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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: johnson359@llnl.gov
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Abstract

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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 FlowsCompressible Flows: Shock waves

Information

Type
Rapids
Information
Journal of Fluid Mechanics , Volume 745 , 25 April 2014, R1
Copyright
© 2014 Cambridge University Press 

References

Becker, R. 1922 Stosswelle und Detonation. Z. Phys. 8, 321–362.Google Scholar
Hayes, W. D. 1960 Gasdynamic Discontinuities. Princeton University Press.Google Scholar
Iannelli, J. 2013 An exact nonlinear Navier–Stokes compressible-flow solution for CFD code verification. Intl J. Numer. Meth. Fluids 72, 157–176.Google Scholar
Johnson, B. M. 2013 Analytical shock solutions at large and small Prandtl number. J. Fluid Mech. 726, R4.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Butterworth-Heinemann.Google Scholar
Morduchow, M. & Libby, P. A. 1949 On a complete solution of the one-dimensional flow equations of a viscous, heat conducting, compressible gas. J. Aeronaut. Sci. 16, 674–684.Google Scholar
Taylor, G. I. 1910 The conditions necessary for discontinuous motion in gases. Proc. R. Soc. Lond. A 84, 371–377.Google Scholar
Thomas, L. H. 1944 Note on Becker’s theory of the shock front. J. Chem. Phys. 12, 449–452.Google Scholar
Zel’dovich, Ya. B. & Raizer, Yu. P. 2002 Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover.Google Scholar