Mode-to-mode energy transfers in convective patterns

We investigate the energy transfer between various Fourier modes in a low-dimensional model for thermal convection. We have used the formalism of mode-to-mode energy transfer rate in our calculation. The evolution equations derived using this scheme is the same as those derived using the hydrodynamical equations for thermal convection in Boussinesq fluids. Numerical and analytical studies of this model show that convective rolls appear as the Rayleigh number R is raised above its critical value R _c. Further increase of Rayleigh number generates rolls in the perpendicular directions as well, and we obtain a dynamic asymmetric square pattern. This pattern is due to Hopf bifurcation. There are two sets of limit cycles corresponding to the two competing asymmetric square patterns. When the Rayleigh number is increased further, the limit cycles become unstable simultaneously, and chaotic motion sets in. The onset of chaos is via intermittent route. The trajectories wander for quite a long time almost periodically before jumping irregularly to one of the two ghost limit cycles.     

Heteroclinic behavior in rotating Rayleigh-Bénard convection

We investigate numerically the appearance of heteroclinic behavior in a three-dimensional, buoyancy-driven fluid layer with stress-free top and bottom boundaries, a square horizontal periodicity with a small aspect ratio, and rotation at low to moderate rates about a vertical axis. The Prandtl number is 6.8. If the rotation is not too slow, the skewed-varicose instability leads from stationary rolls to a stationary mixed-mode solution, which in turn loses stability to a heteroclinic cycle formed by unstable roll states and connections between them. The unstable eigenvectors of these roll states are also of the skewed-varicose or mixed-mode type and in some parameter regions skewed-varicose like shearing oscillations as well as square patterns are involved in the cycle. Always present weak noise leads to irregular horizontal translations of the convection pattern and makes the dynamics chaotic, which is verified by calculating Lyapunov exponents. In the nonrotating case, the primary rolls lose, depending on the aspect ratio, stability to traveling waves or a stationary square pattern. We also study the symmetries of the solutions at the intermittent fixed points in the heteroclinic cycle. Received 10 June 1999     

Elastic convection in vibrated viscoplastic fluids

We observe a new type of behavior in a shear-thinning yield stress fluid: freestanding convection rolls driven by vertical oscillation. The convection occurs without the constraint of container boundaries, yet the diameter of the rolls is spontaneously selected for a wide range of parameters. The transition to the convecting state occurs without hysteresis when the amplitude of the plate acceleration exceeds a critical value. We find that a nondimensional stress, the stress due to the inertia of the fluid normalized by the yield stress, governs the onset of the convective motion.     

Weakly nonlinear convection: Galerkin model,numerical simulation,and amplitude equation

The quality of three-mode Galerkin models for describing convective rolls in a fluid layer with rigid horizontal boundaries is investigated close to onset. Comparisons are made with the correct linear stability properties of the conductive state, known weakly nonlinear perturbation results for convection, and high-precision numerical solutions of the full field equations.     

Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium

By using the mathematical formalism of absolute and convective instabilities we study the nature of unstable three-dimensional disturbances of viscoelastic flow convection in a porous medium with horizontal through-flow and vertical temperature gradient. Temporal stability analysis reveals that among three-dimensional (3D) modes the pure down-stream transverse rolls are favored for the onset of convection. In addition, by considering a spatiotemporal stability approach we found that all unstable 3D modes are convectively unstable except the transverse rolls which may experience a transition to absolute instability. The combined influence of through-flow and elastic parameters on the absolute instability threshold, wave number and frequency is then determined, and results are compared to those of a Newtonian fluid.     

The fate of self-aligned rolls in gravity modulated magnetoconvection

We explore the modifications brought about by gravity modulation in self-aligned two-dimensional (2D) stationary rolls and higher-order convective instabilities with horizontal magnetic field. The Chandrasekhar number plays no role in determining critical values of the forcing parameters for convection to begin. The onset is in the form of 2D rolls with periodically varying intensity. The external magnetic field is solely responsible for aligning the rolls while the induced magnetic field comes into play after three-dimensional convection sets in. Gravity modulation destabilizes the system and results in chaotic flow much earlier above onset.     

