Dimensional transition in Darcy-Rayleigh-Taylor mixing

When a fluid is confined in a thin layer, it undergoes a dramatic change in its dynamical and/or statistical properties, which occurs when the confining scale is of the order of some characteristic scale of the flow. Here we study this dimensional transition in the framework of Rayleigh-Taylor mixing in porous media. By means of extensive direct numerical simulations of the Darcy-Rayleigh-Taylor model, we demonstrate the existence of a transition from three-dimensional (3D) to 2D phenomenology when the horizontal width of the convective structures becomes larger than the confining scale. At variance with the case of turbulent flows, in which the transition is continuous and a coexistence of the 2D and 3D regimes is observed, the transition in porous media is sharp. We investigate the effects of the transition on the evolution of the mixing process and on the fluctuations of the density field. In the 2D regime, we observe a speedup in the growth rate of the mixing layer with an increase of the inhomogeneities of the density field.

Figure 1

Sections of the density field in the $xz$ plane at time $t=4.9\tau$ for the simulations at $R=0.58$ (left), $R=4.61$ (middle), and $R=36.9$ (right). The colors scale ranges from the density values $\rho =-\mathrm{\Delta }\rho /2$ (black) to $\rho =+\mathrm{\Delta }\rho /2$ (white).

Figure 2

(a) Mean density profiles $\overline{\rho }(z,t)$ at time $t=3.5\tau$. (b) Temporal evolution of the width $h\left(t\right)$ of the mixing layer. Inset: Slope $\alpha$, obtained from a fit of $h\left(t\right)$ with $\alpha w_{0}t$ at late times, as a function of $R$. Different colors and symbols represent simulations at different $N_y$: Blue solid line with filled circles is the two-dimensional case $N_y=1$ while black solid line with filled square is the three-dimensional case $N_y=2048$; circles stay for $N_y=2$ (violet), $N_y=4$ (green), $N_y=8$ (light blue); full triangles stay for $N_y=16$ (orange); full diamonds for $N_y=24$ (yellow); squares stay for $N_y=32$ (blue), $N_y=64$ (red), $N_y=128$ (violet), $N_y=256$ (green), and $N_y=512$ (light blue).

Figure 3

(a) Time evolution of the rms vertical velocity $v_z$ at different values of $R$. Inset (b): rms vertical velocity $v_z$ at $t=5\tau$ versus $L_y$ as a function of $R$. Inset (a): Transition time $t_T$, computed from the vertical velocity as the time at which $v_z\left(t_T\right)=0.25w_0$, as a function of $R^2$. (b) Time evolution of the horizontal correlation scale ${\lambda }_x\left(t\right)$ at different values of $R$. Inset (b): ${\lambda }_x\left(t\right)$ compensated with the diffusive scaling ${\left(Dt\right)}^{1/2}$. Colors and symbols are the same of Fig. 2.

Figure 4

(a) Time evolution of density variance ${\sigma }^2={\langle {\langle {\rho }^2\rangle }_{xy}-{\overline{\rho }}^2\rangle }_z$, made dimensionless with its maximum theoretical value, at different $R$. Inset (a): The final value of ${\sigma }^2$ plotted as a function of $R$. (b) Nusselt number as a function of the Rayleigh number for the simulations at different values of $R$. Inset (b): The final value of Nu plotted as a function of $R$. Colors and symbols are the same of Fig. 2. Error bars represent the standard deviation over the different realizations.

Figure 5

Horizontal sections of the density field at $z=0$ and $t=3.5\tau$. The four cases correspond to the simulations with $R=4.61$ ($N_y=16$), $R=6.92$ ($N_y=24$), $R=9.22$ ($N_y=32$), and $R=148$ ($N_y=512$) from top to bottom (in the plot $x$ is along the horizontal and $y$ along the vertical). For visualization clarity, since $L_x\gg L_y$, we plot only half of the section in the $x$ direction, i.e., the region $0\le x\le L_x/2$. The color scale ranges from the density values $\rho =-\mathrm{\Delta }\rho /2$ (black) to $\rho =+\mathrm{\Delta }\rho /2$ (white).

Figure 6

Time evolution of the two-dimensional index $\chi$ for different values of $N_y$ (see labels). Colors and symbols are the same of Fig. 2.