Spontaneous Singularity Formation in Converging Cylindrical Shock Waves

Abstract

We develop a nonlinear, Fourier-based analysis of the evolution of a perturbed, converging cylindrical strong shock using the approximate method of geometrical shock dynamics (GSD). This predicts that a singularity in the shock-shape geometry, corresponding to a change in Fourier-coefficient decay from exponential to algebraic, is guaranteed to form prior to the time of shock impact at the origin, for arbitrarily small, finite initial perturbation amplitude. Specifically for an azimuthally periodic Mach-number perturbation on an initially circular shock with integer mode number q and amplitude proportional to ε ≪ 1, a singularity in the shock geometry forms at a mean shock radius Ru,c ∼(q2ε)-1/ b1 where b1 (γ) < 0 is a derived constant and γ the ratio of specific heats. This requires q2ε ≪ 1, q ≫ 1. The constant of proportionality is obtained as a function of γ and is independent of the initial shock Mach number M0. Singularity formation corresponds to the transition from a smooth perturbation to a faceted polygonal form. Results are qualitatively verified by a numerical GSD comparison.

Department(s)

Mechanical and Aerospace Engineering

Keywords and Phrases

Compressible flows; Shock waves; Structural properties; Pressure effects

International Standard Serial Number (ISSN)

2469-990X

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2018 American Physical Society (APS), All rights reserved.

Publication Date

23 Jul 2018

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