Low-frequency resolvent analysis of the laminar oblique shock wave / boundary layer interaction

Abstract

Resolvent analysis is used to study the low-frequency behaviour of the laminar oblique shock wave / boundary layer interaction (SWBLI). It is shown that the computed optimal gain, which can be seen as a transfer function of the system, follows a first-order low-pass filter equation, recovering the results of Touber and Sandham (JFM, 2011). This behaviour is understood as proceeding from the excitation of a single stable, steady global mode whose damping rate sets the time scale of the filter. Different Mach and Reynolds numbers are studied, covering different recirculation lengths $L$. This damping rate is found to scale as $1/L$, leading to a constant Strouhal number $St_L$ as observed in the literature. It is associated with a breathing motion of the recirculation bubble. This analysis furthermore supports the idea that the low-frequency dynamics of the SWBLI is a forced dynamics, in which background perturbations continuously excite the flow. The investigation is then carried out for 3D perturbations for which two regimes are identified. At low wave numbers of the order of $L$, a modal mechanism similar to that of 2D perturbations is found and exhibits larger values of the optimal gain. At larger wave numbers of the order of the boundary layer thickness, the growth of streaks, which results from a non-modal mechanism, is detected. No interaction with the recirculation region is observed. Based on these results, the potential prevalence of 3D effects in the low-frequency dynamics of the SWBLI is discussed.