Head-on interaction of oblique shock waves

::0 ₀ C» "'C

<

"'C

.

m ;iOO (1)0 m() C ;iO~ >m ""C Z _{- i} ""C> O~ ;iOm

TECHNISCHE

HOGESCHOOL DElFT

VLiEGTUI-';BOtJVI" r .DE Michiel de [ Ilyferweg 10 - OLlFT

7

19

1

HEAD-ON INTERACTION OF OBLIQUE

SHOCK WAVES

by

"

Sannu Molder

SEPT. 1960 TECHNICAL NOTE NO. 38

... rT'I t"'"'I

<~

r- V'\ - t"'"'I m:r ~rT'I c _{- : r}

°0

o:JG"'o 0"", C Vt :Er-. ",,:J: c a z a 0 " ' " mo rn

,....

_..,.., -4

SEPT. 1960

HEAD -ON INTERAC TION OF OBLIQU E SHOCK WAVES

by

"

Sannu MOlder

ACKNOWLEDGEMENTS

The author wishes to thank Dr. G. N. Patterson for the

opportunity to pursue this investigation.

The problem was suggested and subsequently supervised

by Dr. 1.1. Glass. Ris numerous helpful suggestions are gratefully

acknowledged.

This project was supported by the Defence Research

SUMMARY

When two oblique shock waves that face in opposite directions intersect in a uniform supersonic floW, they give rise in the case of a

regular interaction, to a pair of recedin~ shoçk waves,which are separated by a contact surlace or slipstr~am. Such shock interactions have been investigated analytically and exp~rimentally. Exact solutions of the regular flow field, for ~ perfect, inviscid gas, based on the

Rankine-Hugoniot equations, have been compute~ on an 1. B. M. 650 Digital Computer, in the Mach number range of 1. 0 to 5.5. It was found that the exact theory was weU substantiated by thEt experim.ental results, which were obtained in the UTIA 16 ~n. x 16 in. Sup~rsonic Wind Tunnel.

.~

( i )

TABLE OF CONTENTS

NOTATION

1. INTRODUCTION

2. THEORETICAL ANALYSIS OF REGULAR SHOCK INTERACTIONS

2. 1 Mathematical Model 2. 2 Governing Equations

2. 3 Exact Solution of Governing Equations 2.4 Graphical Solution of Governing Equations 2. 5 An Approximate Solution

2. 6 An Empirical Solution

2.7 Factors Affecting the Contact Surface Deflection

Page ii 1 1 5 6 6 14 15 17 18 3. EXPERIMENTAL APPROACH 18

3. 1 Experimentation in the UTIA 5" x 7" Supersonic 18 Wind Tunnel

3. 2 Experimentation in the UTIA 16" x 16" Supersonic 18 Wind Tunnel

4. RESULTS AND DISCUSSIONS 20

4. 1 Measure.ments 20

4. 2 Errors 21

5. CONCLUSIONS 24

( ii ) NOTATION

No'

C>

- - - - t

&

flow deflection angle

e

shock wave angle

€ contact surface deflection angle

M Mach number

p

statie pressure

F

density

T

statie ternperature

-~

Subscripts o 1, 2, 3, 4

*

D

**

DD s w ( iii )

initial free stream flow conditions

variables for regions 1, 2, 3, and 4 respectively

deflection angle at which the flow behind the

reflected shock become s sonic.

deflection angle at whi,ch the reflected shock detaches

i. e. Mach reflection starts

deflection angle at which the flow behind the incident shock becomes sonic

deflection angle at which the incident shock detaches

strong shock solution weak shock solution

-1-1. INTRODUCTION

Ernst Mach (REi. 1) first recognized two possible con-figurations of shock reflections. Regular reflection exhibits supersonic flow everywhere, with straight shocks bounding regions of constant flow velocity, and no shear layers or slipstreams. The other type of re-flection, Mach Rere-flection, exhibits mixed supersonic arrl subsonic flow regions, curved shocks bounding regions of variabIe flow velocity and a shear or slipstream layer.

No theory has yet been found to satisfactorily describe

Mach interactions in steady or unsteady flow. Existing theories are

wanting, especially for weak interactions (Ref. 2, 3). Experimentally, Mach interactions in shock-tube flow have been weU surveyed (Refs. 2,

3, 4).

The theory of the regular reflection process in one-dimen-sional flow, as developed by von Neumann (Ref. 5) and Polacheck and

Seeger (Ref. 6), has been verified by Smith's shock-tube measurements

(Ref. 2). Two-dimensional interactions of equal shock waves have been investiga ted by Lean (Ref. ,7) f9r both strong and weak shocks in a steady flow.

The present report covers the interaction of unequal oblique shock waves in a steady flow at a free-stream Mach number Mo

=

2.48. Previous quantitative experiments of this type do not appear to have been done.

2. THEORETICAL ANALYSIS OF REGULAR SHOCK INTERACTIONS

Shock Reflection

Consider a compressive corner (or wedge) placed in a supersonic flow (Fig. 1). The corner creates a shock wavE1, which:

1) deflects the flow parallel to the deflected wall, 2) changes the flow properties behind the shock.

(

-2-Fig. 1 Compression of Supersonic Flow by

Turning Through an Angle ~

If now the shock impinges on a wall, which is parallel to the original flow (Fig. 2) then the supersonic flow in region 1 will re gard the wall as another compressive corner, This causes a shock to be reflected from the wall tUrning the flow back parallel to the wall (Ref. 8).

/ '

-3-Acoustic reflection can not occur at the wall unless

S'I

·

is infinitesimal.

If at a given Mach number Mo the deflection angle

SI

is increased we

find that the shock strengths increase from that of a Mach wave until at

a deflection ~It- the Mach number behind the reflected shock Jie:coines

unity, and the reflected shock becornes curved. If the flow is deflected

furiher to

Sc

then the reflected shock detaches and Mach reflection

results. The difference between ~It and ~D is very small, usuallyof

the order of O. 5 degrees (Ref. 9). A further increase in

b

to ~u·

results in a curved incident shock with subsonic flow behind it, and then

as

b

increases to

b

DD , the incident shock detaches from the corner.

Figure 3 shows ~* ,

So'

~k-lt" and ~I>D as functions of Mo (Ref. 9).

