Distortion of a shock wave traversed by al vortex

1967

DISTORTION OF A SHOCK WAVE TRAVERSED BY A VORTEX

by

L • T. Filotas

~.

/

TECHNISCHE

HOGfS(I!OOl DELfT

V

lIEGTUIGB$;U'';/l:.UNtJloe

B

lbliOlliEfK

...

:,

.

DISTORTIGN OF A SHOCK WAVE TRAVERSED BY A VORTEX

by

L. T. Fi10tas

ACKNOWLEDGEMENT

This problem was suggested by

Dr. H. S.

Ribner, whose previous

work provides the basis for the analysis.

His supervision and helpful

sugges-tions are gratefully aCknowledgeQ.

I would also like to thank

P.

Hughes and

N.

Ellis for many

in-valuable discussions of the mathematical details, and

W.T.

Chu and

T.E.

Siddon

for their continued interest.

The research was supported by the Air Force Office of Scientific

Research, Office of Aerospace Research, United States Air Force, under AFOSR

Grant

No.

672-64 and by the National Research Council of Canada, under NRC

Grant

No.

A2003.

SUMMARY

A theoretical treatment of the sound field produced upon the

interaction .of a vort ex and a normal shock wave was presented some time ago in

UTIA Report No.61 (H.S. Ribner, 1959). The techniques of this analysis are now

used to derive expressions for the shape of the distorted shock front.

It is found that outside of the section cut off by the cylindrical

acoustic wave em,

anating from the

·

vortex center, the shock remains essentially

straight. The portion within this section has curvature, resulting in a finite

displacement of the two straight segments.

These theoretical predictions are

in qualitative agreement with the features noted by experimental investigators of

the same phenomenon.

As no other presently published theory yields the shock wave

shape, this work demonstrates the generality of the approach used.

1.

11.

111.

IV.

TABLE OF CONTENTS

NOTATION

INTRODUCTION

FORMULATION OF THE PROBIEM

MATHEMATICAL DEVELOPMENT

3.1

3.2

Interaction With a Single

Radial

Component

Interaction With the Complete Vortex

3.2.1

Evaluation of Ia

3.2.2

Evaluation of Ib

3.2.3

Final Result - Upper and Lower Bounds

COMPARISON

WITH EXPERIMENT

REFERENCES

FIGURES

APPENDIX

v

1

1

2

2

3

4

6

8

8

10

a,b

fl(Z

)

and

f 1(z)

gl(x) and

g2(x)

k

1 and m

m

s

r

t

u

m

y

y'

y

x

A(t)

B(t)

D

H(x)

NOTATION

parameters for shock wave perturbation (from Ref.

1)

functions defined below

Eq.

(11)

functions defined by

Eq.

(7)

wave

number

"dummy" variables used in

Eq.

(4)

velocity ratio across shock

(UA/U

constant

(~"I;

1

(m-1)

, tan

~cr

radius vector from vortex center

vortex

core radius

tan

~

(

= m tan

e )

maximum

velocity

in vortex

(r/2nr )

o

"I ;

1

MA2

1

₊

"I

_""2

-1

M

_A

2

of Ref.

1

)

distance along shock, origin on shock (see Fig. 2)

)

distance along shock, origin at virtual vortex

·

center

nondimensiona1

distance along shock (my/D)

downstream distance from shock,

or

"dummy"

variab1e used in defining functions

parameter (D/rO

)

(ro/D)z

function defined

b

y

Eq.

(10)

function defined by

Eq.

(14)

distance

of

virtual

vortex center downstream of

shock

Heavyside function, defined by

E, F. G,

L, M, N,

P,

Q., R,T

5(x)

e

function defined by Eq. (14)

constants defined in the Appendix

Mach number upstream of shock

defined by Eq.

(8)

Bessel function of first kind and first order

stream velocity downstream of

.

Shock

stream velocity upstream of shock

velo city component (cose/u

A

) see Fig. 2

local shock displacement in the streamwise direction

Dirac delta function, defined by

5(x)

0

E(x)

=

1

:fÇ

f

0

net displacement between two extremes of shock

angle of elementary wave with stream

.4 '."L.-I

cp

angle of transmitted elementary wave with stream

(el

in Ref. 2)*

TJA

r

local slope of shock wave

inclined rectangular coordinate (see Fig.2)

vortex circulation

Subscript g pertains to single radial component of vortex

Prime on functions indicates differentiation with respect to the argument.

e.g.

f'

(x)

=

df(x)

dx

*

Ref.

1

shows that the relation between

e

and

cp

is:

tan

e

- tan

1

cp

I. INTRODUCTION

An understanding of the role of shock waves in the product ion

of jet noise is of considerable theoretical and

practical

interest. For

example,

it is known that flow disturbances (vorticity, temperature spottiness) interact

with shock waves to generate intense noise. The specific interaction of a single

columnar vortex with a plane shock front is

particularly

well suited to both

theoretical and experimental investigation.

Experimental studies have been carried out by Hollingsworth

and Richards (Ref.

5)

and by Dosanjh and Weeks (Ref.

4).

The most pronounced

features revealed by these studies are the existance

of

a growing cylindrical

acoustic

wave

centered on the transmitted vort ex and a distortion of the initially

plane shock front.

These details are evident

in

Fig. 1 which is a schlieren

photograph taken in a shock tube (reproduced from Ref.

5).

From the theoretical point of view, an heuristic analysis

of

the

cylindrical sound wave and associated pressure field was first formulated by

Hollingsworth and Richards (Ref.

6).

A quantitative theory was

later

advanced

by Ribner (Ref.

2).

Starting

with

the known result for the

interactio

n

of the

shock wave with a single sinusoidal shear wave

(Ref.l),

Ribner

obtains

the

pressure field due to the complete vortex through a synthesis in a Fourier

Inte-gral.

To the present time attent ion has not been focussed on the shape

of the

distorted

shock.

However, the techniques developed by Ribner in his

treatment of the acoustic wave are also directly applicable to t

his aspect

of

the problem.

In the present work, a synthesis of the effect of individual

sinu-soidal

shear waves,

patterned af ter the development of Ref. 2, is used to

con-struct a solution for the shock distortion. This leads to expressions for upper

and lower bounds which de fine the shape

of

the distorted

sh

ock

quite accurately,

except in the immediate vicinity of the point of maximum curvature.

II. FORMULATION OF

THE

PROBLEM

From Ref.

1

it is known that if a single shear wave of the form

is convected through a normal shock wave, the resulting perturbation in the

local inclination

of

the shock

is

also sinusoidal, and of the form

~

.

~

-

E

[

a sin

(k

y' cos

e)

+

b cos (k y' cose )]

(1)

The details

are

shown in Fig.

2.

The amplitudes a and bare functions

of

the

velocity ratio across the shock m, and the shear wave angle e only.

The

functional forms are given in Ref.

1

.

For the present purpose, the expressions

of

Ref. l have been

re-cast to a form showing the dependenee on

e

explicitly and are presented in the

Appendix as Eq. (Al) and (A2).

Note that here

e

is defined positive

in

the

opposite senSe to that used in Ref. l. Also it is to be noted that in the

reference the expressions for a and b have different functional forms depending

on whether

e

is greater or less than a critical value.

In (Al) and (A2)

how-ever, the Heavyside function H(x) has been used to construct functions valid

for the whole range of

e

>

O.

Symmetry considerations show that b is an even

function of

e

whereas a is an odd function of that variable.

Reference 2 shows that the velocity field

of

a vortex can be

de-composed through a Fourier Integral

into

"elementary" sinusoidal shear waves

having the form

k

r

o

d k de ) sin k.r

An integration of this 'elementary' velocity wave over all wave lengths

( 2)

(0

<

k <

00)

yields a shear wave with the special velocity profile shown in

Fig: 3.Ä f

.

urther integration over all angles

(-rr/2

~

e

~

rr/2)

results in the

well-known expression for the velocity field induced by a columnar vortex of

strength rand core radius rO'

Thus the special profile shear wave of Fig. 3

may be interpreted as a single radial component of the vortex flow field.

It follows that the expression

cos

e

r

cr

=

-UA

7f2

d k de

(a sin k

~A

+

b cos k

~A)

when integrated over all wavelengths and all angles gives the local

inclina-tion of the shock wave due to interacinclina-tion with a complete vortex.

A further

indefinite integration with respect to distance along the shock gives the actual

deflection.

The order of integration may of course be interchanged to facilitate

the analysis.

The remainder of this paper is mainly concerned with the

manipu-lations required to evaluate the triple integral. However, the physical

interpretation of each stage of the analysis is presented as far as possible

.

