Scaling of Rayleigh-Taylor mixing in porous media

Pushing two fluids with different density one against the other causes the development of the Rayleigh-Taylor instability at their interface, which further evolves in a complex mixing layer. In porous media, this process is influenced by the viscous resistance experienced while flowing through the pores, which is described by the Darcy's law. Here, we investigate the mixing properties of the Darcy-Rayleigh-Taylor system in the limit of large Péclet number by means of direct numerical simulations in three and two dimensions. In the mixing zone, the balance between gravity and viscous forces results in a non-self-similar growth of elongated plumes, whose length increases linearly in time while their width follows a diffusive growth. The mass-transfer Nusselt number is found to increase linearly with the Darcy-Rayleigh number supporting a universal scaling in porous convection at high Ra numbers. Finally, we find that the mixing process displays important quantitative differences between two and three dimensions.

Figure 1

Two snapshots of the density fields for the 3D simulation at $\text{Pe}=1.6\times {10}^4$ at times $t=2\tau$ (left) and $t=4\tau$ (right). Colors represent the relative density.

Figure 2

Panel (a): Mean density profiles $\overline{\rho }(z,t)$ at times $t=0.8\tau$ (red), $t=1.6\tau$ (blue), $t=2.4\tau$ (magenta), $t=3.2\tau$ (green), $t=4.0\tau$ (violet), and $t=4.8\tau$ (black). Upper inset: density standard deviation $\sigma (z,t)={(\overline{{\rho }^2}-{\overline{\rho }}^2)}^{1/2}$ at times $t=1.6\tau$ (blue), $t=3.2\tau$ (green), and $t=4.8\tau$ (black). Lower inset: rms vertical velocity profile ${\overline{w}}_{\text{rms}}(z,t)$ at times $t=1.6\tau$ (blue), $t=3.2\tau$ (green), and $t=4.8\tau$ (black). Simulation in 3D at $\text{Pe}=3.1\times {10}^4$. Panel (b): Time evolution of the width of the mixing layer $h\left(t\right)=2z_1\left(t\right)$ for the simulations at $\text{Pe}=0.8\times {10}^4$ (red), $\text{Pe}=1.6\times {10}^4$ (blue), and $\text{Pe}=3.1\times {10}^4$ (black). The black dashed line represents the two-dimensional case with $\text{Pe}=3.1\times {10}^4$. Inset: Time derivative $dh/dt$ normalized with $w_0$ to give the value of the coefficient $\alpha =(dh/dt)/w_0\simeq 0.59$.

Figure 3

Panel (a): Nusselt number as a function of the Rayleigh number for the simulations at $\text{Pe}=1.6\times {10}^4$ (red open circles) and $\text{Pe}=3.1\times {10}^4$ (blue open triangles). Blue filled triangles represent the results for the 2D case $\left(\text{Pe}=3.1\times {10}^4\right)$. The black line represents the linear dependence $\text{Nu}=0.031\text{Ra}-124.9$. Inset: derivative $d\left(\text{Nu}\right)/d\left(\text{Ra}\right)$ for the two three-dimensional cases (symbols as above). Panel (b): Time evolution of the horizontal correlation scale $h_x\left(t\right)$ for the simulations at $\text{Pe}=0.8\times {10}^4$ (red), $\text{Pe}=1.6\times {10}^4$ (blue), and $\text{Pe}=3.1\times {10}^4$ (black). The black dashed line represents the results for the two-dimensional case with $\text{Pe}=3.1\times {10}^4$. Inset: Square horizontal scale $h_x^2$ rescaled with $D\tau$ demonstrating diffusive growth.