Turbulent dissipation-rate equation for compressible flows

The turbulent dissipation rate in a compressible flow is modeled in terms of its solenoidal and dilatational parts. In the k-ε framework the solenoidal dissipation rate is computed using the modeled dissipation-rate equation for incompressible flows. This is based on the assumption that the solenoidal dissipation rate for compressible flows follows the same dynamics as the dissipation rate for incompressible flows. We test this assumption by comparing the exact transport equations of these two quantities, both analytically and using direct numerical simulation data of a Mach 4 boundary layer. The two equations are found to be equivalent except for an additional term caused by the variation of fluid viscosity in the compressible case. This forms the rigorous basis for using the incompressible modeled dissipation-rate equation in a compressible boundary-layer flow provided the extra term caused by variation of fluid viscosity is included. Finally, the implications of the dissipation-rate analysis on the k-ω turbulence model are pointed out.