞ +∞ +∞
∫ ∫ ∫
−∞
−
ve
1
[ (u − u )
2
2 RT i
+ (v − v
)2
+ w2
] du dv dw
0 −∞
(7.2.20) on integration, it reduces to v RT i − 2 R T i + e 2π 2
N i = ni
v 2 R Ti
π 1 + erf
2 R Ti (7.2.21) v
The number of molecules in the velocity range U to U + dU, V to V + dV and W to W + dW that are reemitted from the surface of unit area, in unit time is
183
184
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
n r (− V ) f dU dV dW
(7.2.22)
These molecules are coming from the surface with no mass velocity. Hence, their velocity components are due to thermal motion only. The total number of such molecules in all velocity ranges is +∞
N r = nr
0
+∞
∫ ∫ ∫
−∞ −∞ −∞
1 (− V ) π 2 R Tr
3 2
−
e
1 2 R Tr
(U 2 +V 2 +W 2 )
dU dV dW
(7.2.23) The result of the integration is
N r = nr
R Tr
(7.2.24)
2π
Since Ni = Nr , equating the two expressions of Eqns. 7.2.21 and 7.2.24.
n r = n i β r v
π (1 + erf v β ) +
1 β
e−β
2
v2
(7.2.25)
Substituting for nr in Eqn. 7.2.18
1 1 = mn i (a x u + a y v + a z w ) v π (1 + erf v β ) + e − β 2 v 2 dA β 2 π ay ay 1 − β2 v 2 + (1 + erf v β ) + v π 1 + erf v β + e 2 β β 4 4β r
dF
(7.2.26) Let bx, by and bz be the direction cosines between the mass velocity and the x, y and z axes respectively. Then, u = b x q , v = b y q and w = b z q
(7.2.27)
Introducing a dimensionless quantity s known as the speed ratio, which is the ratio of the mass velocity to the most
Gas Dynamics of Rarefied Flows
q
probable random speed of the molecules (s =
2 RTi
Eqn. 7.2.26 becomes dF
= mn i
dA
q2 2
{(a x b x
+ a y by + a z b z
)
1 − (b s )2 e y b y (1 + erf (b y s )) + πs ay + (1 + erf (b y s )) 2s2 +
ayβ by 2β r s
π (1 + erf (b y s )) +
β
Since, mni = ri and
1 s
(7.2.28) e
− (b y s )2
Tr
=
βr
) the
, Eqn. 7.2.28 can be written Ti
as dF dA
= ρi
q2 2
{(a
x bx
+ a y by + a z b z
)
ay 1 − (b s )2 e y (1 + erf (b y s )) b y (1 + erf (b y s )) + + 2 πs 2s +
Tr b y π 1 − (b s )2 1 + erf (b y s ) + 2 e y Ti s s
ay 2
(7.2.29)
To get the force in the non-dimensional coefficient form we 2 divide the above by ρ q A . The result is i ref 2
{(
dC 1 = dA A ref ay + 2s 2 ay + 2
a xb x + a yby + a zb z
(1 + erf Tr Ti
( ))
b y 1 + erf b y s
)
(b s ))
+
1
πs
e
( )
− bys
2
y
b y π (1 + erf s
(b s )) + s1 y
2
e
( )
− bys
2
(7.2.30)
185
186
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
This is an exact expression for the force coefficient under the assumption of diffuse reflection. All the quantities in the above equation are known except T r. For this, we have to know the energy transfer to the surface. If the body is a good thermal conductor, the surface temperature T w will be the same all over the surface. If the thermal accommodation coefficient for the surface material is known then T r can be determined from Eqn. 7.2.1. We can remove the restriction of complete diffuse reflection by introducing the normal and tangential accommodation coefficients in the force coefficient equation. From Eqn. 7.2.2 τ r = (1 − σ t )τ i
(7.2.31)
and from Eqn. 7.2.3
(
p r = 1−σ n
)pi
+σn pw
(7.2.32)
The total shear stress on the elemental surface is τ = τ i − τ r = τ i − (1 − σ t
)τi
= σt τi
(7.2.33)
and the total normal stress on the surface is p = p i + p r = p i + (1 − σ n ) p i + σ n p w = (2 − σ n ) p i + σ n p w
(7.2.34) Substituting the above in the force coefficient Eqn. 7.2.30 we have dC dA
=
1 Aref
2 − σ n a y b y + σ t a x b x + a z b z
(2 − σ n )a y 1 − (b s )2 e y [1 + erf (by s )] + b y (1 + erf (b y s )) + 2s 2 s π +
σn a y 2
Tr b y π 1 − (b s )2 ( 1 + erf (b y s )) + 2 e y Ti s s
(7.2.35)
By integrating the results of this relation over the entire surface of the body, the aerodynamic forces and moments can be calculated for any arbitrary shaped body. Since a body in free
Gas Dynamics of Rarefied Flows
molecule flow is assumed not to disturb the flow, the complex vehicle configuration is resolved into a combination of simple subshapes, such as flat panels, cylinders, cones, conical frustums, spheres and spherical segments. Each sub-shape is subdivided into an arbitrary number of elemental areas and is integrated over its area. For each elemental area set up by the integration routine, it is necessary to determine if the velocity vector intersects the elemental area directly without first intersecting any portion of the sub-shape or any of the other composite sub-shapes. If it does, then it is considered to be in the shadow region and this has to be treated appropriately. 7.3
AERODYNAMIC FORCES FOR TYPICAL BODIES
Aerodynamic force coefficients in free molecule flow for simple body shapes can be determined by the application of the results of the previous section. In the analysis it is assumed that the body temperature Tw , the tangential and normal momentum accommodation coefficients and the temperature Tr of the molecules reflected from the surface are all constant over the entire surface. 7.3.1 Flat Plate CASE 1: ONLY THE FRONT SIDE EXPOSED TO THE FLOW The flat plate is at an angle of attack a to the incoming velocity q. The elemental surface coordinates x, y and z are as shown. It is to be remembered that while deriving the basic relation for the force on an elemental surface, positive y axis was specified as inward normal to the surface. X and Y are the directions in which the force coefficients are sought, viz., the tangential and normal force coefficients.
