Rayleigh-Darcy convection with hydrodynamic dispersion

We investigate the effect of hydrodynamic dispersion on convection in porous media by performing direct numerical simulations (DNS) in a two-dimensional Rayleigh-Darcy domain. Scaling analysis of the governing equations shows that the dynamics of this system are not only controlled by the classical Rayleigh-Darcy number based on molecular diffusion, ${\text{Ra}}_m$, and the domain aspect ratio, but also controlled by two other dimensionless parameters: the dispersive Rayleigh number ${\text{Ra}}_d=H/{\alpha }_t$ and the dispersivity ratio $r={\alpha }_l/{\alpha }_t$, where $H$ is the domain height and ${\alpha }_t$ and ${\alpha }_l$ are the transverse and longitudinal dispersivities, respectively. For $\mathrm{\Delta }={\text{Ra}}_d/{\text{Ra}}_m>O\left(1\right)$, the influence from the mechanical dispersion is minor; for $\mathrm{\Delta }\lesssim 0.02$, however, the flow pattern is determined by ${\text{Ra}}_d$ while the convective flux is $F\sim c\left({\text{Ra}}_d\right){\text{Ra}}_m$ for large ${\text{Ra}}_m$. Our DNS results also show that the increase of mechanical dispersion, i.e., decreasing ${\text{Ra}}_d$, will coarsen the convective pattern by increasing the plume spacing. Moreover, the inherent anisotropy of mechanical dispersion breaks the columnar structure of the megaplumes at large ${\text{Ra}}_m$, if ${\text{Ra}}_d<5000$. This results in a fan-flow geometry that reduces the convective flux.

Figure 1

Time-averaged horizontal-mean concentration profile $\langle \overline{C}\rangle$ and snapshots of the concentration field $C$ from DNS at ${\text{Ra}}_m=20000$ and $r=10$ for different ${\text{Ra}}_d$. The domain aspect ratio is $L=5$. In panel (a), only half of $\langle \overline{C}\rangle$ is shown due to its antisymmetry about the midplane, and the $z$ values on the horizontal axis are nonuniformly spaced to clearly show the structure near the wall. Increasing mechanical disperison (decreasing ${\text{Ra}}_d$) thickens the diffusive boundary layer, coarsens the flow pattern, and stabilizes the flow. Moreover, the convection transitions to a fan-flow structure at ${\text{Ra}}_d<5000$.

Figure 2

Averaged DNS results of convection at ${\text{Ra}}_m=20000$ and $r=10$ for different ${\text{Ra}}_d$. The domain aspect ratio is $L=5$. The dashed lines denote the results in the absence of mechanical dispersion and the dash-dotted line separates the fan-flow and the columnar-flow regions. Relatively weak mechanical dispersion slightly enhances the convective transport. However, as convection transitions to a fan-flow structure, the transport flux is significantly reduced. Nevertheless, in the strong-dispersion limit, the flow is stabilized and the flux is increased again due to the large magnitude of the effective diffusion coefficient.

Figure 3

Time-averaged horizontal-mean concentration profile $\langle \overline{C}\rangle$ and snapshots of the concentration field $C$ from DNS at ${\text{Ra}}_d=1000$ and $r=10$ for different ${\text{Ra}}_m$. For ${\text{Ra}}_m\le 20000$, the domain aspect ratio is $L=5$, while for ${\text{Ra}}_m>20000$, DNS are performed in a small unit $L=0.5$ where there only exists a single rising and descending megaplume but the turbulent convection still sustains itself. In panel (a), only half of $\langle \overline{C}\rangle$ is shown due to its antisymmetry about the midplane, and the $z$ values on the horizontal axis are nonuniformly spaced to clearly show the structure near the wall. For fixed ${\text{Ra}}_d=1000$ and $r=10$, the averaged and instantaneous concentration fields become nearly invariant at ${\text{Ra}}_m\gtrsim 50000$ (i.e., $\mathrm{\Delta }\lesssim 0.02$).

