  A practical application of universal wall scalings is near-wall turbulence
modeling. In this paper, we exploit temperature's semi-local scaling [Patel,
Boersma, and Pecnik, {Scalar statistics in variable property turbulent channel
flows}, Phys. Rev. Fluids, 2017, 2(8), 084604] and derive an eddy conductivity
closure for wall-modeled large-eddy simulation of high-speed flows. We show
that while the semi-local 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 semi-local 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, 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 semi-local scaling but is modeled in the
equilibrium wall model. We incorporate aerodynamic heating in semi-local
scaling and show that the new scaling successfully collapses the high-speed DNS
data. We also show that incorporating aerodynamic heating or not, the
semi-local scaling gives rise to the exact same eddy conductivity, thereby
answering the outstanding question.