Y y j i α
z OUT OF PAGE
x
z OUT OF PAGE
q
Figure 7.2. The flat plate exposed on the front side only
X
187
188
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
The force coefficient from Eqn. 7.2.35 is dC dA
=
1 Aref
2 − σn
a b + σ a b + a b y y t x x z z
b y 1 + erf b y s + 2 − σ a n y + 2 2s σn a y 2
1 + erf
1 πs
− b y s e
2
b s + y
2 − b y s Tr b y π 1 1 + erf b y s + 2 e Ti S s
(7.3.1) In the above ax, ay and az are the direction cosines between the local x, y and z axes and the desired force direction, bx, by and bz are the direction cosines between the local x, y and z axes and the mass velocity vector. The unit vector in the direction of the mass velocity is q = i cos α + j sin α
For the front surface x = i ; y = j and z = k
The direction cosines between the mass velocity and the x, y and z axes are bx = q . x = cos α; by = q . y = sin α and bz = q . z = 0 Normal Force For the force in the normal direction, the direction cosines are ax = x . j = 0; ay = y . j = 1 and az = z . j = 0 Substituting the values of ax , ay , az and bx , by , bz in Eqn. 7.3.1 and basing the reference area as the area of the front
Gas Dynamics of Rarefied Flows
surface of the flat plate, the expression for the normal force coefficient is dC N front = (2 − σ n
)
2 sin αe − ( s sin α ) 1 sin 2 α + (1 + erf (s sin α )) + 2s 2 s π
+
σn
Tr
2s 2
Ti
[e
− ( s sin α )2
+
]
πs sin α (1 + erf (s sin α ))
(7.3.2) Axial Force The axial force is in the X direction. The direction cosines between the x, y and z axes and the X axis are given by ax = 1, ay = 0 and az = 0 as before bx = cos α, by = sin α and bz = 0. Substituting the above in Eqn. 7.3.1, we obtain for the axial force coefficient as
dC A front
= σ t sin α cos α
CASE 2:
e − (s sin α ) + (1 + erf s sin α ) π s sin α 2
(7.3.3)
REAR SIDE OF THE FLAT PLATE ONLY EXPOSED TO THE FLOW Y Z IN TO PAGE
j x i X
α
Z OUT OF PAGE
q y
Figure 7.3. Flat plate exposed on the rear side only
The elemental coordinate system for this case is as shown, with the y axis pointing downwards (inward normal to the rear surface). As before, X and Y are the directions of axial and normal forces sought respectively.
189
190
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
x = i ; y = − j and z = − k
Normal Force The direction cosines in this case are ax = 0, ay = -1 and az = 0 and bx = cos α, by = - sin α and bz = 0 Substituting the values in Eqn. 7.3.1, we obtain the normal force coefficient as dC N rear = (2 − σ n
)
sin α e − (s sin α )2 1 − sin 2 α + 2 s2 s π −
σn
Tr
2s 2
Ti
[e
− ( s sin α )2
−
(1 − erf (s sin α ))
]
π s sin α (1 − erf (s sin α ))
(7.3.4) Axial Force The corresponding direction cosines in this case are ax = 1, ay = 0 and az = 0 and bx = cos α, by = - sin α and bz = 0. Substituting in the Eqn. 7.3.1 we get
dC Arear
CASE 3.
2 e − (s sin α ) = σ t sin α cos α − (1 − erf (s sin α )) π s sin α (7.3.5)
BOTH SURFACES OF THE PLATE EXPOSED TO THE FLOW
dC N = dC N front + dC N rear
Adding the expressions of Eqns. 7.3.2 and 7.3.4 we get the normal force coefficient as
Gas Dynamics of Rarefied Flows
dC N
= (2 − σ n )
1 2 sin α e − ( s sin α ) 2 sin 2 α + 2 erf (s sin α ) + s s π
+ σ n sin α
π s
2
Tr Ti
(7.3.6) From Eqns. 7.3.3 and 7.3.5, the axial force coefficient is given by
dC A
e − (s sin α ) = 2 σ t cos α sin α erf (s sin α ) + s π
2
(7.3.7)
If one is interested in the lift and drag coefficients, they can be determined by the relation CL = CN cos α - CA sin αá CD = CN sin α + CA cos α 7.3.2 INFINITE RIGHT CIRCULAR CYLINDER AT AN ANGLE OF ATTACK, α The cylinder being axially symmetric, we can orient the body axis system in such a way that the velocity vector is in the X–Y plane. The elemental area chosen for integration is dA = r dϕ dL where, dL is the elemental length along the cylinder symmetry axis. A reference body coordinate system X, Y and Z is chosen such that the axes X and Y represent the directions of desired tangential and normal forces respectively and Z axis normal to them as shown. Let x, y and z be the local elemental surface coordinates with y axis radial inwards and x and z axes normal to each other and tangential to the surface element. The unit vector in the direction of the mass velocity is q = i cos α + j sin α
191
192
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows Y j y
∅
z
α
x
q
i
r
∅
X
k Z
Figure 7.3. Infinite right circular cylinder at an angle of attack, α
The unit vectors in the direction of the elemental area axes system are x =i y = j cos ϕ − k sin ϕ z = j sin ϕ + k cos ϕ
The direction cosines bx, by, bz are bx = q . x = cos α by = q . y = sin α cos ϕ bz = q . z = sin α sin ϕ Normal Force For the normal force, the direction coefficients are ax = x . j = 0 ay = y . j = cos ϕ az = z . j = sin ϕ The integration limits for ϕ are from 0 to 2π.
C N cyl . =
1 Aref
2π L
∫∫ 0
(dC ) r dϕ dL
0
Substituting the direction cosine values in the expression for dC from Eqn. 7.3.1 and integrating over the surface area of the cylinder, the normal force coefficient for the cylinder, based on the frontal projected area 2rL as the reference area Aref is
Gas Dynamics of Rarefied Flows
CN =
σ n sin α 4s πe
−
π3
2
Tr Ti
s 2 sin 2 α 2
+ sin 2 α (4 + σ t − 2 σ n
)
f (s sin α )
(7.3.8)
where 1 1 + s sin α I o f (s sin α ) = α 2 3 s sin 1 s sin α + + I1 3 6 s sin α
and
s 2 sin 2 α 2
s 2 sin 2 α 2
(7.3.9)
s 2 sin 2 α Io = Bessel function of first kind and zero order 2 s 2 sin 2 α I1 = Bessel function of first kind and first order 2
Axial Force The direction cosines for the axial direction force are ax = x . i
=1
ay = y . i
=0
az = z . i
=0
Substituting the direction cosine values in the expression for dC in Eqn. 7.3.1, and integrating over the cylindrical surface gives the result
C Acyl. =
σt
πe
−
(s sin α )2 2
s
(s sin α )2 cos α 2 1 + (s sin α ) I o 2
(s sin α )2 2 + (s sin α ) I 1 2
[
]
(7.3.10) where, Io and I1 are Bessel functions mentioned earlier
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Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
7.3.3 SPHERE An expression for the drag coefficient of an element of area dA when both of its sides are exposed to the flow can be obtained from the normal and axial force relations for a flat plate discussed earlier, by the relation CD = CN cos α + CA sin α Multiplying Eqn. 7.3.6 by cos α and Eqn. 7.3.7 by sin α, and adding the two expressions we get
dC D =
1 Aref +
σn
2 e − (s sin α )2 s π π sin 2 α s
(2 − σ n
[ (2 − σ
Tr Ti
) sin 2 α +
n
) sin 2 α + σ t cos 2 α ]
+ [2 sin α erf (s sin α ) ] 1 + σ t cos 2 α dA 2 2s
(7.3.11) Element of area for sphere dA = 2πR2 cos α dα Choose an elemental ring surface of area dA = 2π R 2cos θ on the front and rear side of the sphere as shown in Fig. 7.3.4. Each dα α
q
R cos α
194
R
Figure 7.3.4. Figure showing drag coefficient of an element of area dA exposed on both sides to the flow.
