TY - JOUR

T1 - Spontaneous singularity formation in converging cylindrical shock waves

AU - Mostert, W.

AU - Pullin, D. I.

AU - Samtaney, R.

AU - Wheatley, V.

N1 - Funding Information:
This research was supported by the KAUST Office of Sponsored Research under Award No. URF/1/2162-01.
Funding Information:
This research was supported by the KAUST Office of Sponsored Research under Award No. URF/1/2162-01.
Publisher Copyright:
© 2018 American Physical Society.

PY - 2018/7

Y1 - 2018/7

N2 - 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 ϵ1, 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 q2ϵ1, 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.

AB - 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 ϵ1, 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 q2ϵ1, 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.

UR - http://www.scopus.com/inward/record.url?scp=85051138830&partnerID=8YFLogxK

U2 - 10.1103/PhysRevFluids.3.071401

DO - 10.1103/PhysRevFluids.3.071401

M3 - Article

AN - SCOPUS:85051138830

VL - 7

JO - Physical Review Fluids

JF - Physical Review Fluids

SN - 2469-990X

IS - 3

M1 - 071401

ER -