Supersonic turbulent boundary layer on a plate. II. Flow in the wall region and the Crocco integral

Abstract

We develop an asymptotic theory of the compressible turbulent boundary layer on a flat plate, in which the mean velocity and temperature profiles can be obtained as exact asymptotic solutions of the boundary-layer 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. In this part of the study, we consider the near-wall region that consists of viscous and logarithmic sublayers. The solution is constructed in the form of expansions in a small parameter ε that is proportional to the Mach number formed with the friction velocity and the speed of sound on the wall. Three characteristic flow regimes are possible in the viscous sublayer, which occur at small (including zero), moderate, and large negative wall heat flux. For the first two regimes, the flow is incompressible to the first approximation, while the compressibility is significant in the viscous sublayer on a strongly cooled plate. The Crocco integral in the logarithmic region is obtained, which in the zeroth-order approximation in ε gives the Waltz equation, but in contrast to it, the new relation describes well the dependence of temperature on velocity for any heat flux on the wall. Along with the constants known for incompressible flow, the theory contains two new universal constants, which are determined from a comparison with direct numerical simulation data for velocity and temperature.