Multiple flow transitions in a box heated from the side in low-Prandtl-number fluids

Periodic orbits in tall laterally heated rectangular cavities

This study elucidates the origin of the multiplicity of stable oscillatory flows detected by time integration in tall rectangular cavities heated from the side. By using continuation techniques for periodic orbits, it is shown that initially unstable branches, arising at Hopf bifurcations of the basic steady flow, become stable after crossing Neimark-Sacker points. There are no saddle-node or pitchfork bifurcations of periodic orbits, which could have been alternative mechanisms of stabilization. According to the symmetries of the system, the orbits are either fixed cycles, which retain at any time the center symmetry of the steady flow, or symmetric cycles involving a time shift in the global invariance of the orbit. The bifurcation points along the branches of periodic flows are determined. By using time integrations, with unstable periodic solutions as initial conditions, we determine which of the bifurcations at the limits of the intervals of stable periodic orbits are sub- or supercritical.

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.

Quasiperiodic routes to chaos in confined two-dimensional differential convection

The complete cascade of bifurcations from steady to chaotic convection, as the Rayleigh number is varied, is considered numerically inside an air-filled differentially heated cavity. The system is assumed to be two-dimensional and is invariant under a generalized reflection about the center of the cavity. In the neighborhood of several codimension-two points, two main routes emerge, characterized by different symmetries of the first oscillatory eigenstate. Along these two competing routes, different sequences of bifurcations and symmetry breakings lead from the steady base flow to the hyperchaotic regime. Several families of two- and three-frequency tori have been identified via the computation of the leading Lyapunov exponents. Modal structures extracted from time series reveal the occurrence of slow internal oscillations in the center of the cavity and faster wall modes confined to vertical boundary layers. Further quasiperiodicity windows have been detected on each route. The different regimes eventually disappear in a boundary crisis in favor of a single, globally symmetric, hyperchaotic regime.

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 θ 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 (θ≈0∘), 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 θ≈0∘ disappear, eventually resulting in the single steady roll solution found in the heated-from-the-sides configuration (θ=90∘). We confirm the existence of critical angles during the transition θ:0∘→90∘, and we demonstrate that such angles are a consequence of either singularities or collisions of bifurcation points in the Rayleigh-number-θ parameter space. We finally derive the most important critical angles corresponding to any Newtonian fluid of Prandtl number greater than that of air.

Instabilities in the Rayleigh-Bénard-Eckart problem

This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the dependence of the instability threshold on the normalized acoustic beam width H(b) for an isothermal fluid layer. The critical curve, given by the critical values of the acoustic streaming parameter, A(c), has a minimum for a beam width H(b) ≈ 0.32. This curve, which corresponds to the onset of oscillatory instabilities, compares well with that obtained for a two-dimensional cavity of large aspect ratio [A(x) = (length/height) = 10]. For a fluid layer heated from below subject to acoustic waves (the Rayleigh-Bénard-Eckart problem), the influence of the acoustic streaming parameter A on the stability threshold is investigated for various values of the beam width H(b) and different Prandtl numbers Pr. It is shown that, for not too small values of the Prandtl number (Pr > Pr(l)), the acoustic streaming delays the appearance of the instabilities in some range of the acoustic streaming parameter A. The critical curves display two behaviors. For small or moderate values of A, the critical Rayleigh number Ra(c) increases with A up to a maximum. Then, when A is further increased, Ra(c) undergoes a decrease and eventually goes to 0 at A = A(c), i.e., at the critical value of the isothermal case. Large beam widths and large Prandtl numbers give a better stabilizing effect. In contrast, for Prandtl numbers below the limiting value Pr(l) (which depends on H(b)), stabilization cannot be obtained. The instabilities in the Rayleigh-Bénard-Eckart problem are oscillatory and correspond to right- or left-traveling waves, depending on the parameter values. Finally, energy analyses of the instabilities at threshold have indicated that the change of the thresholds can be connected to the modifications induced by the streaming flow on the critical perturbations.

Instabilities in a cylindrical cavity heated from below with a free surface. I. Effect of Biot and Marangoni numbers

Convective instabilities in a cylindrical cavity heated from below, with a free surface at the top, are numerically investigated using a spectral-element code. Both buoyancy and surface tension forces are taken into account, and heat exchange is considered at the upper surface. This configuration corresponds to the Bénard-Marangoni situation. The primary thresholds associated with azimuthal eigenmodes and corresponding to the onset of convection are first given as a function of the aspect ratio of the cavity A (radius/height), the Biot number Bi, and the Marangoni number Ma. Particular attention is paid to the influence of the Biot and Marangoni numbers: a stabilizing surface tension effect (Ma>0) induces an increase of the primary thresholds, which is magnified for small values of Bi, but may also change the flow structure by creating counter-rotating rolls near the free surface. The nonlinear evolution of the convection beyond its onset is given through bifurcation diagrams for A=1.5. Two different branches of axisymmetric solutions, either with upflow or downflow at the center, emerge at the onset. The destabilization of these solutions and the further dynamical evolution of the flow has been highlighted for widely varying Biot numbers.

Influence of acoustic streaming on the stability of a laterally heated three-dimensional cavity

The flows induced by acoustic streaming in a three-dimensional side-heated parallelepiped cavity of length Ax representative of crystal growth configurations are numerically studied. Both the structure of the flows and their stability properties are determined. The flows have different symmetries, belonging to the group D4 for pure streaming, Z2xZ2 for pure buoyancy, and Z2 for the mixed case, but these symmetries are generally broken at the first bifurcation points. Bifurcation diagrams are obtained which show that the flows become oscillatory periodic at a Hopf bifurcation, either directly on the primary steady solution branch, or on a secondary branch which bifurcates from the primary branch at a steady bifurcation point. The critical Grashof numbers for these bifurcation points are calculated as a function of the cavity length Ax, the Prandtl number Pr and the acoustic streaming parameter A. The thresholds are generally found to increase when the acoustic streaming contribution is enhanced, which indicates a stabilizing effect induced by acoustic streaming and may explain the observed improvement of the crystal quality when ultrasound waves are applied during the growth process. Destabilization effects are, however, found in some parameter range.