Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number

Abstract

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/pw, ρ′rms/ρ, T′rms/T) and turbulence Mach numbers (Mt, 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.

Mach-Number-Invariant Mean-Velocity Profile of Compressible Turbulent Boundary Layers

“…In previous studies, they were found to vary with M in the entrainment or intermittent layer [1,8]. In our plotting, however, they are M invariant up to vu at a given Re vw , agreeing with Morkovin's scaling.…”

“…The two assumptions give rise to the M invariance of ffiffiffiffiffiffi ffi " þ p @ " u þ =@y þ and further to VDT. The first assumption has been confirmed and referred to as Morkovin's scaling [4,8]. The second is not thoroughly studied [6], however.…”

“…In the bulk region, one finds quasiconstant a u i u j [1,8], but farther away from the wall, the M effect is significant. We argue that the observed M effect is due to improper 99 and Re .…”

“…Comparing Figs. 1(a) and 1(b), one finds that the M-invariant region extends to vu % 1:2 vw , significantly above 99 , where vu is the thickness of a vv ¼ a uu (above vu , an apparent M effect appears, possibly because the intermittency there is influenced by the Mach cone of flow disturbances [1,8]). Note that the significance of vw over 99 becomes more obvious if one notices that it is not the only M-invariant thickness, but so are several other layer thicknesses, including the viscous sublayer, the buffer layer, the bulk flow region, and the entrainment or intermittent layer, which are made up of an M-invariant four-layer structure that we believe is a general feature of the CTBL.…”

DNS of a spatially evolving hypersonic turbulent boundary layer at Mach 8

“…The results demonstrate that Morkovin's hypothesis and the strong Reynolds analogy (SRA) are still valid for 6  Ma  for a different wall temperature. Recently, Martin [8] and Duan et al [9][10][11] proposed a series of investigations on the compressible turbulence boundary layer over a flat-plate by the temporal evolving DNS, to assess the effects of wall temperature, Mach number and high enthalpy on the Morkovin's hypothesis and SRA. In general, when Ma ∞ =5, Morkovin's hypothesis is still considered valid for different wall temperatures.…”