Direct simulation of turbulent supersonic boundary layers by an extended temporal approach

The present paper addresses the direct numerical simulation of turbulent zero-pressure-gradient boundary layers on a flat plate at Mach numbers 3, 4.5 and 6 with momentum-thickness Reynolds numbers of about 3000. Simulations are performed with an extended temporal direct numerical simulation (ETDNS) method. Assuming that the slow streamwise variation of the mean boundary layer is governed by parabolized Navier-Stokes equations, the equations solved locally in time with a temporal DNS are modified by a distributed forcing term so that the parabolized Navier-Stokes equations are recovered for the spatial average. The correct mean flow is obtained without a priori knowledge, the streamwise mean-flow evolution being approximated from its upstream history. ETDNS reduces the computational effort by up to two orders of magnitude compared to a fully spatial simulation. We present results for a constant wall temperature T-w chosen to be equal to its laminar adiabatic value, which is about 2.5 T-infinity, 4.4 T-infinity and 7 T-infinity, respectively, where T-infinity is the free-stream temperature for the three Mach numbers considered. The simulations are initialized with transition-simulation data or with re-scaled turbulent data at different parameters. We find that the ETDNS results closely match experimental mean-flow data. The van Driest transformed velocity profiles follow the incompressible law of the wall with small logarithmic regions. Of particular interest is the significance of compressibility effects in a Mach number range around the limit of M-infinity similar or equal to 5, up to which Morkovin's hypothesis is believed to be valid. The results show that pressure dilatation and dilatational dissipation correlations are small throughout the considered Mach number range. On the other hand, correlations derived from Morkovin's hypothesis are not necessarily valid, as is shown for the strong Reynolds analogy.