TY - JOUR

T1 - Evolution of a shock generated by an impulsively accelerated, sinusoidal piston

AU - Shen, Naijian

AU - Pullin, D. I.

AU - Samtaney, Ravi

AU - Wheatley, V.

N1 - KAUST Repository Item: Exported on 2020-12-03

PY - 2020/11/26

Y1 - 2020/11/26

N2 - 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. 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.

AB - 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. 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.

UR - http://hdl.handle.net/10754/666154

UR - https://www.cambridge.org/core/product/identifier/S0022112020007752/type/journal_article

U2 - 10.1017/jfm.2020.775

DO - 10.1017/jfm.2020.775

M3 - Article

VL - 907

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -