Cmjv03i01p0013

J. Comp. & Math. Sci. Vol.3 (1), 13-18 (2012)

A Study of Shockwave in Spherical Flow T. N. SINGH1 and A. K. RAY2 1

Department of Applied Sciences, B.B.S. College of Engineering and Technology Phaphamau, Allahabad-211014, U.P., India 2 Department of Mathematics, M. G. P. G College Gorakhpur, U.P., India (Received on: 30th November, 2011) ABSTRACT In this paper, we consider a spherical shell, whose thickness is very small as compared to its radius. An explosive (Pentaerythritoltetranitrate PETN) is deposited at the outer surface of the spherical shell. As we initiate the material, the initiation energy is negligible as compared to the chemical energy of the explosive, the shock is formed. The velocity of the imploding shock and detonation wave increases as they come closer to the axis of the shell, but they never reach the axis. Operator splitting technique has been used to analyze the flow. Imploding shock can produce ultra high temperature, pressure and density. Such rare conditions can be used to synthesize the material e.g. from graphite to diamond.

Keywords: Blast waves, instantaneous energy, variable energy, Spherical Imploding shock Spherical shock wave.

INTRODUCTION First we talk about imploding shock, the shock that converges towards a point or an axis is called imploding shock. Imploding shock problem have been studies in fluid dynamics, applied physics and engineering, Gaudery3 solved such problem, in which the similarity solution was presented and self amplifying character of wave was suggested.

Experimental demonstration of such shock wave was given by Perry and Kantrowitz10.

Denney and Wilson2 produced shock waves in air by electrically exploding thin metallic

film on the inner surface of glass spherical shell. The trajectories of the shock waves were compared to the similarity solution.

Lee5, Glass4, Saito and Glass11 and Matsuo et. al.8 have done works in this field. In fluid dynamics, implosion problems have been used to test the numerical methods e.g. random choice method. Lee1 and Matsuo7 have taken the practical side of such problem. The initial propagation of shocks generated by an instantaneous energy deposition at the spherical surface was analyzed by using perturbation technique.

Matsuo6 gave the global solution; the

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

14

T. N. Singh, et al., J. Comp. & Math. Sci. Vol.3 (1), 13-18 (2012)

solution which describes the whole history of the fluid motion, i.e. from initial stage to focusing stage, of such problem using the method of integral relation. The behavior of the problem was compared in detail with the similarity solution. It was shown that the shock trajectory approaches quickly the similarity solution but the spatial distribution of flow properties approaches it very slowly and never reaches the self similar implosion limits. Saito and glass adopted the random choice method to analyze the explosive driven hemispherical imploding shock. The imploding shock is assumed to propagate through the combustion products; the motion is prescribed by similarity solution. The estimated temperature was compared with spectroscopic ally measured temperature.

Van Dyke and Guttmann13 solved the spherical and cylindrical problem using series expansion in powers of time. Matsuo9 simulated strong imploding shocks travelling through atmospheric air.

From the application point of view, imploding shock and detonation waves can be used to produce ultra high temperature, pressure and density. The most important application is expected in synthesizing the material, e.g. diamond from graphite.

Srivastava and Srivastava12 have studied on spherical imploding shock. In the present work the flow induced by the generation and propagation cylindrical shells. The problem is closely associated with the explosion chamber developed by Matsuo for producing an extreme condition of ultra high pressure and temperature (35000 K). Operator splitting method has been used to simulate the problem.

FORMULATION OF PROBLEM We construct a spherical explosive shell of thickness d and radius R0 we assume that the thickness d of the shell is sufficiently small as compared to the radius R0 . The field inside the shell is air. The shell explosions are initiated and initiation energy is negligible as compared to the chemical energy of the explosive. The explosive is Pentaerythritoltetranitrate (PETN) in this problem. After the shell explosion, a spherical shock wave propagates towards the axis of the shell and finally converges there. We do not take into account the multiple interactions and propagation through the air. Therefore the detonation front forms a given boundary with known flow properties. The conservation lows may be written as follows.

∂U ∂F + +W = 0 ∂t ∂r ρ   Where U =  ρ u  , E  

(1)

 ρ   2  F =  ρu + p  u ( E + p )   

 ρu    r    ρu 2  and W =   r   u ( E + p)    r  

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

T. N. Singh, et al., J. Comp. & Math. Sci. Vol.3 (1), 13-18 (2012)

Where t and r are independent variables, that represent time and space coordinates respectively and ρ , u , p and E are density, particle velocity, pressure and total energy per unit volume. Equation (1) has been applied to air and combustion gas.

Both the gases air and combustion gas are assumed to be in viscid and none conducting.

