Mixing by Shaking

Résumé: We study the mixing characteristics of two miscible fluids under the action of a harmonically varying vertical gravity force. The two fluids initially meet at a sharp but continuous vertical interface, and the time-dependent Boussinesq equations are solved numerically and the evolution of the interface is observed and characterized. The three important are the Grashof number, Gr based on the frequency; the Schmidt number, Sc; and the aspect ratio of the domain, A. For small values of Gr, the interface oscillates about the vertical centerline without any deformation. For intermediate Gr, the interface folds onto itself, and the propagation of these folds in time and space is observed to be self-similar in time with a scaling that is consistent with a Leveque type mechanism for vorticity propagation. Instabilities develop for higher values of Gr, which lead to enhanced mixing and which are explained on the basis of combined Kelvin-Hemhotz and Rayleigh-Taylor instabilities. For still higher values of Gr, the flow becomes disordered and transiently turbulent. The study is extended to include both correlated and uncorrelated stochastic gravity modulation. Ensemble averages of realizations exhibit dispersive spreading of the interface at a rate which is amplified by Gr, and which is larger for correlated jitter. Many of the phenomena occurring for deterministic jitter also occur in the stochastic case, but at a lower equivalent Gr. Accordingly, the rate of mixing for stochastic jitter is higher than that for deterministic harmonic jitter.