GUGGENHEIM AERONAUTICAL LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY ...

110 Pages·2015·13.29 MB·English

RESEARCH REPORT
GUGGENHEIM AERONAUTICAL LABORATORY
/
CALIFORNIA INSTITUTE OF TECHNOLOGY
?'
HYPERSONIC RESEARCH PROJECT
Memorandum No. 40
June 25, l957 APR ?, 1958
INVESTIGATION- OF FLOW
AROUND SIMPLE BODIES IN HYPERSONIC FLOW
by
T osh i Kubota
ARMY ORDNANCE CONTRACT NO. DA-04-495-0rd-19
GUGGENHEIM AERONAUTICAL LABORATORY
CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California
HYPERSONIC RESEARCH PROJECT
Memorandum No. 40
June 25, 1957
INVESTIGATION OF FLOW
AROUND SIMPLE BODIES IN HYPERSONIC FLOW
by
Toshi Kubota
ARMY ORDNANCE CONTRACT NO. DA-04-495-0rd-19
Army Project No. 5B0306004
Ordnance Project No. TB3-0118
OOR Project No. 1600-PE
ABSTRACT
A theoretical analysis of the flow aro.und slender blunt-nosed
bodies was made by applying the flow similarity concept to the hyper-
sonic small-disturbance equations. The flow field around a class of
bodies of the form rb....., xm exhibits a certain similarity in the sense
that the pressure, density and transverse velocity are described by
=
relations of the form Q(x, r)/Q(R) f(r /R), where R is the distance
from the axis to the shock wave. This similarity holds when the
Mach nuinber is infinitely large, and when the exponent in the equation
i
defining the body shape lies in the range < m ~ 1 for axially-symmetric
bodies and in the range 2/3 < m ~ 1 for two-dimensional bodies. For
large but finite Mach numbers a second approximation was obtained
2 2
by expanding solutions in series of powers of xZ(l-m)/M 0 •
00
An experimental investigation of the flow around 11 similar-
flow" bodies of revolution was conducted at Mach number 7. 7 in the
GALCIT hypersonic wind tunnel. The surface pressure distributions
agreed closely with the theoretical predictions, after a simplified
correction was made for the boundary-layer displacement effect.
The results indicated that the boundary layer interaction effect needs
a further investigation.
iii
TABLE OF CONTENTS
PART PAGE
Acknowledgments ii
Abstract iii
Table of Contents iv
List of Figures vi
List of Symbols viii
I. Introduction 1
II. Theoretical Investigation 6
A. Equations of Hypersonic Small Disturbance
Theory for Two-Dimensional and Axially
Symmetric Flow 6
B. First Approximation: Similar Flow 8
1. Similar Flow Field 8
2. Drag and Energy 11
3. Mathematical Study of Similarity Solution 15
4. Solution for Constant-Energy Case 19
5. Solutions by Numerical Integration 22
C. Successive Approximations 24
1. Expansion Procedure 24
26
2. Second Approximation
III. Experimental Study 31
A. Description of the Experiment 31
1. Models and Equipment 31
2. Test Procedure 33
3. Data Reduction 34
iv
4. Discussion of Accuracy 37
B. Results and Discussion 37
1. Hemisphere-Cylinder 37
2. "3/4- and 2/3- Power Bodies" 41
3. Paraboloids of Revolution 43
4. Foredrag Coefficient 45
5. Boundary Layer Effect 46
IV. Summary of Results 49
A. Theoretical Investigation 49
B. Experimental Investigation 50
1. Hemisphere-Cylinder 50
2. 11 Power-law'' Bodies 50
3. Paraboloids of Revolution 51
References 52
Tables 54
Figures 56
v
LIST OF FIGURES
NUMBER PAGE
1 Typical Model Installation 56
z Schlieren Photograph o£ Hemisphere Cylinder 57
3 Schlieren Photograph of "3/4-Power" Body 58
" Schlieren Pho~ograph of "2/3-Power" Body 59
5 Schlieren Photographs of "1/2-Power" Body 60
6 Hypersonic Inviscid Flow over a Blunt•Nosed
Slender Body 61
7 Integral Curves in s-t Plane, k =- 0 62
Sa Solutions in s-t Plane, k =- 0 (Schematic) 63
=
8b Solutions in the Physical Plane, k 0 (Schematic) 64
9 Velocity Function 65