Ll.O

Ma

3.0 1 - - - + -2..0 -, O ~ ________ L -_ _ _ _ _ _ _ _ L -_ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ ~ _ _ _ _ _ ;) IC) 2.0 \. (J ) o ' .... iLCj. ~,-l ₄₀

Fig. 3 Upstream Mach Number for: (A) Regular Reflection, (B) Subsonic Flow Behind the Reflected Shock,

(C) Mach Reflection, (0) Non-Uniform Flow Behind Incident Shock, (E) Oetached Incident Shock

Figure 2 is an idealized picture of shock reflection. In reality, es-pecially for strong shocks and high Mach numbers (Ref. 8), the wall boundary layer modifies the reflected shock structure. The interaction with the boundary layer is shown schematically in Fig. 4 (Ref. 8, see Fig. 29 for an illustration from the present experiments).

-4

-d1

~

"

---.

~c.OIv\PRc:.SS\ON

~ L - - - WAVES /' ____ SE:PARAT10N ~ BuBBI..t: BOUNDARY LAye; " , "" , / " 77777////7/ ;;;/7/7//71;;////7/ ///7llll/l11

/ll/Ilill

Fig. 4 Shock-Wàve Boundary Layer Interaction Shock Interaction

The foregoing description of shock reflection applies also, with slight rnodifications, to shock interactions. Consider two equal shock waves, originating from oppositely placed wedges (Fig. 5), pro -ducing a regular interaction at B. The flow is similar above and below

c

M

_o c::=:=!>

Fig. 5 Interation of Two Equal Oblique Shocks

the plane ABC, and ABC rnay thus be replaced by a plane of syrnrn etry or a wall. The shocks may now be considered as reflecting instead of interac.ting at the point B. An important difference between the re-flection of shock waves from walls and the interaction of equal shock waves is that the latter is free frorn viscous effects at the point of reflection (interaction). . .. " _. , . -J

-5-However, if the incident shocks are of unequal strength, then their reflected waves must be such as to make the flow directions and static pressures in regions 3 and 4 equaL These two boundary

conditions, when applied to the Rankine-Hugoniot equations, fprrn the

basis for the theoretical treatment of regular shock interactions. For

unequal incident shocks ABC is no longer a plane of symmetry but BC

is inclined to AB at an angle E (Fig. 6). The angle E. thus represents

the deflection of the initial free-stream flow as it passes through the

shocks. The flow direction and static pressure above and below

Be

are

equal; but, in general, the Mach number velocity, and density are not.

Fig. 6 Interaction of Unequal Oblique Shocks

The discontinuity in the flow speed across BC gives rise to a high

vor-ticity layer called a vortex sheet or slipstream. The density discontin

-uity will make t1'1= vortex sheet visible to schlieren photogr~phy (Ref. 8).

I

The interaction of unequal shocks may still be considered

as a reflection problem, the only difference being that at the point of

reflection (or impingern ent) the wall (in tl?-is case the streamline through

the interaction point) is deflected towards or away from the upstream

flow.

2. 1 Mathematical Model

If two shock wa yes are made to interact then there results,

in general, two reflected shocks and a slip surface (contact surface,

vortex sheet or slipstream). Figure 6 shows a Mo = 2. 48 flow being

deflected by '~I

=

80 _and

_è,

₌

₁₆0

. The resulting shock waves interact

at B producing two .reflected shocks and a contact surface. The flow

directions in regions 3 and 4 are equal and parallel to BC and ther e is

no pressure difference across BC. These two conditions, when

ap-plied to the oblique shock relations, form the basis for the mathematical

-6-2. 2 Governing Equations

Consider the supersonic flow pictured in Figure 6. The known quantitatives are ~, ) ~z. and Mo; from these we may find MI, M2,

9, and

ez..

(Ref. 9). If in addition E were known then all flow quam~·.

tities af ter the reflected shocks could be evaluated.

The condition of unchanged flow direction across the con

-tact surface is expressed by

~3

-

~I

+ E. (1)

(2)

and that of no pressure change across the vortex sheet by,

(3)

In a theoretical analysis of thE problem, the above three boundary con

-ditions must be met for regular interaction. It is the purpose ei. the

next sections to develop and examine several methods for finding E in

terms of ~I , ~1.. and Mo'

2.3 Exact Solution of Governing Equations

By the term "exact" we imply the application of the oblique shock equations in their exact form (as derived from the Rankine-Hugoniot equations).

Dividing equation (3) by

po

(

4)

(5)

The static pressure ratios across the incident shocks are given by (Ref. 9),

h=

2

'l(

t'-1

~ ~

'

IM..2.

el - ('( - ')

(6) 1='0 ~+l

_.

-h=

_po

7

\'v1;;-

~g, - \ for '( = 1. 4 {7 ) b and

t.=

7 M;; /;)i.1A.L.9~ - I (8)

b

-7-For the reflected shocks,

P::

7

M

,2.

~

~

"'-2.

83 -_\

F'

(,

(9)

-.h _

7

(J\~

f<n..,..L84 -

l

_

and

r~

-

b

(10)

Substituting (7), (8), (9), and (10) into Equaticn (5) we get:

(11)

7rl:

A~~8~-1

7H?',-n~93-1

On the left hand side of this equation:

i) Mo is given

ii) h.~

e,

and 1o\",l~2. can be calculated fr om the known

quan-tities Mo,

<b,

and ç;~.

Thus for a given Mach number and for given flow deflections the left hand side of Eq. (11) is fixed. On the right hand side:

i) M12 and M22 are found from Mo, and

~,

and

~l­

respectively.

ii) However

/h',}9

4 is a function of M2 and

~-4

where

~~= ç2. - ~ similarly fo\.\.o\.l..9~ is a function of Ml and ~l:Ç,+€-.

The right hand side would noW contain only

E

as the unknown, pro-vided we had found an expression for ~~\A.le in terms of M and ~ Unfortunately no such convenient relationship exists. The reciprocal relationship (Ref. 9),

c.JÇ.

=

-tOM-

g[

b

KL-

D -

IJ

(12)

S"(ML/:)~:g-I

upon rearrangement, gives the following cubic equations in A~~

e

(Ref. 9), where

3

b

= _

tJ\

'2.. + 2- _

Y

~V\..2. ~

M2-S.

C

=-

2.

M

1 -+-

I-+-[

('r

+

t

4- '(

-11 ;\

~

.. : (

~4

4

tv\'L'J

J..

-=: _

~2.~

H+

(13) (14) (15 ) (16)

-8-For the case of three real roots of a cubic,

'S L

t

+

fL

<

0 (17)

where q = c - b2 :must be negative (18)

and r

=

t

(3bc - d) - b 3 must be pOf)itive (19) The three roots of the cubic are:

~~ ~I

==

2.

