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 ω 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.