Analysis of an unstarting shock train with a modal method suited for non-stationary phenomena

Modal analyses of scale-resolved simulations provide significant insights into the physics of statistically stationary turbulent flows. For non-stationary turbulent problems, however, a time-mean steady state is ambiguous, and results from traditional modal techniques obscure the underlying dynamics. This work develops a method to recast the time-resolved non-stationary data in a form that recovers the utility of such decompositions. The problem of interest is a shock train undergoing unstart in an isolator duct, where a spatially distributed sequence of shock/turbulent boundary layer interactions collectively evolves and propagates upstream. The basic procedure uses time-domain information from empirical mode decomposition (EMD) to determine a suitable moving reference frame in which key elements of statistical stationarity are retained. Recorded snapshots, recast (or preconditioned) in the new moving frame of reference, may then be subjected to traditional modal decomposition methods without modification. Since the focus is on physically relevant frequencies, here we employ dynamic mode decomposition to reveal features that are concealed when directly applied to the unconditioned data. Application to the shock train is initiated by applying EMD to the time signal of the leading shock foot location, effectively separating the bulk upstream motion from the oscillatory components of the shock foot. In the translating reference frame, low-frequency separation bubble breathing motion becomes apparent at Strouhal numbers StL∼0.1 based on inflow velocity and the mean separation length, L. Furthermore, breathing timescales correlate well against the most energetic harmonic motions observed in the non-stationary shock foot signal.