~~

CV-~

Lt

«-,;1

~

] -

b

(20)

V

~-'t1:.

(kL,~e)L

=

2-

{1

c,,,>

[~

<-<_,-I

ft

..

~1-

*'

(21)

(A.,~e).

=2-/1

~G

L~\

v7

-1-

ii

-b

(22)

The smallest root (Eq. 20) corresponds to a decrease in

eritropy and must be discarded in accordance with the second law of thermo-dynamics. The largest root (Eq. 22) gives the strong shock solution,

leaving the :middle root (Eq. 21) to give the weak shock solution (Ref. 9)*.

It is ·apparent that if

/:):""t61.

were found from <Eq.(21) in terms of Ml and

~~

=-

~

I +E. then the unknown

E

would appear transcendentally in the expression for ~~~l.

&-'1

.

The same argument applies to /I)",~ e~

Upon substitution of .è~~

9

1 and ~ .... 'l e4 in :Eq. (11), this equation

will contain

é

as the only unknown. However, because of its trancen-'."

dental appearance,

f

can not be found explicitly. To overcome this difficulty, an iteration method was used in finding the exact solution to Eq. (11). .

The computer perforrned the following iterations, i) assume a value for

€.

,

ii) find

~

'

l\f:-e~

and

1ó\,2Q~

from

~~(: ~I+E)

,

M1 and

~<4(::))-~J

M2 respectively,

iii) test

~~~

91.

and

Iö~"':

e

4 in Eq. (11),

iv} change E. until Eq. (11) is satisfied.

The solid lines in Figs. 7, 8 and 9 show typical results for Mo = 2.00, 3. 00 and 5.00.

The figures are plots of f versus

~I

with

çl.

as a para-meter. The lower left region in the diagrams represents regular

inter-*Although this report does not consider the strong shock solution, it is conceivable that, for certain values of back pressure, it may occur.

22 20 18 16 k - - - ' " 141---~--3Iió.: 10 8 k---.~-6 4 k---~ 2 0 0 2 4 6 FIG. 7

_E

VERSUS

S

I -r--'~-

,-

.

_

_

.

.

;

I

i

-r---\-

-

--

-_

.

.

.

! REGION OF MACH INTERACTION 8 10 12 14

8

₁ ( degrees) AND

~2

_{FOR Mo}

=

2.00 16

E 10 -34

'"

~I

32

~,~

N~

REGION OF MACH INTERACTION 30 28 26 24 exact solution 22

~2~~r~

~ ~'\1

'"

~24f'-~

- - l i n e a r solution -_ -_ -_ empirica 1 solution 20

~~~-L

~~

. ,

~

~-~o~q-18 16 14

~,~

\

o~+{J5f! ~

1 1 6 ,"8

~

2~

_l~

·~~-I~K1\

I

18 20 22

o

2 4 6 8 10 12 14 16

8

(degrees) 1

FIG. 8

€

VERSUS

~I

AND

~2.

FOR Mo

=

3.00

4 6

o

2 14 16 18 20

12

8

(degrees) 1

FIG. 9 E. VERSUS

0l

<'

AN

D~,

C" 2. FOR M 0

=

5.00

--12 -~o ~o .;)

~:

0 Mo _{'- \}

e

MI M₂

@",

3. 000 0.000 0.000 0. 000 2. 999 2. 999 19.47 19.47 3. 000 2.000 0. 000 2.000 2. 999 2.898 20.86 20. 18 3. 000 4. 000 0. 000 4. 000 2. 999 2. 798 22. 35 20. 93 3. 000 4. 000 2. 000 2. 000 2.898 2.798 23.08 22.35 3. 000 6. 000 0. 000 6. 000 2. 999 2. 700 23.93 21. 73 3. 000 6. 000 2. 000 4. 000 2.898 2. 700 24. 67 23:_16 3. 000 6. 000 4. 000 2. 000 2.798 2. 700 25.46 24.68 3. 000 8. 000 0. 000 8.000 2. 999 2.603 25. 61 22.59 3. 000 8. 000 2. 000 5.996 2.898 2. 603 26. 36 24.04 3.000 8.000 4. 000 3. 995 2. 798 2. 603 27. 15 25.58 3. 000 8. 000 6. 000 1. 996 2. 700 2. 603 28. 00 27. 21 3. 000 10. 00 0. 000 10.00 2. 999 2.505 27. 38 23.53 3. 000 10.00 2. 000 7 _ 993 2.898 2. 505 28. 13 25.00 3.000 10. 00 4. 000 5. 985 2.798 2.505 28. 93 26.56 3.000 10.00 6. 000 3. 985 2. 700 2.505 29. 80 28. 21 3.000 10. 00 8. 000 1. 990 2.603 2. 505 30. 74 29. 95 3.000 12.00 0. 000 12. 00 2. 999 2. 405 29. 25 24. 56 3. 000 12.00 2.000 9. 983 2. 898 2.405 30. 00 26.06 3. 000 12. 00 4. 000 7.969 2.798 2.405 30.81 27. 65 3. 000 12. 00 6.000 5. 964 2.700 2.405 31. 68 29. 33 3. 000 12. 00 8.000 3. 964 2.603 2.405 32.64 31. 10 3.000 12. 00 10.00 _, 1. 980 2.505 2.405 33. 72 32. 96 3. 000 14. 00 0.000 14. 00 2. 999 2. 305 31. 21 25.70 3.000 14. 00 2. 000 11. 97 2.898 2.305 31.97 ~7. 24 3. 000 14. 00 4. 000 9. 949 2. 798 2. 305 32.78 28. 87 3. 0-00 14. 00 6. 000 7. 933 2. 700 2. 305 33. 66 30.59 3. 000 14.00 8. 000 5.931 2.603 2.305 34. 64 32.40 3. 000 14. 00 10. 00 3. 938 2.505 2. 305 35. 74 34.30 3.000 14.00 12. 00 1. 965 2.405 2. 305 36. 99 36.30 3. 000 16. 00 0.000 16.00 2. 999 2. 203 33. 28 26. 99 3. 000 16.00 2. 000 13. 95 2.898 2. 203 34. 04 28. 57 3. 000 16.00 4.000 11. 92 2.798 2. 2rl3 34. 85 30. 25 3. 000 16. 00 6. 000 9.896 2. rlOO 2. 203 35. 74 32. 01 3. 000 16. 00 8.000 7.882 2.603 2. 203 36. 73 33.88 3. 000 16.00 10. 00 5. 886 2.505 2. 203 37. 86 35.84 3. 000 16. 00 12. 00 3. 905 2. 405 2. 203 39. 16 37. 91 3. 000 16. 00 14. 00 1. 947 2.305 2. 203 40. 66 40. 09 Table 1 , Computer Output for Mo = 3.00

-13-ÇQ

ç~

..

