Vortex Dynamics of Oscillating Flows

We employ the method of multiple scales (two-timing) to analyse the vortex dynamics of inviscid, incompressible flows that oscillate in time. Consideration of distinguished limits for Euler’s equation of hydrodynamics shows the existence of two main asymptotic models for the averaged flows: strong vortex dynamics (SVD) and weak vortex dynamics (WVD). In SVD the averaged vorticity is ‘frozen’ into the averaged velocity field. By contrast, in WVD the averaged vorticity is ‘frozen’ into the ‘averaged velocity + drift’. The derivation of the WVD recovers the Craik–Leibovich equation in a systematic and quite general manner. We show that the averaged equations and boundary conditions lead to an energy-type integral, with implications for stability.