Similarity Solutions for Strong Shocks in an Ideal Gas

Abstract

The method of Lie group invariance is used to obtain a class of self-similar solutions to the problem of shocks in an inhomogeneous gaseous medium and to characterize analytically the state-dependent form of the medium ahead for which the problem is invariant and admits self-similar solutions. Different cases of possible solutions, known in the literature, with a power law, exponential, or logarithmic shock paths are recovered as special cases depending on the arbitrary constants occurring in the expression for the generators of the transformation. Particular cases of plane rising shocks in an exponential medium and collapse of an imploding shock are worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of the flow variables, and comparison is made with the known results.

Similarity solutions for two-dimensional weak shock waves

“…Finally we point out that (13)- (15) are invariant under a change of coordinates from (α, β) to (α, β) where d β = ψ(β)dβ by absorbing the extra factor ψ(β) into u such that u(α, β) = u(α, β)ψ(β) = u(α, β)d β/dβ and u(α, β) = u(α, β). In (15), when h = 0, the extra factor in terms of ψ(β) can be absorbed into Z (β). Under this new change of coordinates, u and β do not represent cross-sectional area and arclength, respectively.…”

“…For the class of power-law-type solutions we have obtained ten solutions that take into account the effects of nonlocal nonlinearity. We have also pointed out that the governing equations (13)- (15) for an initially single hump are invariant under a certain change of coordinates. This is particularly useful for obtaining separable solutions and could simplify the computations.…”

“…3-29, 269-291], [11, pp. 367-407] and for more recent applications see [12][13][14][15] and [16, pp. 65-105].…”

Numerical study of shock waves in non-ideal magnetogasdynamics (MHD)

“…A theoretical study of the imploding shock wave near the center of convergence, in an ideal gas was first investigated by Guderley [1]. Several authors contributed to this investigation and we mention the contributions of, Hafner [2], Manganaro and Oliveri [3], Sharma and Radha [4], Hunter and Ali [5], Sharma and Arora [6], Stanyukovich [7], Chisnell [8], Lazarus and Richtmyer [9], Ramu and Ranga Rao [10], Madhumita and Sharma [11], Sen [12], who presented high accuracy results and alternative approaches for the investigation of implosion problem. The propagation of shock waves under the influence of strong magnetic field is of great interest to many researchers in various fields such as astrophysics, nuclear science, geophysics, and plasma physics.…”

Asymptotical Solutions for a Vibrationally Relaxing Gas

“…There has been widespread interest in the nonlinear wave phenomena. The work of Whitham [28], Moodie et al [18], He and Moodie ([9,10]), Shtaras [27], Kalyakin [13], Krylovas and Čiegis ( [14,15]), Sharma and Radha [24], Sharma and Srinivasan [25], Sharma and Arora [23], Arora and Sharma [3], and Arora ( [1,2]) is worth mentioning in the context of nonlinear wave propagation in gas dynamic media.…”