Evolution of a shock generated by an impulsively accelerated, sinusoidal piston
Embargo End Date2021-05-26
Permanent link to this recordhttp://hdl.handle.net/10754/666154
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AbstractWe consider the evolution of a shock wave generated by an impulsively accelerated, two-dimensional, almost planar piston with a sinusoidally corrugated surface of amplitude. 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 , and follows a scaling inversely proportional to 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.
CitationShen, N., Pullin, D. I., Samtaney, R., & Wheatley, V. (2020). Evolution of a shock generated by an impulsively accelerated, sinusoidal piston. Journal of Fluid Mechanics, 907. doi:10.1017/jfm.2020.775
PublisherCambridge University Press (CUP)
JournalJournal of Fluid Mechanics