Diffraction of Oblique Shock Wave Past a Bend of Small Angle (Subsonic Case)

Full text

(1)

Diffraction of Oblique Shock Wave Past a Bend of Small

Angle (Subsonic Case)

R.S. Srivastava

Formerly of Defence Science Centre, New Delhi (India)

Sanjay Srivastava, Bechtel, 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 case of diffraction of oblique shock waves (Consisting of incident and reflected shock wave). Srivastava obtained the curvature of reflected diffracted shock wave when the relative outflow behind reflected shock is sonic. In the present investigation a qualitative estimate of curvature of the reflected diffracted shock wave is obtained when the relative outflow behind reflected shock wave is subsonic.

Keyword:Curvature, Diffraction, Reflection, Subsonic.

I. INTRODUCTION

Lighthill (1949) considered the diffraction of a normal shock wave past a bend of small angle . The analogous problem of 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 oblique shock wave diffraction when the relative outflow behind 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 before diffraction is supersonic. Srivastava (2016) gave the results concerning curvature of the reflected diffracted shock when the relative outflow behind reflected shock before diffraction is sonic. In the present paper we have obtained curvature results for the reflected diffracted shock when the relative outflow behind reflected shock before diffraction is subsonic. Reference may be made to the book by Srivastava (1994).

II. MATHEMATICAL FORMULATION

Let the velocity, pressure, density and sound speed ahead of the shock wave be denoted by in the intermediate region by and behind the reflected shock wave by Let U denote the velocity of the point of intersection of the incident and reflected shock,  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

(2)





0 2 2 2 0 0 1

sin

1

sin

6

5

U

a

U

q

- (1)





7

sin

6

5

02

0 2 2 0 1

a

U

p

- (2)





0 2 2 2 0 0 1

sin

5

1

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

a

q

U

q

q

- (4)





7

6

5

2 12

1 * 1 2

a

q

U

p

- (5)

2

1 * 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

a

p

U

U

Also we have

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

and S2. Following Lighthill (1949), by the use of 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

p

y

x

p

x

y

y

x

x









- (7) where

t

a

Y

y

t

a

t

q

X

x

q

a

p

p

p

2 1 1 2 2 2 2 2

,

,

- (8)

(3)

It would be necessary here to discuss about the axes. X,Y are the axes in which X axis is along the original wall produce, Y is the axis normal to the leading edge of the wedge. In (x,y) axes, the origin and x axis lies on the original wall produced and y axis normal to it. The origin with respect to x,y coordinates is at

     

 ,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

,

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

x

2

y

2

1

,

reflected diffracted shock and the wall.

Following Srivastava (1968) and Srivastava and Chopra (1970), the equation of the straight portion of the reflected shock after diffraction is given by

2

cot

y

k

x

where

2 2

a

q

U

k

- (10)

The equation of the diffracted shock may therefore be written as

 

y

f

y

k

x

cot

2

- (11)

where

f

 

y

is small.

From the equation (11), curvature

is given by

 

y

f

dx

dy

dx

y

d







3 2

2 3 2

2 2

sin

1

- (12)

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

 

B

A

G

y

  

f

y

y

p



2

2 - (13)

where

G

  

y

q

2

sin

2

cos

2

a

2

y

and B2, and A2 are constants. Combining (12) and (13) we obtain

 

y

p

y

G

A

B

3 2

2 2

sin

1

- (14)

Equation (14) can be written as

 

dy

x

x

p

y

G

A

B

1 1 2 3 2

2

sin

1

- (15)

(4)

cos

2

sin

2

tan

θ

y

, 2

2 2

sin

a

q

U

- (16)

In (16)





1

1

θ

tan

2 2

Z

Z

,

1

2

The relationship between Z and from Srivastava (1968) is

  

1

1

1

1

2

1

1

bz

bz

bz

bz

z

- (17)

where 2 1 2 2 2 2

cos

sin

cos

sin





b

From equation (17), we obtain

   

1

1

1

1

1

2 1 1 2 1 1

z

z

z

z

b

z

- (18)

1 1 1

x

iy

z

and on the real axis

z

1is replaced by

x

1. From (16) we have

1

1

sin

cos

2 2 2 2

Z

Z

y

- (19)

In (19), Z is substituted in terms of

z

1 (in terms of

x

1 as on the real axis

y

1

0

,

z

1 being equal to

x

1

iy

1

then we obtain

1

dx

dy

.

The values for the calculation are

0 2

0 1

0

0

,

39

.

97

,

32

.

97

p

p

(obtained through calculation).

These data produces

0

.

94699

1

2

2

a

q

U

(subsonic)

Following Srivastava (1968)

 

 1 1 12 1

1

1

0

.

5

05767

.

0

2

25

.

0

1

05779

.

0

4

51698

.

1

1

1

x

x

e

x

x

p

 x

cos

1

1

1

2

75

.

0

1

18751

.

0

4

2 1 0 1 0 1 1

1



x

x

x

x

x

D

C

x

(5)

where 1 1

1 1

1

1

tan

z

x

t

x

x

p

y

p





- (21)

and

0

.

73441

0

.

26559

tan

θ

 

0

.

12908

0

.

04215

tan

θ

0

.

02592

tan

θ

θ

tan

01038

.

0

θ

tan

02355

.

0

θ

tan

03072

.

0

16149

.

0

2 2

1 2

3 2

1

1

x

p

y

p

- (22)

1

x

p

is given by (20) and

1

dx

dy

is given by (19) and so

(curvature) is known from (15). This gives the final

expression for curvature.

From (16) we have

1

1

θ

tan

2

2

Z

Z

where

tan

θ

z

so from the relation (16)

we have

2 2

sin

cos

y

So

1

sin

cos

2 2

y

when

b

z

1

, then we have

2

2

2

cot

1

1

θ

tan

b

b

So from the relation (16)

cos

2

sin

2

tan

y

We have

y

0

or

0

sin

cos

2 2

y

So finally

z

z

1

1

ie

x

1

1

,

1

sin

cos

2

2

y

and

1

z

1

,

x

1

,

b

z

cos

2

sin

2

0

y

From (15) and (20) it could be seen that

0

at

cos

2

sin

2

1

y

(6)

at

0

sin

cos

2 2

y

ie at the wall and shock wave intersection. The variation of

between

0

sin

cos

2

2

y

to

1

sin

cos

2

2

y

is expected to be that as proposed in the case

1

2 2

a

q

U

(sonic case) given by Srivastava (2016).

The point of inflexion in the curvature of the diffracted shock is given by when from (20)

x

1

x

0

1

0

D

- (23)

or when

D

x

x

1

0

1

- (24)

From the calculations we have

60388

.

0

0

x

and

D

0

.

52589

when these values of substituted in (24) we get

505418

.

2

1

x

This means that at the point

x

1

2

.

50548

,

there is a point of inflexion. So in the reflected diffracted shock there is a point of inflexion. The curvature is infinite, then it passes through point of inflexion and finally it becomes zero. This is a qualitative estimate of the curvature.

III. CONCLUSION

The results for curvature distribution

for the sonic case by Srivastava (2016) and present results for subsonic case are very important contribution on the subject and possibly first attempt in this direction. The results will be useful in the area of aeronautics.

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. Diffraction of oblique shock wave past a bend of small angle. International Journal of Science Technology and Management, Vol. No.5, Issue No.01, January 2016 (www.ijstm.com).

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

[6.] Srivastava, R.S. Starting point of curvature for reflected diffracted shock wave, AIAAJ33, 2230-2231 (1995). [7.] Srivastava, R.S. On the starting point of curvature for reflected diffracted shock wave (subsonic and sonic cases),

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