Characteristic Scales in Shock–Turbulence Interaction

Abstract

A semi-empirical estimate of the time-averaged thickness of a planar shock embedded in a turbulent mean flow is presented in an effort to quantify the characteristic time and length scales for turbulence modeling. Simplified Favre-averaged Navier–Stokes equations are reformulated and combined with the Rankine–Hugoniot relations to obtain an equation that relates the shock structure and its characteristic thickness to the upstream turbulent and mean flow quantities. The only mean flow and turbulence quantities needed to compute the shock thickness or velocity profile through the shock are upstream Mach number Mu, turbulent Mach number Mt, and the Taylor-based Reynolds number Reλ. The accuracy of method for the shock structure is validated against direct numerical simulations of shock–turbulence interaction and shows a very good agreement over a wide range of Mach, Reynolds, and turbulent Mach numbers. Finally, the obtained ratio of characteristic shock time scale to the upstream turbulence time scale is shown to be proportional to the turbulent kinetic energy amplification through the shock and supported through direct numerical simulations.