High Rayleigh number convection in a three-dimensional porous medium

Abstract

High-resolution numerical simulations of statistically steady convection in a threedimensional porous medium are presented for Rayleigh numbers Ra 2 × 10 4 . Measurements of the Nusselt number Nu in the range 1750 Ra 2 × 10 4 are well fitted by a relationship of the form Nu = α 3 Ra + β 3 , for α 3 = 9.6 × 10 −3 and β 3 = 4.6. This fit indicates that the classical linear scaling Nu ∼ Ra is attained, and that Nu is asymptotically approximately 40 % larger than in two dimensions. The dynamical flow structure in the range 1750 Ra 2 × 10 4 is analysed, and the interior of the flow is found to be increasingly well described as Ra → ∞ by a heat-exchanger model, which describes steady interleaving columnar flow with horizontal wavenumber k and a linear background temperature field. Measurements of the interior wavenumber are approximately fitted by k ∼ Ra 0.52±0.05 , which is distinguishably stronger than the two-dimensional scaling of k ∼ Ra 0.4 .

A short review on porous media convection

“…The direct numerical simulations (DNS) of [36] show that the time-average inter-plume spacing in the 2D Rayleigh-Darcy system scales as δ ∼ Ra −0.4 and the interior flow can be modeled using a single horizontal Fourier-mode 'heat-exchanger' solution. Subsequently, they perform three-dimensional (3D) DNS in [38] and find that the high-Ra convection still retains the columnar structure in the 3D system, but with a different scaling for the time-average interplume spacing, i.e. α ≈ 0.5.…”

“…α ≈ 0.5. To explore the mechanism for this nonlinear scale selection, [36,[38][39][40] perform linear stability analysis of the heat-exchanger solution and the nonlinear steady solution. They find that the mean inter-plume spacing δ observed at large Ra results from an interplay between two types of instability: when δ is too small, a bulk mode controls the instability, causing plume merger and coarsening of the convective pattern; when δ is too large, a wall-mode instability dominates, causing small plumes generated from the walls to split the wider plumes into narrower ones.…”

“…Although an extra freedom in 3D may lead to more complicated convective patterns and affect the transport efficiency, the studies by [38,53,54] show that in 3D the flow still retains a similar columnar structure at large Ra in horizontal porous layers, and the convective flux increases linearly with Ra but with a slightly larger prefactor. Therefore, the analysis in 2D does, indeed, shed light on the real 3D system.…”

“…Moreover, porous media convection is also used as a classical example to study instabilities, bifurcations, pattern formation, and spatiotemporally chaotic dynamics [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

Effect of layered heterogeneity of a porous medium on density-driven natural convection

“…二酸化炭素気泡には界面張力に伴うトラップが働き，多孔質内部にトラップされる Suekane et al, 2008)． さらに， 二酸化炭素は徐々に地下水へと溶解する (Gilfillan et al, 2009;Iglauer, 2011;Lindeberg and Wessel-Berg, 1997;Lindeberg and Bergmo, 2003;Riaz and Cinar, 2014 Ghesmat et al, 2010;Hewitt et al, 2014;Hidalgo and Carrera, 2009;Pau et al, 2010;Xie et al, 2011)や実験 (Backhaus et al, 2011;Huppert et al, 1986;Neufeld et al, 2010) Fig. 2 Evolution of three-dimensional finger structure associated with the Rayleigh-Taylor convection in a porous medium with the layered structure.…”