Response of second-mode instability to backward-facing steps in a high-speed flow

Abstract

Stability in a Mach 4.5 boundary layer over backward-facing steps (BFSs) is investigated using numerical methods. Two types of cases are considered with different laminar inflow conditions, imposed with single-frequency or broadband-frequency modes, respectively. Compared with the typical K-type transition over a flat plate, the boundary layer transition initiated by 90 kHz-frequency second mode appears to follow the same pattern but with a noticeable delay over the step. A larger step height leads to a better inhibition of the downstream Λ-vortices and thus a later transition, providing the step height is smaller than the local boundary layer thickness. Moreover, both the frequency weighted power spectral density and the root mean square of the streamwise velocity indicate the presence of Kelvin–Helmholtz (K–H) instability when the step height is equivalent to the thickness of the nearby boundary layer. There may exist an optimal step height for suppressing single-frequency (90 kHz) mode without exciting significant K–H modes. Similar to the previous studies on roughness, BFS can act as an amplifier for the low-frequency second modes and a suppressor for the high-frequency second modes. The critical frequency is equal to that of the unstable mode whose synchronization point is exactly located at the step corner. Additionally, the correction effects of the step induce the change of the phase speed of the fast mode, which correspondingly results in the movement of the synchronization point. Generally, the BFS is not able to completely alleviate the transition initiated by the broadband-frequency second modes but can still delay the boundary layer transition in a certain degree by suppressing the high-frequency unstable waves.