Low-order stochastic modelling of low-frequency motions in reflected shock-wave/boundary-layer interactions

Abstract

A combined numerical and analytical approach is used to study the low-frequencyshock motions observed in shock/turbulent-boundary-layer interactions in theparticular case of a shock-reflection configuration. Starting from an exact formof the momentum integral equation and guided by data from large-eddy simulations,a stochastic ordinary differential equation for the reflected-shock-foot low-frequencymotions is derived. During the derivation a similarity hypothesis is verified for thestreamwise evolution of boundary-layer thickness measures in the interaction zone. Inits simplest form, the derived governing equation is mathematically equivalent to thatpostulated without proof by Plotkin (AIAA J., vol. 13, 1975, p. 1036). In the presentcontribution, all the terms in the equation are modelled, leading to a closed form ofthe system, which is then applied to a wide range of input parameters. The resultingmap of the most energetic low-frequency motions is presented. It is found that whilethe mean boundary-layer properties are important in controlling the interaction size,they do not contribute significantly to the dynamics. Moreover, the frequency of themost energetic fluctuations is shown to be a robust feature, in agreement with earlierexperimental observations. The model is proved capable of reproducing available lowfrequencyexperimental and numerical wall-pressure spectra. The coupling betweenthe shock and the boundary layer is found to be mathematically equivalent to afirst-order low-pass filter. It is argued that the observed low-frequency unsteadinessin such interactions is not necessarily a property of the forcing, either from upstreamor downstream of the shock, but an intrinsic property of the coupled system, whoseresponse to white-noise forcing is in excellent agreement with actual spectra