A macroscopic two-length-scale model for natural convection in porous media driven by a species-concentration gradient

Abstract

The modelling of natural convection in porous media is receiving increased interest due to its significance in environmental and engineering problems. State-of-the-art simulations are based on the classic macroscopic Darcy–Oberbeck–Boussinesq (DOB) equations, which are widely accepted to capture the underlying physics of convection in porous media provided the Darcy number, $Da$ , is small. In this paper we analyse and extend the recent pore-resolved direct numerical simulations (DNS) of Gasow et al. (J. Fluid Mech, vol. 891, 2020, p. A25) and show that the macroscopic diffusion, which is neglected in DOB, is of the same order (with respect to $Da$ ) as the buoyancy force and the Darcy drag. Consequently, the macroscopic diffusion must be modelled even if the value of $Da$ is small. We propose a ‘two-length-scale diffusion’ model, in which the effect of the pore scale on the momentum transport is approximated with a macroscopic diffusion term. This term is determined by both the macroscopic length scale and the pore scale. It includes a transport coefficient that solely depends on the pore-scale geometry. Simulations of our model render a more accurate Sherwood number, root mean square (r.m.s.) of the mass concentration and r.m.s. of the velocity than simulations that employ the DOB equations. In particular, we find that the Sherwood number $Sh$ increases with decreasing porosity and with increasing Schmidt number $(Sc)$ . In addition, for high values of $Ra$ and high porosities, $Sh$ scales nonlinearly. These trends agree with the DNS, but are not captured in the DOB simulations.

Effects of pore scale and conjugate heat transfer on thermal convection in porous media

“…Furthermore, the boundary layer thickness from DNS results is determined by the pore size ( m / H ) for general cases, which is consistent with recent works on mass convection (Gasow et al. 2020, 2021). For the special cases at low Ra and high σ in group 2, the boundary layers were thicker due to the decrease in convection strength ( Ra f is considerably smaller for these cases).…”

“…When the dimensionless coordinate y/H is rescaled by φ/Ra, the temperature profiles in the boundary layer were shown to collapse for a wide range of inputs (Ra = 1000 to 20 000 at different φ and σ values). Furthermore, the boundary layer thickness from DNS results is determined by the pore size (m/H) for general cases, which is consistent with recent works on mass convection (Gasow et al 2020(Gasow et al , 2021. For the special cases at low Ra and high σ in group 2, the boundary layers were thicker due to the decrease in convection strength (Ra f is considerably smaller for these cases).…”

“…A recent work by Gasow et al. (2021) included the effects of momentum dispersion through a ‘two-length-scale diffusion’ model. By considering the pore-scale momentum transport through an effective diffusion term, they were able to obtain more accurate Sherwood numbers than with the traditional DOB equations.…”

“…Their work showed that increasing the mechanical dispersion would increase the plume spacing, thus showing that the grain/pore size plays a role in shaping the plumes and flow pattern. A recent work by Gasow et al (2021) included the effects of momentum dispersion through a 'two-length-scale diffusion' model. By considering the pore-scale momentum transport through an effective diffusion term, they were able to obtain more accurate Sherwood numbers than with the traditional DOB equations.…”

Prediction of spontaneous imbibition with gravity in porous media micromodels

“…The above studies are carried out on the basis of irregular geometries, while regular geometries composed of orderly pores and throats are also encountered quite often in both experimental and numerical researches (Corapcioglu, Chowdhury & Roosevelt 1997;Sun et al 2016;Wu et al 2016;Gasow et al 2021;Xie et al 2021;Yang et al 2021). To examine whether the developed prediction equations are general enough to describe various geometries, we investigate imbibition in a regular micromodel.…”