Multi-frequency Craik–Criminale solutions of the Navier–Stokes equations

Abstract

An exact Craik–Criminale (CC) solution to the incompressible Navier–Stokes (NS) equations describes the instability of an elliptical columnar flow interacting with a single Kelvin wave. These CC solutions are extended to allow multi-harmonic Kelvin waves to interact with any exact ‘base’ solution of the NS equations. The interaction is evaluated along an arbitrarily chosen flowline of the base solution, so exact nonlinear instability in this context is locally convective, rather than absolute. Furthermore, an iterative method called ‘WKB-bootstrapping’ is introduced which successively adds Kelvin waves with incommensurate phases to the extended CC solutions. In illustrating WKB bootstrapping, we construct a succession of extended nonlinear CC solutions consisting of a circular columnar flow interacting progressively with one (Kelvin 1887), two (Fabijonas & Lifschitz 1996), or three waves with incommensurate phases. The phase of each wave packet is frozen into the previous flow and we examine the exact nonlinear convective instability induced at each stage (primary, secondary, tertiary). At each stage, the flow becomes progressively more unstable.