The stability of three-dimensional time-periodic flows with spatially uniform strain rates

Abstract

Unbounded incompressible fluid in solid-body rotation is subjected to spatially uniform strain rates that are sinusoidal in time and of arbitrarily large amplitude. The exact governing equations for the evolution of plane-wave disturbances to this time-periodic flow are linear, as for related steady flows. Attention focuses mainly on the in viscid problem, since incorporation of viscosity is straightforward.Plane-wave disturbances to axisymmetric flows are governed by a Hill's equation, or equivalently, a pair of first-order equations, to which Floquet theory is applied. Analytical and computational results show several instability bands, the first few of which can exhibit large growth rates. The exact governing equations for plane-wave disturbances to non-axisymmetric flows are similarly derived; but, as these are not singly periodic, results are given only for small-amplitude periodic forcing. As the non-axisymmetric strain produces a periodic elliptical distortion of the flow, a modified elliptical-instability mechanism joins that present in axisymmetric cases.Despite necessary idealizations, the analysis and results shed light on the stability of periodically strained vortices in a turbulent environment and in geophysical contexts.