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Closed-form shock solutions

Published online by Cambridge University Press:  25 March 2014

B. M. Johnson
B. M. Johnson*
Affiliation:
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
*
†Email address for correspondence: [email protected]
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Abstract

It is shown here that a subset of the implicit analytical shock solutions discovered by Becker and by Johnson can be inverted, yielding several exact closed-form solutions of the one-dimensional compressible Navier–Stokes equations for an ideal gas. For a constant dynamic viscosity and thermal conductivity, and at particular values of the shock Mach number, the velocity can be expressed in terms of a polynomial root. For a constant kinematic viscosity, independent of Mach number, the velocity can be expressed in terms of a hyperbolic tangent function. The remaining fluid variables are related to the velocity through simple algebraic expressions. The solutions derived here make excellent verification tests for numerical algorithms, since no source terms in the evolution equations are approximated, and the closed-form expressions are straightforward to implement. The solutions are also of some academic interest as they may provide insight into the nonlinear character of the Navier–Stokes equations and may stimulate further analytical developments.

JFM classification

Compressible Flows: Compressible Flows Compressible Flows: Shock waves
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 745 , 25 April 2014 , R1
Copyright
© 2014 Cambridge University Press 

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