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

The present paper addresses the direct numerical simulation of turbulent ze ro-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 l ayer is governed by parabolized Navier-Stokes equations, the equations solv ed locally in time with a temporal DNS are modified by a distributed forcin g term so that the parabolized Navier-Stokes equations are recovered for th e spatial average. The correct mean flow is obtained without a priori knowl edge, the streamwise mean-flow evolution being approximated from its upstre am history. ETDNS reduces the computational effort by up to two orders of m agnitude compared to a fully spatial simulation. We present results for a constant wall temperature T-w chosen to be equal t o its laminar adiabatic value, which is about 2.5 T-infinity, 4.4 T-infinit y and 7 T-infinity, respectively, where T-infinity is the free-stream tempe rature for the three Mach numbers considered. The simulations are initializ ed with transition-simulation data or with re-scaled turbulent data at diff erent parameters. We find that the ETDNS results closely match experimental mean-flow data. The van Driest transformed velocity profiles follow the in compressible 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 th at pressure dilatation and dilatational dissipation correlations are small throughout the considered Mach number range. On the other hand, correlation s derived from Morkovin's hypothesis are not necessarily valid, as is shown for the strong Reynolds analogy.