Supersonic turbulent boundary layer on a plate. III. Laws of the wall for velocity and temperature

We develop an asymptotic theory of compressible turbulent boundary layers on a flat plate, in which the mean velocity and temperature profiles can be obtained as exact asymptotic solutions of the boundary-layer and energy equations, which are closed using functional relations of a general form connecting the turbulent shear stress and turbulent enthalpy flux to mean velocity and enthalpy gradients. The laws of the wall for velocity and temperature are constructed in the form of expansions in a small parameter that is proportional to the Mach number formed with the friction velocity and the speed of sound on the wall. The leading term of the expansion for velocity coincides with the Van Driest formula; however, the law of the wall also contains a term of order one, the presence of which explains the discrepancy between the Van Driest formula and experimental and calculated data. The formulation of the law of the wall for temperature takes into account the fact that in the case of a cooled wall, the temperature varies non-monotonically across the boundary layer and has a local maximum in the logarithmic sublayer. Along with the constants known for incompressible flow, the theory contains three new universal constants, which are determined from a comparison with direct numerical simulation data for velocity and temperature.