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Universality in the run-up of shock waves to the surface of a star

Published online by Cambridge University Press:  08 April 2011

C. GUNDLACH  and
R. J. LEVEQUE
C. GUNDLACH*
Affiliation:
School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK
R. J. LEVEQUE
Affiliation:
Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA 98195-2420, USA
*
†Email address for correspondence: cjg@soton.ac.uk
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Abstract

We investigate the run-up of a shock wave from inside to the surface of a perfect fluid star in equilibrium and bounded by vacuum. Near the surface we approximate the fluid motion as plane-symmetric and the gravitational field as constant. We consider the ‘hot’ equation of state P = (Γ − 1)ρe and its ‘cold’ (fixed entropy, barotropic) form P = K0ρΓ (the latter does not allow for shock heating). We numerically find that the evolution of generic initial data approaches universal similarity solutions sufficiently near the surface, and we explicitly construct these similarity solutions. The two equations of state show very different behaviour because shock heating becomes the dominant effect when it is allowed. In the barotropic case, the fluid velocity behind the shock approaches a constant value, while the density behind the shock approaches a power law in space, as the shock approaches the surface. In the hot case with shock heating, the density jumps by a constant factor through the shock, while the sound speed and fluid velocity behind the shock diverge in a whiplash effect. We tabulate the similarity exponents as a function of the equation of state parameter Γ and the stratification index n∗.

JFM classification

Mathematical Foundations: Computational methods Compressible Flows: Gas dynamics Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 676 , 10 June 2011 , pp. 237 - 264
Copyright
Copyright © Cambridge University Press 2011

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References

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