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

Shen, N. and Pullin, D. I. and Samtaney, R. and Wheatley, V. (2021) Evolution of a shock generated by an impulsively accelerated, sinusoidal piston. Journal of Fluid Mechanics, 907 . Art. No. A35. ISSN 0022-1120. doi:10.1017/jfm.2020.775.

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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 ϵ. 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 ϵ, and follows a scaling inversely proportional to ϵ 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.

Item Type:Article
Related URLs:
URLURL TypeDescription
Shen, N.0000-0002-0533-8081
Samtaney, R.0000-0002-4702-6473
Wheatley, V.0000-0002-7287-7659
Additional Information:© The Author(s) 2020. Published by Cambridge University Press. Received 30 April 2020; revised 1 August 2020; accepted 9 September 2020. Published online by Cambridge University Press: 26 November 2020. This work was supported by the KAUST Office of Sponsored Research under Award No. URF/1/3418-01. The authors also thank Professor H. G. Hornung for helpful discussions. The authors report no conflict of interest.
Funding AgencyGrant Number
King Abdullah University of Science and Technology (KAUST)URF/1/3418-01
Subject Keywords:gas dynamics, shock waves, nonlinear instability
Record Number:CaltechAUTHORS:20201216-084823910
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Official Citation:Shen, N., Pullin, D., Samtaney, R., & Wheatley, V. (2021). Evolution of a shock generated by an impulsively accelerated, sinusoidal piston. Journal of Fluid Mechanics, 907, A35. doi:10.1017/jfm.2020.775
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:107115
Deposited By: Tony Diaz
Deposited On:16 Dec 2020 17:18
Last Modified:16 Nov 2021 18:59

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