Interaction of Vortical and Acoustic Waves: From General Equations to Integrable Cases

The equations of the $(2+1)$-dimensional boundary-layer perturbation split into eigenmodes: a vortex wave and two acoustic waves. We assume that the equations of state (Taylor series approximation) are arbitrary. We realize a mode definition via local-relation equations extracted from the linearization of the general system over the boundary-layer flow. Each such link determines an invariant subspace and the corresponding projector. We examine the nonlinear equation for a vortex wave using a special orthogonal coordinate system based on streamlines. The equations for the orthogonal curves are linked to the Laplace equations via Laplace and Moutard transformations. The nonlinearity determines the proper form of the interaction between vortical and acoustic boundary-layer perturbation fields fixed by projecting to a subspace of the Orr–Sommerfeld equation solutions for the Tollmienn–Schlichting (linear vortical) wave and by the corresponding procedure for the acoustic wave. We suggest a new mechanism for controlling the nonlinear resonance of the Tollmienn–Schlichting wave by sound via a four-wave interaction.