DIFFRACTION OF OBLIQUE SHOCK WAVE PAST A BEND OF SMALL ANGLE

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199 | P a g e

DIFFRACTION OF OBLIQUE SHOCK WAVE PAST A BEND OF SMALL ANGLE

R.S. Srivastava

Formerly of Defence Science Centre, New Delhi, (India)

ABSTRACT

Lighthill considered the diffraction of normal shock wave past a small bend of angle . Srivastava and Srivastava and Chopra extended the work of Lighthill to the diffraction of oblique shocks (consisting of incident and reflected shock wave). In the present investigation the curvature of the reflected diffracted shock and vorticity distribution of a particle over the reflected diffracted shock have been obtained

Keyword: Curvature, Diffraction, Reflection, Vorticity.

I. INTRODUCTION

Lighthill (1949) considered the diffraction of a normal shock wave parallel to a wall and meeting a corner where the wall turns through a small angle . The analogous problem for a plane shock wave hitting the wall obliquely together with the associated reflected shock has been considered by Srivastava (1968) and Srivastava and Chopra (1970). Srivastava (1968) has developed the mathematical theory of diffraction when the relative outflow from the reflected shock wave before diffraction is subsonic and sonic. Srivastava and Chopra (1970) completed the theory when the relative outflow behind the reflected shock wave before diffraction is supersonic.

Reference may be made to the book by Srivastava (1994). This paper gives new results concerning curvature of the diffracted reflected shock and vorticity distribution over the same.

II. MATHEMATICAL FORMULATION

In Figure-1, the velocity, pressure, density and sound speed are denoted by subscripts 0 ahead of incident shock wave, 1 in the intermediate region and 2 behind the reflected shock wave. U in the figure denotes the velocity of intersection of the incident and reflected shock and  is the angle of bend. 0 is the angle of incidence and 2 is the angle of reflection. The Rankine-Hugoniot equations across the incident and reflected shock for

  1 . 4

( being the ratio of specific heats) are given as follows (Srivastava 1995, 2003)

(2)

Fig.-1 Schematic drawing for oblique shock configuration passing over a small bend before diffraction

Across the incident shock

 

 

 

0 2 2

2 0 0

1

sin 1 sin

6 5

 

U U a

q - (1)

 

 

 

 sin 7

6

5

02

0 2 2 0 1

U a

p  

- (2)

 

 

 

0 2 2

2 0 0 1

sin 1 5

6

 

U

a - (3)

0 0

0

p

a

Across the reflected shock:

2

1

* 2 1 1

* 1

2

1

6 5

q U q a

U q

q

- (4)

  

 

  

 6 7

5

2 12

1

* 1 2

q a U

p

- (5)

1

2

* 2 1 1

2

5

1 6

q U

a

 

 

 - (6)

Where

q

2

q

2

sin 

2

, q

1

  q

1

cos  

0

 

2

1 1 1

2

*

sin ,

ap U

U  

Also from Figure-1, we have

(3)

201 | P a g e

2 , cos

cos

2 2 0 2

1

 

   q     q

Let the pressure, density, velocity and entropy behind the reflected diffracted shock be denoted by

2 2 2

, , q

p     

and S2. Using small perturbation theory and conical field transformations, the flow equations can be linearized and they yield a single second order differential equation in p, namely

2 2 2 2

1 y

p x

p y

y p x x p y y

x x

 

 

 

 

 

 

 

 

 

- (7)

where

t a y Y t a

t q x X

q a

p p p

2 2

2 2

2 2

2

2

 ,   , 

 

 

u v

q

q

2

 

2

1  ,

, - (8)

It may be pointed out that in Figure-1 and 2 for X,Y coordinates the origin is at the corner and in Figure-2 for x,y coordinates the origin lies on the original wall produced. The origin with respect to x,y coordinates is

 

 

  , 0

2 2

a

q

and the coordinates of the point of intersection of incident and reflected shock is



 

  , 0

2 2

a q

U

.

The characteristics of the differential equation (7) are tangents to the unit circle

x

2

y

2

 1 ,

consequently the region of disturbance will be enclosed by the arc of the unit circle

x

2

y

2

 1 ,

reflected diffracted shock and the wall.

A schematic drawing after diffraction is shown in Figure-2. The problem is to determine the curvature and vorticity over diffracted shock as indicated in Figure-2.

