On the role of discrete and continuous modes in a cooled high-speed boundary layer flow

Abstract

The disturbance flow field for a Mach $6$ flat-plate boundary layer flow with a wall-to-free-stream temperature ratio of $0.5$ is studied using direct numerical simulation (DNS), linear stability theory (LST) and biorthogonal decomposition. In the second-mode instability region, the DNS flow field can be reconstructed using a single LST mode, namely $F^+_1$ . However, when the supersonic mode emerges, none of the discrete modes nor the continuous branches alone can precisely match the DNS data. A superposition of the pair of discrete supersonic modes $F^pm _1$ and the slow acoustic continuous spectrum is required to reproduce the disturbance amplitude distributions and the amplification rates observed in the DNS. Another finding is that the supersonic mode $F^-_1$ , which should be decaying following LST, in actuality is amplified based on the projected DNS data.