Pattern formation in rayleigh-Benard convection in a cylindrical container

Spiral Thermal Plumes in Water under Conventional Heating: Numerical Results on the Effect of Rotation

In this paper, we study numerically the effect of rotation within a sample of water in a cylindrical container subject to rotation which is heated with a constant temperature at the bottom and lateral wall. We analyze the temporal behavior of temperature and flow velocity of the solvent. The thermal plumes developed at lower levels, already observed in the case without rotation, begin to spiral spreading outwards by the effect of rotation, increasing the azimuthal velocity of the fluid. No significant increases in the radial and vertical velocity components are observed which do not favor the mixture of hotter and colder flows in the sample and a faster heating of the solvent. In the rotation range studied, the state loses the axisymmetry and becomes fully 3D earlier in time as the rotation rate increases. To perform simulations, we use a 3D temporal model that couples momentum and heat equations and is based on spectral element methods.

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.

Pattern formation of Rayleigh-Bénard convection of cold water near its density maximum in a vertical cylindrical container

In order to understand the onset of convective instability and multiple stable convection patterns of buoyancy-driven convection of cold water near its density maximum in a vertical cylindrical container heated from below, a series of three-dimensional numerical simulations were performed. The aspect ratio of the container was 2 and Prandtl number of cold water was 11.57. The sidewall was considered to be perfectly adiabatic, and the density inversion parameter was fixed at 0.3. The result shows that the density inversion phenomenon in cold water has an important effect on the critical Rayleigh number for the onset of convection and the pattern formation at higher Rayleigh numbers. When the Rayleigh number varies from 3×10(3) to 1.2×10(5), eight stable, steady convection patterns are obtained under different initial conditions. The coexistence of multiple stable steady flow patterns is also observed within some specific ranges of the Rayleigh number.

Onset of Rayleigh-Bénard convection in cylindrical containers

We determined the critical Rayleigh numbers Ra{c} for the onset of convection in cylindrical containers with aspect ratios 1 approximately <Gamma [ triple bond] D/L approximately <9 ( D is the diameter and L the height) and the patterns that form just above Ra{c}, both from experiment and by direct numerical simulation (DNS). Results for Ra{c} agree well with the linear stability analysis by Buell and Catton for containers with finite sidewall conductivity. For Gamma<or=1.58+/-0.10 , we found that the patterns correspond to an azimuthal Fourier mode with mode number m=1 , corresponding to a single convection roll. For 1.58 approximately < Gamma approximately <3.26+/-0.02 , the pattern was a concentric roll, corresponding to m=0 . For 3.26<or=Gamma approximately >4, an m=1 mode was found again, but near Gamma=4 either m=1 or m=2 was observed in different runs. These results are consistent with the marginal stability curves calculated by Buell and Catton in the sense that the mode that is the first as a function of Ra to acquire a positive growth rate is the one that is observed. For Gamma approximately >4, the theoretical marginal curves for the four lowest modes lie very close together. There we found patterns near onset that corresponded to various modes, including m=2 and 4. At relatively large Gamma approximately > 6, we observed parallel straight rolls quite close to onset. Our patterns agree with several DNS investigations by others, but at some Gamma values differ from those observed experimentally by Stork and Müller. Some results for the pattern evolution with increasing Ra are reported as well.

Extreme multiplicity in cylindrical Rayleigh-Bénard convection. I. Time dependence and oscillations

Rayleigh-Bénard convection in a cylindrical container can take on many different spatial forms. Motivated by the results of Hof [Phys. Fluids 11, 2815 (1999)], who observed coexistence of several stable states at a single set of parameter values, we have carried out simulations at the same Prandtl number, that of water, and a radius-to-height aspect ratio of two. We have used two kinds of thermal boundary conditions: perfectly insulating sidewalls and perfectly conducting sidewalls. In both cases we obtain a wide variety of coexisting steady and time-dependent flows.

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.

Multiplicity of steady states in cylindrical Rayleigh-Bénard convection

Three-dimensional steady Rayleigh-Bénard convection in a vertical cylinder is investigated by numerical simulation and bifurcation analysis. The complex pattern formation beyond the onset of the convection is presented by a bifurcation diagram. The coexistence of multiple stable states is observed near the threshold of the first bifurcation and two group symmetries are summarized for the corresponding primary branches. The first stable target pattern originates through a subcritical bifurcation. A multiplicity of flow states for the Rayleigh number of 14200 is validated numerically in comparison with the experiment, and a four-spoke pattern is observed.