Compressibility Modification to RANS Model for Hypersonic Cold-Wall Turbulent Boundary Layers

Severe compressibility and cold-wall effects in hypersonic turbulent boundary layers (HTBLs) cause issues in Reynolds-averaged Navier–Stokes (RANS) simulations with the Spalart–Allmaras model, including inaccurate velocity profiles, temperature peak overshoot, and failure to simulate wall-normal pressure variations. This paper analyzes the physical reason behind these issues based on direct numerical simulations (DNSs) and proposes novel effective modifications. First, the compressible law of the wall (CLW) for turbulent eddy viscosity is incorporated into the Spalart–Allmaras model and improves velocity profile accuracy in RANS simulation of HTBLs. Second, a CLW for the correlation between turbulent kinetic energy and Reynolds shear stress is discovered, and anisotropic distribution functions for second-eddy-viscosity coefficients are proposed, which enable accurate closure of Reynolds normal stresses and overcome the defect of the Spalart–Allmaras model in simulating wall-normal pressure variations in HTBLs. Finally, our studies reveal that the essential reason for the temperature overshoot is the neglect of molecular diffusion and turbulent transport in the RANS energy equation. These terms are closed by employing Wilcox’s model, and model coefficients are recalibrated using DNS results, resolving the temperature overshoot issue. RANS studies incorporating these modifications show that after accurately simulating mean profiles, prediction errors for skin friction and wall heat flux in HTBLs are reduced to under 3.5%.