Universality in the run-up of shock waves to the surface of a star
by Carsten Gundlach and Randall J Leveque. J. Fluid Mech., 676 (2011), pp. 237 - 264 . DOI: http://dx.doi.org/10.1017/jfm.2011.42

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 polytropic equation of state $P=K_0\rho^\Gamma$ and the related ideal gas equation of state $P=(\Gamma-1)\rho e$. We find numerically that the evolution of generic initial data approaches universal similarity solutions sufficiently near the surface, and we construct these similarity solutions explicitly. The two equations of state show very different behaviour, because shock heating becomes the dominant effect in the ideal gas case. In the polytropic case, the fluid velocity behind the shock approaches a constant value, while the density behind the shock approaches a power law. In the ideal gas case, 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 $\Gamma$ and the stratification index $n_*$.

Preprint: arXiv:1008.2834v1 [astro-ph.SR]

Simulations and code to accompany this paper

bibtex entry:

  author="C. Gundlach and R. J. LeVeque",
  title="Universality in the run-up of shock waves to the surface of a star", 
  journal="J. Fluid Mech.",

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