Spectral analysis of turbulence kinetic and internal energy budgets in hypersonic turbulent boundary layers

The evolution of turbulence kinetic and internal energy plays a critical role in aero-dynamics and thermodynamics. Following the previous study on the framework of energy exchange in compressible turbulent flows [Fan et al., Phys. Rev. Fluids 7, L092601 (2022)], we rigorously derive the spectral transport equations of turbulence kinetic and internal energy in terms of the two-point correlations of velocity and a sound-speed-like variable, respectively. With the equations, the spectral distributions of the budget terms, including production, dissipation, pressure strain, heat-conduction action, spatial and interscale trans-fer, are comprehensively examined in the hypersonic turbulent boundary layers at Mach number 5.86 and friction Reynolds number 420. Special attention is paid to the effect of wall cooling. The results enlighten the scale dependence of the energy transport across space and scales, and among components, helping us to clarify the evolution of turbulence kinetic and internal energy in high-speed turbulent flows.