Theory of vortex nutation and amplitude oscillation in an inviscid shear instability

Abstract

In the first part of this paper the nonlinear development of the most unstable mode is numerically studied for a bounded shear layer with a hyperbolic-tangent profile. It is found that the vortex nutation, discovered by Zabusky & Deem (1971) for a jet profile, is a manifestation of strongly coupled oscillations in the vortex amplitude and the phase. In the second part, with the aid of the numerical result we devote ourselves to deriving coupled nonlinear equations that describe the amplitude oscillation, the vortex nutation and the momentum transport. The approximate oscillatory solution for the vortex amplitude and phase in the nonlinear stage is compared with the numerical solution and agreement is found.