Stationary localized solutions in binary convection in slightly inclined rectangular cells.

We analyze numerically the effect of a slight inclination in the lowest part of the snaking branches of convectons that are present in negative separation ratio binary mixtures in two-dimensional elongated rectangular cells. The exploration reveals the existence of novel stationary localized solutions with striking spatial features different from those of convectons. The numerical continuation of these solutions with respect to the inclination of the cell unveils the existence of even further families of localized structures that can organize in closed branches. A variety of localized solutions coexist for the same heating and inclination, depicting a highly complex scenario for solutions in the lowest part of the snaking diagrams for moderate to high heating. The different localized solutions obtained in the horizontal cell are discussed in detail.

Effect of small inclination on binary convection in elongated rectangular cells.

We analyze the effect of a small inclination on the well-studied problem of two-dimensional binary fluid convection in a horizontally extended closed rectangular box with a negative separation ratio, heated from below. The horizontal component of gravity generates a shear flow that replaces the motionless conduction state when inclination is not present. This large-scale flow interacts with the convective currents resulting from the vertical component of gravity. For very small inclinations the primary bifurcation of this flow is a Hopf bifurcation that gives rise to chevrons and blinking states similar to those obtained with no inclination. For larger but still small inclinations this bifurcation disappears and is superseded by a fold bifurcation of the large-scale flow. The convecton branches, i.e., branches of spatially localized states consisting of counterrotating rolls, are strongly affected, with the snaking bifurcation diagram present in the noninclined system destroyed already at small inclinations. For slightly larger but still small inclinations we obtain small-amplitude localized states consisting of corotating rolls that evolve continuously when the primary large-scale flow is continued in the Rayleigh number. These localized states lie on a solution branch with very complex behavior strongly dependent on the values of the system parameters. In addition, several disconnected branches connecting solutions in the form of corotating rolls, counterrotating rolls, and mixed corotating and counterrotating states are also obtained.

Time-dependent patterns in quasivertical cylindrical binary convection.

This paper reports on numerical investigations of the effect of a slight inclination α 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 α≈0.036rad=2^{∘} 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.

Natural convection in a horizontal cylinder with axial rotation.

We study the problem of thermal convection in a laterally heated horizontal cylinder rotating about its axis. A cylinder of aspect ratio Γ=H/2R=2 containing a small Prandtl number fluid (σ=0.01) representative of molten metals and molten semiconductors at high temperature is considered. We focus on a slow rotation regime (Ω<8), where the effects of rotation and buoyancy forces are comparable. The Navier-Stokes and energy equations with the Boussinesq approximation are solved numerically to calculate the basic states, analyze their linear stability, and compute several secondary flows originated from the instabilities. Due to the confined cylindrical geometry-the presence of lateral walls and lids-all the flows are completely three dimensional, even the basic steady states. Results characterizing the basic states as the rotation rate increases are presented. As it occurred in the nonrotating case for higher values of the Prandtl number, two curves of steady states with the same symmetric character coexist for moderate values of the Rayleigh number. In the range of Ω considered, rotation has a stabilizing effect only for very small values. As the value of the rotation rate approaches Ω=3.5 and Ω=4.5, the scenario of bifurcations becomes more complex due to the existence in both cases of very close bifurcations of codimension 2, which in the latter case involve both curves of symmetric solutions.

Localized pinning states in closed containers: Homoclinic snaking without bistability.

Binary mixtures with a negative separation ratio are known to exhibit time-independent spatially localized convection when heated from below. Numerical continuation of such states in a closed two-dimensional container with experimental boundary conditions and parameter values reveals the presence of a pinning region in Rayleigh number with multiple stable localized states but no bistability between the conduction state and an independent container-filling state. An explanation for this unusual behavior is offered.

Spatiotemporal dynamics near the onset of convection for binary mixtures in cylindrical containers.

Pattern selection near the onset of convection in a cylindrical container heated from below is investigated numerically for a water-ethanol mixture, with parameter values and boundary conditions relevant to experiments. The Boussinesq three-dimensional equations for binary fluid convection are simulated for cylinders of aspect ratio Gamma=11 and 10.5 (Gamma identical with R/d, where R is the radius of the cell and d its height). The onset of convection occurs via a subcritical Hopf bifurcation in which the critical mode is strongly influenced by small variations of the aspect ratio of the cell. During the linear regime, an m=1 azimuthal mode consisting of radially traveling waves grows in amplitude in the Gamma=11 cell, while an m=0 azimuthal mode is selected in the Gamma =10.5 cylinder. As convection evolves, simulations for subcritical and supercritical Rayleigh numbers reveal differences in the dynamics. Very close to the critical value, convection is erratic and focuses along one or more diameters of the cell; growths and collapses of the convection amplitude take place, but convection eventually dies away for subcritical values and persists for slightly supercritical values. For larger supercritical values, convection grows progressively in amplitude, and patterns consist of traveling-wave regions of convection initially focused near the cell center, though expanding slowly until a large-amplitude state is reached. Depending on the reduced Rayleigh number, the final state can be a nonsteady state filling the cell or a disordered confined state.

Complex dynamical states in binary mixture convection with weak negative Soret coupling.

Binary convection in large-aspect-ratio annular containers heated from below is studied numerically for a water-ethanol mixture. High-resolution numerical tools based on spectral methods are used to solve the hydrodynamic equations in the two-dimensional approximation. The weakly nonlinear states arising very close to the onset of convection, the strongly nonlinear bursts of amplitude that precede the small-amplitude states, and the dispersive chaotic states encountered further above onset in experiments for mixtures with a weak negative Soret coupling are analyzed in detail in extended domains of aspect ratio 80. Steady localized states surrounded either by quiescent fluid or by small-amplitude waves are also obtained, and the role they play in the dynamics is elucidated.

Simulations of localized States of stationary convection in 3He-4He mixtures.

Simulations of convection in 3He-4He mixtures with a negative separation ratio in two-dimensional containers with realistic boundary conditions and moderately large aspect ratio reveal, at supercritical Rayleigh numbers, the existence of "convectons," i.e., localized states of stationary convection, separated by regions of no convection. The origin and properties of these states are described.

Blue sky catastrophe in double-diffusive convection.

A global bifurcation of the blue sky catastrophe type has been found in a small Prandtl number binary mixture contained in a laterally heated cavity. The system has been studied numerically applying the tools of bifurcation theory. The catastrophe corresponds to the destruction of an orbit which, for a large range of Rayleigh numbers, is the only stable solution. This orbit is born in a global saddle-loop bifurcation and becomes chaotic in a period-doubling cascade just before its disappearance at the blue sky catastrophe.

Simulations of oscillatory binary fluid convection in large aspect ratio containers.

Direct numerical simulations of chevrons, blinking states, and repeated transients in binary fluid mixtures with a negative separation ratio heated from below are described. The calculations are performed in two-dimensional containers using realistic boundary conditions and the parameter values used in the experiments of Kolodner [Phys. Rev. E 47, 1038 (1993)]. Particular attention is paid to the multiplicity of states, and their dependence on the applied Rayleigh number and the aspect ratio of the container. Quantitative agreement with the experiments is obtained, and a mechanism explaining the origin and properties of the repeated transients observed in the experiments is proposed.