Oscillatory convection and chaos in a Lorenz-type model of a rotating fluid

A four-mode model of convection in a rotating fluid layer is studied. The model is an extension of the Lorenz model of Rayleigh-Bénard convection, the extra mode accounting for the regeneration of vorticity by rotation. Perturbation theory is applied to show that the Hopf bifurcations from conductive and steady convective solutions can be either supercritical or subcritical. Perturbation theory is also used at large Rayleigh numbersr to predict novel behavior. Supercritical oscillatory convection of finite amplitude is found by numerical integration of the governing equations. The general picture is of a series of oscillatory solutions stable over larger intervals, interspersed by short bursts of chaos.     

Large-scale patterns in Rayleigh-Bénard convection

Rayleigh-Bénard convection at large Rayleigh number is characterized by the presence of intense, vertically moving plumes. Both laboratory and numerical experiments reveal that the rising and descending plumes aggregate into separate clusters so as to produce large-scale updrafts and downdrafts. The horizontal scales of the aggregates reported so far have been comparable to the horizontal extent of the containers, but it has not been clear whether that represents a limitation imposed by domain size. In this work, we present numerical simulations of convection at sufficiently large aspect ratio to ascertain whether there is an intrinsic saturation scale for the clustering process when that ratio is large enough. From a series of simulations of Rayleigh-Bénard convection with Rayleigh numbers between 10⁵ and 10⁸ and with aspect ratios up to 12π, we conclude that the clustering process has a finite horizontal saturation scale with at most a weak dependence on Rayleigh number in the range studied.     

A diagrammatic derivation of (convective) pattern equations

In complicated bifurcation problems where more than one instability can arise at onset, reasonably sound derivations of the equations that govern the amplitudes of the nearly marginal modes have been developed when the spectrum of the modes is discrete. The basis of these derivations lies in the center manifold theorem of dynamical systems theory. But when the spectrum of the modes is continuous and we no longer have that theorem to fall back on, there is nevertheless an equation (the Swift-Hohenberg equation) that well describes the patterns seen in Rayleigh-Bénard convection. Indeed, several ‘derivations’ of the S-H equation have been offered and here we describe how to obtain the S-H equation using Bogoliubov’s method. We suggest that this procedure clarifies and simplifies (though it does not make rigorous) the derivation of the S-H equation.Looking ahead to the derivation of pattern equations for more complicated problems with continuous spectra, we also describe a diagrammatic procedure that, once mastered, is useful in performing the complicated perturbative developments that are needed in such derivations. Here we illustrate the proposed combination of the ideas of Bogoliubov and Feynman for the standard form of the Rayleigh-Bénard convection problem.The resulting pattern equation is nonlocal but it reduces without approximation to the 1-D Swift-Hohenberg equation in the case of 2-D convection. Like the S-H equation, the nonlocal version admits a Lyapunov functional and we briefly indicate its utility in pattern selection both for the Swift-Hohenberg equation and its nonlocal extension. We conclude by describing the kinds of problems for which we intend the combined method but reserve the exhibition of the required calculations for a future festschrift.     

A hydrodynamically correct thermal lattice Boltzmann model

A three-dimensional lattice-Boltzmann model which yields correct hydrodynamics at the Navier-Stokes level of the Chapman-Enskog expansion requires a minimum of 26 velocities. We present results for a model with one additional velocity, determined by maximizing the equilibrium entropy. For compressible Rayleigh-Bénard convection the model is more accurate but considerably less stable, than a previous, approximate 21-speed model.     

Theoretical and numerical study of a thermal convection problem with temperature-dependent viscosity in an infinite layer

A convection problem with temperature-dependent viscosity in an infinite layer is presented. This problem has important applications in mantle convection. The existence of a stationary bifurcation is proven together with a condition to obtain the critical parameters at which the bifurcation takes place. A numerical strategy has been developed to calculate the critical bifurcation curves and the most unstable modes for a general dependence of viscosity on temperature. An exponential dependence of viscosity on temperature has been considered in the numerical calculations. Comparisons with the classic Rayleigh-Bénard problem with constant viscosity indicate that the critical temperature difference threshold decreases as the exponential rate parameter increases. The vertical velocity of the marginal mode exhibits motion concentrated in the region where viscosity is smaller.     

Wavelength selection in Rayleigh-Bénard convection

Rayleigh-Bénard convection is considered in the presence of a weakly nonuniform temperature distribution and weakly nonuniform layer height with their gradients perpendicular to the (parallel) convection rolls. In the infinite Prandtl number limit the phase equation that describes the slow spatial and temporal evolution of the local wave number is derived. Evaluating the equation near threshold and for stress-free boundary conditions we find nonuniversal wavenumber selection, forced phase diffusion with moving patterns, and other measurable effects.     

