Non-linear wave interactions from transient growth in plane-parallel shear flows

Based on the normal velocity-normal vorticity (nu-eta) formulation for the development of 3D disturbances in plane-parallel shear flows, the non-linear terms in the governing equations are derived as convolution integrals of the Fourier-transformed variables. They are grouped in three categories: nu-nu, nu-eta and eta-eta terms, and are expressed in a simple geometric form using the modulus of the two wave-vectors (k' and k '') appearing in the convolution integrals. and their intervening angle (chi). The non-linear terms in the nu-equation involving eta are all weighted by sin chi (or sin(2) chi). This confirms the known result that non-linear regeneration of normal velocity, necessary for a sustained driving of 3D disturbances, is not possible for stream-wise elongated structures (alpha = 0). only. It is therefore suggested how transiently amplified eta can interact with decaying 2D waves to activate (oblique) waves which may be less damped than the 2D wave. This is shown to be possible for Blasius flow. In the eta-equation, non-linear effects are possible for elongated structures resulting in shorter spanwise scales appearing at a shorter time-scale than the (linear) transient growth. A numerical example shows the details of this process in plane Poiseuille flow. From an inspection of the y-dependency (wall-normal direction) of the non-linear terms it is suggested that higher y-derivatives may give rise to non-linear effects in the inviscid development of perturbations. Also, a result for the y-symmetry of the non-linear terms is derived, applicable to plane Poiseuille flow. (C) 2008 Elsevier Masson SAS. All rights reserved.