Fig.-2 Schematic drawing for oblique shock configuration passing over a small bend after Diffraction

Following Srivastava and Chopra (1970) the equation of the straight portion of the reflected shock after diffraction is given by

cot 

2

y k

x  

where

2 2

a q

kU

- (9)

The equation of the diffracted shock may therefore may be written as

  y f y

k

x   cot 

2

- (10)

(4)

Where

f   y

is small

From (10) we obtain

  cot

2

1

 

f y dx

dy

- (11)

and

 

 

2

3

2 2

cot 

 

 

y f

y f dx

y d

The curvature

is therefore given by

  y f

dx dy

dx y

d   



 



 

 

 

 

3 2

32 2

2 2

sin 1

- (12)

by neglecting higher order terms of

f   y

.

Following Srivastava (1968) and Srivastava and Chopra (1970), on the reflected diffracted shock we have the relation

B A G   yf   y y

p   

2

2 - (13)

where

G   yq

2

sin 

2

cos 

2

a

2

y

- (14)

In (13), B2, A2 are constants.

From the equation (13) and (12) we have

  y

p y

G A

B

 

 

3 2

2 2

1 sin 

- (15)

Equation (15) can be put in the form

  dy

x x

p y

G A B

1 1 2 3 2

2

1 sin 

 

 

 

 

- (16)

This is the mathematical expression giving the curvature of the reflected diffracted shock wave for all the three cases (when the relative outflow is supersonic, sonic and subsonic). In the present paper we take the case when the relative outflow is sonic.

We have the relation on the diffracted shock (Srivastava, 1968) as

 cos 

2

 sin 

2

tan θ 

K y

cos sin tan θ

sin 

2

2

 

2

- (17)

where

K K Z

Z

 

 

 

1 θ 1

tan

2

2

 

 

 

1 cot

2

1

2

2

Z

Z

- (18)

(5)

203 | P a g e

 

 

 

2

1

sin

cos 1

h Z

z

- (19)

From (18) and (19), y can be expressed in the form

 

 

 

 

 

 

 



 



 

   

 



 



 

 

cos 1 1 sin

1 cos

1 sin

cos cos

sin

2

1 1 2

2 2

2 1 1 2

2 2

2 2

2

z h

z y h

- (20)

 

 

 

 

 

 



 



 

   

 



 



 

  

 

cos 1 1 sin

cos 1 1 sin

cos cos

sin

2

1 1 2

2 2

2

1 1 2

2 2

2 2

2

x h

x h

- (21)

on the real axis From (21) we obtain

1

2 2

1 2

2 2 2

1 1 -1 2

2

1

cos

sin

1 1

cot cos 4

 

 

 

 

x x h

x h dx

dy

 

- (22)

The values for the calculation are

00009 . 1

, 29 . 31

, 91 . 39

, 0

2 0 2 2

0 0

1

0

 

a

q U p

p  

(Sonic)

For these calculations Bleakney and Taub (1949) paper has been used.

For these values Srivastava (1968) gives

   

 

   

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

cos 1

1 1

2 75

. 0 1

58272 . 0 4 5

. 0 1

31151 . 0 2

25 . 0 1

09153 . 0 4 1 2

2 1 0 1

0 1 1

1 1

1 12

1 1

1

x x x

x x D C x x

x e x x x

p

x

where 1 1

1 1

1

1

tan z

t x x x

p y

p     

 

 

 

 

 

  

 

 

- (23)

 

0 . 73114 0 . 5010 0 . 26885 0 . 03039 tan

2

tan θ  

12

θ 0 . 11300 0 . 0215 tan 0 . 3685 θ tan 0 . 00930 θ 0 . 02617 tan θ tan

2

θ

3 2

1

1

  

 

 

 

  

 

 

  

x p y

p

- (24)

(6)

 

1 1

θ cot

tan

2

2

2

 

Z

Z

- (25)

and

 

 

 

2

1

sin

1 cos Z

h

z

- (26)

dx

1

dy

is known from (22) and

x

1

p

is known from (23). Substituting these values in (16), curvature

is

known.

On the reflected diffracted shock combining (17) and (18) we have

1

2

sin

2

2

2

 

Z Z y

- (27)

From (19) when

Z   , z

1

 1

and when

Z0 z ,

1

 

From (27)

1  , 1 

2

sin y

2

Z   z

1

and

  

1

  

2

, 0 2 0

sin y Z z

. The variation of

2

2

sin 

y

on the diffracted shock is therefore

0 2 sin

2

y

(Point of intersection of diffracted shock and wall)

to

1

2 sin

2

y

(Point of intersection of diffracted shock and unit circle).