•

9~

Mo

..

.

.

E.

M-1 :_{M 2}

el

2-

,

3.000 18.00 0.000 18. 00 2.999 2. 099 35.46 28. 44 3.000 18.00 2.00 15. 94 2.898 2. 099 36.21 30.08 3. 000 18. 00 4. 000 13.89 2.798 2.099 37.02 31. 81 3. 000 18. 00 6. 000 11. 85 2.700 2.099 37.92 33.65 t' 3. 000 18.00 8.000 , 9.825 2.603 2.099 38. 94 35.58 3. 000 18.00 10:00 7.-817 2. 505 2.099 40. 10 37. 62 3. 000 18.00 12.00 ,,5_833 2. 405 2.099 41. 45 39. 77 3. 000 18. 00 14. 00 3. 867 2.305 2.099 43.03 42. 07 3. 000 18. 00 16.00 1. 928 2. 203 2. 099 44. 92 44. 52 3.000 20.00 0.000 20. 00 2. 999 1.994 37.76 30.10 3. 000 20. 00 2.000 17.92 2. 898 1. 994 38.50 31. 81 3. 000 20. 00 4. 000 15. 85 2.798 1. 994 39.32 33. 62 3. 000 20. 00 6. 000 13. 79 2.700 1.994 40. 24 35. 54 3. 000 20. 00 8.000 11. 75 2. 603 1.. 994. 41. 27 37. 57 3. 000 20. 00 10. 00 9.737 2.505 1.994 42.48 39. 72 3.000 20.00 12. 00 7.745 2.405 1. 994 43.90 42. 00 3. 000 20. 00 14.00 5.776 2. 305 1. 994 45.59 44.45 3.000 20.00 16. 00 3. 835 2. 203 1. 994 47.64 47. 11 3.000 20.00 18. 00 1. 916 2.099 1. 994 50.22 50.09 3.000 22. 00 0. 000 22. 00 2. 999 1. 885 40. 19 32.02 3.000 22.00 2. 000 19. 90 2.898 1.885 40.93 .33. 83 3.000 22.00 4. 000 17.81 2.798 1. 885 41. 75 35.74 3.000 22.00 6.000 15.73 2.700 1. 885 42.69 37. 77 3. 000 22.00 8. 000 13. 68 2. 603 1. 885 43.77 39. 93 3.000 22. 00 10. 00 11. 65 2.505 1. 885 45. 03 42.23 3. 000 22.00 12. 00 9. 648 2.405 1. 885 46.55 44.71 3. 000 22.00 14. 00 7. 677 2. 305 1. 885 48.40 47. 41 3. 000 22.00 16.00 I 5.741 2.203 1. 885 50. 72 50.43 3.000 22.00 18. 00 3. 833 2,099 1. 885 53.81 54. 00 3. 000 22.00 20. 00 1. 947 1. 994 1.885 58.62 58.98 3. 000 24.00 0. 000 24.00 2. 999 1. 774 42.77 34.30 3. 000 24. 00 2. 000 21. 87 2. 898 1. 774 43.51 36.23 3. 000 24.00 4. 000 19.76 2.798 1. 774 44.35 38.28 3. 000 24.00 6.000 17.66 2.700 1. 774 45.32 40.47 3. 000 24.00 8. 000 15.59 2.603 1. 774 46.46 42.82 3. 000 24. 00 10.00 13.55 2.505 1. 774 47.82 45.36 3. 000 24.00 12.00 11. 55 2.405 1. 774 49.50 48.14 3. 000 24.00 14. 00 9.598 2. 305 1. 774 51. 64 51. 27 3. 000 24.00 16.00 7.691 2. 203 1. 774 54.52 55.03 3. 000 24.00 18.00 5.869 2.099 1. 774 59. 25 60.50

Table 2 (Continued) Table of Computer Output for Mo = 3.00

action,· and the upper right represents Mach interaction. It may be noted that if, at a given Mach number,

ç

I is made to approach ~ 2

-then €. beco.mes smaller; also that if ~ \ ~ 0 th en E. '"' ç~ and if ~ 1 =- 0 then E.:~,; the latter two cases expressing He regular interaction of a

shock and a Mach wave. Furthermoreif ~,=~~ then ~ : o;this represents

the interaction of two equal shocks. Table 1 shows the computer output

for Mo = 3. O. Column three contains ~I which is allowed to increase by two degree intervals until it is two degrees less than

è:i..

The value of

ç ..

is then increased by two degrees and the process repeated. In

column four ~ remains equal to the difference in Ç"1.. and ~ \ with an

accuracy of four significant figures up to ~z..

=

8,000- and ~ I

=

O. 000.

The accuracy then worsens until at ~~ = 24.000 _and ~, _{= 12.00}0_{, E}

equals 11.55 deg. Columns 5 and 6 show the variaticn of MI and M2. It is interesting to note that when ~ I is increased by two degrees then the corresponding Mach number is decreased by approximately 0.1. That is, for an initia 1 Mach number of 3.0 and for deflection angles ~ .... and

~ \ equal to 18 degrees and 10 degrees respectively we may expect the

corresponding values of M 2 and MI to be 3. 0 - 1. 8/2 = 2.1 and

3. 0 - 1. 0/2

=

2.5. An examination of Table 1 shows these Mach numbers

to have the va lues 2. 099 and 2.505 indicating a fair correspondence. It should be emphasized that this empirical relation:

tv1.

_...

'

:.~

_{I - I ()}

_h

_{2-0 w ere} h · 1 2 _{1 = , ,} is approximately valid only for Mo ne ar 3. Columns 7 and 8 show

values of the reflected shock angles

91

and

e

+

~ It should be noted

that the difference in g~ and

e-4

is from two to four time s smaller than the difference in 'é, land

b2...

