The effects of harmonic and stochastic gravity modulation on fluid mixing

We study the effects of zero-mean harmonic and stochastic vertical gravity modulations on the mixing characteristics of two miscible Boussinesq fluids initially separated by a thin horizontal diffusion layer. For harmonic modulation, the main parameter governing the flow is Grashof number, $Gr$, based on the viscous length scale, $l_{\nu}=\sqrt{\frac{\nu}{\omega}}$, where $\omega$ is the frequency of the modulation. Contrary to the case of constant gravity, for which the arrangement is unstable, we observe a critical $Gr$ for the occurrence of Rayleigh-Taylor instability. This is explained on the basis of earlier work by Gresho \& Sani (1970). As $Gr$ is increased, we observe that the flow-field becomes chaotic. We investigate the route to chaos and compute various metrics to characterize it. The stochastic modulation is characterized by an exponentially damped cosine auto-correlation, $\langle g(t) g(t+\tau)\rangle/\langle g^{2}(t)\rangle=e^{-\lambda\tau}cos(\omega\tau)$, and has a power spectrum which is a Lorentzian with width $\lambda$ and peak at $\omega$, on which the Grashof number is based. We find that stochastic modulation leads to Rayleigh-Taylor instabilies at smaller equivalent Gr than harmonic modulation.