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The effects of gravity modulation on fluid mixing. Part 1. Harmonic modulation

Published online by Cambridge University Press:  14 August 2006

V. K. SIDDAVARAM  and
G. M. HOMSY
V. K. SIDDAVARAM
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106-5070, USA
G. M. HOMSY
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106-5070, USA
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Abstract

We study the effects of gravity modulation on the mixing characteristics of two interdiffusing miscible fluids initially in two vertical regions separated by a thin diffusion layer. We formulate the case of general gravity modulation of arbitrary orientation, amplitude $g$ and characteristic frequency $\omega$. For harmonic vertical modulation in two dimensions, the time-dependent Boussinesq equations are solved numerically and the evolution of the interface between the fluids is observed. The problem is governed by six parameters: the Grashof number, $\hbox{\it Gr}\,{=}\,({\Delta\rho}/{\bar{\rho}})g({l^{3}_{\nu}}/{\nu^{2}})$, based on the viscous length scale, $l_{\nu}\,{=}\,\sqrt{{\nu}/{\omega}}$; the Schmidt number, $\hbox{\it Sc}\,{=}\,{\nu}/{D}$; the aspect ratio, $A$; the non-dimensional length of the domain, $l$; the steepness of the initial concentration profile, $\delta$; and the phase angle of the harmonic modulation, $\phi$. When $\phi\,{=}\,0,\;\pi$, we observe four different flow regimes with increasing $\hbox{\it Gr}$: neutral oscillations at the forcing frequency; successive folds which propagate diffusively; localized shear instabilities; and both shear and convective instabilities leading to rapid mixing. In the last regime, the flow is disordered but not chaotic. By varying $\hbox{\it Sc}$, it was determined that the mechanism for the formation of these shear and convective instabilities is inertial. When $\phi \neq 0$ or $\pi$, the flow is similar to a modulated lock exchange flow.

JFM classification

Convection: Buoyancy-driven instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 562 , 10 September 2006 , pp. 445 - 475
Copyright
© 2006 Cambridge University Press

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Siddavaram and Homsy supplementary movie

Movie 1. Evolution of the concentration field when two miscible fluids, whose densities are $\bar{\rho}+\Delta\rho$ and $\bar{\rho}$, are subject to vertical harmonic gravity modulation of the form $g \cos({\omega}t+\phi)$, where $\phi$ is the phase angle. The main parameter governing the evolution is Grashof number, defined as $Gr = (\Delta\rho/{\bar{\rho}})g(l^3_{\nu}/\nu^2)$ where $l_\nu = (\nu/\omega)^{1/2}$ is the viscous length, $\omega$ is the frequency of the gravity modulation, and $\nu$ is the kinematic viscosity. In here we plot $C$, defined as the concentration of the heavier fluid, which is initially (i.e. at $t=0$) on the left side of the domain. Therefore, at $t=0$, $C$ varies smoothly from 1 on the left side to 0 on the right side. This movie illustrates the formation of folds for $Gr=10$.

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Siddavaram and Homsy supplementary movie

Movie 2. Evolution of the interface, taken as the contour line, C = 0.5, for the same conditions as movie no. 1 (Gr = 10). C is defined in the caption for movie 1. This representation shows the fold formation more clearly.

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Siddavaram and Homsy supplementary movie

Movie 3. Evolution of the concentration field for Gr = 14 showing Kelvin-Helmholtz (KH) instabilities which occur due to the nearly parallel shear flow between the folds. These KH instabilities lead to the detachment of concentration pockets which can be clearly seen between x ~ 100 – 200 and y ~ 300 – 350 from t ~ 9 – 11 periods.

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Siddavaram and Homsy supplementary movie

Movie 4. Evolution of the interface for the same parameter values as movie no. 3 Gr = 14). The KH instabilities leading to the break up of the interface can be more clearly seen in this representation in the region whose coordinates are given in movie no. 3.

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Siddavaram and Homsy supplementary movie

Movie 5. Evolution of the concentration field showing both KH and Rayleigh-Taylor (RT) instabilities for Gr = 22. The mushroom structure associated with the RT instability can be clearly seen between x ~ 50 – 150 and y ~ 300 – 350 at t ~ 5 periods. We do not show the interface representation for this value of Gr because it is highly disordered.

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Siddavaram and Homsy supplementary movie

Movie 6. The concentration fields for a phase angle of π/2, where the phase angle, φ, is defined in the caption for movie 1. (a) Gr = 10 - smooth gravity current (b) Gr = 14 - Kelvin-Helmholtz instability (c) Gr = 20 - Kelvin-Helmholtz and Rayleigh-Taylor instabilites.

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Siddavaram and Homsy supplementary movie

Movie 7. Evolution of the concentration fields, averaged over the full range of phase angles from 0 to 2π, for Gr = 10 and Gr = 14.

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