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Analytical shock solutions at large and small Prandtl number

Published online by Cambridge University Press:  14 June 2013

B. M. Johnson
B. M. Johnson*
Affiliation:
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
*
†Email address for correspondence: johnson359@llnl.gov
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Abstract

Exact one-dimensional solutions to the equations of fluid dynamics are derived in the $\mathit{Pr}\rightarrow \infty $ and $\mathit{Pr}\rightarrow 0$ limits (where $\mathit{Pr}$ is the Prandtl number). The solutions are analogous to the $\mathit{Pr}= 3/ 4$ solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large-$\mathit{Pr}$ solution is very similar to Becker’s solution, differing only by a scale factor. The small-$\mathit{Pr}$ solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite-$\mathit{Pr}$ equations is $\mathit{O}({\mathit{Pr}}^{- 1} )$ as $\mathit{Pr}\rightarrow \infty $ and $\mathit{O}(\mathit{Pr})$ as $\mathit{Pr}\rightarrow 0$.

JFM classification

Compressible Flows: Compressible Flows Compressible Flows: Shock waves
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , R4
Copyright
©2013 Cambridge University Press 

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