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

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.

Mode selection in trailing vortices: harmonic response of the non-parallel Batchelor vortex

In the present study, the response of model trailing vortices subjected to a harmonic forcing is studied. To this purpose, a globally stable non-parallel Batchelor vortex is considered as the baseflow. Direct numerical simulations (DNS) show that a large variety of helical responses can be excited and amplified through the domain when a harmonic inlet forcing is imposed. The spatial shape of the responses strongly depends on the forcing frequency, with the appearance of modes with progressively higher azimuthal wavenumber $m$ as the frequency increases. The mode-selection mechanism is shown to be directly connected to the local stability properties of the flow, and is simultaneously investigated by a WKB (Wentzel, Kramers, Brillouin) approximation in the framework of weakly non-parallel flows and by the global resolvent approach. In addition to the excellent agreement between the two (local and global) approaches for the computation of the linear response to harmonic forcing at the inlet, the usual WKB analysis is extended to a suitably chosen type of harmonic body forcing, showing also good agreement with the corresponding global results. As expected, the gain of the nonlinear response is significantly lower than that of the linear response, but the mode selection observed in the DNS as a function of the forcing frequency can be predicted fairly accurately by the linear analysis. Finally, by comparing the linear and nonlinear results in terms of energy content for different $m$, we suggest that the origin of the meandering observed in trailing-vortex experiments could be due to a nonlinear excitation stemming consistently at $m=1$ from the competition between the leading linear modes.