Gas Dynamics of Rarefied Flows
point on this ring element will be at the same angle of attack α and the expression for dCD can be applied. In general, the temperature Tw varies from one surface element to another. In the special case where the body is a perfect thermal conductor, Tw is constant over the entire surface, the above expression can be integrated over the surface to get the drag coefficient of the sphere based on the frontal projected area. π
C D sph ere
1 = A ref
2
∫dC
D
dA
(7.3.12)
0
The result of the integration is
C D sphere
2 − σn + σt = s3 +
2 σn 3 s
π
4 s 4 + 4 s 2 −1 e− s erf s + 4s π
2
2 1 s + 2
Tr Ti (7.3.13)
7.3.4 CONE FRUSTRUM Closed form solutions for the force coefficients are not available for other body geometries and one has to go in for numerical integration. The analysis for a cone at an angle of attack has been outlined in detail5,7. The final results for the normal and axial force coefficients taken from Sentman7 is reproduced below. It is assumed that the accommodation coefficients σn and σt are equal to one. L2
Y
L1 δ α
j y δ
q
z
i
∅
Z
X
k
x Figure 7.3.5. Figure depicting normal and axial force coefficients for a cone frustrum.
195
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Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
We consider only the conical surface omitting the front and rear ends of the cone. From the figure r = L tan δ and dr = dL tan δ The element of area considered is dA = r dϕ ds (ds)2 = (dL)2 + (dr)2 = (dL)2 (1 + tan2δ ) = (dL)2 sec2 δ ds = sec δ dL dA = L
tan δ cos δ
dL dϕ
The unit vectors in the direction of the local axis system are x = − j sin ϕ − k cos ϕ y = i sin δ + j cos δ cos ϕ − k cos δ sin ϕ z = i cos δ − j sin δ cos ϕ + k sin δ sin ϕ
The body is oriented in such a way that the velocity vector is in the X-Y plane. The unit vector in the direction of mass velocity is q = i cos α + j sin α
The direction cosines bx, by and bz are b x = x . q = − sin α sin ϕ b y = y . q = cos α sin δ + sin α cos δ cos ϕ b z = z . q = cos α cos δ − sin α sin δ cos ϕ
Normal Force For the normal force the direction cosines are a x = x . j = − sin ϕ a y = y . j = cos δ cos ϕ a z = z . j = − sin δ cos ϕ
Gas Dynamics of Rarefied Flows
Substitution of the above direction cosine values in the Eqn. 7.3.1 and integrating the resulting expression for ϕ from 0 to 2 π we get for the normal force coefficient L 22 − L12 tan δ
C N cone =
cos δ
2 Aref
{2 π sin α cos α sin δ + 2 sin α cos α sin δ
π
π
1 erf (b y s ) dϕ + cos δ 2 sin 2 α + 2 cos ϕ erf (b y s ) dϕ s o o
∫ +
∫
2 sin α s π
π
∫e
− (b y s )2
o
dϕ +
3 Tr π 2 sin α cos 2 δ + Ti 2s
π cos δ s
π
(cos α sin δ ∫ cos ϕ erf (b y s ) dϕ o
π
+ sin α cos δ cos ϕ erf (b y s ) dϕ ) +
∫
2
o
cos δ s
2
π
∫ cos ϕ e
− (b y s )2
o
dϕ
(7.3.14) Axial Force For this case the direction cosines are ax = x . i = 0 a y = y . i = sin δ
a z = z . i = cos δ
Substituting the above in Eqn. 7.3.1 and integrating for ϕ between the limits 0 to 2 π we get for the axial force coefficient
C Acone
=
L 22 − L12 tan δ 1 2 π sin δ 2 cos α + 2 2 Aref cos δ s π
1 + sin δ 2 cos 2 α + 2 erf (b y s ) dϕ s o
∫
π
+ 2 sin α cos α cos δ cos ϕ erf (b y s ) dϕ +
∫ o
2 cos α s π Contd ...
197
198
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
π
∫
e
− (b y s )2
dϕ +
o
π sin δ
Tr π 3 2 cos α sin 2 δ + Ti s
(cos α sin δ
s π
π
∫ erf (b s ) dϕ + sin α cos δ ∫ cos ϕ erf (b s )dϕ ) y
y
0
+
o
sin δ s
2
π
∫
e
− (b y s )2
o
dϕ
(7.3.15)
7.3.5 SPHERICAL SEGMENT Most of the space vehicles have spherical caps in front of the body. It is therefore useful to have relations about the normal and axial pressure coefficients for a spherical segment. The details of the analysis are given by Sentman7 and the final results from it are reproduced below. The back side of the spherical segment is not considered. The element area of the sphere chosen for integration is
dA = r
2
sinθ d θ d ϕ
The integration limits are 0 to θ1 for θ and 0 to 2 π for ϕ. Substituting the relevant direction cosines in the expression for the force on an element Eqn. 7.3.1 one obtains the following expressions for the normal and axial force coefficients. The normal force coefficient is given by
Gas Dynamics of Rarefied Flows
Y j
θ1 ϕ θ
z
α
Z
i
k
X
y
q
x
Figure 7.3.6. Normal and axial pressure coefficients for a spherical segment.
