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Evolution of a shock generated by an impulsively accelerated, sinusoidal piston

Published online by Cambridge University Press:  26 November 2020

N. Shen Open the ORCID record for N. Shen [Opens in a new window] ,
D. I. Pullin ,
R. Samtaney Open the ORCID record for R. Samtaney [Opens in a new window]  and
V. Wheatley
N. Shen*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA91125, USA
D. I. Pullin
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA91125, USA
R. Samtaney
Affiliation:
Mechanical Engineering, Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal23955-6900, Saudi Arabia
V. Wheatley
Affiliation:
School of Mechanical and Mining Engineering, The University of Queensland, St Lucia, QLD4072, Australia
*
†Email address for correspondence: nshen@caltech.edu
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Abstract

We consider the evolution of a shock wave generated by an impulsively accelerated, two-dimensional, almost planar piston with a sinusoidally corrugated surface of amplitude $\epsilon$. We develop a complex-variable formulation for a nonlinear theory of generalized geometrical shock dynamics (GGSD) (Best, Shock Waves, vol. 1, issue 4, 1991, pp. 251–273; Best, Proc. R. Soc. Lond. A, vol. 442, 1993, pp. 585–598) as a hierarchical expansion of the Euler equations that can be closed at any order. The zeroth-order truncation of GGSD is related to the equations of Whitham's geometrical shock dynamics (GSD), while higher-order corrections incorporate non-uniformity of the flow immediately behind the piston-driven shock. Numerical solutions to GGSD systems up to second order are coupled to an edge-detection algorithm in order to investigate the hypothesized development of a shock-shape curvature singularity as the rippled shock evolves. This singular behaviour, together with the simultaneous development of a Mach-number discontinuity, is found at all orders of the GGSD hierarchy for both weak and strong shocks. The critical time at which a curvature singularity occurs converges as the order of the GGSD system increases at fixed $\epsilon$, and follows a scaling inversely proportional to $\epsilon$ at sufficiently small values. This result agrees with the weakly nonlinear GSD analysis of Mostert et al. (J. Fluid Mech., vol. 846, 2018, pp. 536–562) for a general Mach-number perturbation on a planar shock, and suggests that this represents the universal behaviour of a slightly perturbed, planar shock.

JFM classification

Compressible Flows: Gas dynamics Compressible Flows: Shock waves Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , A35
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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