The effects of gravity modulation on fluid mixing. Part 2. Stochastic modulation

Abstract

We study numerically the effects of zero-mean stochastic gravity modulation on the mixing characteristics of two interdiffusing miscible Boussinesq fluids initially separated by a thin diffusion layer. The gravity modulation has a Gaussian probability distribution and is characterized by an exponentially damped cosine autocorrelation function, i.e. $\langle g(t) g(t+\tau)\rangle/\langle g^{2}(t)\rangle\,{=}\,{\rm e}^{-\lambda \tau} \cos (\omega \tau)$. The associated power spectrum is a Lorentzian with peak at ω and width λ. The flow is found to depend on the following parameters: the Grashof number, Gr, based on the viscous length scale, $l_{\nu}\,{=}\,\sqrt{{\nu}/{\omega}}$; the Schmidt number, Sc; the correlation exponent, λ; and other geometric parameters. Even for extremely small Gr, we observe the propagation of gravity currents, Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instabilities. This is in contrast to the case of harmonic modulation considered in Part 1 (Siddavaram & Homsy J. Fluid Mech. vol. 562, 2006, p. 445) wherein these phenomena occur sequentially as Gr increases. The mixed volume is found to vary non-monotonically with the correlation exponent, λ, with narrow-band modulation having the largest mixed volume followed by harmonic modulation and then broadband modulation. This non-monotonicity of the mixed volume with λ is explained on the basis of the competition between the effects of excitation of lower frequencies, which lead to higher mixing, and the effects of the reduction in the energy content at the dominant frequency, which leads to reduced mixing. The value of the correlation coefficient, λ, at which the mixed volume is the largest is found to be independent of Gr. To understand the finer details of the mechanisms, we consider two- and three-frequency modulations. We find that increasing the amplitude of the secondary component when its frequency is smaller than that of the primary component leads to the occurrence of KH and RT instabilities at smaller Gr than that for the case of single-frequency modulation. We have understood the non-monotonic variation in the mixed volume by considering a three-frequency modulation, where one of the frequencies is smaller than the characteristic frequency and the other larger.