CN =
θ1 π 2 π α α θ + α α sin cos sin 2 sin cos sin θ cos θerf by s dϕdθ 1 Aref o o
r2
(
∫∫
1 + 2 sin 2 α + s2 +
+
2 sin α s π
θ1 π
∫ ∫ sin θ e
θ1 π
∫ ∫ sin
2
(
)
)
θ cos ϕ erf b y s dφ dθ
o o
(
)
− by s 2
dϕ dθ
o o
cos 3 θ1 2 Tr π 3 2 − cos θ1 + sin α 3 3 Ti 2 s +
θ1 π π cos θ sin 2 θ cos ϕ erf b y s dϕ dθ cos α s o o θ1 π
+ sin α
+
(
∫∫
1 s2
sin 3 θ cos 2 ϕ erf b y s dϕ dθ o o θ1 π − (b y s )2 sin 2 θ cos ϕ e dϕ dθ o o
∫∫
(
∫∫
where, by = cos α cos θ + sin α sin θ cos ϕ
)
)
(7.3.16)
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200
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
The axial force coefficient is given by
C A sph . sg
r2 π 2 2 sin 2 θ1 = π cos α sin θ1 + 2 Aref 2s 1 + 2 cos 2 α + 2 s
θ1 π
∫ ∫ sin θ cos θerf (b s )dϕ dθ y
o o
θ1 π
+ 2 sin α cos α
∫ ∫ sin
2
θ cos ϕerf (b y s )dϕ dθ
o o
+ +
+
2 cos α s π
θ1 π
∫ ∫ sin θ e
dϕ dθ
o o
Tr π 3 2 cos α 1 − cos 3 θ1 Ti 3 s
(
π s
θ1 π
(cos α ∫ ∫ sin θ cos
+ sin α
∫ ∫ sin o
1 s2
2
)
θ erf (b y s ) dϕ dθ
(7.3.17)
o o
θ1 π
+
− (b y s )2
2
θ cos θ cos ϕerf (b y s )dϕ dθ )
o
θ1 π
∫∫ o o
sin θ cos θ e
− (b y s )2
dϕ dθ
where, by = cos α cos θ + sin α sin θ cos ϕ as before. 7.4
AERODYNAMIC FORCES IN SLIP & TRANSITIONAL FLOWS
Theoretical methods based on Thirteen Moment and Burnett equations to approximate the integro-differential equation of Boltzman have been utilised to analyse the flow problems in the near free molecule or transitional flow conditions. Only a few special cases such as shock structure, couette flow, etc., have been studied and these are of no practical use in the determination of the aerodynamic forces or heat transfer of bodies in this flow regime. Computer oriented Direct Simulation Monte Carlo Technique is the only known method to give reliable quantitative aerodynamic force and heat transfer parameters all the way from free molecule to
Gas Dynamics of Rarefied Flows
continuum. The involved computer programming, with separate analysis for each flow condition does not allow this method to be of use in the preliminary analysis stage. Simple empirical formulae have been suggested for quick determination of the flow parameters in the transitional flow regime. It has been experimentally observed that the aerodynamic force parameters vary monotonically from free molecule to continuum flows. Utilizing this aspect, several authors have suggested so-called bridging formulae. Some of these are,
C FTRAN = C FCONT
1 1 2 + C F FM − C FCONT sin π + log 10 Kn 3 6
(7.4.1) where, CF is the force coefficient and the subscripts TRAN, CONT and FM refer to transition, continuum and free molecule flow regimes respectively. The continuum values are usually based on the Newtonian analysis. The above equation and its correlation with the experimental data are discussed by Lott8. Another similar empirical bridging formula suggested by Blanchard9 is C FTRAN = C FCONT + C F FM − C FCONT sin 2 w
(7.4.2)
where w = π ( 3 + log 10 Kn ) / 8
Based on considerations of fluid dynamic simulation appropriate to hypersonic viscous flows over blunt nosed bodies, Potter10, 11 has suggested a method for estimating the lift and drag coefficients in the transitional flow regime. According to him, a simulation parameter suitable for scaling viscous, hypersonic flow effects was devised. The characteristic length in this parameter was taken to be a shape factor modified wetted length in the streamwise direction. To quote Potter, “this approach was suggested by the empirical data showing that projected area and wetted area override detailed body shape variations in determining overall forces in hypersonic, transitional flows.”
201
202
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
The suggested method for correlating the lift and drag coefficients in hypersonic rarefied transitional flows are as follows: A normalized form of drag coefficient is defined as C D − C Di
_
CD =
(7.4.3)
C D fm − C D i
For a variety of practical vehicle shapes, the above is correlated with a simulation parameter defined as
Pn D
=
U υ
H∞ s * * ∞ H
ω
1
2
(7.4.4)
where U = free stream velocity υ = kinematic viscosity 1
s * =s (P F A /W A )2
ω = 0.63
H∞ H s
*
=
H∞ ( 0.2 H o + 0.5 H w )
= streamwise wetted length
PFA = projected frontal area WA = wetted area 2nD = drag parameter
CD = drag coefficient
C D i = inviscid drag coefficient Similarly for the lift case _
CL =
C L − C Li C L fm − C L i
2nL = 2nD (PPA /PFA)1/ 4, the lift parameter
where, PPA is the planform projected area.
