Nonlinear interaction between two second-mode disturbances with the same frequency in a hypersonic boundary layer over a swept blunt plate

Abstract

Zero- or low-frequency crossflow instabilities play a dominant role in the hypersonic boundary layer over a swept blunt plate. However, high-frequency second-mode disturbances dominate and lead to transition due to the wall cooling effect. The nonlinear interaction of high-frequency disturbances generates zero- or low-frequency disturbances. This paper investigates the relationship between the generated disturbances and the crossflow instability disturbances through nonlinear parabolized stability equations (NPSE) and direct numerical simulations. It is found that a special zero-frequency disturbance is generated by two second-mode disturbances with the same frequency but different spanwise wavenumbers, defined as a forced solution in this paper. This differs from the so-called eigenvalue solution obtained by linear stability theory, with the growth rate of the former being much larger than that of the latter. Additionally, the profile of the forced solution is similar to that of the second-mode disturbances, but not the crossflow-mode disturbances. The evolution of the forced solution is determined by the initial amplitude, spanwise wavenumber, and frequency of the second-mode disturbances, and is independent of the initial amplitude of crossflow-mode disturbances. If the second-mode disturbances begin to attenuate, the forced solution cannot maintain itself and transitions to the eigenvalue solution. The amplitude of the forced solution can be predicted using magnitude analysis through the amplitude of the second-mode disturbances, and the theoretical predictions are consistent with those of the NPSE when the second-mode disturbances are unstable. The results demonstrate that the forced solution is a mechanism for the second-mode disturbances to promote transition.