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

Published online by Cambridge University Press:  07 March 2011

EMILE TOUBER  and
NEIL D. SANDHAM
EMILE TOUBER*
Affiliation:
Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, University Road, Southampton, SO17 1BJ, UK
NEIL D. SANDHAM
Affiliation:
Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, University Road, Southampton, SO17 1BJ, UK
*
†Present address: Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK. Email address for correspondence: e.touber@imperial.ac.uk
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Abstract

A combined numerical and analytical approach is used to study the low-frequency shock motions observed in shock/turbulent-boundary-layer interactions in the particular case of a shock-reflection configuration. Starting from an exact form of the momentum integral equation and guided by data from large-eddy simulations, a stochastic ordinary differential equation for the reflected-shock-foot low-frequency motions is derived. During the derivation a similarity hypothesis is verified for the streamwise evolution of boundary-layer thickness measures in the interaction zone. In its simplest form, the derived governing equation is mathematically equivalent to that postulated without proof by Plotkin (AIAA J., vol. 13, 1975, p. 1036). In the present contribution, all the terms in the equation are modelled, leading to a closed form of the system, which is then applied to a wide range of input parameters. The resulting map of the most energetic low-frequency motions is presented. It is found that while the mean boundary-layer properties are important in controlling the interaction size, they do not contribute significantly to the dynamics. Moreover, the frequency of the most energetic fluctuations is shown to be a robust feature, in agreement with earlier experimental observations. The model is proved capable of reproducing available low-frequency experimental and numerical wall-pressure spectra. The coupling between the shock and the boundary layer is found to be mathematically equivalent to a first-order low-pass filter. It is argued that the observed low-frequency unsteadiness in such interactions is not necessarily a property of the forcing, either from upstream or downstream of the shock, but an intrinsic property of the coupled system, whose response to white-noise forcing is in excellent agreement with actual spectra.

JFM classification

Compressible Flows: Compressible boundary layers Nonlinear Dynamical Systems: Low-dimensional models Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 417 - 465
Copyright
Copyright © Cambridge University Press 2011

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References

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