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A Family of Wave‐Mean Shear Interactions and Their Instability to Longitudinal Vortex Form
Author(s) -
Phillips W. R. C.,
Shen Q.
Publication year - 1996
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1996962143
Subject(s) - inviscid flow , instability , vortex , physics , wavenumber , mean flow , classical mechanics , amplitude , shear flow , mechanics , shear (geology) , rayleigh wave , turbulence , surface wave , optics , geology , petrology
The inviscid instability of O ( ε ) two‐dimensional periodic flows to spanwiseperiodic longitudinal vortex modes in parallel O (1) shear flows of the form ū = ± | z | q is considered. Here the mean velocity ū is relative to the wave and q is a constant. Such shear flows admit neutral Rayleigh waves with amplitudes that either diminish or diverge with | αz |; both are considered. Of particular interest are streamwise α and spanwise l wavenumbers in the range l 2 ≫ α 2 , α = O (1), as it is here that the most analytical progress can be made. A generalized Lagrangian‐mean formulation is used to describe the effect of fluctuations upon the mean state and, because the developing mean flow acts to distort the waves, a further equation, the Rayleigh‐Craik equation, is employed to complete the specification. It is shown that instability to longitudinal vortex form is likely for both classes of waves in many physically interesting situations, from simple mixing layers to atmospheric boundary layers over undulating surfaces.