111. MATHEMATICAL DEVELOFMENT

3.l Interaction With a Single Radial Component

It is convenient to integrate equation (3) with respect to k

first. This results in an expression for the local slope of the shock wave

due to interaction with a single radial component of the vortex field, of the

type discussed earlier (Fig. 3).

Making use of the known definite integrals (special cases of the

Weber-Schafheitlin integral, see for example Ref

.

3):

g i ( ! )

fa

Jl(mx)

0

(m ::: ,e)

cos (,ex)

d.x

x

Jl

(,e/m) 2

(m

2:

,e)

(4)

g2(~){·

Jl(mx)

,e

Jl

~

·

_{_.}

_{(mJe )}

~

_(m

_:5

_,e)

sin (,ex)

d.x

-

m

·

{l

x

the slope function becomes

The

CT

g

.r::...

_7r2

corresponding

5

g

local displacement

y'

J

'

CT

_g

dy'

,e

m

in the x direction is then

r

[a g2

(Y'~:se)

+

b

y'cose

) J

-~

gl

_r

where

.11

7r/2

2

x

2

x

· -1

+

sln

x

(x :::

1)

(x

2:

1)

(x :::

1)

+

cosh- l lxi (x

~

1)

The shape functions gl and g2 are sketched in Fig.

4.

3.2 Interaction With the Complete Vortex

(m

2:

,e)

given by:

'-(6)

The total effect of the vort ex is obtained by summation of the effect

due to shear waves of the velocity profile sketched in Fig. 3. (this was obtained

in the previous section) disposed at all possible

ang~es

and intersecting at

the instantaneous position of the vortex center.

This is accomplished by an intergration of Eq.

(6)

from

e

= -

7r/2 to

e

=

+

7r/2.

As

y'

in Eq.

(6)

is a function of

e

it must first be replaced

by its equivalent

It is also convenient to change the variable of integration to

t

=

=

m tane

Then the shock deflection due to the complete vortex is given by:

where

u

r/(2rrr )

m

00

Ia

(y:)

₌

100

_,

_g2

.

00

Ib

(y)

100

gl

my

D

3.2.1 Evaluation of Ia

5

2

TT

, maximum velo city

(+

~

- t

)

J

t2+m2

(-l-

Y -

t

)

J

t2+m2

( 8)

in the vortex.

m a(tL

_dt

t2

_{+ m}

2

m

b~tL

_dt

t2

_{+ m}

2

Convergence of the improper integral Ia is assured as the integrand

is proportional to t- 3 as t

~

±

00 .

Let

A(t)

=

l t

o

m a(xL

2

2

x

+

m

dx

(10)

A(t) may be evaluated analitically; the explicit form is given in the Appendix.

An

integration by parts of Eq. (9a) gives

I

a

[ A( t)

g2

CD

r

J

"!! -

2

t

2

0

o

t

+m

J

oo

Joo

A(t) g'

(~

y-

t \

~(~

y-

t

·

~t

_00 -

00

~

.

.

~

J

t2+m

2)

dt r

Jt 2+m2 /

Since a is an odd function ot, A(t) must be

even.

of its argument, the first term in (11) vanishes.

second term, the change of variable

z

=

D

_r

o

(11)

Since g2 is an even function

In

·

the

of the

is useful. It must be noted that since t is a double valued

'

function of z,

the integration has to be subdivided into two parts such that for the range of

the limits in each part, zand t are mutually single valued.

For the range

<

t :::: _

m

2

/y-corresponding to

t

and

A(t)

say.

2

Whereas for the range -m

/y

<

t

<

+

corresponding to

J(y/m)2

+1

<

Z

<

-1

- z

~(1-z7)

-2

t

y

+ Y

1 - Z

2

and

A(t)

f

2

(z)

where

_r

-

Z

D

Z

With this change of variable, Eq. (11) becomes

I

a

_

J

D/ro

.J<y/m)

2

D/ro

+

1

f

l

(z)g2(z)dz

-J

-D/ro

Dir

~r-( y-I-m-') 2"'-'+-1

(12)

By noting that fl(-z)

=

_{f2(z) and f 2(-z)}

to the form:

fl(z), Eq. (12) may be condensed

Ia

=

J

D/ro

1ly/m)2

+

o

1

[fl(Z)- f

2

(z)] g2(z) dz

(13)

It will

.

be shown later that the major contribution to the shock deflection comes

from Ib. Thus it will be sufficient to calculate a suitable upper bound for Ia·

This is necessary as Eq. (13) is too complex to be evaluated analytically. The

upper bound is obtained by replacing the integrand

in

(13) by its maximum value.

