
Physics 2014
Closedform shock solutionsDOI: 10.1017/jfm.2014.107 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 closedform solutions of the onedimensional compressible NavierStokes 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 closedform expressions are straightforward to implement. The solutions are also of some academic interest as they may provide insight into the nonlinear character of the NavierStokes equations and may stimulate further analytical developments.
