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Uniformly valid analytical solution to the problem of a decaying shock wave

  • V. D. Sharma (a1), Rishi Ram (a1) and P. L. Sachdev (a2)

Abstract

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.

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References

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Ardavan-Rhad, H. 1970 The decay of a plane shock wave. J. Fluid Mech. 43, 737.
Friedrichs, K. O. 1948 Formation and decay of shock waves. Commun. Pure Appl. Maths 1, 211.
Lighthill, M. J. 1950 The energy distributions behind decaying shocks. Phil. Mag. 41, 1101.
Meyer, R. E. 1960 Theory of characteristics of inviscid gasdynamics. In: Handbuch der Physik (ed. S. Flügge), vol. IX. Springer.
Meyer, R. E. & Ho, D. V. 1963 Notes on non-uniform shock propagation. J. Acoust. Soc. Am. 35, 1126.
Pert, G. J. 1980 Self-similar flows with uniform velocity gradient and their use in modelling the free expansion of polytropic gases. J. Fluid Mech. 100, 257.
Pillow, A. F. 1949 The formation and growth of shock waves in the one-dimensional motion of a gas. Proc. Camb. Phil. Soc. 45, 558.
Stanyukovich, K. P. 1960 Unsteady motion of continuous media. Pergamon.
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Uniformly valid analytical solution to the problem of a decaying shock wave

  • V. D. Sharma (a1), Rishi Ram (a1) and P. L. Sachdev (a2)

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