2.4 Graphical Solution of Governing Equations

A graphical solution is outlined in Ref. (10). It makes use of a shock polar diagram of pressure versus deflection angle, which can be derived from Eqs. {6} and (12). Figure 10 shows such a plot of

tLo\.,

P'/'Po

against

~

.

s

\ "!'S~r-I(i. "!'S~OC.K$

wE:AK ~+-loc.K:S

Fig. 10 Shock Diagram of Pressure Versus Deflection Angle for a

- -- - - ,

-

15-As an example consider an initial flow, denoted by subscript zero, deflected through an angle ~ I ' From Fig. (10) we get two values of

LV\..

f';~o

'

one for the strong and the other for the weak shock. The shock interaction problem is shown in Fig. (11). Here the flow is deflected by ~I and

S:'-

to M1 and M2. Flows 1 and 2 are then again deflec.ted by <bol and g~ respectively in such a manner that

P1:'

p~

and ~4~ ~z. -

e

and ~1:: <b, +é.. The slipstream deflection is now given by

the point (3, 4). The point {_3',

4')

also fulfills the above conditions but

it represents tIE strong shock solution of shocks ~ and 4.

(0)

- - - - -

-

--è

Fig. 11 Superposition of (

~) ~

) Diagrams Illustrating UnequalShock Interation for a Given Flow Mach Number Mo.

This method is rather cumbersome since it requires one graph for each

downstream Mach number M1 and M2. and in addition it demands the

superpositioning of three graphs.

2. 5 An Approximate Solution

The difficulties encountered in attempting to find an exact

explicit solution and a convenient graphical solution for

=.

make it desirable to find a first order solution. Such a solutiCll is possible when it is based on the assumption that the shocks are weak, i. e., that

H,

ç

,and

M

~ );1- are small co.mpared to unity. For such small deflections (Ref. 9),

.h

_-

₊

tK~ ~

I -

-po

V

t-102- -

t

(23) ~I

I

+ Vi;I

~!--Çl

-

_t-1

_.

_o

-~

H

02. - \ (24)

I

i

I

•

i

82

/8.

20 I I I I I I I

I

,20/0 I

±îFtti:f

2/2 € 9 8 -< ~ 4 € 3 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

FIG. 12 VARIATION OF

Ë

WITH MO FOR SEVERAL COMBINATIONS

OF

b.

AND

b2.·

-16-The boundary conditions remain as

and

11

= p~

~l ~

ç,

-+

é

~""

c

~~

- €.

l As in the exact solution,

(25)

Substituting

E.

-This equation will henceforth be referred to as the linear solution.

Equation 28 is not really lipear i n ' but it contains

ç

2.. terms in the

numeratqr. Usual superso nic linearization considerations would start

with

instead of Eq. (23), and end with the neglection of all teI"ms of order two

and higher in .~ . Here, however, in order to find a workable explicit

solution for €. , use was made of Eq. (23). The second-order term s

were not dropped in Eq. (23), because it was found that the resulting

relation gave a poorer agreement with the exact solution. Equation (28)

was computed for several Mach numbers with ~,~ ranging from 0 to 200 ,

and the results are shown in Figs. 7, 8 and 9. The resulting values for

f.

are less than the exact values, the discrepancy increasing with

increasing Mo and ~1. . For a given Mo and ) ~ the discrepancy is

lar-gest when

bi

=

~a./1-

.

The following table gives the difference between

-17-Mo ilE.°

2. 00 8.29

3. 00 0.33

5.00 0.59

Table 3 Differences in E. Between Approxima te and Exact Solutions

The approximate solution was not carried beyond ~1..;o 2-Do because it was

noted that the so-called empirical solution E:::~, -

S; \

was becoming more

accurate than the approximate solution (Sec. 2.6).

2.6 An Empirical Solution

A v~ry simple approximation for € is given by

(29)

This empirical equation gives an

E..

which is larger than the exact €...

(See Figs. 7, 8 and 9): The accura.~y of Eq. (29) was first suspected

after a few hand calculations of €. had been performed. Later

exten-sive computer calculations showed that Eq. (29) holds very cIa:> ely for

a large range of Mach numbers and deflection angles.

Again the discrepancy increases with Mo and is largest,

for given Mo and ~ 2-when ~I: b~ . At ~~ =2...Ö and ~I

=

10° the

dis-crepancies are given in the following table.

Mo

.

~Eo

2. 00 0.01

3.00 Q. Z6

5. 00 0.67

Table 4. Differences in € Between Empirical and Exact Solution

The error in the empirical relation varies with Mo in the following

manner.

f

L

-18-Max. Error in E Mo Range

.4% Below 2.0

2% 2.0 ~ 2.6

4% 2.6

-

3.0 10% 3. 0

-

4. 0 18% 4.0

-

5. 0

Table 5. Percent Error in Empirical For mula

2.7 Factors Affecting the Contact Surface Deflection

Figure 12 shows the variation of E. with Mo for several

combinations of

8,

and

Sz

The three sets of curves are for values

of

b

2 - 81 equal to 200, 100 and 40 • It is interesting to note that for a

given

b,

and

8

2- )

(

decreases with increasing Mo, and also that for

a given Mo, 6 decreases as

6,

and

82..

are increased by equal amounts.

3. EXPERIMENTAL APPROACH

3. 1 Experimentation in the UTIA 5" x 7" Supersonic Wind Tunnel

When this project was started it was planned to perform all experiments in the UTIA 5" x 7" Supersonic Tunnel. Two shock

gen-erators or flat steel plates (Fig. 13), whose angle of attack was

contin-uously variable from the outside of the tunnel, were constructed and

subsequently mounted in the test section. Unfortunately, the pressure

gradient, induced at the wall boundary layer by the shocks from the shock generators, was strong enough to feed upstream along the wall boundary layer and thereby separate the boundary layer at a point upstream from

the shock generators. This led to the formation of separation shocks

which severely disturbed the flow at the shock generators. Several

attempts were made to overcome this difficulty but none was successful.

It was then decided to perform the experiments in the 16" x 16" Tunnel.

3.2 Experimentation in the UTIA 16" x 16" Supersonic Wind Tunnel

A description of this tunnel and its associated schlieren system is contained in Ref. (10). The Mach number of the tunnel is

statie pressure density statie temperature flow velocity

-19-= .

9 psi = 3. 30 x 10- 4 slugs/ft3 = 2430 R = 1660 ft/sec

Shocks were generated by two steel wedges mounted on .etings in the

test section (Fig. 14). The wedge angles were 100 and 250. However,

flow deflections were not limi ted to these angles but could be varied by inclining the wedges at the rotatable fittings (Fig. 15). Flow photo

-graphs were taken by a schlieren system consisting cf a mercury arc

light source, two 2411

diameter spherical mirrors, a graded filter (in

place of the usual knife edge) and a 4" x 5" camera (Ref. 10).

