The interaction of an oblique shock wave and a laminar boundary layer developing over a flat plate is investigated by means of numerical simulation and global linear-stability analysis. Under the selected flow conditions (free-stream Mach numbers, Reynolds numbers and shock-wave angles), the incoming boundary layer undergoes separation due to the adverse pressure gradient. For a wide range of flow parameters, the oblique shock wave/boundary-layer interaction (OSWBLI) is seen to be globally stable. We show that the onset of two-dimensional large-scale structures is generated by selective noise amplification that is described for each frequency, in a linear framework, by wave-packet trains composed of several global modes. A detailed analysis of both the eigenspectrum and eigenfunctions gives some insight into the relationship between spatial scales (shape and localization) and frequencies. In particular, OSWBLI exhibits a universal behaviour. The lowest frequencies correspond to structures mainly located near the separated shock that emit radiation in the form of Mach waves and are scaled by the interaction length. The medium frequencies are associated with structures mainly localized in the shear layer and are scaled by the displacement thickness at the impact. The linear process by which OSWBLI selects frequencies is analysed by means of the global resolvent. It shows that unsteadiness are mainly associated with instabilities arising from the shear layer. For the lower frequency range, there is no particular selectivity in a linear framework. Two-dimensional numerical simulations show that the linear behaviour is modified for moderate forcing amplitudes by nonlinear mechanisms leading to a significant amplification of low frequencies. Finally, based on the present results, we draw some hypotheses concerning the onset of unsteadiness observed in shock wave/turbulent boundary-layer interactions.
In recent years, the stability analysis in inhomogeneous flows in at least two spatial directions took a essort very important. However, these analyzes are restricted mostly to incompressible flows in relatively simple geometries. The extension of these methods in compressible regime poses a number of problems particularly for the boundary conditions because they must be compatible with the existence of acoustic waves propagating in the far field. One way to solve the problem is to extract the Jacobian of the discrete equations of motion. Consideration of more complex geometry requires on the use of a standard CFD solver, and a method of approximation of the Jacobian matrix because the complete extraction is unfeasible due to large memory requirement reasons. In addition, the standard linearization of a CFD code can be a complex and difficult step. A method of automatic differentiation is possible to approximate the Jacobian matrix. This method does not modify in depth the CFD code and allows to study the stability of flow in complex configurations. The objective of this paper is to present a procedure for calculating global modes using a standard CFD code. These methods will be illustrated by several examples: cylinder flow, 2-D bump flow and shockwave boundary-layer interaction flow.
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