Figure 4

Averaged DNS results of convection at ${\text{Ra}}_d=1000$ and $r=10$ for different ${\text{Ra}}_m$. $L$ is as in Fig. 3. For fixed ${\text{Ra}}_d$ and $r$, the concentration field $C$, the interplume spacing $\delta$ and the buoyancy flow velocity $\mathbf{u}$ become invariant at sufficiently large ${\text{Ra}}_m$. Hence, as ${\text{Ra}}_m\to \infty ,$ the flux by molecular diffusion, $F_m=\langle {\partial }_z\overline{C}\rangle {|}_{z=1}$, becomes constant, while the flux by mechanical dispersion, $F_d=\langle {\text{Ra}}_m/{\text{Ra}}_d\overline{\left|u\right|{\partial }_{z}C}\rangle {|}_{z=1}$, increases linearly with ${\text{Ra}}_m$.

Figure 5

Time-averaged horizontal-mean concentration profile $\langle \overline{C}\rangle$ and snapshots of the concentration field $C$ from DNS at ${\text{Ra}}_m=20000$ and ${\text{Ra}}_d=1000$ for different $r$. The domain aspect ratio is $L=5$. In panel (a), only half of $\langle \overline{C}\rangle$ is shown due to its antisymmetry about the midplane, and the $z$ values on the horizontal axis are nonuniformly spaced to clearly show the structure near the wall. When mechanical dispersion is the dominant dissipative mechanism at ${\text{Ra}}_d=1000$, i.e., $\mathrm{\Delta }\ll 1$, the high-${\text{Ra}}_m$ convection in porous media remains a columnar structure at $r=1$, but transitions to a fan-flow structure at $r>1$.

Figure 6

Averaged DNS results of convection at ${\text{Ra}}_m=20000$ and ${\text{Ra}}_d=1000$ for different $r$. The domain aspect ratio is $L=5$. In the absence of mechanical dispersion, the flux $F\approx 138$ at ${\text{Ra}}_m=20000$, so the mass transport is enhanced after adding isotropic, velocity-dependent dispersion ($r=1$). Increasing $r$ enlarges the interplume spacing and decreases the convective flux and buoyancy velocity. For $r\ge 10$, the averaged results become nearly invariant.

Figure 7

Variations of the prefactors (a) $c_1$ and (b) $c_2$ as a function of ${\text{Ra}}_d$ in the limit of ${\text{Ra}}_m\to \infty$ at $r=10$. The solid lines denote the best power-law fit to the DNS data (circles). For each ${\text{Ra}}_d$, DNS are performed up to ${\text{Ra}}_m\sim 1000{\text{Ra}}_d$ to determine $c_1$ and $c_2$.

Figure 8

Schematics showing the distribution of the hydrodynamic dispersion tensor in the form of ellipses. (a) Columnar flow in the absence of mechanical dispersion; (b) fan flow with mechanical dispersion. The arrows denote flow direction. In panel (a), ${\mathbf{D}}^{*}=D_m\mathbf{I}$ is homogeneous and isotropic; in panel (b), the anisotropy of the hydrodynamic dispersion leads to an asymmetry between the rising and the descending megaplumes near the walls.

Figure 9

Variations of control parameters ${\text{Ra}}_m$ and ${\text{Ra}}_d$ as a function of (a) grain size $d$ and (b) density difference $▵\rho$ in laboratory experiments. In panel (a), $▵\rho =9.3\mathrm{kg}/{\mathrm{m}}^3$; in panel (b), $d=3$ mm. The dash-dotted lines separate different regimes and the dots denote the experiments in Ref. [36] in terms of ${\text{Ra}}_m$ and ${\text{Ra}}_d$. In regime I, $\mathrm{\Delta }\ge 10$, ${\text{Ra}}_h\sim {\text{Ra}}_m$, and molecular diffusion dominates the dissipation; in regime II, $0.02<\mathrm{\Delta }<10$; and in regime III, $\mathrm{\Delta }\le 0.02$, ${\text{Ra}}_h\sim {\text{Ra}}_d$, and mechanical dispersion dominates the dissipation. In experiments, varying $d$ changes both ${\text{Ra}}_m$ and ${\text{Ra}}_d$, while varying $▵\rho$ only changes ${\text{Ra}}_m$. Moreover, most of the experiments in Ref. [36] are in the dispersion-dominant regime.