The experimental work was limited to a schlieren study

of the geometrie properties of the flow. Three sets of experiments

were performed at Mo = 2. 48 (See Figs. 17 to 28).

i)

~'L~

120 with

~,

increasing from 00 to 1-20 in steps of 20

ii) ~2. ~ 160 with ~t increasing from 00 to 160 in steps of 20 iii)

~2-~

200 with

ç

I increasing from 00 to 200 in steps of 20

In Fig. (17) the flow is from right to left. The upper

sur-face of the lower wedge is inclined at an arigle of 11. 5 degrees toward§

the flow, and the lower surface of the uprer wedge is inclined at 1. 0 degree towards the flow. This then r.epresents the interaction of a

shock wave. emanating from the lower wedge, with a weak shock wave

from the upper wedge. Note that the weak wave is deflected as it passes

through the shock whereas the strong shock remains straight af ter the

interaction.

In Fig. ( 8) the flow is deflected through 11. 5 degrees by the lower wedge and 5.5 degrees by the upper wedge. Here both waves are

bent by the interaction. Af ter the interaction, the wave from the upper

wedge passes through the rarefaction fan, originating at the trailing

edge of the lower wedge, and then disappears into the wake region below

the arefaction fan. Both the rarefaction fan am the shock are bent by

their mutual interaction. The dark portion of the rarefaction fan is

caused by "overloading" of the schlieren system i. e., strong density

gradients.

Figure (19) shows an interaction similar to that in Fig.

U

.

7>:

except here the deflection angles are

o.

5 and 15.6 degrees. Also in this

-20-boundary layer at the lower surface of the upper wedge (see Fig. 4). In the previous three figures t1::e density gradient across the contact sur-face was not large enough to make the surface visible in the schlieren records. Figure 20 is the first which shows traces of a contact surface downstream of the point of interaction. In Fig. (20)

S,

=

3.50 and

~ 2- = 15. 5 degrees. Figure (21) with ~ I = 9. 6 degrees and S; 1- = 15. 5

degrees, shows a more pronounc ed contact surface. Figure (22), wi th

~,

=

16.1 degrees and );1.. = 15.5 degrees pictures the interaction of two equal shocks. It is worthwhile drawing attention to a three-dimen-sional effect that causes an apparent thickening of the shock waves

especially noticable for the lower shock. The thickening may be explained as foll~ws: At the leading edge of the wedge the entire shock wave is dim ensional, but as we proceed downstrea.m along the shock, the two-dimensional central partion becomes progressively shorter as it is over-taken by the c onical tip shock. It is t1::e curvature of this conical shock that causes the seeming thickeniDg of the two-dimensional wave.

Figure 23 shows a shock wave interacting with a Mach wave and a boundary layer. The next figure, Fig. 24, where ~I

=

4.60

and <is l..

=

20 degrees, shows the interaction of two unequal shock waves.

In Fi.g. 25 the reflected shocks have become curved implying that we are approaching Mach interaction. Fig. (26) is at the start of Mach inter-action. Note the curved reflected shocks and the pronounc ed contact region.

Figures 27 and 28 show two stages of Mach interaction. Note the curved reflected shocks and the two dissipating contact surfaces. Figure 29 shows a strong shock wave interacting with a boundary layer. The boundary layer is lifted from the sUl' façe of the wedge creating a com-pression region in front of the interaction (light area). At the

inter-action there originates a"dark" rarefinter-action fan which is imm ediately followed by another compression region (light area). Both

com-pression regions coalesce into shocks. In this figure the boundary layer detaches, forms aseparation bubble, and then re-attaches af ter the inter-acUon. However, in Fig. (30) reattachment did not occur; instead the flow was very unsteady after the interaction. The last two pictures

were taken in an effort to show shock reflection off a wall, this , however I

was foum impossible at this relatively high Mach number because of the strong shock-boundary layer interaction.

4. EXPERIMENTAL RESULTS AND DISCUSSION

4. 1 Measurem ents

Shock wa~e and flow inclinations were measured from the schlieren negatives using a Hilger T. 500 Universal Measuring Pro-jector. All angular measurements were m crle with respect to the up-stream flow direction, which was assumed to be parallel to the stings.

-

21-In most pictures both the boundary layer and portions of the underlying wedge surface are visible. For the thinner wedge this is usually brought about by the action of the shock from the l~ger wedge, separating the boundary layer thus exposing the wedge surface in the separation bubble (Fig. 23). The position of the surface of the larger wedge may be inferred from an examination of the bourrlary layer spillage at the trailing edge of the wedge (Fig. 23). Measurements of wedge inclination ( ~I and ~l- ) were made with respect to the leading edge, which was usually well defined, and another point located at the intersection of the base line and the tangent drawn to the lower surface of the spilling boundary layer.

In many cases the contact surface was only barely disting

-uishable, or some times even totally invisible in the schlieren photo

-graphs. Figure (23) is an example of this; calculation shows that in this instance the density ratio across the contact surface (

f:3./

P4 ) is 1. 029. This resulted in poor or no readings of the contact surface inclination. In these instances the reflected shock angles

el

and e~ were measured and compared with theory. Seven measurements were made from each negative ~I ,

çJ,.,

€ , (dl •

G-z..

,

G].,

and

g",

.

The measured values of

bi

and ~

z.

were then fed into the computer and from them the theo retical values of € ,

G

I

9z..'

9

1 ,and

9

~ were

computed.

Table 6 compares the experimental and the theoretical values of

Q

I ,

el. '

E. ,

G

1 ' and

e"

.

The first two columns of Table 6 contain ~I and

ç).,

b,-

is kept approxima tely constant at 11. 5, 15.5 and 20. 0 degrees while

ç,

is increased from a value near .

zero to

b2.

in roughly 20 _{steps. The difference between experimental}

and theoretical values of € found in column 3 is always less than 2 degrees and usually less than 1 degree. Blank portions of column 3 correspond to unde~ectable contact surfaces. Negative values of Ë

are due to

bi

exceeding

hz..

The shock angle

9,

corresponding to the flow deflection

ç

I increases from the Mach angle of 23.8 degrees to

the shock angle corresponding to ~2- • Columns 6 and 7 show the re~

flected shock angles

9

₁ and

94 .

The values of ~l and @" range from 30.7 degrees to 54.6 degrees. It is interesting to note that,

although

ç

land

ç'l..

and

G

land

G '-

differ by as m uch as 20 degrees,;

93.

and

9

₄ never differ by more than 10 degrees. Correspondence between experimental and theoretical values of the shock angles is within 2 degrees.

