We present spatially developing direct numerical simulations (DNS) of turbulent boundary layers at Mach 3 and Mach 7 with Reτ ≈ 600. In this work we make an explicit distinction in the wall thermal boundary condition, which, to our knowledge, has not been addressed in the literature. Namely, we deem "weakly adiabatic" walls as those whose temperature is fixed at the recovery temperature, and "strongly adiabatic" walls as those that enforce null heat transfer in the local and instantaneous sense. These two boundary conditions are bracketing cases for real materials that have finite, non-zero thermal diffusivities. Using scaling arguments, we propose a dimensionless quantity, the "fluctuation Nusselt number," as the relevant similarity parameter describing the thermal damping at the wall. Furthermore, we demonstrate that this parameter can vary by many orders of magnitude due to different thermal diffusivities of relevant wall materials and different edge Mach numbers. By design, the "weakly adiabatic" boundary condition damps near wall temperature fluctuations, which helps to enforce the assumption of weak total temperature fluctuations built into much of the theory for compressible boundary layers. Adopting a "strongly adiabatic" wall will place greater strain on these assumptions and may be more relevant to flight conditions of interest. Here we present data at Mach 3 for both boundary conditions, and at Mach 7 for the "strongly adiabatic" case. The simulations are spatially developing and have large domains to prevent unphysical forcing due to the inflow and spanwise boundary conditions, as discussed in Beekman, Priebe, Kan & Martin. 1 For all three data sets we present basic turbulence statistics and note that a non negligible effect is observed due to the differences in wall boundary condition
