Viscous structure of plane waves in spatially developing shear flows

Abstract

This paper is concerned with the propagation of linear plane waves in incompressible, two-dimensional weakly nonparallel shear flows for large Reynolds numbers. Waves are analyzed for arbitrary complex frequency ω and local wave number k when nonparallel effects are assumed to be due to weak viscous diffusion. The inviscid approximation is shown to correctly describe, at leading order, the cross-stream variations of local plane waves at all stations where they are locally amplified in a frame of reference moving at the local phase speed ℜeω/ℜek, i.e., at stations where the temporal growth rate σ≡𝔉mω−𝔉mk ℜeω/ℜek remains positive. This result also holds as long as the local phase speed lies outside the range of values reached by the basic velocity profile. By contrast, the inviscid approximation fails to represent cross-stream variations in the critical layers when waves are locally neutral (σ=0), and in large viscous regions when they become damped (σ<0). Uniformly valid WKBJ approximations are derived in these regions and the results are applied to the description of forced spatial waves and self-excited global modes. © 1995 American Institute of Physics.