(7.4.5) (7.4.6)
Gas Dynamics of Rarefied Flows
The available experimental data of lift and drag in transitional flow was normalised as above and plotted against 2nL and 2nD . From the graph it was found that one analytical curve was able to adequately fit the lift and drag data. A simple form of such a curve is _
CL
_
CD
2. 6 = 2 . 6 + Pn1L.6
(
2. 6 = 2 . 6 + P 1. 6 nD
(
)
0.5
0.5
)
(7.4.7)
(7.4.8)
In view of the apparently good agreement with full scale flight data, Potter10, feels that Eqns. 7.4.7 and 7.4.8 represent useful tools for preliminary design studies. 7.5
ENERGY TRANSFER IN FREE MOLECULE FLOW
The molecular energy transfer with the surface element involved by the incident and reflected molecules can be determined in a manner very similar to that of momentum transfer studied earlier. Incident energy on the front side of an element The translational energy incident on the front side of an element of are dA due to incident molecules in unit time is
(dE ) i tr
=
f
1 2
+∞ +∞ +∞
m
∫ ∫ ∫
−∞
0
−∞
n i u 2 + v 2 + w 2
(
v f u ,v , w
)
du dv dw dA (7.5.1) The result of the integration is
(
dE = ρ i R T i i tr f 2π
)
3 2
5 + s2 + 2
2 s + 2
2 e − s (s sin α )
π s sin α 1 + erf s sin α dA
[
(
)]
(7.5.2)
203
204
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
If the gas is monotomic, then the total incident energy is given by the above expression. However, if the gas is composed of diatomic molecules, then each molecule carries an additional amount of energy called the internal energy. By the principle of equipartition of energy, the amount of energy carried by each molecule is (½) j mRTi, where j is the number of degrees of freedom. j is related to the ratio of specific heats γ by the relation. j = (5 – 3 γ)/(γ – 1)
(7.5.3)
In contrast to the translational and rotational degrees of freedom, the process of vibrational energy exchange is inefficient as thousands of collisions are required for effective energy transfer. For this reason, at normal temperatures the translational and rotational degrees of freedom are considered active, and vibration as an inert degree of freedom. At normal temperatures, the value of j is approximately 2 for air. The internal energy of molecules striking the front surface in unit time is j m R T i N i f dA ( dE int ) f = (7.5.4) 2 where, Nifront is the number of molecules striking the front surface in unit time and is given by the Eqn. 7.2.21, viz.,
{e 2π
RT i
N i f = ni
− ( s sin α )2
+
π (s sin α )
}
( 1 + erf s sin α ) dA (7.5.5)
Subsituting Eqn. 7.5.5 in Eqn. 7.5.4 we have
( dE int ) f =
j 2
{e
m (R T i
− ( s sin α )2
ni
)3 2
2π π (s sin α )
+
}
( 1 + erf s sin α ) dA
(7.5.6)
The total incident energy is (dEi)f = (dEtr)f + (dEint)f
(dE ) i
(RT )
3 2
f
= ρi
+
i
2π
(
− ( s sin α )2 e
π s sin α
s2 + 2 + j 2
) (1 + erf s sin α ) s
2
+
5 2
+
j dA 2 (7.5.7)
Gas Dynamics of Rarefied Flows
Similarly, for the rear surface
(dE i )r = ρ i −
(RT i )3 2 2π
e
π (s sin α )
− ( s sin α )2
j 2 s + 2 + 2
5
j
( 1 − erf s sin α ) s 2 + + dA 2 2
(7.5.8)
Energy carried by the reflected molecules The energy dEw carried by the reflected molecules that are in Maxwellian equilibrium with the surface can be calculated in a similar manner. +∞
dE = 1mn w tr wf f 2
∞
0
∫ ∫ ∫ U
2
+V
2
+W
− ∞ −∞ −∞
2
(
V f U ,V ,W
)
dU dV dW dA (7.5.9) On integration dE = 2 ρw w tr f
(RT ) w
2π
3 2
dA
(7.5.10)
From Eqn. 7.2.25
ρwf = ρi
T i − (s sin α )2 + e Tw
[
)]
(
π s sin α 1 + erf s sin α (7.5.11)
Substituting for ρw in Eqn. 7.5.10 we get f
3 dE = 2 ρ i R r tr f 2π
2
Tw f
Ti
[
(
)]
− (s sin α )2 + π s sin α 1 + erf s sin α dA e (7.5.12)
The internal energy of reemitted molecules is
205
206
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
(dE ) int
f
j
=
(7.5.13)
m R T w f N w f dA
2
Since N w f = N w i , the above reduces to
j ρ i R3
(dE int ) f =
{e
2
Tw f
2 2π − ( s sin α
)2
+
Ti
}
π s sin α [1 + erf (s sin α )] dA
(7.5.14) The total energy of reflected molecules in equilibrium with the surface is
(dE w )f tr
(
+ dE w int
)f
= (dE w ) f
3 j ρi R
2
(dE w ) f = 2 + 2
Tw f
(7.5.15)
Ti
2π
{e
− ( s sin α )2
(7.5.16)
}
+ π s sin α [ 1 + erf (s sin α )] dA Similarly, for the rear side of the surface
3 j ρi R
(dE w )r = 2 + 2
{e
− ( s sin α ) 2
2
Tw r
2π
Ti
}
− π s sin α [ 1 − erf (s sin α )] dA (7.5.17)
For specular reflection, the accommodation coefficient α = 0, and there is no energy transfer to the body. If α ≠ 0 and if there is no internal and radiation heat transfer, then the elemental surface will reach a temperature known as the Equilibrium Temperature. This value can be determined by equating the incident energy to the reflected energy.
Gas Dynamics of Rarefied Flows
7.5.1 Equlibrium Temperatures for Simple Shapes Flat Plate: Front surface only exposed to the flow The energy relations derived above for an elemental surface refers to the case of a flat plate at angle of α to the free stream. The following expression gives the ratio of the equilibrium temperature to the free stream temperature when only the front surface is exposed to the flow.
(T ) wequ
f
Ti
=
1 j 2 + 2 j − (s sinα )2 2 5 j 2 + s + + π s sin α [1 + erf (s sin α)] s + 2 + e 2 2 2 − ( s sin α )2 e + π s sin α [1 + erf (s sin α)] (7.5.18)
Similarly for the rear surface
(T )
wequ r
Ti
=
1 j 2 + 2 j − (s sinα )2 2 5 j 2 − s + + π s sin α [1 − erf (s sin α)] s + 2 + 2 e 2 2 − (s sinα )2 − π s sin α [1 − erf (s sin α)] e (7.5.19)
When both the front and rear surfaces are in perfect thermal contact Twequ 1 = j Ti 2 + 2
(
)
j − (s sin α )2 2 5 j 2 + s + + π s sin α erf (s sin α ) s + 2 + 2 e 2 2 2 − (s sin α ) + π s sin α erf (s sin α ) e (7.5.20)
207
208
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Sphere If the temperature Tw is constant over the entire surface of the sphere then the expressions for incident and reflected energies on an element of the sphere can be integrated to obtain the equilibrium temperature. The elemental area chosen is identical to the momentum calculation case. The result of the integration is
Tweq . Ti
1 = j 2+ 2
j 1 3 j 2 5 j − s2 3 s + + e + s + s 3 + + + 2 2 2 s 4 4 2 1 π erf (s ) e − s + s + s 2
π erf (s )
(7.5.21) For a monatomic gas j = 0 and j =2 for a diatomic gas. Cylinder For the case of a cylinder having a constant surface temperature, the energy transport equations for an elemental area can be applied and integrated over the cylinder surface to get the equilibrium temperature. The elemental area chosen for integration is similar to the case of momentum transfer. The result is
Tweq Ti
s2 4 2 7 j s2 j 5s2 j s2 + I o s + s + + 2 + + I 1 s 4 + 2 2 2 2 2 2 1 2 = j s2 2 s2 2 2+ I o s +1 + I 1 2 s 2 2 2 (7.5.22)
(
)
where, Io and I1 are modified Bessel functions of first kind, zero and first order respectively. 7.5.2 Heat Transfer for Typical Bodies in Free Molecule Flow The two non-dimensional parameters that are normally used in heat transfer calculations are the thermal recovery factor and the Stanton number. They are defined by r =
Tw eq . − Ti Ttot . − Tw
(7.5.23)
Gas Dynamics of Rarefied Flows
St =
Q
(
Aρ i V i c p Tw eq . − Tw
)
(7.5.24)
where, Q is the total heat transfer, Ttot. is the adiabatic stagnation temperature, Ti is the free stream temperature, Tw is the body surface temperature, pi is the free stream density, Vi is the free stream velocity and cp is the specific heat at constant temperature and A is the total heat transfer surface area. dQ = (incident energy – reflected energy ) = dEi - dEr The thermal accommodation coefficient α from Eqn. 7.2.1 is
α =
dE i − dE r dE i − dE w
From the above dQ = dEi – dEr = α (dEi – dEw) Eqns. 7.5.7 and 7.5.16 give the expressions for dEi and dEw for a front-surface element. When dQ is zero
Tw = T eq.
