Reynolds-averaged closure modelling of the energy equation for compressible wall turbulence based on the bold italic k minus bold italic omega k − ω $\boldsymbol{k-\omega}$ equations

Wall cooling in hypersonic turbulent boundary layers produces an off-wall temperature maximum at which the turbulent heat flux vanishes, and the normally secondary turbulent kinetic-energy transport becomes locally important. We formulate an algebraic closure for the corresponding kinetic-energy contributions within a Reynolds-averaged k minus omega k − ω $k-\\omega$ framework. The transported variable k k $k$ remains in the turbulence equations and eddy-viscosity relation, whereas an algebraically reconstructed quantity script upper K K $\\mathcal{K}$ is used in the physical kinetic-energy terms of the momentum and energy equations. A local turbulent-Prandtl-number relation is obtained separately by regularised ensemble-Kalman inversion of two compressible-channel direct numerical simulation cases. Tests in compressible channels and zero-pressure-gradient boundary layers with free-stream Mach numbers from 2 to 14 show that the script upper K K $\\mathcal{K}$ reconstruction accounts for most of the reduction in the overpredicted temperature maximum; the new italic probability Subscript t Pr t $\\textit{Pr}_t$ relation provides a smaller but also effective additional correction, where italic probability Subscript t Pr t $\\textit{Pr}_t$ is the turbulent Prandtl number. Tests with the italic SST minus omega 0 SST − ω 0 $\\textit{SST}-\\omega _0$ and italic EARSM minus omega 0