Lower branch of second mode neutral curve determination base on APSE in hypersonic boundary layers

In hypersonic boundary layers, the lower branch of the neutral curve for the second Mack mode needs to consider the non-parallel effect and the modal synchronization mechanism. The linear parabolized stability equations (LPSE) are firstly used to find the lower branch of the neutral curve in hypersonic boundary layers, but face two major challenges: First, it is difficult to give the discrete spectrum as the initial condition at the inlet because the discrete mode is stable there and an iterative method is hard to be adopted; Second, due to the synchronization of the fast mode and the slow mode, the evolution of these modes should be computed separately to determine the lower branch of the neutral curve, resulting in the calculation efficiency relatively low. To solve the above problems, a new method for calculating the stability of non-parallel boundary layers based on the adjoint parabolized stability equations (APSE) is proposed in the present study, and it can be used to determine the lower branch with the consideration of the receptivity. This method starts from the unstable region of the Mack mode, as a result, the initial condition of the calculation is easier to give compared to LPSE. We compare the results given by the present method to those by both the linear stability theory (LST) and LPSE for various conditions. The prediction for the dominant mode upstream given by our new method is consistent with that by LPSE, but our new method does not rely on the selection of the initial value at the inlet, thus the efficiency is higher. The present method has the potential to further predict transition in hypersonic boundary layers.