10 Pressure Function 66
11 Density Function 67
12& Shock Location 68
12b Shock Location 69
13a Pressure Coefficient 70
13b Pressure Coefficient 71
14 Details of Typical Model Construction 72
15 Test Section Mach Number as Function of
Supply Temperature 73
16 Pressure on Hemisphere-Cylinder 74
17 Shock Shape for Hemisphere-Cylinder at Mach
No. 7. 7 75
18 Impact Pressure Profile for Hemisphere-Cylinder 76
19 Flow Quantity Profiles for Hemisphere-Cylinder 77
20 Static Pressure Distribution for Hemisphere
Cylinder 78
2la Surface Pressure Distribution 79
2lb Surface Pressure Distribution 80
2.2.a Shock Wave Location 81
22.b Shock Wave Location 82
23 Impact Pressure Profiles 83
24 Density Profile from Similarity Theory 84
25 Static Pressure Profile from Similarity Theory 85
26 Flow Quantity Profiles 86
27 Surface Pressure Distribution 87
28 Shock Wave Location 88
29 Impact Pressure Profiles 89
30 Flow Quantity Profiles 90
31 Foredrag Coefficients of Test Bodies 91
vii
LIST OF SYMBOLS
a speed of sound
a coefficient in expression for shock shape
1
c specific heat at constant volume
v
i 2
CD drag coefficient, D/ ( p u (1r/4 )k d l+kJ
00 00
AI 2
Cp pressure coefficient, (p- p )tlz p u )
00 00 00
d diameter, or thickness
D drag
E · energy
f, F, f function of z appearing in expression for pressure
1
k geometric index, 0 for two-dimensional flow
1 for axially-symmetric flow
L body length
m
m exponent, rb,..., x
M Mach number
p pressure
Q any physical quantity
r distance normal to body axis or chord line (x- axis)
R distance of shock wave from x- axis
L/f'
ReL Reynolds number, p u
00 00 00
s distance along body surface measured from forward
stagnation point
,..,
2
s distorted distance = (s p LJ u r k ds
Jo e r e e
t time
T absolute temperature
u,v velocity components parallel and normal to x- axis
X distance along body axis or chord line measured from forward
stagnation point
viii
y r- rb
-;-m
z r x
o. exponent, (1 - m)/m
j3 shock wave angle
pressure gradient parameter,
1
ratio of specific heats
6 thickness ratio of bodyi or shock shape parameter
6* viscous boundary layer displacement thickness
g angle between x- axis and tangent to body surface; also flow
deflection
g c semi-vertex angle of cone
g shock wave angle
w
fA
viscosity coefficient
p density
t'jJ, 4'. ~ functions of z appearing in expression for velocity
t.f, cf; functions of z appearing in expression for density
( )b quantity evaluated at body surface
( )e quantity evaluated at edge of boundary layer
( ) quantity evaluated at shock wave
8
( ) quantity evaluated at free stream
00
(l
non-dimensionalized quantity
ix
I. INTRODUCTION
The flow around a slender body traveling in a stream of a gas
at a uniform very high speed may be adequately described by the
hyper sonic small disturbance theory. The basic idea of this approxi-
mation is that the flow perturbation along the stream caused by the
body is an order of magnitude smaller than that transverse to the
stream. This fact may be deduced from a study of shock wave
relations.
The relation between Mach number M, wave angle ~. and
deflection angle Q for an oblique shock wave may be written in the
form
1
2 . 2 = 'I + 1 sin sin Q
M s1n ~ - 1 2 cos ~ - Q)
For small values of Q, and large values of M, such that MQ ~ 1, ~is
also the same order of magnitude as Q for a branch of the solution
usually called that for weak waves (the part of the shock polar curve
lying outside,the sonic circle), and hence the above relation reduces to
. 't + 1
=
2
or
! = 'I + 1 +
4
The components u, v of the velocity behind the wave are related to the
upstream velocity u by
1
2 2
2(M sin ~- 1)
= 1 -
2
( ~ + 1) M