The study of thermal convection in porous media is of both fundamental and practical interest. Typically, numerical studies have relied on the volume-averaged Darcy–Oberbeck–Boussinesq (DOB) equations, where convection dynamics are assumed to be controlled solely by the Rayleigh number ( Ra ). Nusselt numbers ( Nu ) from these models predict Nu – Ra scaling exponents of 0.9–0.95. However, experiments and direct numerical simulations (DNS) have suggested scaling exponents as low as 0.319. Recent findings for solutal convection between DNS and DOB models have demonstrated that the ‘pore-scale parameters’ not captured by the DOB equations greatly influence convection. Thermal convection also has the additional complication of different thermal transport properties (e.g. solid-to-fluid thermal conductivity ratio k s / k f and heat capacity ratio σ ) in different phases. Thus, in this work we compare results for thermal convection from the DNS and DOB equations. On the effects of pore size, DNS results show that Nu increases as pore size decreases. Mega-plumes are also found to be more frequent and smaller for reduced pore sizes. On the effects of conjugate heat transfer, two groups of cases (Group 1 with varying k s / k f at σ = 1 and Group 2 with varying σ at k s / k f = 1) are examined to compare the Nu – Ra relations at different porosity ( ϕ ) and k s / k f and σ values. Furthermore, we report that the boundary layer thickness is determined by the pore size in DNS results, while by both the Rayleigh number and the effective heat capacity ratio, $\bar{\phi } = \phi + (1 - \phi )\sigma$ , in the DOB model.
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Effects of pore scale on the macroscopic properties of natural convection in porous media
Natural convection in porous media is a fundamental process for the long-term storage of CO 2 in deep saline aquifers. Typically, details of mass transfer in porous media are inferred from the numerical solution of the volume-averaged Darcy–Oberbeck–Boussinesq (DOB) equations, even though these equations do not account for the microscopic properties of a porous medium. According to the DOB equations, natural convection in a porous medium is uniquely determined by the Rayleigh number. However, in contrast with experiments, DOB simulations yield a linear scaling of the Sherwood number with the Rayleigh number ( $Ra$ ) for high values of $Ra$ ( $Ra\gg 1300$ ). Here, we perform direct numerical simulations (DNS), fully resolving the flow field within the pores. We show that the boundary layer thickness is determined by the pore size instead of the Rayleigh number, as previously assumed. The mega- and proto-plume sizes increase with the pore size. Our DNS results exhibit a nonlinear scaling of the Sherwood number at high porosity, and for the same Rayleigh number, higher Sherwood numbers are predicted by DNS at lower porosities. It can be concluded that the scaling of the Sherwood number depends on the porosity and the pore-scale parameters, which is consistent with experimental studies.
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- Award ID(s):
- 1642262
- NSF-PAR ID:
- 10179708
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 891
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)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.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.164