Reentrant hexagons in non-Boussinesq Rayleigh-Bénard convection: effect of compressibility

We present experimental studies of a new pattern sequence observed in non-Boussinesq convection in a compressible fluid near its gas-liquid critical point (CP). Besides the known hysteretic transitions among conduction state, hexagons, and rolls, another hysteretic transition from rolls to hexagons at higher values of the control parameter is found. This reentrance phenomenon is observed in a rather narrow range of about 60- to 100-microm cell heights and is attributed to large compressibility of a fluid near the CP.     

Dynamics of zero-Prandtl number convection near onset

We present a detailed bifurcation scenario of zero-Prandtl number Rayleigh-Be?nard convection using direct numerical simulations (DNS) and a 27-mode low-dimensional model containing the most energetic modes of DNS. The bifurcation analysis reveals a rich variety of convective flow patterns and chaotic solutions, some of which are common to that of the 13-mode model of Pal et al. [EPL 87, 54003 (2009)]. We also observed a set of periodic and chaotic wavy rolls in DNS and in the model similar to those observed in experiments and numerical simulations. The time period of the wavy rolls is closely related to the eigenvalues of the stability matrix of the Hopf bifurcation points at the onset of convection. This time period is in good agreement with the experimental results for low-Prandtl number fluids. The chaotic attractor of the wavy roll solutions is born through a quasiperiodic and phase-locking route to chaos.     

Linear stability analysis of the natural convection in inclined rotating parallel plates

The linear stability analysis of the natural convection in a rectangular tilted infinite cavity filled with a Boussinesq fluid subject to Coriolis force is presented. The bottom and top surfaces have fixed temperatures. Both unstable and stable thermal conditions are studied (heated from below and heated from above respectively). The rotation axis passes through the center and it is orthogonal to the hot and cold surfaces. The stability equations were solved using the Tau–Chebyshev spectral method. The critical Rayleigh number and critical wave number were obtained for several rotation rates and different orientation of convective oblique rolls in a range of inclination of the cavity from 0 to 120 degrees. The stability analysis show that rotation rate affects the basic velocity profile, onset of convection, wave number and critical orientation of convective rolls.     

Convective roll dynamics in liquid 4He near the onset of convection

We present results of experiments on Rayleigh-Bénard convection in liquid 4He at several temperatures. We show visually that with carefully defined boundary conditions the basic convection state consists of parallel rolls which are aligned in one of two directions, the angle thus defined as being temperature dependent and we attempt to explain this behavior. We also show directly the skew-varicose instability acting on the basic state and correlate it with fluctuations in the temperature difference across the fluid layer.     

Localized waves without the existence of extended waves: oscillatory convection of binary mixtures with strong soret effect

Spatially confined solutions of traveling convection rolls are determined numerically for binary mixtures such as ethanol-water. The appropriate field equations are solved in a vertical cross section of the rolls perpendicular to their axes subject to realistic horizontal boundary conditions. The localized convective states are stably and robustly sustained by strongly nonlinear mixing and complex flow-induced concentration redistribution. We elucidate how this enables their existence for strongly negative separation ratios at small subcritical heating rates below the saddle node of extended traveling convection rolls where the quiescent fluid is strongly stable.     

