Secondary instability of stationary Gortler vortices originating from first/second Mack mode

This work investigates the origination of the secondary instability in Gortler vortices using the linear stability theory, BiGlobal analysis, three-dimensional linear parabolized stability equations (3DLPSEs), and direct numerical simulation (DNS). The flow over a concave wall suffering from the Gortler instability and first/second Mack mode instability is selected. Furthermore, this work simulates the evolution of infinitesimal Mack mode disturbance in a flow perturbed by finite-amplitude Gortler vortices by using DNS and 3DLPSE methods. The 3DLPSE approach accurately predicts the process of Mack mode disturbance evolving into the secondary instability of Gortler vortices, and a perfect agreement with results by DNS is obtained. The results indicate that the secondary instability of stationary Gortler vortices can originate from the first/second Mack mode. The evolutions of first/second Mack mode with different spanwise wavenumbers are performed based on 3DLPSE and compared against the BiGlobal method. The results show that the shape functions and growth rates of disturbances always tend to the results of dominant modes obtained by the BiGlobal method. Because the dominant mode might shift from one to another, the overall evolution cannot be predicted only by the BiGlobal method based on a fixed mode. According to our computations, it is deduced that the Mack modes with the same frequency and symmetric characteristics would finally develop into the secondary instability with similar shapes.