The maximum may be shown to occur at the point z=l (g2 has a peak at this point).

Then

A

(-Y-

~o J~2+m2)

J

(14)

Since the functional form of A is

.

known, Eq. (14) represents an

explicit

upper bound. However a great simplification can be achieved througb

the following

approximation:

Using

(10), Eq

.

(14) can

·

be rewritten as:

dx

Since

the range of the integrand is always

small

with respect to y/m, the

inte-gr-.al

:'

above

'.

may be approximated by the va1ue of the integrand at the mid point

of the limits times the length of the interval. This results in the relation:

( 15)

When

y/m

«

1

,

all

the

va

-

lues take

.

n

on by the integrand in

(15)

will

be small.

As a(x) is proportional to

x

for small x, the approximation becomes

asymtotically exact as

y

~O

.

When y/m»

1,

the integrand varies as x- 3 ; it

may be shown that in

this

case the approximation is again exact, if terms of the

order (roD)2 and higher

are

neglected

.

However at the point x - s, a(x)

exhibits a very sharp peak (Fig

.

7)

and the approximation is not valid in the

neighbourhood of this point

.

The difference between Eqs. (14) and (16) is

only appreciab1e at this point. The value of the approximation lies in the

great simplification of form it brings about and in the fact that it gives an

upper bound which is independent of raD.

3.2.2

Eva1uation of Ib

Consider the function

~

1

1

H(x)

=;

gl(~)

+

2

The relation

between H(x) and the usua1 Heavyside function is illustrated in

Fig.

5

for the argument

When

yl

~t2

+ m2 is considered as the variable,

H(~

y - t

·

)

r

A.j

t2 + m2

rises from zero to its maximum value in a distance of 2

ro/D,

independently of

t

:

Thus for ro« D (convected vort ex many core radii downstream of the shock),

the approximation

is valid.

Since for any positive constant k,

H(kx)

=

H(x)

H

(y -

t)

Hence from

(16)

and

(17),

with the definition of H:

g

(~

y -

t)

=

I [

2

H(y -

t) -

1 ]

1

r

Af t2+m2

2

Using

(18)

and the fact that b

0

for Itl

~ s

Let

t

B(t)

1

mb(t)

dx

t

2

+m

2

dt

(16)

(18)

(20)

B(t) may be evaluated analytically; the resulting expression is given in the

Appendix.

Then af ter an integration by parts, (19) becomes:

Ib

=

.

[~[2

H(y-t) -1 J. B(t)

·

Is+

1:

B(t) .5 (y-t) dt

Since b is an even function of t, B is odd; whence

or

the above in a different form

( Iy I

>

s)

(

11'

I

<

s)

(22)

Since b is always negative, Ib is always positive for

l'

>

0

and

negative for

l'

<

O.

For all

IYI~

s, Ib is constant whereas Ia is significant

only for values of

l'

near s. Hence the previous remark that Ib makes the major

contribution to the shock deflection.

3.2.3 Final Result - Upper and Lower Bounds

The total deflection

5

2

7T

is bounded by the following limits:

UA

2B(Y:)

<

--

u

m

5(1")

<

r

o

2

+

(23)

The upper and lower bounds represented by Eq. (23) are shown for an upstream

Mach number of

1.5

in Fig.

6.

The value of Ia(1')max represented by Eq.

(14)

for

rolD

= 10

is shown as asolid line while its value from Eq.

(16)

is shown

broken. It is seen that the two upper bound curves differ only in the

neigh-bourhood of

Y

=

s.

IV.

COMPARISON WITH EXPERIMENT

The upper and lower bounds for the shock deflection obtained in

the foregoing analysis define the shape of the perturbed shock wave within

narrow limits, except in the neighbourhood of

y

=

s. The effect of Ib is to

displace the segments outside of these points relative to each other, in the

direction of the circulation of the vortex.

Ia gives a slight curvature to each

of the displaced segments.

Maximum curvature occurs at

l'

=

s.

In Ref. 2 it was shown how the individual sound waves produced at

the shock by the "elementary" sinusoidal components of the vort ex combine to form

an envelope.

This envelope has the form of a growing cylindrical acoustic wave

which intersects the shock at

l'

=

!

s.

It is evident from the photograph in

Fig.

1

that experimentally the maximum shock curvature occurs at these points.

