Influence of upstream disturbances on the primary and secondary instabilities in a supersonic separated flow over a backward-facing step

Abstract

The development of primary and secondary instabilities is investigated numerically for a supersonic transitional flow over a backward-facing step at Ma = 1.7 and Re δ 0 = 13 718. Oblique Tollmien-Schlichting (T-S) waves with properties according to linear stability theory (LST) are introduced at the domain inlet with zero, low, or high amplitude (cases ZA, LA, and HA). A well-resolved large eddy simulation (LES) is carried out for the three cases to characterize the transition process from laminar to turbulent flow. The results for the HA case show a rapid transition due to the high initial disturbance level such that the non-linear interactions already occur upstream of the step, before the Kelvin-Helmholtz (K-H) instability could get involved. In contrast, cases ZA and LA share a similar transition road map in which transition occurs in the separated shear flow behind the step. Case LA is analyzed in detail based on the results from LST and LES to scrutinize the evolution of T-S, K-H, and secondary instabilities, as well as their interactions. Upstream of the step, the linear growth of the oblique T-S waves is the prevailing instability. Both T-S and K-H modes act as the primary mode within a short distance behind the step and undergo a quasilinear growth with a weak coupling. Upon pairing of the large K-H vortices, subharmonic waves are produced, and secondary instabilities begin to dominate the transition. Simultaneously, the growth of T-S waves is retarded by the slow resonance between subharmonic K-H and secondary instabilities. The vortex breakdown and reattachment downstream further contribute to the development of the turbulent boundary layer.