Eqn. 7.5.18 gives the expression for the equilibrium temperature of a surface element. The Stanton number can be obtained by integrating the dQ expression over the total surface of the body of interest. Teq . − T w Similarly, for the thermal recovery factor. The results thus obtained for various simple shaped bodies are given by Schaff & Talbot6 and are presented below. Front Face of a Flat Plate at an Angle of Attack α
r =
γ
(γ + 1)s
2
2 2s + 1 − 1+
π ( s sin α )[1 + erf (s sin α )]e − (s sin α ) (7.5.25) 1
2
209
210
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
{e π
α (γ +1)
St =
4 γs
− ( s sin α 2 )
}
π (s sin α ) [1 + erf (s sin α )]
+
(7.5.26)
Rear Side of the Flat Plate at an Angle of Attack α For the case of the rear side of the flat plate exposed to the flow, the expressions are obtained by replacing α by –α in the above equations, viz., γ
r =
(γ + 1)s 2
1 2 2s + 1 − 2 1 − π ( s sin α ) [1 − erf (s sin α )]e (s sin α )
(7.5.27) St =
α (γ + 1 ) 4 γs π
{e (
− s sin α )2
−
}
π (s sin α ) [1 − erf (s sin α )]
(7.5.28) Front and Rear Surfaces of the Flat Plate It is assumed that the front and rear surfaces are in perfect thermal contact and the plate temperature Tw is the same throughout.
r =
γ
( γ + 1)s 2
2 2s + 1 − 1+
π ( s sin α ) erf (s sin α ) e (s sin α ) 1
2
(7.5.29) St =
{e π
α ( γ + 1) 4 γs
− ( s sin α )2
}
π (s sin α ) erf (s sin α )
+
(7.5.30)
Infinite Circular Cylinder at an Angle of Attack α
r =
γ
( γ + 1)s 2
2 ξ2 2 2 + ξ 2 1 + 2s 2 s I ξ + + ξ 2 1 o 2 2 ξ2 ξ 2 2 I I + ξ + ξ 1 1 o 2 2
[
(
)]
(
)
[ (
)] I 1 ξ2
2
(7.5.31)
Gas Dynamics of Rarefied Flows
St =
(γ + 1 )
α
4γs π
e
−
ξ2 2
ξ2 ξ2 2 2 + ξ I1 1+ ξ I o 2 2
(
)
(7.5.32)
ξ2 ξ2 and I 1 are modified Bessel where, ξ = s sin α and I o 2 2 functions of the first kind, zero and first order respectively. The area considered being the total curved surface area π L d where L is the length of the cylinder. Sphere 2 2s + 1 γ r = γ +1 s2
(
−1 erf ( s ) 2s 1 1 ( ) erf s 1 + ierfc (s ) + 2 s 2s
) 1 + 1s ierfc (s ) + 2s
2
2
(7.5.33) St =
α
( γ +1 ) 8γs
2
2 s + s ierfc ( s
)+
1 2
erf
( s )
(7.5.34) where, ierfc = ∫ ∫ {1 – erf (s)} dA is the complimentary error function integrated over the total heat transfer area, πd 2. Cone at an angle of attack α Only the conical surface considered neglecting the base.
Q = L 2 α ρ i R Ti +
R Ti tan δ 2 π cos δ
π
∫ o
γ γ + 1 Tw 2 s + − γ − 1 2 ( γ − 1 ) Ti
π (s sin θ ) [1 + erf (s sin θ )] −
2 e − ( s sin θ ) 2
− ( s sin θ )2 e
1
(7.5.35)
where, L is the height of the cone, δ the semi-vertex angle of the cone and sin θ = – cos δ cos ϕ sin α + cos α sin δ
211
212
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
Integrating the above and substituting the value in the Eqn. 7.5.24, the Stanton number can be obtained. When there is no heat transfer, by equating Q = 0, the equilibrium temperature is obtained, as Tw will be equal to T w ( eq .)
Tw ( eq . ) Ti
=1 + r
( γ − 1) (7.5.36)
γ s2
From the above relation the recovery factor can be determined. 7.5.3 Heat Transfer In Slip & Transitional Flow Regimes Due to the complexity of interactions between the viscosity, rarefaction and compressibility, it has not been possible to theoretically analyze the heat transfer parameters in the slip and transitional flow regimes, although some approximate theories have been advanced for limited cases particularly in the slip flow regime. The Navier-Stokes equations with modified boundary conditions for slip and temperature jump are the best theoretical approach in the slip flow regime. Only the computer oriented Direct Simulation Monte Carlo technique is able to give the aerodynamic and heat transfer parameters all the way from free molecular to continuum. For approximate preliminary analysis, recourse is made to empirical relations. One such relation is listed below. The near free molecular heat flux (qnfm ) to a surface is given in terms of the free molecular heat flux by the following relationship described by Caruso and Naegeli12
q nfm q fm
Tw = 1 + 2 T∞
1 − 0.1414 s ∞ 2 s∞
T∞ Tw
12
1 Kn ∞ (7.5.37)
where the subscript nfm refers near free molecule or transitional flow conditions. For the recovery factor in the slip and transitional flow regime the suggested formula is
Gas Dynamics of Rarefied Flows
r =
rc + Kn r fm 1 + Kn r fm
(7.5.38)
where, rc is the recovery factor in continuum flow and rfm the recovery factor in free molecule flow. REFERENCES 1. 2. 3.