Stability of impulsively-driven natural convection with unsteady base state: implications of an adiabatic boundary

Stability conditions of a quiescent, horizontally infinite fluid layer with adiabatic bottom subject to sudden cooling from above are studied. Here, at difference from Rayleigh-Bénard convection, the temperature base state is never steady. Instability limits are studied using linear analysis while stability is analyzed using the energy method. Critical stability curves in terms of Rayleigh numbers and convection onset times were obtained for several kinematic boundary conditions. Stability curves resulting from energy and linear approaches exhibit the same temporal growth rate for large values of time, suggesting a bound for the temporal asymptotic behavior of the energy method.     

Stabilization of thermal lattice Boltzmann models

A three-dimensional thermal lattice-Boltzmann model with two relaxation times to separately control viscosity and thermal diffusion is developed. Numerical stability of the model is significantly improved using Lax-Wendroff advection to provide and adjustable time step. Good agreement with a conventional fiitedifference Navier-Stokes solver is obtained in modeling compressible Rayleigh-Bénard convestion when boundary conditions are treated similarly.     

A lattice gas model for thermohydrodynamics

The lattice gas model is extended to include a temperature variable in order to study thermohydrodynamics, the combination of fluid dynamics and heat transfer. The compressible Navier-Stokes equations are derived using a Chapman-Enskog expansion. Heat conduction and convection problems are investigated, including Bénard convection. It is shown that the usual rescaling procedure can be avoided by controlling the temperature.     

Bound on vertical heat transport at large Prandtl number

We prove a new upper bound on the vertical heat transport in Rayleigh-Bénard convection of the form under the assumption that the ratio of Prandtl number over Rayleigh number satisfies where the non-dimensional constant c₀ depends on the aspect ratio of the domain only. This new rigorous bound agrees with the (optimal) bound (modulo logarithmic correction) on vertical heat transport for the infinite Prandtl number model for convection due to Constantin and Doering [P. Constantin, C.R. Doering, Infinite Prandtl number convection, J. Stat. Phys. 94 (1) (1999) 159-172] and Doering, Otto and Reznikoff [C.R. Doering, F. Otto, M.G. Reznikoff, Bounds on vertical heat transport for infinite Prandtl number Rayleigh-Bénard convection, J. Fluid Mech. 560 (2006) 229-241]. It also improves a uniform (in Prandtl number) bound for the Nusselt number [P. Constantin, C.R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996) 1049-1060] in the case of large Prandtl number.     

Scaling in convective evaporation and sidewall boundary layer

Turbulent convection at aspect ratios from 0.06 to 2 is investigated in the laboratory with evaporation experiments from vertical cylinders having different diameters and liquid levels. With alcohol, only diffusive evaporation takes place. With water, for small diameters, evaporation proceeds by diffusion whereas convective evaporation develops when the diameter is increased. This onset can be effectively interpreted in terms of a viscous sidewall boundary layer, whose thickness δ varies with respect to the available height h according to δ/h = 3.4 Ra^-0.28±0.01 versus Rayleigh number Ra. The Sherwood number Sh, analog of the Nusselt number, exhibits a power law variation Sh = 0.6 Ra^0.27±0.02 for Ra varying from 10⁴ to 3 ×10⁸. The scaling observed in this case of an open boundary is thus similar to the scaling measured in confined Rayleigh-Bénard convection.     

Note on the hydrodynamic eigenmodes of Couette flow

The axisymmetric eigenmodes for the velocity and pressure disturbances in the incompressible Couette flow between two concentric rotating cylinders with no-slip boundary conditions are computed numerically and plotted. As found previously in the narrow-gap approximation as well as for the Rayleigh-Bénard system, damped propagating viscous modes are present for wide ranges of parameters. Closed-form solutions for the special case of corotating cylinders show that the time constants for the decay of the eigenmodes then become insensitive to the ratio of the radii of the cylinders.     

Nambu力学系统的Lie对称性及其守恒量

讨论了Nambu力学系统的Lie对称性;建立了系统Lie对称性的确定方程;得到了该对称性引起的守恒量;研究了Lie对称性逆问题. 并以Euler方程为例说明了本文的主要结果. 关键词： Nambu力学系统 Lie对称性 守恒量     

Real time observations by neutron scattering of fluctuations near the Rayleigh-Bénard instability

Using long time-series counting in a neutron scattering experiment, hydrodynamic fluctuations were studied near the Rayleigh-Bénard instability in para-azoxyanisole. Critical enhancement was observed and the coupling to inner orientational fluctuations of the nematic liquid crystal was demonstrated through the response of the signal to an external magnetic field. The similarity of the experimental curves to those of critical scattering in equilibrium physics lends support to the phase-transition analogy of hydrodynamic instabilities.Work performed in partial fullfilment of the cand.real.-degree at the Institute of Physics, University of Oslo     

Fluctuations in a fluid under a stationary heat flux II. Slow part of the correlation matrix

