Consistent energy-based framework of amplification mechanisms for the second mode in hypersonic boundary layers

Abstract

The second mode is of general interest in hypersonic boundary layer flows due to its underlying responsibilities for transition to turbulence. However, a long-term debate exists on the detailed energy sources that sustain the modal exponential growth. Currently, three influential energy-based approaches appear to show different significant energy sources due to dissimilar mathematical formulations, including the momentum potential theory, the inviscid Lagrangian energy analysis, and the relative phase analysis. In this study, these three fundamental approaches are employed and examined in conjunction with direct numerical simulations. The purpose is to seek a possible unified explanation of the source terms that dominate the exponential evolution of the second mode. In the considered Mach 6 flow state, all three approaches consistently point to the same local energy amplification route driven by two pronounced source terms: the dilatation term in the near-wall region and the Reynolds thermal stress term or heat exchange term across the outer layer region, depending on the selection of the specific energy norm. The mathematical forms of the corresponding sources are derived or discussed explicitly. Theoretical and simulation results provide a unified understanding of the local energy amplification mechanisms of the second mode.