Dynamic linear response of a shock/turbulent-boundary-layer interaction using constrained perturbations

Abstract

Comprehensive experimental and computational investigations have revealed possible mechanisms underlying low-frequency unsteadiness observed in spanwise homogeneous shock-wave/turbulent-boundary-layer interactions (STBLI). In the present work, we extend this understanding by examining the dynamic linear response of a moderately separated Mach 2.3 STBLI to small perturbations. The statistically stationary linear response is analysed to identify potential time-local and time-mean linear tendencies present in the unsteady base flow: these provide insight into the selective amplification properties of the flow at various points in the limit cycle, as well as asymmetry and restoring mechanisms in the dynamics of the separation bubble. The numerical technique uses the synchronized large-eddy simulation method, previously developed for free shear flows, significantly extended to include a linear constraint necessary for wall-bounded flows. The results demonstrate that the STBLI fosters a global absolute linear instability corresponding to a time-mean linear tendency for upstream shock motion. The absolute instability is maintained through constructive feedback of perturbations through the recirculation: it is self-sustaining and insensitive to external forcing. The dynamics are characterized for key frequency bands corresponding to high–mid-frequency Kelvin–Helmholtz shedding along the separated shear layer $(St_{L}\sim 0.5)$, low–mid-frequency oscillations of the separation bubble $(St_{L}\sim 0.1)$ and low-frequency large-scale bubble breathing and shock motion $(St_{L}\sim 0.03)$, where the Strouhal number is based on the nominal length of the separation bubble, $L$: $St_{L}=fL/U_{\infty }$. A band-pass filtering decomposition isolates the dynamic flow features and linear responses associated with these mechanisms. For example, in the low-frequency band, extreme shock displacements are shown to correlate with time-local linear tendencies toward more moderate displacements, indicating a restoring mechanism in the linear dynamics. However, a disparity between the linearly stable shock position and the mean shock position leads to an observed asymmetry in the low-frequency shock motion cycle, in which upstream motion occurs more rapidly than downstream motion. This is explained through competing linear and nonlinear (mass depletion through shedding) mechanisms and discussed in the context of an oscillator model. The analysis successfully illustrates how time-local linear dynamics sustain several key unsteady broadband flow features in a causal manner.