A new mathematical model of rigid boundary in shear flows

Abstract

Abstract The present investigation is devoted to the revealing of the physical essence of the action of the rigid boundary on the linear dynamics of perturbations in plane shear flows. A new mathematical approach is proposed which accurately and completely replaces the horizontal rigid boundary by the actions of certain localized sources placed on the plane of the boundary (z = 0) in the original/canonical shear flow dynamical equations for wave and vortex mode perturbations, i.e. for perturbations with zero and nonzero potential vorticity. The approach is elaborated on the example of the well-known problem of geophysical hydrodynamics—the linear dynamics of perturbations in semi-infinite ( z ⩾ 0 ) flow of stratified rotating fluid with a vertical shear of velocity, {U} 0 ( A z , 0 , 0 ) , zero beta, β = 0, and horizontal rigid boundary at z = 0—which analytical wave and vortex mode solutions are well-known and serve as the reference solutions of our model equations. From the beginning, we have replaced the rigid boundary with localized sources for vortices and wave modes separately. The sources are the sum of the delta function and its first derivative the coefficients in front of which were unknown at this stage. In what follows, we accurately calculated these coefficients and thus determined the exact expressions for localized sources. Subsequent mathematical analysis showed that the localized sources give rise to new perturbations which can be labeled as secondary. The revealed physical essence of the action of the rigid boundary is as follows: being the external (localized) source of the analyzed open complex flow system, the rigid boundary puts an additional energy into perturbation harmonics that becomes an important power supplier supporting the linear dynamics of the system under study. Namely, the rigid boundary rearranges the vortex component via exciting new/secondary perturbations, which, superimposing with the main/primary perturbations of the flow, ensures zeroing of perturbations in the area z < 0 and, at the same time, provides the growth of the wave component in the area z ⩾ 0 coinciding with the well-known analytical solution of the canonical approach. From a general point of view, the following is important: In the existing/classical trend of studying the dynamics of perturbations in bounded shear flows, the actions of the flow boundary/boundaries and the flow shear are not delimited from each other. Whereas, the proposed new mathematical approach delimits them highlighting the action of the boundary on the linear dynamics of wave and vortex modes of perturbations.