Time-dependent patterns in quasivertical cylindrical binary convection

This paper reports on numerical investigations of the effect of a slight inclination $\alpha$ on pattern formation in a shallow vertical cylindrical cell heated from below for binary mixtures with a positive value of the Soret coefficient. By using direct numerical simulation of the three-dimensional Boussinesq equations with Soret effect in cylindrical geometry, we show that a slight inclination of the cell in the range $\alpha \approx 0.036\mathrm{rad}=2^{\circ }$ strongly influences pattern selection. The large-scale shear flow (LSSF) induced by the small tilt of gravity overcomes the squarelike arrangements observed in noninclined cylinders in the Soret regime, stratifies the fluid along the direction of inclination, and produces an enhanced separation of the two components of the mixture. The competition between shear effects and horizontal and vertical buoyancy alters significantly the dynamics observed in noninclined convection. Additional unexpected time-dependent patterns coexist with the basic LSSF. We focus on an unsual periodic state recently discovered in an experiment, the so-called superhighway convection state (SHC), in which ascending and descending regions of fluid move in opposite directions. We provide numerical confirmation that Boussinesq Navier-Stokes equations with standard boundary conditions contain the essential ingredients that allow for the existence of such a state. Also, we obtain a persistent heteroclinic structure where regular oscillations between a SHC pattern and a state of nearly stationary longitudinal rolls take place. We characterize numerically these time-dependent patterns and investigate the dynamics around the threshold of convection.

Figure 1

Sketch showing the geometry of the cell and the choice of axis orientation. The component of gravity along the inclination of the cell, $g_x$, is much smaller than the vertical component, $g_z$.

Figure 2

The LSSF obtained for $\mathrm{Ra}=1610$ and $\alpha =0.024\mathrm{rad}$. (a) Isosurfaces of temperature deviation from a vertical linear profile, $\mathrm{\Theta }$. (b) Arrow plot of the $u–w$ velocity field (top panel), contour plots of constant temperature fluctuation $\mathrm{\Theta }$ (middle panel), and contour plots of concentration of the denser component $C$ (lower panel) in the vertical plane $y=0$ along the direction of inclination of the cell. In these views, the more elevated part of the cell is on the left. (c) Arrow plot of the $u–v$ velocity field in a horizontal plane at height $z=0.8$. (d) Contour plots of constant temperature fluctuation $\mathrm{\Theta }$ at midplane $z=0.5$. (e) Contour plots of constant concentration of denser component $C$ at $z=0.5$. In the colormap used here and throughout the paper yellow (light) corresponds to the higher values of the field and blue (dark) to the lower values.

Figure 3

Contour plots of constant temperature at $z=0.5$ for eight time instants within a period showing the spatiotemporal structure of the superposition of the LSSF and the eigenfunction associated to the pair of leading eigenvalues in the Hopf instability of the LSSF that takes place at ${\mathrm{Ra}}_c=1695$ with ${\omega }_c=2.19$. This bifurcation is subcritical.

Figure 4

The SHC obtained for $\mathrm{Ra}=1610$ and $\alpha =0.024\mathrm{rad}$. (a) Isosurfaces of constant temperature deviation from a vertical linear profile, $\mathrm{\Theta }$. (b) Arrow plot of the $u–w$ velocity field (top panel), contour plots of constant temperature fluctuation $\mathrm{\Theta }$ (middle panel) and contour plots of constant concentration of the denser component $C$ (lower panel), in a vertical plane along the direction of inclination of the cell. (c) Arrow plot of the $u–v$ velocity field at midplane $z=0.5$. (d) Contour plots of constant temperature fluctuation $\mathrm{\Theta }$ at $z=0.5$. (e) Contour plots of constant concentration of denser component $C$ at $z=0.5$.

Figure 5

The SHC obtained for $\mathrm{Ra}=1610$ and $\alpha =0.024\mathrm{rad}$. (a) At top, contour plots of constant temperature fluctuation $\mathrm{\Theta }$ at $z=0.5$ and $t=0$, and space-time plot showing the evolution of $\mathrm{\Theta }$ in a cell diameter parallel to the direction of inclination (dotted line in the contour plot) with time. At bottom, contour plots of constant temperature fluctuation $\mathrm{\Theta }$ at $z=0.5$ and $t=0$, and space-time plot showing the evolution of $\mathrm{\Theta }$ in an off-diameter line (dotted line in the contour plot) with time. Within each line, the location of the cooler regions of descending fluid forms a wave traveling downhill, while the location of the warmer regions of ascending fluid forms a wave propagating uphill. (b) Time series showing the vertical velocity $w$ at $(r,\theta ,z)=(4.9,0.04,0.8)$ and the kinetic energy $E$ of the periodic solution.

Figure 6

(a) Contour plot of constant temperature and arrow plot of the $u$–$v$ velocity field at midplane $z=0.5$ in time instant $t=0$. Detail of the contour plots of constant temperature and arrow plots of the $u$–$v$ velocity field at (b) $z=0.45$ and (c) $z=0.55$ for four time instants within a period showing the spatiotemporal structure of a SHC state in the central part of the cell. The isolines of temperature are $\mathrm{\Theta }=\pm 0.015$, which correspond to $\pm 0.35{\mathrm{\Theta }}_{\max }$.

Figure 7

Contour plots of constant (a) temperature fluctuation $\mathrm{\Theta }$ and (b) concentration $C$ at midplane $z=0.5$ showing the formation of a SHC pattern for $\mathrm{Ra}=1610, \alpha =0.024\mathrm{rad}$, when simulations are initiated with an homogeneous state. The time instants correspond to $t=1.5,2.5,10,145,250,350,371,385,500$ (nondimensional units). (c) Concentration profile at midplane along a cell diameter parallel to the direction of inclination of the cell at the previous time instants. Whereas the circulatory flow is rapidly established in the cell, the horizontal concentration gradient is slowly generated by mass diffusion at a rate two orders of magnitude smaller than heat diffusion.

Figure 8

Persistent oscillations between a superhighway convection solution and nearly stationary longitudinal rolls (SHC–LHC oscillations) obtained for $\mathrm{Ra}=1680$ and $\alpha =0.024\mathrm{rad}$. (a) At top, contour plots of temperature fluctuation $\mathrm{\Theta }$ at $z=0.5$ and $t=30$, and space-time plot showing the evolution of $\mathrm{\Theta }$ in a cell diameter parallel to the direction of inclination (dotted line in the contour plot) with time. At bottom, contour plots of temperature fluctuation $\mathrm{\Theta }$ at $z=0.5$ and $t=80$, and space-time plot showing the evolution of $\mathrm{\Theta }$ in an off-diameter line (dotted line in the contour plot) with time. The contour plots of $\mathrm{\Theta }$ at midplane at two time instants, $t=30$ and $t=80$, show the spatial structure of the two different patterns between which the system oscillates. (b) Vertical velocity $w$ at $(r,\theta ,z)=(4.9,0.04,0.8)$ and kinetic energy $E$ of the periodic solution as a function of time $t$.