International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 4, Oct 2013, 79-86 © TJPRC Pvt. Ltd.

A STUDY ON PROPAGATION OF SPHERICAL SHOCK WAVES IN THE ATMOSPHERE PRASHANT CHAUHAN & MUKESH CHANDRA Department of Mathematics, IFTM University, Moradabad, Uttar Pradesh, India

ABSTRACT In the present paper, we shall study the propagation of spherical shock waves in the atmosphere. The propagation of shock waves in the earth's atmosphere is in the vertical direction. It was found that the shock velocity increase as the shock front propagates in the vertically upward direction but in the horizontal direction the density is constant and the shock velocity remains constaint.

KEYWORDS: Shock Waves, Mach Number, Standard Velocity, Propagation 1. INTRODUCTION Singh [8] studied the propagation of the shock waves in the atmosphere of the earth in a vertical direction, in which the model of [5] was used with minor change. It was found that the shock velocity increases as the shock front propagates in the vertically upward direction, but in the horizontal direction, the density is constant and the shock velocity remains constant. In the isothermal part of the atmosphere, the increase in shock velocity is smaller, but larger in layers where the temperature decreases. In the present paper, a spherical shock wave is assumed and its propagation in the atmosphere of the earth is studied by using Whitham's Rule [10]. Curvature of the earth is neglected. So that the density, pressure and temperature of the atmosphere are varying in planes parallel to the surface of the earth. Let us assume that the component of the shock front making angle parameters are inclined at an angle

to the vertical, so that the planes of variation of thermo dynamical

to the shock surface which is considered to be plane. In the last it is found that the

rate of increase of shock velocity decreases as

increases, being the maximum in the vertical direction.

2. BASIC FUNDAMENTAL EQUATIONS We have assume the following notations: u

=

Shock velocity of fluid

p

=

Pressure of the fluid

=

Density of the fluid

Cs

=

Standard velocity

=

Constant ratio of the specific heat

g

=

Acceleration due to gravity at a height

M

=

Mach number

=

Angle made by the radius vector to the vertical

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Prashant Chauhan & Mukesh Chandra

Let us consider that the explosion occur at a height HE from the surface of the earth A spherical shock front will propagate outward from the point of explosion. If we take the point of explosion as origin 'O',r 0 be the radius vector,

be

the angle made by the radius vector tothe vertical and go be the acceleration due to gravity at a distance r 0 from the origin 'O', then

(2.1) Where gs be the value of g0 at the surface of the earth and a0 be the radius of the earth. Let us consider that there is an explosion at the origin 'O', which is on the surface of the earth. The shock wave created by the explosion propagates in all directions. Let us assume that the component of the shock which propagates along the radius vector r0 and inclined at an angle to the vertical direction. Let us assume that p0,

0 , T0, u0 and g0 be the pressure, density, temperature, fluid velocity and the acceleration

due to gravity at a height and Ps, Ps, Ts sand gs be the corresponding values at the surface of the earth. Let t0, r0 be the time and distance and U be the shock velocity, the dimensionless parameters can be defined by

p

u=

p0 ps

u0

(2.2a)

cs

0 s

r

r0 HE

a

a0 HE

(2.2b)

t

t0 cs HE

g

g0 gs

U

And

U0

(2.2c)

cs R

R0 HE

A Study on Propagation of Spherical Shock Waves in the Atmosphere

81

Where the standard velocity cs is defined as

p cs s s

(2.3)

The equation of motion in terms of dimensionless parameters are given by

u u 0 t r r

(2.4)

u u 1 p a u cos 0 t r r a r cos 2

(2.5)

And

p t

p u r

0

(2.6)

The equation of state in non-dimensional parameter is given by

P 1

(2.7)

The variation of pressure in the atmosphere in equilibrium condition is given by

(2.8) The equation (2.7) can be written as

p1 /

(2.9)

By virtue of (2.8) and (2.9), we get

(2.10) Integrating the equation (2.10) from p = 1 to p and r = 0 to r and after some simplification, we obtain an expression for pressure 'p'

(2.11) As a consequence of (2.9) and (2.11), we obtain an expression for density ' '

(2.12) Using the relations (2.11) and (2.12) in the relation T = p / , we get

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Prashant Chauhan & Mukesh Chandra

(2.13) Therefore, the equations (2.11), (2.12) and (2.13) give the variations of the pressure, density and temperature at any point, whose radial co-ordinate be (r, ).

