Temperature scaling for high-Mach-number flows above adiabatic walls*

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.

However, 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 scaling 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.