4. 2 Errors

There are four factors that may contribute to errors in the measured values of ~I' Ç, l - ' €:: ,

et

and

9

₄ . They are variations in the test section M.ach number, initial flow inclination, boundary layer effects on the wedges, and errors in measuring the wedge de

Fig.

~I

~~

E.

E

G,

el

G2.

Q~

G~

Q'!,

_e4-

G

4

No. _{Exp. Th. Exp. Th. Exp. Th,_ . Exp. Th. Exp. Th .}

•

17 :.1. 00 11. 48 10.47 24.6 24.3 34. 1 33.6 34.9 34.0 :.31:".0 30. 7 1. 50 i1. 70 10. 20 25.5 24.6 34.0 33. 9 35.3 34.5 .32.1 31.3 3.55 11.53 7.96 27.0 26. 3 34.3 33.7 37.0 35.4 34.. 9 32.9 18 5.50 11. 55 6. 03 28.4 27. 9 S4. 1 33.7 38.5 36.6 .36.3 34.7 8.87 11.75 2.86 31. 4 30. 9 34. 1 33.8 40. 1 39.0 .40.0 38.2 11. 93 11. 72 - .208 33.0 33. 9 34. 1 33.8 42.2 41. 5 . -.42. 2 41. 5 19 .48 15.58 15.09 24.6 23.8 38.2 37,8 38.7 38. 1 L • • 33.8 33.5 ::.1. 72 15.72 13.98 25.5 24.8 38.2 37. 9 39.4 38. 9 .3,6. 6 34. 7 20 3.52 15. 53 11. 9 11. 97 26.8 26.3 38.2 37.8 40.8 39.7 .3B.1 36. 2 5.52 15.55 9.98 9.97 28.6 28.2 38.1 37.8 42.2 41. 0 . ~ .3.9. 8 38.2 7.48 15.56 7.48 8.01 30.0 30.0 38. 2 37.8 43.6 42.4 _.~ . .41. 6 40.3 21 9. 62 15.53 3.57 5.84 31. 9 31. 9 38. 1 37.8 45.3 44. 1 , . . 43.8 42.7 11. 55 15.47 2.68 3.86 33. 9 33.6 38.2 37.8 47.4 45.8 .46. 2 45.0 I ~ 13.70 15.53 1. 90 1. 80 36.0 35. 9 38.1 37.8 49.7 48.4 48.8 48.1 ~ I 22 16. 05 15.52 -1. 00 - .52 38. 7 38.2 38.2 37.8 53.3 51. 8 52.7 51. 9 23 .32 2.0. 13 19.79

25, 1

33.7 43.8 43. 1 43. 8 43.5 38.3 38.3 1. 47 20. 08 18.57 2r?,4 24. 9 43.8 43. 1 44.3 44. 1 41. 6 39.4 24 4.63 20.07 14.35 15.32 -28.1 27.3 43.8 43. 1 46.7 46.2 43.8 42. 9 6.40 20.00 12.16 1.3' •. 46 29.4 29.0 43.8 43.0 47.9 47.4 46.0 44. 9 8.00 20.08 11. 72 12.00 30,7 30.3 43.5 43. 1 49.3 48,9 48. 1 47.5 10.17 19. 92 9.71 32.5 32. 2 43.6 43.0 51. 6 51. .1 49.7 49. 1 25 12.08 19.51 7.40 33.$ 33.4 43.3 42.4 53.0 52.5 52.0 51. 5 14.22 19.52 5.02 36.4 36. 0 43. 1 42.5 54.6 54.4 53.0 52.7

Note: All quantities in the above table are in degrees

-23-A test section calibration (Ref. 10) shows that the ab

-solute variation in test section Mach number is of the order of

j: O. 01. This would cause an error of ± O. 2 degrees in eland

tiz..,

. 001 degrees in €. and

±

0. 3 degrees in 9'1 and

e,\ .

A calibra-tion of the flow direccalibra-tion is not available, but it may be assumed that the flow does not deviate more than 1 degree from a direction

parallel to the wall. Thi~ assumption was substantiated by flow de

-flection - shock angle measurements. Ca1culations showed th at a one degree change in initial flow direction introduces a one degree change in €. and a ~ degree change in

9'1.

and

e",

.

(See Table 7 below for Mo = 2. 48).

,

blo

_~:

9,·

e

O 0

go

e4~

Flow Directiön j _€.

z... _~

Parallel to Wall 10 20 32. 1 42. 8 9.85 50.96 50.03 Inclined 10 to Wall 11 19 30. 0 41. 6 8. 87 51. 53 î 50. 63

Table 7. Effect of Initial Flow Deflection on Shock Angles Calculations (Ref. 12) as weU as schlieren photographs show that the boundary layer thickness

h

at the trailing edge of the wedges is approxima tely 0. 1 inches. This leads to a displacement thickness

*'

~

=

O. 03 in.

A deflection of 0. 03 inches in 6 inches causes an angular flow deflection of O. 25 degrees. The table below shows the effect of a O. 25 additional flow deflection on original deflections of 60 and 160 at Mo

=

.~. 48

~I ~2- E- g.

eL

93

_e""

60 160 9. 93 28.5 38. 0 41. 8 39. 2

6. 25 16. 25 9. 93 28.7 38. 4 42. 3 39.8 Table 8. Effect of Boundary Layer Displacement on Shock Angles A check on the repeatability of the wedge deflection measurements

revealed a maximum deviation of O. 1 degrees. This introduces a possible error of 0.2 into E , 0.1 into

9,

and

92,..

,

and 0.3 into @l

and e~

.

The following table is a summary of the possible errors together with their causes. When these errors are applied to the exper -imental values it is seen that all theoretical values lie wel! within

-

24-Source of Error I1N.N O~ ITtJbE Ë _G1)'l.

9

₁,4

ERROR.

Uncertainty in Flow

!

.01 Negl.

t.

2 -+ -: . 3

Mach No.

Possible Flow Inclination

t

10

!

1. 00

t

1. 00

±

1. 50

Boundary Layer "Effect + .25° Negl. +.20 +.50

Errors in Wedge Inclin-

t

.

10

~

.20

~

. 10

t

.30

ation Measurement

Maximum Possible Errors +1. 2 0 +1. 5 0 + 2. 60

Table 9. Summary of Possible Errors

By way of improving this experiment it is suggested that the wedges should be made wider so as to provide a longer disturbed path for the schlieren system and thereby improve the outline of the slipstreams on

the photographs. This widening would also decrease three-dimensional

effects as seen on the schlieren photographs.