3. SOLUTION OF THE PROBLEM Assume that an explosion at height H = 1 due to which a spherical shock wave moves in all the directions at time t, let R be the shock position of the shock front, the point of explosion being the pole. When the shock created at the point O reaches the distance R, the jump conditions across the shock front at r = R are given by [7] and [8]

U 1 (U u1 )

(3.1)

u1U p1 p

(3.2)

(3.3) Where u1,p1 and

1 are the values of u, p and behind the shock front and is the adiabatic gas constant. The

quantities behind the shock front are functions of R and t and ahead the shock front they are functions of R alone, R be the shock position. It is considered that the fluid velocity u in front of the shock is zero. Solving the equations (3.1), (3.2) and (3.3) for

u , p , 1

1

1

in terms of Mach number, we obtain

(3.4)

(3.5) And

(3.6) Wherein

(3.7)

g ( M ) 1M 2 2

(3.8)

(3.9) And

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A Study on Propagation of Spherical Shock Waves in the Atmosphere

(3.10) To stabilize the problem we must know four unknown variables relate these four variables u1,p1,

u , p , p , and M in terms of p and 1,

L

1

ď ˛ . To

ď ˛1 and M we have three jump conditions (3.4), (3.5) and (3.6). One extra relation

between these four variables is required, which we obtain from the rule derived by Whitham's Rule [10]. According to the rule, we write the equations of motion just behind the shock front along the positive characteristic axis. The equation of motion along the positive characteristic axis, just behind the shock front is given by

(3.11) Now, differentiating the equations (3.4) and (3.5), we obtain

(3.12) And

(3.13) Using (3.5) and (3.6) in equation (3.10), we obtain

(3.14) Further, making use of the relations (3.4), (3.12), (3.13) and (3.1 4) in the relation (3.17) and after a bit calculation, we obtain

(3.15) Where

(3.16) The equation (3.15) gives the variation of M, if p and c are known. Logarithmically differentiating the equations (2.11) and (2.7) at r = R, we obtain

(3.17) And

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Prashant Chauhan & Mukesh Chandra

(3.18) Further more, using the equations (3.17) and (3.18) into the equation (3.15), and after some simplification, we obtain

(3.19) where

(3.20) And

(3.21)

If

then the equation (3.20) can be written as

(3.22) Table 4.1: Variation of K3(M) and M Mach Number (M) 1 2 3 4 5 6 7 8 9 10 15 20

K3(M) 0.0000 0.4135 0.5069 0.5436 0.5617 0.5718 0.5780 0.5821 0.5850 0.5869 0.5917 0.5934

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A Study on Propagation of Spherical Shock Waves in the Atmosphere

Figure 1.1 It is easy to verify by the figure 1.1 that K3(M) increases sharply as M increases from 1 to 3, having 0 and 0.5069 at M= 1 and 3 respectively. When M increases from 3 onwards, it's variation becomes negligible. We take the value of M at the surface of the earth as equal to 4. Thus if we take K3(M), which no doubt is a function of M, as a constant and evaluate M from equation (3.22). In figure (1.1), we have drawn the value of K3(M) versus M, for

= 1.4. If we found that the variation of

K3(M) is small for M 3, as compared to the variation of M. Integrating the equation (3.22) from M = M1 to M and taking K3(M) as a constant during integration, we obtain

(3.23) where M1 is the value of M at R = 0. By virtue of (3.9) and (3.23), we obtain the shock velocity U, is given by

(3.24) Thus the equations (3.23) and (3.24) give the variation of Mach number and the shock velocity as the shock propagates along the radial distance at an angle e to the vertical. Remark 3.1 If we take U =

R , then the equation (3.24) can be reduces in the form t

1 aR cos R 1 / 2 M 1 a R cos 1

1 K 3 ( M ) 1 2 ( 1)

.t

From the equation (3.25) we can compute the distance R at a time t.

(3.25)

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Prashant Chauhan & Mukesh Chandra

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Hayes, W.O.: The vorticity jump across a gas dynamics discontinuity. J. Fluid Mech, 2(1957), p.597-600.

2.

Kanwal,R.P.: Propagation of curved shocks in pseudo [1] stationary three dimensional gas flows. Illinois; J.Math, 2( 1958), p.129-136.

3.

Kanwal, R.P.: Flows behind shock waves in conducting gases, Proc. Roy. Soc., 257( 1960), p.263268.

4.

Lighthill, MJ.: Dynamic of dissociating gas part I, Equilibrium How. J. Fluid Mech., 2(1957), p.I-32.

5.

Mitra, S. K.: Upper atmosphere, Asiatic Society, Calcutta ( 1952).

6.

Pant, J.C. and Mishra, R.S.: Existence and uniqueness of flows behind three-dimensional unsteady and pseudo stationary curved shock waves III conducting gases, to appear in J. of Indian. Math. Soc., India

7.

Pai, S. l.: Introduction to the theory of compressible flow, D.Van. Nostrand Co. N.Y., (1959).

8.

Singh, V.P: Propagation of shock waves in the earth's atmosphere, Indian J.Physics, 43( 1969), p.519-527.

9.

Singh, V.P. and Kumar, P.: Propagation of oblique shock waves in the troposphere, Indian J.Physics, 44( 1970), p.554-560.

10. Whitham, G. B.: On the propagation of shock waves through region of non-uniform area or flow, J.Fluid Mech., 4(1958), p.337-360.