大尺度环形液池内双层热毛细对流不稳定性

假设双层流体的交界面不发生变形,热毛细力作用于此交界面,三维数值研究了大尺度环形液池中双层流体系统在内外壁面温差加热下的热毛细对流不稳定性,其中外壁面维持高温,内壁面维持低温。计算结果显示,上下层流体的流动特性受Marangoni效应和浮力效应的影响;热毛细对流的振荡产生于内壁面附近,并沿着温度梯度的方向传播;随着温差的增大,热毛细对流的振荡逐渐增强,温度振荡波数增大。     

Pattern formation in inclined layer convection

We report experiments on thermally driven convection in an inclined layer of large aspect ratio in a fluid of Prandtl number sigma approximately 1. We observed a number of new nonlinear, mostly spatiotemporally chaotic, states. At small angles of inclination we found longitudinal rolls, subharmonic oscillations, Busse oscillations, undulation chaos, and crawling rolls. At larger angles, in the vicinity of the transition from buoyancy- to shear-driven instability, we observed drifting transverse rolls, localized bursts, and drifting bimodals. For angles past vertical, when heated from above, we found drifting transverse rolls and switching diamond panes.     

Different localized states of travelling-wave convection in a rectangular container

We have performed numerical simulations of localized travelling-wave convection in a binary fluid mixture heated from below in a long rectangular container. Calculations are carried out in a vertical cross section of the rolls perpendicular to their axes. For a negative enough separation ratio, two types of quite different confined states were documented by applying different control processes. One branch of localized travelling waves survives only in a very narrow band within subcritical regime, while another branch straddles the onset of convection existing both in subcritical and supercritical regions. We elucidated that concentration field and its current are key to understand how confined convection is sustained when conductive state is absolutely unstable. The weak structures in the conducting region are demonstrated too.     

Ergodic stream-lines in steady convection

Stream-lines of steady Rayleigh-Bénard convection with square planform are displayed using Poincaré maps. As the second order mode becomes more important the flow becomes ergodic from the boundaries inward, like a perturbed, integrable hamiltonian system.     

Drifting convection cells in rotating fluid layers heated from below

It is shown that hexagonal convection cells in a rotating horizontal fluid layer heated from below will in general exhibit a drift in contrast to convection rolls except in the case of a vertical axis of rotation. The direction of the drift is prograde (retrograde) for cells with rising (descending) motion in the center of the convection cell. In addition a mean flow generated by convection is derived. An application to solar convection is discussed.     

Reentrant and whirling hexagons in non-Boussinesq convection

We review recent computational results for hexagon patterns in non-Boussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a vertical axis the transition from hexagons to rolls is replaced by a Hopf bifurcation to whirling hexagons. For weak non-Boussinesq effects they display defect chaos of the type described by the two-dimensional (2D) complex Ginzburg–Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and localized bursting of the whirling amplitude is found. In this regime the coupling of the whirling amplitude to (small) deformations of the hexagon lattice becomes important. For yet stronger non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly disordered states characterized by whirling and lattice defects are obtained.     

Anisotropic Diffusion in Driven Convection Arrays

We numerically investigate the transport of a Brownian colloidal particle in a square array of planar counter-rotating convection rolls at high Péclet numbers. We show that an external force produces huge excess peaks of the particle’s diffusion constant with a height that depends on the force orientation and intensity. In sharp contrast, the particle’s mobility is isotropic and force independent. We relate such a nonlinear response of the system to the advection properties of the laminar flow in the suspension fluid.     

竖壁传热对方腔内Benard对流的影响

本文采用具有QUICK差分格式的SIMPLE算法对边界竖壁传热的方腔内空气Benard对流进行了数值计算,根据计算结果探讨了竖壁传热对Benard对流的影响.计算表明,在所考虑的几何和物理条件下,所有竖壁绝热时,腔内流体形成平行与短轴的多个涡卷;竖壁存在传热时,腔内流体形成平行于长轴的两个涡卷,并且平行于长轴竖壁的传热热流方向相反时,涡卷的旋转方向也相反;垂直于长轴蛏壁的传热对近壁附近的流动有一定影响.     

On thermal convection in slowly rotating systems

The dynamical properties of convection patterns in a fluid layer heated from below and rotating slowly about a horizontal axis are reviewed. Applications to the equatorial regions of planetary and stellar atmospheres are emphasized. Attention is drawn to the wavelike drift of hexagonal convection cells in the azimuthal direction and to the mean flow generated by all convection patterns except for rolls aligned with the axis of rotation.