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

Based on the normal velocity–normal vorticity (v–ηv–η) 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: v–vv–v, v–ηv–η and η–ηη–η terms, and are expressed in a simple geometric form using the modulus of the two wave-vectors (k′k′ and k″k″) appearing in the convolution integrals, and their intervening angle (χ). The non-linear terms in the v-equation involving η   are all weighted by sinχsinχ (or sin2χsin2χ). 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 (α=0α=0), only. It is therefore suggested how transiently amplified η 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 η-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.