Analytical shock solutions at large and small Prandtl number

TL;DR

This paper derives exact analytical shock solutions in fluid dynamics for large and small Prandtl numbers, revealing different shock structures and providing insights into ideal gas behavior across these regimes.

Contribution

It introduces new analytical solutions for shock profiles at extreme Prandtl numbers, extending Becker's work and including effects of radiation-conducted heat transfer.

Findings

Large-Pr solution closely resembles Becker's solution with a scale change.

Small-Pr solution features an embedded isothermal shock at a critical Mach number.

Analytical solutions have maximum errors of O(1/Pr) for large Pr and O(Pr) for small Pr.

Abstract

Exact one-dimensional solutions to the equations of fluid dynamics are derived in the large-Pr and small-Pr limits (where Pr is the Prandtl number). The solutions are analogous to the Pr = 3/4 solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large-Pr solution is very similar to Becker's solution, differing only by a scale factor. The small-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-Pr equations is O(1/Pr) for large Pr and O(Pr) for small Pr.