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Application of an inner expansion method to plane, inviscid, compressible flow stability studies

The inviscid compressible flow stability problem is mathematically similar to that of sound propagation in a sheared flow field. This similarity has been exploited by applying an inner expansion technique to study the effect of finite shear gradients on free parallel flow instabilities. This technique had previously been used to investigate the effect of thin boundary layers on sound propagation in ducts. The expansion, which is applicable to flow profiles involving thin, but finite, shear layers separating regions of uniform flow, offers a significant computational advantage over the numerical methods commonly employed to determine the stability of continuous mean flow profiles. Although equally applicable to three-dimensional and to spatially growing hydrodynamic instabilities, the procedure is demonstrated by application to the eigenvalue problem for temporal instabilities of shear layers and jets in plane inviscid compressible flow. For the case of vanishingly thin shear layers, the eigenvalue equations derived here reduce to those obtained by Miles (1958) for parallel flows bounded by vortex sheets. The series solution of Graham & Graham (1969), valid for linear shear-layer profiles of arbitrary thickness, provides a basis of comparison for the expansion-method results. Unstable-mode eigenvalues obtained using the two methods are found to be in good agreement for a significant range of values of the ratio of shear-layer thickness to axial wavelength.