Analysis of the two-dimensional dynamics of a Mach 1.6 shock wave/transitional boundary layer interaction using a RANS based resolvent approach

Abstract

A two-dimensional analysis of the resolvent spectrum of a Mach 1.6 transitional boundary layer impacted by an oblique shock wave is carried out. The investigation is based on a two-dimensional mean flow obtained by a RANS model that includes a transition criterion. The goal is to evaluate whether such a low cost RANS based resolvent approach is capable of describing the frequencies and physics involved in this transitional boundary layer/shock-wave interaction. Data from an experiment and a companion large eddy simulation (LES) are utilized as reference for the validation of the method. The flow is characterized by a laminar boundary layer upstream, a laminar separation bubble (LSB) in the interaction region and a turbulent boundary layer downstream. The flow exhibits low amplitude unsteadiness in the LSB and at the reflected shock wave with three particular oscillation frequencies, qualified as low, medium and high in reference to their range in Strouhal number, here based on free stream velocity and LSB length ($S_{t}=0.03{-}0.11$, 0.3–0.4 and 2–3 respectively). Through the resolvent analysis this dynamics is found to correspond to an amplifier behaviour of the flow. The resolvent responses match the averaged Fourier mode of the time dependent flow field, here described by the LES, with a close agreement in frequency and spatial distribution, thereby validating the resolvent approach. The low frequency dynamics relates to a pseudo-resonance process that sequentially implies the amplification in the separated shear layer of the LSB, an excitation of the shock foot and a backward travelling density wave. As this wave hits back the separation point the amplification in the shear layer starts again and loops. The medium and high frequency modes relate to the periodic expansion/reduction of the bubble and to the turbulent fluctuations at the reattachment point of the bubble, respectively.