A practical application of universal wall scalings is near-wall turbulence modeling. In this paper, we exploit the semilocal scaling [Patel, Boersma, and Pecnik, Phys. Rev. Fluids, 2, 084604 (2017)] and derive an eddy conductivity closure for wall-modeled large-eddy simulation of high-speed flows. We show that while the semilocal scaling does not collapse high-speed direct numerical simulation (DNS) data, the resulting eddy conductivity and the wall model work fairly well. The paper attempts to answer the following outstanding question: why the semilocal scaling fails but the resulting eddy conductivity works well. We conduct DNSs of Couette flows at Mach numbers from $M=1.4$ to 6. We add a source term in the energy equation to get a cold wall, a close-to-adiabatic wall, and a hot wall. Detailed analysis of the flows' energy budgets shows that aerodynamic heating is the answer to our question: Aerodynamic heating is not accounted for in Patel et al.'s semilocal scaling but is modeled in the equilibrium wall model. We incorporate aerodynamic heating in the semilocal scaling and show that the new scaling successfully collapses the high-speed DNS data. We also show that incorporating aerodynamic heating or not, the semilocal scaling gives rise to the exact same eddy conductivity, thereby answering the outstanding question.

The mean velocity follows a logarithmic scaling in the surface layer when normalized by the friction velocity, i.e. a velocity scale derived from the wall-shear stress. The same logarithmic scaling exists for the mean temperature when one normalizes the temperature with the friction temperature, i.e. a scale derived from the wall heat flux. This temperature normalization poses challenges to adiabatic walls, for which the wall heat flux is zero, and the logarithmic temperature scaling becomes singular. This paper aims to establish a temperature transformation that applies to both isothermal walls and adiabatic walls. We show that by accounting for the diffusive flux, both the Van Driest transformation and the semi-local transformation (and other transformations alike) apply to adiabatic walls. We also show that the classic Walz equation works well for adiabatic walls because it models the diffusive flux, albeit in a rather crude way. For validation/testing, we conduct direct numerical simulations of supersonic Couette flows at Mach numbers $M=1$ , 3 and 6, and various Reynolds numbers. The two walls are adiabatic, and a source term is included to cancel the aerodynamic heating in the domain. We show that the adiabatic wall data collapse onto the same incompressible logarithmic law of the wall like the isothermal wall data.