Closed-form shock solutions

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 condu...

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Bibliographic Details
Published in:Journal of fluid mechanics 2014-04, Vol.745, p.np-np, Article R1
Main Author: Johnson, B. M.
Format: Article
Language:English
Subjects:
Algebra
Constants
Exact solutions
Fluid mechanics
Kinematic viscosity
Kinematics
Mach number
Mathematical analysis
Mathematical models
Navier-Stokes equations
Rapids
Shock waves
Thermal conductivity
Velocity
Viscosity
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Summary: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.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2014.107