4.
5.
6.
7.
8.
9.
10.
11.
Tsien, H.S. Superaerodynamics mechanics of rarefied gases. J. Aero. Sci. 1946, 13(12). Stalder, J.R. & Jukoff, D. Heat transfer to bodies travelling at high speed in the upper atmosphere. NACA, 1949. Rep. 944. Stalder, J.R.; Goodwin, G. & Creager, M.O. A comparison of theory and experiment for high speed free molecule flow. NACA, 1950. TN 2244. Stalder, J.R. & Zurik, V.J. Theoretical aerodynamic characteristics of bodies in free-molecule flow field. NACA, July 1951.TN- 2423. Schaaf, S.A. & Chambre, P.L. Flow of rarefied gases, Section H, In Fundamentals of Gas Dynamics, Vol. III. High Speed Aerodynamics and Jet Propulsion, Princeton University Press, 1958. Schaff, S.A. & Talbot, L. Mechanics of rarefied gases In Handbook of Supersonic Aerodynamics, Section 16. NAVORD, 1959. Report 14885. Sentman, L.H. Free molecule flow theory and its application to the determination of aerodynamic forces. Lockheed Missile & Space Company Report, October 1961. LMSC - 448514. Lott, R.A. Aerodynamic characteristics for the saturn SA-6 vehicle and the SA-5 after orbital breakup. Lockheed Missiles and Space Company, Huntsville, Ala. Feb. 1964. LMSC/ HREC TM 54/01-44. Blanchard, R.C., Rarefied flow lift-to-drag measurements of the shuttle orbiter, Presented at the 15th ICAS Conference, London, 1986. Paper No. ICAS –86-2.10.2. Potter, J.L. Transitional, hypervelocity aerodynamic simulation and scaling. AIAA 20th Thermophysics Conference, Williamsburg, VA, 1985. Potter, J.L. Procedure for estimating aerodynamics of three dimensional bodies in transition flow. In Rarefied Gas Dynamics. Edited by Muntz, E.P., Weaver, D.P. & Campbell, D.H. Progress in Astronautics and Aeronautics. Vol. 118.
213
214
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
12.
Caruso, P.S. Jr. & Naegeli, C.R. Theoretical and empirical low perigee aerodynamic heating during orbital flight of an atmosphere explorer. Proc. of the 1976 Heat Transfer and Fluid Mechanics Institute, Stanford University Press.
APPENDIX
THE MARK N SUPERSONIC HYPERSONIC ARBITRARY BODY The Mark IV Supersonic-Hypersonic Arbitrary Body Program known as SHABP or HABP for short, is a digital computer program package that is capable of calculating the supersonic hypersonic aerodynamic characteristics of complex arbitrary 3-dimensional shapes. This program was developed by A.E. Gentry and his group at Douglas Aircraft Corporation. In literature it is sometimes referred to as Gentry's Program. Complete documentation of the work is presented in three volumes as Wright-Patterson Air Force Base Flight Dynamics Laboratory, Technical Reports, (AFFDL-TR-73-159, Vols. 1,2 and 3 , November 1973). The outstanding features of this program are to quote from its documentation page: "Itsflexibility in covering a wide variety of problems and the multitude of program options available. The program is a combination of techniques and capabilities necessa ry in performing a complete aerodynamic analysis of supersonic and hypersonic shapes. These include: vehicle geometry preparation: computer graphics to check out the geometry; analysis techniques for defining vehicle component flow field effects; surface streamline computations; the shielding of one part of the vehicle by other; calculation of surface pressures using a great variety of pressure calculation methods including embedded flow field effects; and computation of skin friction forces and wall temperature. Although the program primarily uses local-slopepressure calculation methods that are most accurate at hypersonic speeds, its capabilities have been extended down into supersonic speed range by the use of embedded flow field concepts. This pennits the first order effects of component interference to be accounted for." The functional organization of th; program is given in Fig. A. 1. The program is written in Fortran.
2 16
Aerodynamic Predictive Methods and their Validationin Hypersonic Flows
Based on the program formulation and listing given in the above mentioned AFFDL reports, the aerodynamics group at DRDL, Hyderabad, has replicated the program and made it operational so that it could be used for its in-house work. To test that the replicated program is a copy of the original SHABP program of Gentry et al., and is capable of giving the desired results, a test case was chosen. Wind tunnel measurements of normal force and pitching moment coefficients of body alone, wing alone and wing body combination at Mach No. 6 and their comparisons with theoretical predictions have been reported in NASA TP 2467. Figure A-2 gives details of the model tested and Figures A-3 to A-8, the measured test results and the comparison of the same with the theories. In these figures three theoretical curves are drawn and compared with the experimental data. The theoretical curves are: (a) Theoretical curve from the earlier version of SHABP (known a s the Mark I11 version) a s given in the NASA TP 2467. (b) Theoretical curve a s calculated from the DRDL program which is a duplicate of Mark IV version. (c) Theoretical curve from the DRDL developed program based on the tangent cone method for the body and tangent wedge for the wing. Close agreement with the Mark I11 version results and the DRDL developed SHABP based on the original Mark IV version of SHABP shows that the DRDL replicated programme is a true duplicate of the original version and could be used with confidence for preliminary aerodynamic analysis. Figures A1-A8 are given on the following pages.
EXECUTIVE PROGRAM SERVICE ROUTINES, ERROR HANDLING, ETC.
READ SURFACE PROPERTY DATA CALCULATE STREAMLINE TRAJECTORIES
CALC FRIC
INTEG FRICT
SURFACE PRO· PERTY DATA SAVE
SKIN FR FORCE INTEGRA
FORCE DAT
.
Figure A-l. Functional organisation of Mark IV progr
2 18 Aerodynamic Predictive Methods a n d their Validation in Hypersonic Flows
Figure A-2. Wing body confirmation as given in NASA TP-2467
Appendix 21 9
7 SHABP (DUD\)
-0.04 - - -
I -0.08
I
-15.0
I
-1d.O
I
I
I
I
I
I
-5.0
0.0
I
I I
5.01
ANGLE OF ATTACK (deg)
Figure A-3.Normal force coemcient (body alone) at Mach 6.0
10.