The fact that the segments of the shock outside of these points are very nearly

straight is also evident.

Further it may be noted that the deflection of the

shock is small in comparison with the vortex core radius. This is ensured in

the theory by the factor

~UA

in equation (22).

A

quantitative

check on the theory could be made by comparison

of the values of the total displacement between the ends of the shock and of

the slope at the origin with experimental values.

The theory gives simple

analytic expressions for these quantities.

The total displacement between the extremes of the shock is

obtained from Eq. (22) as:

U

_m

6

= -

4 --

B(s) r

UA

Upper and lower bounds for the slope at the origin are:

Urn

ro

<

I.d5J

UA

D -

ldy y=o

<

2m

2

[ sT

+

~

.sJ

um r

LM

LM

N

UA

D

However, in this report, no quantitative comparison of the theory with experiment

was attempted as it is difficult to determine the appropriate flow constants for

presently published experimental photographs.

1.

Ribner, H.S.

2. Ribner, H.S.

3. Watson, G.N.

4. Dosanjh, D.S.

Weeks, T.M.

5. Hollingsworth,M.A.

Richards, E.J.

6. Hol1ingsworth,M.A.

Richards, E.J.

REFERENCES

Convection of a Pattern of Vorticity Through a Shock

Wave.

NACA Rep. 1164, 1954

(Supersedes TN 3255, July 1954).

The Sound Generated by Interaction of a Single Vortex

with a Shock Wave.

University of Toronto, Institute

of Aerophysics, UTIA

.

Report No. 61, 1959.

A Treatise on the Theory of Bessel Functions.

Cambridge U. Press, 1962.

Interaction of a Starting Vortex as Well as Karman

Vort ex Street with Travelling Shock Wave, AIAA Paper

No. 64-425, 1964.

A Schlieren Study of the Interaction Between a Vort ex

and a Shock Wave in a Shock Tube. A.R.C. 17, 985,

F.M. 2323, 5th Nov., 1955.

\

On the Sound Generated by the Interaction of a Vort ex

and a Shock Wave. A.R.C. 18, 257, F.M. 2371, 29th

Feb., 1956.

a(t)

APPENDIX

EXPLICIT FORMS OF THE DEFLECTION FUNCTIONS

a,

b, A, B

-m

Ft

3

₊

_{Qt - H(t-s) • [R(t}

2

_m)

,.,ft 2

- s2 ]

n

(t2 -

L2)(t 2_M2)

b(t) = m H(s-t)

(Al)

(A2)

(The above definitions are va1id for t

>

0

only; a

and

bare to be taken as odd

and even functions of t respective1y.)

Where

t = tan

= m tan

e

s =

~I'

;

1 (m-1)

H(x) - Heavyside function

P

=

2

EF

-

8G

Q = 2(1'+1)(m-1)(2G-E)

N =

F2

-

8G

R

=

4/m • (1-G2)/G

8s

2

T

=

A/s2+1

b+1

Á

(m-1)

{ F

-

G 2 +

(m-1)J

I'!-1

] }

E

= -

1'+1

-

(1'-1)

m

F

= - +

1'+1

(3-1')

m

G =

2

s2+1

alm and

b/m

are p10tted against t for a representative va1ue

of

m in Fig.

7.

(

_[

a(x)

A

t)

=

m

o

x

2

+m

2

dx

+

B(t)

+ f t m

roT

+

N

m2

M2 ( t 2_L2)

I

L2 (t 2_M2)

I

/m

2+s2 •

_tn

I

m+1

_{(L2_m2 )}

-

_m

b(X2

_dx

x2 + m2

{(L-m)

A/L2_s2

L(L2-m2) (L2+m2 )

[

sin- 1

(t-L) -V(t 2 _s 2 ) (L2_s2) _Lt_s2

(t+L)

4

(t 2 _ s2)(L2_s2) +Lt-s 2

(t-M)

~(t2_s2)(M2_s2)

-

·

Mt - s2

(t+M)

~(t2_s

2)(M2

_s2)

+ Mt

-

s2

s

1m

2

+s

2

~

I}

+ m

t

,4m

2+s2

-

m

Nt 2-s 2

Lt-s

+

.

-1

Lt+s

.

-1

s(t-L)

Sln

s(t+L)

-

2 Sln

(M-m) II/M2 _

s2

M(M2_m2 ) (M2+m2 )

[

1

Mt-s

sin-s(t-M)

+

Sln

.

