LES wall modeling for heat transfer at high speeds

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)10.1103/PhysRevFluids.2.084604] 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 Formula Presented 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.