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Spontaneous singularity formation in converging cylindrical shock waves

Mostert, W. and Pullin, D. I. and Samtaney, R. and Wheatley, V. (2018) Spontaneous singularity formation in converging cylindrical shock waves. Physical Review Fluids, 3 (7). Art. No. 071401. ISSN 2469-990X. http://resolver.caltech.edu/CaltechAUTHORS:20180723-095359785

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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 R_(u,c)∼(q^2ε)^(−1/b1), where b_1(γ)<0 is a derived constant and γ the ratio of specific heats. This requires q^2ε≪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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/PhysRevFluids.3.071401DOIArticle
ORCID:
AuthorORCID
Samtaney, R.0000-0002-4702-6473
Additional Information:© 2018 American Physical Society. Received 28 March 2018; published 23 July 2018. This research was supported by the KAUST Office of Sponsored Research under Award No. URF/1/2162-01.
Group:GALCIT, Graduate Aeronautical Laboratories (Fluid Mechanics)
Funders:
Funding AgencyGrant Number
King Abdullah University of Science and Technology (KAUST)URF/1/2162-01
Record Number:CaltechAUTHORS:20180723-095359785
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20180723-095359785
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88121
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:23 Jul 2018 17:02
Last Modified:09 Oct 2018 23:17

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