5. CONCLUSIONS

The experimental results agree weIl with the exact theory

as derived from the Rankine-Hugoniot oblique shock-wave equations.

The maximum deviation between the experimental and the theoretical results is 2 degrees (5%) in shock angle measurements. Since the

flow Mach number

(>f

2. 48 was used throughout, any conclusions drawn

from the experimental results apply strictly only to flow at this par

"e--ticular Mach number. However, there is little doubt tha t similar results would be obtained at other Mach numbers.

The approximate solution, as formulated in Sec. 2. 5, is

in close agreement with the exact solution, but it is not as close, and

certainly not as effortlessly applied as the empirical relation. The

empirical equation states that E.::::~:l. -~, with a maximum error of 30

up to Mo

=

5. O.

The following qualitative conclusions can be drawn:

1) As Mo - oa ,(; -') () for any given ~I and

~1-2) As ~I -'t ~')... , €: ~ C) for any given Mo.

3) At Mo

=

2.48 and a Reynolds number of 2. 5 x 10 5

per inch, shock wave reflection from a plane wall is strongly influenced by the wall boundary layer.

-

-25-flections, without boundary layer interf~rence is best accomplished by a study of equal shock

interactions .

A cancellation of shock waves by expansion corners rnay also be sirnulated by the interaction of unequal shock waves. Unequal shock interaction provides)through the contact surface>a rnethod of obtaining shear flow free of asolid boundary.

1. Mach, E., 2. Smith, L. G. , 3. Lighthill, M. J . , 4. Fletcher, C. H., 5. von Neumann, J., 6. Seeger, R. J. , and Polachek, H., 7. Lean, G. H., 8. Lieprnann, H. W. , and Roshko, A.,

9. Arnes Research Staft:

10. Fallis, W B. , Johnston, G. W., Lee, J. D , Tucker, N. B. , and Wade, J. H. T. 11. Rockett, J . H. , and Hayes, W. D. , 12. Chapman, D. R., and Rubesin, M. W.,

-

26

-REFERENCES

Vienna Acad. Sitzeingeberichte 77,.

.a

91 (1878)

Photographic lnvestigation of the Reflection of Plane Shocks in Air, OSRD Rep. 6271 (1951)

The Düfraction of Blasts lI, Proc. Roy. Soc., A200 p. 554 (1949)

The Mach Reflection of Weak Shock Waves, Tech. Rep. II - Physics Dept., Princeton University (1950)

Oblique ]~.eflection of Shocks; Explosives

Research Report, No. 12, US Navy, Bur.

Ord., Washington (1943)

Regular Reflections of Shocks in ldeal Gases,

Explosives Research Rep. No. 13, US Navy

Bur. Ord., Washington

Report on Further Experiments on the Re

-flection of lnclined Shock Waves, National

Physics Laboratories, Eng. Div. 193/46

(1946)

Elernents of Gasdynamics, John Wiley ànd Sons lnc., New York (1957)

Equations, Tables and Charts for Cornpre

s-sible Flow, NACA Report 1135

U

953)

Design and Calibration of the lnstitute of Aerophysics 16" x 16" Supersonic Wind Tunnel, UTIA Rep. 15 (1953)

The Method of Characteristics in Cornpres

-sible Flow, Part IC AMC /TR 102 - AC 49/6-100

Ternperature and Velocity Profiles in the

Compressible Laminar Boundary Layer with

Arbitrary Distribution of Surface Temperature,

FIG. 14 WEDGES MOUNTED IN THE UTIA 16" x 16" SUPERSONIC WIND 1UNNEL

•

I

I

I

I

..

,

FIG. 17 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF UPPER WEDGE IS INCLINED 1. 00 deg. TO THE FLOW (

8, ).

UPPER SURFACE OF LOWER WEDGE IS INCLINED 11. 45 deg. TO THE FLOW ( ~2.).

FIG. 18 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF UPPER WEDGE IS INCLINED 5.50 deg. TO THE FLOW.

UPPER SURFACE OF LOWER WEDGE IS INCLINE D 11. 6 deg. TO THE FLOW

FIG. 19 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF UPPER WEDGE IS INC LINED .48 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE IS INCLINED 15.6 deg. TO THE FLOW

FIG. 20 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW

DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF

UPPER WEDGE IS INCLINED 3. 5 deg. TO THE FLOW. UPPER

SURFACE OF LOWER WEDGE IS INCLINED 15.5 deg. TO THE

FIG. 21 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF UPPER WEDGE IS INCLINED 9.6 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE IS INCLINED 15.5 deg. TO lliE FLOW

FIG. 22 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW

DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF

UPPER WEDGE IS INCLINED 16.1 deg. TO THE FLOW. UPPER

SURFACE OF LOWER WEDGE IS INCLINED 15.5 deg. TO 1HE

FIG. 23 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TC LEFT. LOWER SURFACE OF UPPER WEDGE IS INCLINED TO .32 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE IS INC LINED 20. 1 deg. TO THE FLOW

FIG. 24 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF UPPER WEDGE IS INCLINED 4.6 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE IS INCLINED 20.1 deg. TO THE FLOW

.

I

FIG. 25 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF UPPER WEDGE IS INCLINED 12. ·1 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE IS INCLINED 19.5 deg. TO THE FLOW

FIG. 26 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW

DURECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF

UPPER WEDGE IS INCLINED 14 deg. TO THE FLOW. UPPER

'

.

FIG. 27 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW

DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF

UPPER WEDGE IS INCLINED 15 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE IS INCLINED 20 deg. TO THE FLOW

FIG. 28 SCHLIEREN PHOTOGRAPH OF SHOCK INTERACTION. FLOW DIRECTION IS FROM RIGHT TO LEFT. LOWER SURFACE OF UPPER WEDGE IS INCLINED 22 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE IS INCLINED 20 deg. TO '!HE FLOW

FIG. 29 SHOCK BOUNDARY LAYER INTERACTION Mo

=

2.48; SHOCK

FIG. 30 STRONG SHOCK BOUNDARY LAYER INTERACTION Mo :: 2.48. LOWER SURFACE OF UPPER WEDGE INCLINED AT 0.0 deg. TO THE FLOW. UPPER SURFACE OF LOWER WEDGE

INCLINED AT 33 deg. TO THE FLOW