We give below the table showing

,

(

being the curvature and  is the bend) on the diffracted shock.

Table-1

2

2

sin 

y

0 0.30 0.4 0.6 0.8 1

2 . 6  10

55

0 . 09  10

30

 0 . 105  0 . 000534

0

The perturbed velocities uand v behind the diffracted shock are given by

     

 

   

 

 

 

 

 

 

 

 

 

  

 

 

 

 

2 2 1 2 1

2 2

2

2 2 1 2 2 2 2

2 2

2

1 θ cos

sin 2

sin sin

1 sin θ sin

6 cos 5

V q a

U y

f

V y a

f q

y f y a y f q a u

-

(28)

     

 

   

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

2 2 1 2 2 1

2 1 2 2

2

2 2 1 2 2 2 2

2 2

2

1 sin sin 2

cos sin

1 cos sin sin 6 cos

5

V q a

V U a

y f

V y a

f q

y f y a y f q a v

(7)

205 | P a g e

U sin

2

q

1

sin

V  

- (29)

We have

 (Vorticity)



 

 

 

y u x v 2

1

- (30)

We find from (29) that

 0

x

v

as all the terms on the right hand side of (29) are function of y. Therefore using

(30) we have

 

 

 

y

u 2

 1

    

 

 

 

 

 

 

 

 

 

 

 

2 2 1 2 1

2 2

2

2 2

2 2 2 1 2 2 2

1 cos sin 2

sin sin

sin cos 1

1 sin 12

5

V q a

U

q y V a

y a q f

   

 

 

 

  

 

 

 

 

 

 

 

  

 

2 2 1 2 1

2 2

2 2

2 2

2 2 2 1 2

2 2

1 cos sin 2

sin 6 sin

5

sin cos sin 1

6 5 2

1

V q a

q U

q y V a

a y q

f

 

   

 

 

 

 

    

 

 

y q B

a A y

f

2 2 2

2

2

sin 2 cos sin

2 sin 2

1   

- (31)

In (31),

A

and

B

are constants.

From (12)

  sin

3

2

f  y - (32)

From (31) and (32) we have

 

 

  

 

   

 

 

 

y q B

a

A

2 2 2

2 2

2

3

sin 2 cos sin

2 sin sin

2

1   

 

We give below the table showing

( is the Vorticity and  is the bend) over the diffracted shock.

Table-2

2

2

sin 

y

0 0.3 0.4 0.6 0.8 1

2 . 52  10

55

0 . 8  10

30

 0 . 07557  0 . 000297

0

(8)

III. NUMERICAL CALCULATION 3.1 Curvature

The computed curvature for the data indicated in the paper is shown in Table-1. The table shows that the curvature takes an infinite value at the point of intersection of the oblique diffracted shock and the wall. The curvature then falls and takes the negative values while approaching the point of intersection of oblique diffracted shock and Mach circle and finally it becomes zero at the intersection of Mach circle and reflected diffracted shock.

3.2 Vorticity

The computed vorticity distribution is shown in Table-2. Here also we find that vorticity takes an infinite value at the point of intersection of diffracted shock and the wall. The vorticity then falls and take negative values while approaching the point of intersection of shock and Mach circle and finally it becomes zero at the intersection of shock and Mach circle.

IV. CONCLUSION

Development of theory concerning normal shock diffraction has been completely analyzed. But in the case of oblique shock diffraction (consisting of incident and reflected shocks) investigations are lacking. From that point, the work presented in the paper concerning curvature and vorticity assumes importance. It is expected that more work would flow after this work.

REFERENCES

[1]. Lighthill, M.J. The diffraction of blast-1, Proc. Roy. Soc. A, 198,454-470 (1949).

[2]. Srivastava, R.S. Diffraction of blast wave for the oblique case. British Aeronautical Research Council, Current Paper No. 1008 (1968).

[3]. Srivastava, R.S. and Chopra, M.G. Diffraction of blast wave for the oblique case, J. Fluid Mech 40, 4, 821-831 (1970).

[4]. Srivastava, R.S. Interaction of Shock Waves, Kluwer Academic Publishers (1994).

[5]. Srivastava, R.S. On the starting point of curvature for reflected diffracted shock wave (subsonic and sonic cases), Shock Waves, 13, 323-326 (2003).

[6].

Srivastava, R.S. Starting point of curvature for reflected diffracted shock wave, AIAAJ33, 2230- 2231 (1995).

[7].

Bleakney W and Taub, A.H. Interaction of Shock Waves Revs Mod Physs 31, 589-605 (1949).

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