Resolvent-based analysis of hypersonic turbulent boundary layers with/without wall cooling

The ability of the low-rank approximation of hypersonic turbulent boundary layers with/without wall cooling is examined with the linear resolvent operator in a compressible form. The freestream Mach number of the base flow is 5.86, and the friction Reynolds number is 420. The wall-to-recovery temperature ratio is set as 1.0 and 0.25, respectively, corresponding to an adiabatic wall condition and a cold-wall condition. Different from the resolvent analysis of incompressible turbulent boundary layers, the optimal response mode in the wave-parameter space exhibits a relatively subsonic and a relatively supersonic region [Bae et al., "Resolvent-based study of compressibility effects on supersonic turbulent boundary layers," J. Fluid Mech. 883, A29 (2020)], divided by the freestream relative Mach number of unity. The features of energy distribution of the optimal response mode in space and scales are examined, and the energy spectra of streamwise velocity and temperature fluctuations, carried by the optimal response mode, are discussed with typical subsets of streamwise and spanwise wavelengths. This reveals the dynamics of the near-wall small-scale and outer larger-scale motions and the distinction in the relatively subsonic/supersonic region. Moreover, the coherent structures, including the velocity and temperature streaks, quasi-streamwise vortices, and large-scale/very-large-scale motions, are identified in the optimal response mode. Special attention is paid to the effects of wall cooling.