220 Aerodynamic Predidive Methods and their Validationin Hypersonic Hows
ANGLE OF ATTACK (deg)
Figure A-4. Pitching moment coefficient (body alone) at Mach 6.0
Appendix 221
0.20
----
I I 0I
4
---h&kFor---Te-I I I I I I
-I-
- - -f-
1 I I
EXPT (TP246Y) SHABP (TP 2487)
- - - - ySHxB~D(g-qDjj ---T---7---- t T C W (DRDL)1
0.10-
- - - -k-----+--
0.00-
2
-
-0..20
0 I
-15.0
I I I -10.0
I I I
I -5.0
I
i I
I 0.0
1
I
I
I
I
I
I
5.0
ANGLE OF ATTACK (deg)
Figure A-5. Normal force coemcient (wing-alone)at Mach 6.0
I
10.0
222 Aerodynamic Predictive Methods and their Validationin Hypersonic Flows
.---
-----I----
I
I
I -
MACH NO 6 . 0 ~ -
-7----7
I
I EXPT ( ~ 2467) h
I I I
-I I
I I
ANGLE OF ATTACK (deg)
Figure A-6. Pitching moment coemcient (wing-alone)at Mach 6.0
Appendix 223
I I
I I - - - - - _I
r
I MACH NO 6.0 I I I
I
SHABP ( D R D ~ )
I
T P n V (DRDL) I I
I I
-6-7
ANGLE OF ATTACK (deg)
Figure A-7. Normal force coemcient (wing-body)at Mach 6.0
I
INDEX A
C
Adiabatic wall 39 Aerodynamic preliminary analysis II (APAS II) 109 heating at hypersonic speeds 67 performance 3 preliminary analysis II (APAS II) 109 Aerodynamics 175 body-alone 34 wing alone 45 of rarefied gas flows 175 Ailerons 156 Analysis drag buildup 96 APAS 97 code 156 APAS II 109 Aspect ratio 107 Asynchronous elevon deflection 150 Axial pressure coefficient 40
Circular cylinder 45 Compressibility factors 39 Compressible turbulent flow 62 Compression flow 46 Computation of aerodynamic coefficients 98 Cone 16, 211 frustrum 195 Convective heating equation 83 Cylinder 42, 208,
B
Embedded Newtonian flow 8 Empirical equations 65 Energy transfer 203 Enthalpy 68, 73, 74 recovery 70 stagnation 70 empirical 65 Expansion 46
Base pressure coefficient 39 Bessel function 193 Bessel functions 193 Blast wave 18 Boundary layer transition 86
D Dahlem–Buck empirical 109 method 17 Delta wing emperical method 14 Direct simulation monte carlo technique 176 Double circular arc 56
E
226
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
F
Isentropic expansion 11
Flow regimes 176 Fluid dynamic simulation 201 Force axial 37, 190, 197 normal 41, 50, 188, 190 Free molecule flow 176, 203
K
G
Mach number 14, 97 Maxwellian distribution function 183 equilibrium velocity distribution function 179 Method Dahlem–Buck empirical 17 delta wing emperical 14 Hankey flat surface empirical 17 hypersonic impact 96 nonlinear CFD 168 OSU blunt body 17 panel 95 prediction method 95 Quinn & Gong 80 second-order shock-expansion 128, 138 Sommer & Short 62 Spalding & Chi 64, 98,108, 109 subsonic panel 156 tangent wedge, tangent cone 14 tangent-cone empirical 96 tauber 87 Van driest-II 63 vortex lattice 95 MISLIFT 97 MNT 7 Modified Newtonian theory 7 circular arc 56 double wedge 55 Monte carlo technique 200
Gentry program 107 Gentry’s hypersonic arbitrary body aerodynamics 139 program (HABP) 109
H HABP 101, 108, 168, 215 predictions 158 Hankey flat surface empirical method 17 Heat flux 69 transfer analysis 80 transfer coefficient 75 transfer methodology of Tauber 87 Heating analysis 67 rates cone flat plate 89 Hemisphere 45 Hemispherical nose 42 HYFAC 101 Hypersonic impact methods 96 isolation principle 107 Mach numbers 110 rarefied transitional flow 202 speeds 67 research airplane 95, 110, 139
I Inclined cone 109 Inviscid zero lift drag 108
Knudsen number 175
L Laminar flow 66, 78
M
Index
N Newtonian & Prandtl-Meyer model 10 Newtonian theory 6 North American X-15 110
O OSU blunt body method 17
P Pan air 97 Panel method 95 Pitching moment 52 coefficient predictions 149 characteristics 119 Planform area 107 Pointed cone 40, 41, 44 ogive 42, 44 Potential flow theory 110 Prandtl number 71 Prandtl-Meyer expansion 108 flow 96 Prediction methods 95 Preliminary design analysis 158 Program supersonic implicit marching (SIMP) 108
R Ramp surface 9 Real time heating analysis 80 Recovery enthalpy 70 Reynolds analogy factor 77 number 65 Rolling moment 149 Rudder 51
S SHABP 215 Shuttle orbiter 109
SIMP 108 Single parabolic arc 58 wedge 55 Skin friction coefficient 37 forces 61 Slip & transitional flow regime 212 flows 200 Solar radiation 68 Sommer-Short estimate 108 skin friction estimate 108 Space shuttle orbiter 156 configuration 156 Spalding & Chi method 64 Sphere 194, 208 Spherical nose 69 segment 198 Stagnation density 71 enthalpy 70,74 line heat transfer 72 point 71 heat transfer 69,88 heating rate 80 temperature 71 Stanton number 77 Subsonic flow analysis 95 Sutherland law 71 Sutherland relation 79 Swept infinite cylinder 88 Swept wing 72
T Tangent cone 101 wedge 109 Tangential accommodation coefficients 186 Taper ratio 107 Theory 2-D airfoil 25
227
228
Aerodynamic Predictive Methods and their Validation in Hypersonic Flows
theory in hypersonic flows 25 Allen & Perkins viscous cross flow 43 first order 18 linear 108 linear supersonic 96 modified Newtonian 7, 34 Newtonian 6 potential flow 110 Prandtl-Meyer 98 expansion 10 second order shock expansion 19, 168 SOSET 19 Shock expansion 18 tangent-cone 98 unified supersonic-hypersonic small disturbance 128 Van dyke unified 24 small disturbance 116 Thickness ratio 107 Transitional flow 176
Turbulent flow 66, 78
U Unified distributed panel 109
V Validation 95 Vortex lattice method 95
W Walker and Wolowicz 115, 138 Wedge 16 Wing cone configuration 107 leading edge variations 107 sections 52
X X-15 118, 168
Y Yawing moment 138, 149
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