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.

Biorthogonal decomposition of the disturbance flow field generated by particle impingement on a hypersonic boundary layer

“…It is particularly important to understand how modes F and S couple (synchronize) with the other branches of the spectrum (SA, FA, entropy and vorticity waves) because it can lead to the emergence of unstable modes that play a major role in triggering the transition of the boundary-layer flow. Mode coupling can occur, for example, due to non-parallel mean flow effects (Tumin 2020;Saikia et al 2022).…”

“…Mode coupling can occur, for example, due to non-parallel mean flow effects (Tumin 2020; Saikia et al. 2022).…”

“…Moreover, biglobal stability formulation can also be adopted to account for strong non-parallel flow effects as suggested by Forgoston & Tumin (2006) and Theofilis (2003). Recently, Saikia, Hasnine & Brehm (2022) employed BOD to fully reconstruct the disturbance flow field of a Mach 6 flat plate dominated by the supersonic mode. In this work, it was demonstrated that the stability behaviour of the flow could not be predicted by LST due to the presence of the continuous modes in addition to the two discrete supersonic modes.…”

A unified explanation of energy growth sources for unstable modes in flat-plate boundary layers

“…It is also generally accepted that wall cooling stabilises the first mode and destabilises the second mode. If the level of wall cooling is high, a new unstable branch called the supersonic mode can be generated by the synchronisation between the second mode and the slow acoustic wave of the continuous spectrum (Chuvakhov & Fedorov 2016; Saikia, Al Hasnine & Brehm 2022). This supersonic mode may radiate slow acoustic waves outside the boundary layer and also enlarge the unstable frequency band.…”