-1

- -

Mt+s

.

s(t+M)

- sin- 1

:

]

(m+1)

A/m

2

+s 2

(L2+ffi

2

) (M

2+m2 )

t

m

12

iJ

I

FIG.

1 SCHLIEREN PHOTOGRAPH OF SHOCK-VORTEX

INTERACTION

INCIDENT

SHEAR WAVE

y'

- a sin(ky' cos.)

- b cos(ky' cos")

---1~~~~---~~---~x

UNDISTURBED

SHOCK FRONT

FIG. 2

INTERACTION OF SINGLE 'ELEMENTAHY' SHEAR WAVE

VELOCITY PROFILE

OF SINGLE RADIAL

COMPONENT OF

VORTEX

VELOCITY

FIELD

y

y'

UNDISTORTED

SHOCK

'"'

D

ACTUAL

POSITION

VIRTUAL

POSITION

VORTEX CORE

FIG. 3

INTERACTION OF SHOCK WITH SINGLE RADIAL COMPONENT

OF VORTEX

VELOCITY FIELD

-3

-2

-1

1

2

3

x

- .5

-1.

0

H{x)

---~~---~---r_---~

X

-1

+1

ro

~

•

2 -

_D

1

Hl

~

ro

y-t

I

r--it

2+ m 2

_...

'

I

I

H

-.., I

D

y-t

ro

it2

+m2

y

-t

i

_t

2

_+m2

~

t

2

+m2

-3

-2

-1

L

D

4

3

2

1

-1

-2

1

2

LOWER BOUND

EXACT UPPER BQUND

(r

O

/D=O.1)

APPROX. UPPER BOUND

:y

3

u

m

MA

=

1.5

m

=

1.821

s

=

1.017

-5

-4

-3

MA

=

1.5

m

=

1.821

s

=

1.017

2

a(t)

m

1

-2

2

3

4

5

b(t)

m

-1

-2

Security Classification

DOCUMENT CONTROL DATA. • R&D

~ecurity claeeli/cat/on ol t/tl •• body ol abetreet .... d Inde./n, annot.t/on mu.t be ent.rad ""'.n th. overall r.port I. ele •• lllad)

1. ORIGINA TIN G ACTIVI"!Y (Corpora ta author) Za. REPORT SECURITY C LASSIFICATION

Institute for Aerospace Studies,

Unclassified

Uni ver s ity of Toronto, Toronto 5, Ont.

3. REPORT TITL.E

DISTORTION OF A

.

SHOCK WAVE TRAVERSED BY A VORTEX

4. DESCRIPTIVE NOTES (Type ol report and 'nelu.'ve dat •• )

Scientific

Interim

5. AUTHOR(S) (I..aet ... II,.t name. In/tlal)

Filotas, L.T.

•. REPO RT DATE 7a. TOTAL. NO. OF PAGES _{17b . NO.}

6"

January 1967

25

' a. CONTRACT OR GRANT NO. ta. ORIGINATOR'S REPORT NUt,lSER(S)

AFOSR 672-64

b. PRO.JIEC TNO.

9781-02

UTIAS Tech Note

96

e.

61445014

d.

681307

AFO

S R 67'"

0

12 "

l .

Distribution of this document is unlimited.

11. SUPPL.EMENTARY NOTES 12. SPONSORING MIL.ITARY ACTIVITY

(SREM)

Air Force Office of Scientific Re

s

earch

Ä~~O l~t~sOnv~oulevard

_{r in on}

_{irginia 2}

₂₂

13. ABSTRACT

A theoretical treatment of the sound field produced upon the

inter-action of a vort ex and a normal shock wave was presented some time ago

in UTIA Report No. 61 (H.S. Ribner, 1959). The techniques of this

analysis are now used to derive expressions for t he shape of the

dis-torted shock front.

It is found that outside of the section cut off

b

y

the cylindrical acoustic wave emanating from the v

o

rte

,

·

.

center, the shock

remains essentially straight. The porti on within this section has

curva-ture, resulting in a finite displacement of the two straight segments.

These theoretical predictions are in qualitative agreement with the

features noted by experimental investigators of the same phenomenon.

As no other presently published theory yields the shock wave shape,

this work demonstrates the generality of the approach used.

I

I

14.

Security Classification

KEY WOROS

S

hock

W

ave

s

Sho

c

k

-V

ort

e

s

I

nte

racti

on

A

erod

yn

ami

c N

o

i

s

e

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