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High-Rayleigh-number convection of a reactive solute in a porous medium

Published online by Cambridge University Press:  04 November 2014

T. J. Ward ,
O. E. Jensen ,
H. Power  and
D. S. Riley
T. J. Ward
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
O. E. Jensen*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
H. Power
Affiliation:
Faculty of Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK
D. S. Riley
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
*
†Email address for correspondence: Oliver.Jensen@manchester.ac.uk
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Abstract

We consider two-dimensional one-sided convection of a solute in a fluid-saturated porous medium, where the solute decays via a first-order reaction. Fully nonlinear convection is investigated using high-resolution numerical simulations and a low-order model that couples the dynamic boundary layer immediately beneath the distributed solute source to the slender vertical plumes that form beneath. A transient-growth analysis of the boundary layer is used to characterise its excitability. Three asymptotic regimes are investigated in the limit of high Rayleigh number $\mathit{Ra}$, in which the domain is considered deep, shallow or of intermediate depth, and for which the Damköhler number $\mathit{Da}$ is respectively large, small or of order unity. Scaling properties of the flow are identified numerically and rationalised via the analytic model. For fully established high-$\mathit{Ra}$ convection, analysis and simulation suggest that the time-averaged solute transfer rate scales with $\mathit{Ra}$ and the plume horizontal wavenumber with $\mathit{Ra}^{1/2}$, with coefficients modulated by $\mathit{Da}$ in each case. For large $\mathit{Da}$, the rapid reaction rate limits the plume depth and the boundary layer restricts the rate of solute transfer to the bulk, whereas for small $\mathit{Da}$ the average solute transfer rate is ultimately limited by the domain depth and the convection is correspondingly weaker.

JFM classification

Convection: Convection Convection: Convection in porous media Reacting Flows: Reacting Flows

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Papers
Information
Journal of Fluid Mechanics , Volume 760 , 10 December 2014 , pp. 95 - 126
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© 2014 Cambridge University Press 

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Ward et al. supplementary movie

Concentration (left) and streamfunction (right) for fully established convection in a deep domain for parameters as given in figure 2 of the main paper.

Download Ward et al. supplementary movie(Video)
Video 10.1 MB

Ward et al. supplementary movie

Concentration (left) and streamfunction (right) for fully established convection in a deep domain for parameters as given in figure 2 of the main paper.

Download Ward et al. supplementary movie(Video)
Video 6 MB