Isabel Mercader

Spatially localized binary-fluid convection

Multiple states of spatially localized steady convection are found in numerical simulations of water–ethanol mixtures in two dimensions. Realistic boundary conditions at the top and bottom are used, with periodic boundary conditions in the horizontal. The states form by a mechanism similar to the pinning region around a Maxwell point in variational systems, but are located in a parameter regime in which the conduction state is overstable. Despite this the localized states can be stable. The properties of the localized states are described in detail, and the mechanism of their destruction with increasing or decreasing Rayleigh number is elucidated. When the Rayleigh number becomes too large the fronts bounding the state at either end unpin and move apart, allowing steady convection to invade the domain. In contrast, when the Rayleigh number is too small the fronts move inwards, and eliminate the localized state which decays into dispersive chaos. Out of this state spatially localized states re-emerge at irregular times before decaying again. Thus an interval of Rayleigh numbers exists that is characterized by relaxation oscillations between localized convection and dispersive chaos.

Centrifugal effects in rotating convection: axisymmetric states and three-dimensional instabilities

Rotating convection is analysed numerically in a cylinder of aspect ratio one, for Prandtl number about 7. Traditionally, the problem has been studied within the Boussinesq approximation with density variation only incorporated in the gravitational buoyancy term and not in the centrifugal buoyancy term. In that limit, the governing equations admit a trivial conduction solution. However, the centrifugal buoyancy changes the problem in a fundamental manner, driving a large-scale circulation in which cool denser fluid is centrifuged radially outward and warm less-dense fluid is centrifuged radially inward, and so there is no trivial conduction state. For small Froude numbers, the transition to three-dimensional flow occurs for Rayleigh number R ≈ 7.5 × 10 3 . For Froude numbers larger than 0.4, the centrifugal buoyancy stabilizes the axisymmetric large-scale circulation flow in the parameter range explored (up to R = 3.5 × 10 4 ). At intermediate Froude numbers, the transition to three-dimensional flow is via four different Hopf bifurcations, resulting in different coexisting branches of three-dimensional solutions. How the centrifugal and the gravitational buoyancies interact and compete, and the manner in which the flow becomes three-dimensional is different along each branch. The centrifugal buoyancy, even for relatively small Froude numbers, leads to quantitative and qualitative changes in the flow dynamics.

Spectral methods for high order equations

Abstract Solving Navier-Stokes equations by means of scalar potentials leads to higher order equations with low order boundary conditions. This approach usually involves numerical instabilities that do not appear using the primitive Navier-Stokes equations. Such difficulties can be avoided with an explicit evaluation of some of the high order derivatives at the boundary. We present three numerical techniques based on this method and on spectral expansions. We apply them to the linear stability analysis of thermal convection in a cylinder.

Convectons, anticonvectons and multiconvectons 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. Numerical continuation is used to compute such states in the presence of both Neumann boundary conditions and no-slip no-flux boundary conditions in the horizontal. In addition to the previously identified convectons, new states referred to as anticonvectons with a void in the centre of the domain, and wall-attached convectons attached to one or other wall are identified. Bound states of convectons and anticonvectons called multiconvecton states are also computed. All these states are located in the so-called snaking or pinning region in the Rayleigh number and may be stable. The results are compared with existing results with periodic boundary conditions.

Thermal convection in vertical cylinders. A method based on potentials of velocity

Abstract Any solenoidal vector field can be written in terms of two scalar fields. Solving the Navier-Stokes equation for these two so-called potentials is a widely used method, although difficulties have been reported in using this method to describe non-Cartesian confined flows, in particular cylindrical geometries. Difficulties arise because the potentials are coupled at the boundaries but also because of possible numerical instabilities associated with the high order of the equations. This paper shows how all these difficulties can be circumvented: firstly by properly deriving the required boundary conditions, but also through an adequate discretization, which is the final requirement. The use of spectral methods together with a reduction of order for the axisymmetric modes is sufficient for this purpose. In this paper, as a test, a velocity potentials method is used to solve the stability problem for the onset of convection in a fluid confined in a cylindrical container.

Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus-Benjamin-Feir instability

A numerical study of the onset of thermal convection in a rotating circular cylinder of radius-to-depth ratio equal to four is considered in a regime dominated by the Coriolis force where the onset is to so-called wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and descending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dynamics are controlled by an Eckhaus–Benjamin– Feir instability, the most basic features of which had been captured by a complex Ginzburg–Landau equation model. Here, we analyse this instability in rotating convection using direct numerical simulations of the Navier–Stokes equations in the Boussinesq approximation. Several properties of the wall modes are computed, extending the results to far beyond the onset of convection. Extensive favourable comparisons between our numerical results and previous experimental observations and complex Ginzburg–Landau model results are made.

Bifurcations and chaos in single-roll natural convection with low Prandtl number

Convective flows of a small Prandtl number fluid contained in a two-dimensional cavity subject to a lateral thermal gradient are numerically studied by using different techniques. The aspect ratio (length to height) is kept at around 2. This value is found optimal to make the flow most unstable while keeping the basic single-roll structure. Two cases of thermal boundary conditions on the horizontal plates are considered: perfectly conducting and adiabatic. For increasing Rayleigh numbers we find a transition from steady flow to periodic oscillations through a supercritical Hopf bifurcation that maintains the centrosymmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation the system initiates a complex scenario of bifurcations. In the conductive case these include a quasiperiodic route to chaos. In the adiabatic one the dynamics is dominated by the interaction of two Neimark-Sacker bifurcations of the basic periodic solutions, leading to the stable coexistence...

Large‐scale flows and resonances in 2‐D thermal convection

Recent experiments of thermal convection in finite containers of intermediate and large aspect ratios have shown the presence of flows spanning the largest dimension of the container [R. Krishnamurti and L. N. Howard, Proc. Natl. Acad. Sci. 78, 1985 (1981); J. Fluid Mech. 170, 385 (1986)]. Large‐scale flows of this kind computed from two‐dimensional (2‐D) numerical simulations are presented. The marginal stability curves for the bifurcations are computed in the range of aspect ratios L=1,...,6 and for Prandtl number σ =10. The nonlinear dynamics of the bifurcated solution is explored for containers with aspect ratios L=1,2,4. By increasing the Rayleigh number from criticality the system produces different sequences of symmetry breaking, Hopf‐type bifurcations, which finally result in large scale flows, oscillatory net mass flux and chaos. The bifurcation involves different mode resonances with vertical and horizontal couplings, which are modeled using formal group theoretical techniques.

ROBUST HETEROCLINIC CYCLES IN TWO-DIMENSIONAL RAYLEIGH–BÉNARD CONVECTION WITHOUT BOUSSINESQ SYMMETRY

The onset of convection in systems that are heated via current dissipation in the lower boundary or that lose heat from the top boundary via Newton’s law of cooling is formulated as a bifurcation problem. The Rayleigh number as usually dened is shown to be inappropriate as a bifurcation parameter since the temperature dierence across the layer depends on the amplitude of convection and hence changes as convection evolves at xed external parameter values. A modied Rayleigh number is introduced that does remain constant even when the system is evolving, and solutions obtained with the standard formulation are compared with those obtained via the new one. Near the 1 : 2 spatial resonance in low Prandtl number fluids these eects open up intervals of Rayleigh number with no stable solutions in the form of steady convection or steadily traveling waves. Direct numerical simulations in two dimensions show that in such intervals the dynamics typically take the form of a nearly heteroclinic modulated traveling wave. This wave may be quasiperiodic or chaotic.

Instability of swirl in low-prandtl-number thermal convection

In the present paper we examine low-Prandtl-number thermal convection using a highly truncated modal approach. For the horizontal structure we assume a hexagonal planform as in Toomre Gough & Spiegel (1977) but including a vertical vorticity mode. The system develops a non-zero vertical vorticity component through a finite-amplitude instability. Following this, the system displays a Hopf bifurcation giving rise to periodic oscillations. The mechanism for this instability is associated with the growth of swirl in the azimuthal direction. We have found three different types of periodic solutions, possibly associated with subharmonic bifurcations, and their structure has been examined. A large part of the present work is devoted to exploring the cases of mercury and liquid helium - or air - as the best-known examples of low and intermediate-Prandtl-number fluids. Results for mercury are quite satisfactory as far as frequencies and fluxes are concerned and they show reasonable agreement with experimental measurements at mildly supercritical Rayleigh values. On the other hand, for liquid helium or air agreement is poor.