On the Reynolds analogy for high-speed rough-wall flows: implications for wall modelling

We investigate compressible turbulent boundary layers over prism-shaped roughness using direct numerical simulations at Mach 2 and 4, with adiabatic and isothermal walls. Analysis of instantaneous and mean-density fields shows that compressibility effects and wall-thermal conditions significantly influence the boundary-layer structure, although the roughness elements remain submerged within the subsonic layer. While roughness disrupts near-wall momentum–energy coupling within the roughness sublayer, the Reynolds analogy recovers in the outer region where velocity and temperature fields exhibit smooth-wall-like similarity. A systematic comparison of different formulations indicates that a mixed Prandtl number approach provides the most accurate heat-flux predictions among the tested closures. Based on these observations, we develop a wall model that couples drag prediction for small prism-shaped roughness patterns with a compressibility transformation and a temperature–velocity closure. The formulation is modular, allowing the straightforward incorporation of alternative closures and traditional correlations. The model predicts both wall-shear stress and heat flux using only roughness geometry and information from the matching location, and shows good a priori agreement with the present direct numerical simulation data.