Transition from multiplicity to singularity of steady natural convection in a tilted cubical enclosure

The transition from the complex Rayleigh-Bénard convection to the simple heated-from-the-sides configuration in a cubical cavity filled with a Newtonian fluid is numerically studied. The cavity is tilted by an angle $\theta$ around its lower horizontal edge and is heated and cooled from two opposite tilted sides. We first analyze the effect of a marginal inclination angle on quasi-Rayleigh-Bénard convection ($\theta \approx 0^{\circ }$), which is a realistic physical approximation to the ideal Rayleigh-Bénard convection. We then yield the critical angles where multiple solutions that were initially found for $\theta \approx 0^{\circ }$ disappear, eventually resulting in the single steady roll solution found in the heated-from-the-sides configuration ($\theta ={90}^{\circ }$). We confirm the existence of critical angles during the transition $\theta :0^{\circ }\to {90}^{\circ }$, and we demonstrate that such angles are a consequence of either singularities or collisions of bifurcation points in the Rayleigh-number-$\theta$ parameter space. We finally derive the most important critical angles corresponding to any Newtonian fluid of Prandtl number greater than that of air.

Figure 1

Stable solutions found in a cubical cavity filled with air at $\text{Ra}=10000$ in quasi-Rayleigh-Bénard convection ($\theta =0.1^{\circ }$) and heated-from-the-sides configuration. Multiple solutions are found in the former case, whereas a single solution is found in the latter case. The magnitude of the dimensionless velocity is plotted on the mean-isothermal surface $T=(T_H+T_C)/2$.

Figure 2

Bifurcation diagrams showing the multiple stable solutions found in RBC ($\theta =0^{\circ }$, gray lines starting at $P_1$) and QRBC ($\theta =0.1^{\circ }$, color online) in a cubical cavity filled with air. The velocity $w$ at a reference point is plotted as a function of $\text{Ra}$ for $2000\le \text{Ra}\le 6000$ (a) and $\text{Ra}\le 100000$ (b). The insets (c), (d), and (e) show the transition between RBC and QRBC around the main bifurcations. The thick arrows indicate the translation of the critical thresholds as $\theta$ is increased.

Figure 3

Transition from RBC to HSC for a cavity filled with air. (a) The single branch found in HSC ($\theta ={90}^{\circ }$) is depicted by a single solid black line, while the multiple branches found in RBC are depicted by light-gray lines. (b) The loci of the most relevant bifurcations are shown every $\mathrm{\Delta }\theta =1^{\circ }$ (open points) starting from $\theta =0^{\circ }$ (solid points). The circles indicate those bifurcations that evolve from the primary instability threshold $P_1$ at $\theta =0^{\circ }$; the squares indicate the saddle-node points where the stable four-roll solutions evolve from; the triangles indicate those solutions related to the bifurcation point $S_1{B}_{\pm y}$ and $S_1{B}_{\pm z}$. The collisions $\epsilon$ and singularities $\gamma$ ($d{\text{Ra}}_c/d\theta \to \infty$) are plotted using and , respectively.

Figure 4

Collisions ${\epsilon }^{+z}, {\epsilon }^{\mathrm{R}4}$, and singularity ${\gamma }^{-z}$ for any Newtonian fluid of $\text{Pr}$ greater than that of air, i.e., $0.71\le \text{Pr}\to \infty$. The inverse of Prandtl number ${\text{Pr}}^{-1}$ is given in the abscissa. The transition from multiple solutions found in RBC to a single solution found in HSC corresponds to the largest critical angle ${\theta }_c$ at a given value of $\text{Pr}$.