Complex dynamics in double-diffusive convection

The dynamics of a small Prandtl number binary mixture in a laterally heated cavity is studied numerically. By combining temporal integration, steady state solving and linear stability analysis of the full PDE equations, we have been able to locate and characterize a codimension-three degenerate Takens-Bogdanov point whose unfolding describes the dynamics of the system for a certain range of Rayleigh numbers and separation ratios near S=-1.Comment: 8 pages, 5 figure

Time-dependent patterns in quasivertical cylindrical binary convection

This paper reports on numerical investigations of the effect of a slight inclination a 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 a˜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.Postprint (author's final draft

Secondary flows in a laterally heated horizontal cylinder

In this paper we study the problem of thermal convection in a laterally heated, finite, horizontal cylinder. We consider cylinders of moderate aspect ratio (height/diameter approximate to 2) containing a small Prandtl number fluid (sigma < 0.026) typical of molten metals and molten semiconductors. We use the Navier-Stokes and energy equations in the Boussinesq approximation to calculate numerically the basic steady states, analyze their linear stability, and compute some nonlinear secondary flows originated from the instabilities. All the calculated flows and the stability analysis are characterized by their symmetry properties. Due to the confined cylindrical geometry, -presence of lateral walls and lids-, all the flows are completely three dimensional even for the basic steady states. In the range of Prandtl numbers studied, we have identified four different types of instabilities, either oscillatory or stationary. The physical mechanisms, shear or buoyancy, of the corresponding flow transitions have been analyzed. As the value of the Prandtl number approaches sigma = 0.026 the scenario of bifurcations becomes more complicated due to the existence of two different stable basic states originated in a saddle-node bifurcation; a fact that had been overlooked in previous works. (C) 2014 AIP Publishing LLC.Postprint (published version

Pattern selection near the onset of convection in binary mixtures in cylindrical cells

We report numerical investigations of three-dimensional pattern formation of binary mixtures in a vertical cylindrical container heated from below. Negative separation ratio mixtures, for which the onset of convection occurs via a subcritical Hopf bifurcation, are considered. We focus on the dynamics in the neighbourhood of the initial oscillatory instability and analyze the spatio-temporal properties of the patterns for different values of the aspect ratio of the cell, 0.25 less than or similar to Gamma less than or similar to 11 (Gamma equivalent to R/d, where R is the radius of the cell and d its height). Despite the oscillatory nature of the primary instability, for highly constrained geometries, Gamma less than or similar to 2.5, only pure thermal stationary modes are selected after long transients. As the aspect ratio of the cell increases, for intermediate aspect ratio cells such as Gamma = 3, multistability and coexistence of stationary and time-dependent patterns is observed. In highly extended cylinders, Gamma approximate to 11, the dynamics near the onset is completely different from the pure fluid case, and a startling diversity of confined patterns is observed. Many of these patterns are consistent with experimental observations. Remarkably, though, we have obtained persistent large amplitude highly localized states not reported previously.Postprint (published version

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.Postprint (author's final draft

Thermal binary mixture convection in inclined cylindrical containers

Convection in a fluid layer is affected by its orientation with respect to the gravitational field. In the present work, we investigate numerically pattern selection in a cylindrical cell heated from below and inclined against gravity. We are interested in analyzing the effect of inclination on pattern formation near the onset of convection. We will focus on thermophobic mixtures, i.e. mixtures in which the heavier component of the fluid is driven into the direction of lower temperature, where square patterns, roll structures and cross-roll regimes are expected to arise. In agreement with experimental observations, we will show that pattern formation in disklike cylinders is strongly affected even by very small inclinations.Postprint (published version

A 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.Comment: 4 pages, 6 figures, REVTeX, To be published in Physical Review Letter

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.Postprint (published version

Travelling convectons in binary fluid convection

Binary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. With no-slip, fixed-temperature, no-mass-flux boundary conditions at the top and bottom stationary odd- and even-parity convectons fall on a pair of intertwined branches connected by branches of travelling asymmetric states. In appropriate parameter regimes the stationary convectons may be stable. When the boundary condition on the top is changed to Newton’s law of cooling the odd-parity convectons start to drift and the branch of odd-parity convectons breaks up and reconnects with the branches of asymmetric states. We explore the dependence of these changes and of the resulting drift speed on the associated Biot number using numerical continuation, and compare and contrast the results with a related study of the Swift–Hohenberg equation by Houghton & Knobloch (Phys. Rev. E, vol. 84, 2011, art. 016204). We use the results to identify stable drifting convectons and employ direct numerical simulations to study collisions between them. The collisions are highly inelastic, and result in convectons whose length exceeds the sum of the lengths of the colliding convectons.Postprint (published version

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.Postprint (published version