L. Laguarda, S. Hickel, F.F.J. Schrijer, B.W. van Oudheusden (2024)*International Journal of Heat and Fluid Flow* 105: 109234. doi: 10.1016/j.ijheatfluidflow.2023.109234

Wall-resolved large-eddy simulations (LES) are performed to investigate Reynolds number effects in supersonic turbulent boundary layers (TBLs) at Mach 2.0. The resulting database covers more than a decade of friction Reynolds number Re_{τ} from 242 to 5554, which considerably extends the parameter range of current high-fidelity numerical studies. Reynolds number trends are identified on a variety of statistics for skin-friction, velocity and thermodynamic variables. The efficacy of recent scaling laws as well as compressibility effects are also assessed.

In particular, we observe the breakdown of Morkovin’s hypothesis for third-order velocity statistics, in agreement with previous observations for variable-property flows at low Mach number. Special attention is also placed on the size and topology of the turbulent structures populating the TBL, with an emphasis on the outer-layer motions at high Reynolds number. The corresponding streamwise spectra of streamwise velocity fluctuations show a clear separation between inner and outer scales, where energetic peaks are found at streamwise wavelengths of λ_{x} ~ 700 δ^{+} and λ_{x} ~ 6 δ_{99}. The spanwise spacing of the outer-layer structures, in turn, is found to be insensitive to the Reynolds number and equal to λ_{z} ~ 0.7 δ_{99}. It is also found that the integral length-scales in spanwise direction for the temperature, streamwise and spanwise velocity fields appear to progressively collapse with increasing Reynolds number.

The modulating influence that the outer-layer structures exert on the near-wall turbulence is also clearly visible in many of the metrics discussed. In addition, the present LES data is further exploited to assess the Re_{τ}-sensitivity of uniform momentum regions in the flow. We find that the resulting probability density function of the number of zones as well as its evolution with Re_{τ} agrees well with incompressible data. This suggests that uniform zones, which have been associated with outer-layer dynamics, are not strongly influenced by compressibility at the considered Mach number.

When mean-property variation effects are taken into account, temperature and density fluctuations are found to exhibit very similar behavior across the boundary layer (except for the near-wall region). This highlights the close relation between these two variables in compressible TBLs, which is further confirmed by their correlation coefficient being close to throughout the boundary layer. The fact that they are anti-correlated also reveals the non-isentropic nature of compressible wall-bounded turbulence. The resulting fluctuation profiles for the temperature and density are found to be almost flat from the upper end of the buffer layer until the boundary layer edge, where a peak seems to emerge at high Reynolds number. The fluctuating pressure, on the other hand, is not modulated by mean-property variation effects and the corresponding fluctuations do not resemble those associated with the temperature and density. This indicates a lower degree of correlation, i.e., higher decoupling, between them, which is also confirmed by their corresponding covariances. Pressure fluctuations are largest in the near-wall region, where the strongest vorticity fluctuations are found, and progressively decrease along the quasi-logarithmic region and towards the free-stream flow. While their behavior is qualitatively similar for the different TBLs considered, the intensity of the pressure fluctuations is also clearly sensitive to the Reynolds number across the whole boundary layer.

The present database demonstrates the deficiencies of the strong Reynolds analogy (SRA), starting with the presence of non-negligible total temperature fluctuations. The modified-SRA relation of Huang et al. (1995), however, shows clear improvements.