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 Tw chosen to be equal to
its laminar adiabatic value, which is about 2.5 T∞,
4.4 T∞ and 7 T∞, respectively,
where T∞ 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∞ ≃ 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.