A spectral theory for weakly nonlinear instabilities of slowly divergent shear flows

A general theory is proposed for the downstream evolution of nonlinearly interacting waves in a nonparallel shear flow. The waves are represented as Fourier integrals over the spectral range. A multiple scales expansion is employed to predict evolutions of the Fourier mode amplitudes A(x,ω) - where x is the streamwise coordinate and ο is the frequency - by assuming that both nonlinear and nonparallel effects can be expressed in terms of the same expansion parameter ε. The theory produces in the leading order the Orr-Sommerfeld equation, and in the next order a Landau-type integrodifferential equation for A. The linear part of the equation contains only the first-order spatial derivative, and the nonlinear part consists of wave-wave interactions integrated over the entire spectral range of A. The approach is very general in that no Reynolds number (Re) scaling is involved as Re is retained independent of ε throughout and that no Landau constant is involved so that each wave can grow or decay in space or time. The Landau-type equation is too intricate for analytic solution but is suitable for numerical solution. An asymptotic analysis for large x via the method of stationary phase reveals that the subharmonic components provide the dominant contribution to the nonlinear terms, consistent with experimental observations.