Power spectrum analysis in supersonic/hypersonic turbulent boundary layers

Large databases of Direct Numerical Simulation (DNS) of spatially-developing turbulent boundary layers (SDTBL) are scrutinized. The major purpose is to evaluate compressibility effects on power spectra of velocity fluctuations at the supersonic ($M_{\infty}$ = 2.86) and hypersonic ($M_{\infty}$ = 5.0) regime. An incompressible SDTBL is also examined. In all cases, the Reynolds numbers are sufficiently high in order to obtain a noticeable separation of turbulent scales. Adiabatic wall conditions are assumed for the compressible cases; whereas, an isothermal wall condition is prescribed for the incompressible flow (temperature regarded as a passive scalar) over Zero-Pressure Gradient (ZPG) flat plates. Turbulent inflow information is generated via a dynamic rescaling-recycling approach (Araya et al., JFM, 2011), which avoids the use of empirical correlations in the computation of inlet turbulent scales. This approach has been recently extended to compressible wall-bounded flows (Araya et al, SciTech, 2020). In general and based on power spectrum analysis, we observe a reduction in the dimensions of the coherent structures as Mach number increases. Further, as the Mach number grows, the structures become narrower and their downstream influence is notably diminished. The inner peak of the streamwise component of the Reynolds normal stresses, $(u'u')^+$, coincides with the major peak in the pre-multiplied spectra at roughly $y^+=10-15$. A less energetic but obvious secondary peak is found in the region $y^+ \approx$ 100-150 of the pre-multiplied spectra of $u'$. Further, the proposed inflow condition generation method proposed by Araya et al. (2011, 2020) was demonstrated to be highly robust with a minimal development region (in the order of $2.5\delta's$ at most) and an energy spectra resembling that of a fully developed flow. Interestingly, the outflow condition selection was found to have a more detrimental impact on the quality of the energy spectra.