THREE-DIMENSIONAL INSTABILITIES IN QUASI-TWO DIMENSIONAL INVISCID FLOWS.

The three-dimensional inviscid Navier-Stokes equations have a large family of exact steady 'quasi-two-dimensional' solutions, in which the velocity in the x,y plane is determined by a stream function psi (x,y), with the z-velocity and z-vorticity functions of psi (x,y) alone. If the projected streamlines in the x,y plane are closed curves, the flow may be subject to a broad band three-dimensional instability in the form of a wave packet centered on a particular surface of constant psi . The structure of the wave is determined by a Floquet system of ordinary differential equations around the corresponding psi contour in the x,y plane, and the Floquet exponent gives the growth rate. This family of instabilities includes the Rayleigh centrifugal instability, the Leibovich-Stewartson columnar vortex instability, and the secondary instability of finite-amplitude waves in plane shear flows.