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In this paper, we perform direct numerical simulations (DNS) of turbulent boundary layers with nominal free-stream Mach number ranging from 0.3 to 12. The main objective is to assess the scalings with respect to the mean and turbulence behaviours as well as the possible breakdown of the weak compressibility hypothesis for turbulent boundary layers at high Mach numbers ( M > 5). We find that many of the scaling relations, such as the van Driest transformation for mean velocity, Walz's relation, Morkovin's scaling and the strong Reynolds analogy, which are derived based on the weak compressibility hypothesis, remain valid for the range of free-stream Mach numbers considered. The explicit dilatation terms such as pressure dilatation and dilatational dissipation remain small for the present Mach number range, and the pressure–strain correlation and the anisotropy of the Reynolds stress tensor are insensitive to the free-stream Mach number. The possible effects of intrinsic compressibility are reflected by the increase in the fluctuations of thermodynamic quantities ( p ′ rms / p w , ρ′ rms / ρ , T ′ rms / T ) and turbulence Mach numbers ( M t , M ′ rms ), the existence of shocklets, the modification of turbulence structures (near-wall streaks and large-scale motions) and the variation in the onset of intermittency.
Duan et al. (Wed,) studied this question.