Assuming polytrophic gas E is given by

E=

1 2 p ρu + 2 γ −r

Where γ is the ratio of specificheat. The boundary conditions on the front are ρ = ρCJ , u = uCJ , p = pCJ , where ρ CJ , uCJ , pCJ are density, particle velocity and pressure at Chapman – Jouguet detonation front.

These constants values are known when the explosive is specified. Another boundary condition is imposed on the outer surface of explosive, i.e. u = 0 at r = R0 Now, we define the dimensional quantities as follows,

r= u=

r C0t , t = , R0 γ 0 R0 u γ0 C0

, T =

p=

non-

p ρ , ρ= , p0 ρ0

R − Rs T , and e = 0 T0 R0

Where T is temperature, γ 0 , ρ 0 , p0 and

C0 are respectively the ratio of specific heats, density, pressure, velocity of sound

15

and temperature in undisturbed fluid and Rs the coordinate of the shock front.

NUMERICAL CALCUCATION Let ∆r and ∆t be the increments in the variables r and t where ∆r = R0 N , N being the number of mesh which are contained inside the spherical shell r = R0 .

The operator splitting technique applied to the problem is as follows.

U j* − U j n

∆t

= − Fr (U j n )

(2)

and

U j n +1 − U j n

∆t

= − W (U j * )

(3)

Where the subscript r denote the partial differentiation with respect to space coordinate r and

U j n+1 = U ( j ∆r, n∆t )

(4)

From (2) we can say that U * is a solution of the plan flow problem. The increment may be given as Courent–Friedrichs-Lewy condition as

∆t ≤

∆r u +c

(5)

Where c is the velocity of sound.

The condition of symmetry is used for the boundary condition at the rigid wall r = R0 i.e.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

16

ρ p

T. N. Singh, et al., J. Comp. & Math. Sci. Vol.3 (1), 13-18 (2012)

N −

1 2

N −

1 2

=ρ =p

N +

1 2

N +

1 2

,U

N −

1 2

=U

N +

1 2

pCJ = 7.68 ×109 pa

, (6)

Where N indicates the mesh point on the wall, for the present problem the number of meshes N is taken 1000. The explosive Pentaerythritoltetranitrate (PETN) has been taken for calculation. The characteristic values are

ρCJ = 1.34 ×103 kg / m3 uCJ = 1.0 ×103 m / sec

γ = 2.98 And

e = 5.65 MJ / kg

Where uCJ is the Chapman – Jouguet wave velocity and e is the specific chemical energy of PETN. Here γ which is larger then the maximum value that perfect gas can take i.e. 5 3 . It implies that the combustion gas of PETN is not a real perfect gas. The ratio of specific heats of air ( γ 0 ) is assumed to be 1.4.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

T. N. Singh, et al., J. Comp. & Math. Sci. Vol.3 (1), 13-18 (2012)

RESULTS

REFERENCES

As the shock approaches the axis more closely, the distribution of flow properties would becomes steeper and steeper in a very small region just behind the shock front. At a later stage of implosion, the shock wave is accelerated by self amplifying effect as a result of decreasing frontal area. Spatial distribution of flow properties are shown in Figure 1 and 2.

1.

The flow induced by the generation and propagation of spherical imploding shock is numerically analyzed. The shocks are supposed to be generated by detonating explosive shell. The distribution of flow variables is shown. The rate of implosion is faster in case of spherical shock wave.

2.

3.

4.

5.

6.

7.

8.

17

G. G. Bach, and J. H. Lee, J. Fluid Mach. 37, 513 (1969).

R. S. Dennen, L.N. Wilson, Exploding Wires, edited by W.G. Chance and H. K. Mooe, 2, 145 (1962).

Guderely, G. Luftfahrt – Forsch, 19, 302 (1942).

Glass, I.I. Chan, S.K. and Brode, H.L. AIAAJ, 12, 367 (1974).

Lee, J.H., and Lee, B.H.K. Phys. Fluids 8, 2148 (1965).

Matsuo, H., Phys. Fluids 26, 1755 (1983).

Matsuo, H.,Mem. Fac. Eng. Kumamoto Univ, 23, 1 (1977).

Matsuo. H. and Nakamura, Y., J. Appl.

Phys. 52, 4503 (1981).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

18 9.

T. N. Singh, et al., J. Comp. & Math. Sci. Vol.3 (1), 13-18 (2012)

Matsuo, H., Ohya, Y., Fujiwara, K., and Kudoh, H. Comput, Phys. 75,384 (1988).

10. Perry, R.W., and Kantrowitz, A. J. Applied phys. 22,278 (1951).

11. Saito, T. and Glass, I. I. Proc. R, Soc.

London Ser A 384 (1982).

12. Srivastava, R.C. Srivastava R., and Leutlof, D. for East J. Appl. Math 4(3), 371-378 (2000).

13. Van Dyke, M. and Guttmann, A.J., J. fluid Mech. (1982).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)