Molecular hints of two-step transition to convective flow via streamline percolation

Convection is a key transport phenomenon important in many different areas, from hydrodynamics and ocean circulation to planetary atmospheres or stellar physics. However, its microscopic understanding still remains challenging. Here we numerically investigate the onset of convective flow in a compressible (non-Oberbeck-Boussinesq) hard disk fluid under a temperature gradient in a gravitational field. We uncover a surprising two-step transition scenario with two different critical temperatures. When the bottom plate temperature reaches a first threshold, convection kicks in (as shown by a structured velocity field) but gravity results in hindered heat transport as compared to the gravity-free case. It is at a second (higher) temperature that a percolation transition of advection zones connecting the hot and cold plates triggers efficient convective heat transport. Interestingly, this picture for the convection instability opens the door to unknown piecewise-continuous solutions to the Navier-Stokes equations.

Figure 1

Onset of convection. Average hydrodynamic velocity field $\langle \mathbf{u}\left(\mathbf{r}\right)\rangle$ for hard disks measured across the Rayleigh-Bénard instability. Color indicates velocity modulus from $u_{\text{min}}$ (blue) to $u_{\text{max}}$ (red). The $(u_{\text{min}},u_{\text{max}})$ interval changes among panels. Left column: $g=5$ and $T=1.4,2.6,4,10,20$, for which $(u_{\text{min}},u_{\text{max}})=(0.000043,0.016)$, (0.00025, 0.025), (0.00021, 0.04), (0.0016, 0.11), (0.0032, 0.15). Central column: $g=10$ and $T=1.8,4,6,15,20$ for which $(u_{\text{min}},u_{\text{max}})=(0.00018,0.029)$, (0.00017, 0.031), (0.001, 0.045), (0.0087, 0.24), (0.004, 0.3). Right column: $g=15$ and $T=2,6,8,15,20$, for which $(u_{\text{min}},u_{\text{max}})=(0.00029,0.051)$, (0.00019, 0.033), (0.0011, 0.053), (0.0023, 0.25), (0.015, 0.44), all from top to bottom. Note the typical roll structure associated to the Rayleigh-Bénard instability, most apparent for the largest $T$

Figure 2

Temperature field. Color maps for the average temperature field $\langle T\left(\mathbf{r}\right)\rangle$ measured for gravity $g=10$ and different values of the bottom plate temperature $T$, together with the average profiles across the vertical and horizontal directions. Note the nontrivial spatial structure of $\langle T\left(\mathbf{r}\right)\rangle$ for strong enough temperature gradients, as well as the asymmetric, nonlinear shape of temperature profiles along the vertical direction.

Figure 3

Packing fraction field. Color maps for the average packing fraction field $\langle \eta \left(\mathbf{r}\right)\rangle$ measured for gravity $g=10$ and different values of $T$, together with the average profiles across the vertical and horizontal directions. The packing fraction field exhibits a nontrivial spatial structure, with nonlinear profiles and clear boundary effects. Note also the density gradient inversion in the vertical direction for intermediate values of $T$.

Figure 4

Order parameter. Scaled average hydrodynamic kinetic energy, $\langle e\rangle /g^2$, as a function of $T$ for different values of $g$. Solid curves are polynomial fits (see text). Vertical dashed lines locate the critical temperatures obtained from these fits. Inset: Scaled relative fluctuations $N{\sigma }_{\varepsilon }^2/{\langle \varepsilon \rangle }^2$ for the kinetic energy per particle. Solid curves are fits to the data, and vertical dashed lines locate their minima. Error bars are included in both plots.

Figure 5

Convective heat transport. Average reduced heat current $\tilde{r}\langle J_g\rangle$ as a function of $T$ and different $g$, with $\tilde{r}$ the disks radius. Lines are fits to the data, see text. Bottom inset: Excess current compared to the gravity-free case, $\mathrm{\Delta }{J}_g=\langle J_g\rangle -\langle J_0\rangle$, as a function of $T$ and different $g\ne 0$. Vertical lines mark the temperatures $T_J\left(g\right)$, where $\mathrm{\Delta }{J}_g\left[T_J\left(g\right)\right]=0$. Top inset: $T_c\left(g\right)$ ($◯$) and $T_J\left(g\right)$ ($\square$) as a function of $g$. The different transport regimes are highlighted in color.

Figure 6

Active advection zones and streamlines. Right: Average hydrodynamic velocity field $\langle \mathbf{u}\left(\mathbf{r}\right)\rangle$ for $g=10$ and $T=1.8,4,6,20$ (from top to bottom). For $g=10$, we have $T_c=3.4$ and $T_J=5$. Active advection cells ($|\langle \mathbf{u}(\mathbf{r})\rangle |>\sigma \left[\mathbf{u}\right(\mathbf{r}\left)\right]$) are marked with dark vectors. The red lines correspond to ${10}^2$ streamlines sampled from the associated $\langle \mathbf{u}\left(\mathbf{r}\right)\rangle$. Left: Streamline length distribution in each case. Solid lines are fits to the data.

Figure 7

Percolation transition in convection. Main: Log-log plot of the fraction ${\pi }_g\left(T\right)$ of active advection zones as a function of the average streamline length $\langle {\ell }_g\rangle \left(T\right)$ $\forall g>0$. Notice the striking collapse observed for all curves. The thick dashed line is a power law ${\pi }_g\sim {\langle {\ell }_g\rangle }^{1/4}$. Top inset: ${\pi }_g\left(T\right)$ vs $T$ $\forall g>0$. Bottom inset: $\langle {\ell }_g\rangle$ vs $T$ $\forall g>0$. The dashed vertical lines mark $T_J\left(g\right)$ for each $g$ in both insets, while the horizontal lines signal $\langle {\ell }_g\rangle =0.5$ (bottom inset) and the critical covered area fraction ${\varphi }_c=0.666$ (top inset).

Figure 8

Top: Average reduced pressure profiles measured along the vertical direction for a bottom plate temperature $T=1$ (equal to the top plate temperature) and different values of gravity $g$. Full lines correspond to the macroscopic prediction in each case. Inset: Macroscopic local equilibrium and equation of state (EoS) for hard disks. The measured local reduced pressure over the local temperature, $P\left(y\right)/T\left(y\right)$, versus the local packing fraction $\eta \left(y\right)$ for different values of $g$. The line represents the Henderson macroscopic EoS approximation [63]. Bottom: Average packing fraction profiles along the vertical direction for different values of $g$, together with the macroscopic prediction derived from the hydrostatic formula and the Henderson's EoS.

Figure 9

Color maps for the reduced pressure field measured for gravity $g=10$ and different values of the bottom plate temperature $T$, together with the average profiles across the vertical and horizontal directions.

Figure 10

Top: Rayleigh number Ra in the Enskog approximation as a function of the global packing fraction $\rho$ for $T=5$, $T_0=1$ and $g=5$ (bottom blue curve), $g=10$ (middle green curve) and $g=15$ (top black curve). Dashed orange line signals the critical Rayleigh number ${\text{Ra}}_c=27{\pi }^4/4\simeq 657.51$ obtained using Oberbeck-Boussinesq approximation on the Navier-Stokes equations [22]. Bottom: Rayleigh number as a function of the bottom plate temperature $T$ for fixed value of $T_0=1$ and $\rho =0.2$, and for the same values of $g$ on the top panel.