The general formulas, derived in a previous paper, are used to calculate the correlation functions of the hydrodynamic variables in the Rayleigh-Bénard system. The behavior of the correlation functions on a time scale slow compared to that of sound propagation is determined, using systematically nonequilibrium hydrodynamic eigenmodes. These (slow) eigenmodes of the linearized Boussinesq equations in the presence of gravity and a temperature gradient are the viscous and the visco-heat modes. They are determined for ideal heat-conducting plates with stick boundary conditions. The visco-heat modes are found to behave qualitatively different from those obtained with slip boundary conditions. Using these eigenmodes, the slow part of the correlation functions can be determined explicitly. On a small length scale, as probed by light scattering, we recover the same expression for the Rayleigh line as quoted in the literature. On larger length scales, as probed by microwaves, the coupling of gravity to the temperature gradient gives rise to a convective instability (heating form below) or to propagating visco-heat modes (heating from above). The corresponding correlation functions and the Rayleigh line are calculated and discussed.     

Poiseuille-Rayleigh-Bénard流动中对流斑图的分区和成长

采用SIMPLE算法对二维流体力学基本方程组进行了数值模拟,研究了Poiseuille-Rayleigh-Bénard流动中对流斑图的分区、成长及水平流动对不同斑图特征物理量的影响.结果表明,上下临界雷诺数Re_u,Re_l将流动分成三个区域,即行波区、局部行波区、水平流区.Re_u和Re_l随着相对瑞利数r的增大而增大.在对流斑图的成长阶段,三种斑图随时间的成长过程是不同的,但对流圈都是从下游区开始成长;特征物理量随着时间的变化也是不同的,行波对流和局部行波对流的最大垂直流速wmax和努塞尔数Nu经过指数增长阶段后进入周期变化的稳定阶段;水平流斑图的w_(max)和Nu经过缓慢增长后又缓慢降到稳定值.三种斑图的w_(max)和Nu随雷诺数Re增大而减小,不同斑图区域有不同的变化规律.本文给出了Re_u和Re_l随r的变化关系式及不同斑图的w_(max)和Nu随着Re的变化关系式.     

分离比对混合流体Rayleigh-Bénard对流解的影响

混合流体Rayleigh-Bénard对流是研究非平衡对流的非线性动力学特性的典型模型之一.基于流体力学方程组的数值模拟,首先探讨了矩形腔体中具有强Soret效应(分离比Ψ=-0.60)的混合流体行波对流的分叉特性及斑图演化,沿着分叉曲线的上部分支,随着相对瑞利数的增加,此系统依次出现了局部行波对流、具有缺陷的行波对流、行波对流、摆动行波对流及定常对流5种行波对流解.然后,研究了分离比Ψ对对流解的影响,与弱Soret效应(Ψ=-0.11)时的对流解相比较,强Soret效应(Ψ=-0.60)时出现的对流解更丰富.由于有强Soret效应的对流的复杂性,Ψ=-0.60时的对流解与Ψ=-0.20,-0.4时的对流解不同.     

Fluctuations in a fluid under a stationary heat flux. I. General theory

We present the basic formulas for a unified treatment of the correlation functions of the hydrodynamic variables in a fluid between two horizontal plates which is exposed to a stationary heat flux in the presence of a gravity field (Rayleigh-Bénard system). Our analysis is based on fluctuating hydrodynamics. In this paper (I) we show that in the nonequilibrium stationary state the hydrodynamic fluctuations evolve on slow and fast time scales that are widely separated. A time scale perturbation theory is used to diagonalize the hydrodynamic operator partially. This enables us to derive the eigenvalue equations for the nonequilibrium hydrodynamic modes. Therein we take into account the variation of the macroscopic quantities with position. The correlation functions are formally expressed in terms of the nonequilibrium modes. In paper II the slow hydrodynamic modes (viscous and viscoheat modes) will be determined explicitly for ideal heat-conducting plates with stick boundary conditions and used to compute the slow part of the correlation functions; in paper III the fast hydrodynamic modes (sound modes) will be explicitly determined for stick boundary conditions and used to compute the fast part of the correlation functions. In these papers we will also compute the shape and intensity of the lines measured in light scattering experiments.     

On numerical realizability of thermal convection

Astounded at the regularity of convective structures observed in simulations of mesoscale flow past realistic topography, we investigate the computational aspects of a classical problem of flow over a heated plane. We find that the numerical solutions are sensitive to viscosity, either incorporated a priori or effectively realized in computational models. In particular, anisotropic viscosity can lead to regular convective structures that mimic naturally realizable Rayleigh–Bénard cells, which are unphysical for the specified external parameter range. Details of the viscosity appear to play a secondary role; that is, similar structures can occur for prescribed constant viscosities, explicit subgrid-scale turbulence models, ad-hoc numerical filters, or implicit dissipation of numerical schemes. This implies the need for a careful selection of numerical tools suitable for convection-resolving simulations of atmospheric circulations. The implicit large-eddy-simulation (ILES) approach using non-oscillatory schemes is especially attractive, as for under-resolved calculations it reproduces well the coarsened results of finely-resolved boundary layer convection.