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.     

The role of defects in the transition between different symmetries in convective patterns

The properties of the transition between two patterns with different symmetries are studied in an experiment on thermal convection in a fluid layer heated from below. The defect structure is also analyzed and it is found that the unstable phase is present in the defect cores. The experimental results are in agreement with a model of three coupled Ginzburg-Landau equations.     

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.     

用格子Boltzmann模型模拟垂直平板间的热对流

引入一个新的能量分布函数，利用该能量分布函数与粒子速度分布函数耦合来求解一个热流场. 因而，这一能量分布函数与粒子速度分布函数和Boltzmann方程构成了一个新的格子Boltzmann模型. 这一模型满足质量、动量和能量守恒的准则. 用该模型对垂直平板间的狭缝热对流进行了数值模拟，数值结果表明，在Prandtl数为1，Grashof数在1.3×10²—1×10⁶之间时，流场将出现多个旋涡结构的流型. 得出了与Lee相一致的结论. 关键词： 能量分布函数 Boltzmann方程 热对流     

Dependence of magnetic field generation by thermal convection on the rotation rate: A case study

Dependence of magnetic field generation on the rotation rate is explored by direct numerical simulation of magnetohydrodynamic convective attractors in a plane layer of conducting fluid with square periodicity cells for the Taylor number varied from zero to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We observe 5 types of hydrodynamic (amagnetic) attractors: two families of two-dimensional (i.e. depending on two spatial variables) rolls parallel to sides of periodicity boxes of different widths and parallel to the diagonal, travelling waves and three-dimensional “wavy” rolls. All types of attractors, except for one family of rolls, are capable of kinematic magnetic field generation. We have found 21 distinct nonlinear convective MHD attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge. In addition, we have observed a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The system is highly symmetric. We have found two novel global bifurcations reminiscent of the SNIC bifurcation, which are only possible in the presence of symmetries. The universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid on continuing the increase), remains unaltered, but we demonstrate that this “large-scale” picture lacks many significant details.     

Fluctuating hydrodynamics and principal oscillation pattern analysis

Principal oscillation pattern (POP) analysis was recently introduced into climatology to analyze multivariate time series x_i(t) produced by systems whose dynamics are described by a linear Markov process x=Bx + . The matrixB gives the deterministic feedback and is a white noise vector with covariances (t) _j (t*Q _ij (t–t. The POP method is applied to data from a direct simulation Monte Carlo program. The system is a dilute gas with 50,000 particles in a Rayleigh-Bénard configuration. The POP analysis correctly reproduces the linearized Navier-Stokes equations (in the matrixB) and the stochastic fluxes (in the matrixQ) as given by Landau-Lifschitz fluctuating hydrodynamics. Using this method, we find the Landau-Lifschitz theory to be valid both in equilibrium and near the critical point of Rayleigh-Bénard convection.     

A lattice Boltzmann equation for diffusion

The formulation of lattice gas automata (LGA) for given partial differential equations is not straightforward and still requires some sort of magic. Lattice Boltzmann equation (LBE) models are much more flexible than LGA because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements. Here a LBE is presented for diffusion in an arbitrary number of dimensions. The model is probably the simplest LBE which can be formulated. It is shown that the resulting algorithm with relaxation parameter =1 is identical to an explicit finite-difference (EFD) formulation at its stability limit. Underrelaxation (0<<1) allows stable integration beyond the stability limit of EFD. The time step of the explicit LBE integration is limited by accuracy and not by stability requirements.     

Double MRT thermal lattice Boltzmann method for simulating convective flows

A two-dimensional double Multiple Relaxation Time-Thermal Lattice Boltzmann Equation (2-MRT-TLBE) method is developed for predicting convective flows in a square differentially heated cavity filled with air (Pr=0.71). In this Letter, we propose a numerical scheme to solve the flow and the temperature fields using the MRT-D2Q9 model and the MRT-D2Q5 model, respectively. Thus, the main objective of this study is to show the effectiveness of such model to predict thermodynamics for heat transfer. This model is validated by the numerical simulations of the 2-D convective square cavity flow. Excellent agreements are obtained between numerical predictions. These results demonstrate the accuracy and the effectiveness of the proposed methodology.     

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.     

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.     

A lattice Boltzmann model for the generalized Burgers-Huxley equation

In this paper we develop a lattice Boltzmann model for the generalized Burgers-Huxley equation (GBHE). By choosing the proper time and space scales and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation, and the local equilibrium distribution functions are obtained. Excellent agreement with the exact solution is observed, and better numerical accuracy is obtained than the available numerical result. The results indicate the present model is satisfactory and efficient. The method can also be applied to the generalized Burgers-Fisher equation and be extended to multidimensional cases.     

A time-reversal lattice Boltzmann method

In this paper we address the time-reversed simulation of viscous flows by the lattice Boltzmann method (LB). The theoretical derivation of the reversed LB from the Boltzmann equation is detailed, and the method implemented for weakly compressible flows using the D2Q9 scheme. The implementation of boundary conditions is also discussed. The accuracy and stability are illustrated by four test cases, namely the propagation of an acoustic wave in a medium at rest and in an uniform mean flow, the Taylor–Green vortex decay and the vortex pair–wall collision.     

基于总能形式的耦合的双分布函数热晶格玻尔兹曼数值方法

双分布函数热晶格玻尔兹曼数值方法在微尺度热流动系统中得到广泛的应用. 本文基于晶格玻尔兹曼平衡分布函数低阶Hermite展开式, 创新性地提出了包含黏性热耗散和压缩功的耦合的双分布函数热晶格玻尔兹曼数值方法, 将能量场内温度的变化以动量源的形式引入晶格波尔兹曼动量演化方程, 实现了能量场与动量场之间的耦合. 研究了考虑黏性热耗散和压缩功的和不考虑的两种热自然对流模型, 重点分析了不同瑞利数和普朗特数下流场内的流动情况以及温度、速度和平均努赛尔数的变化趋势. 本文实验结果与文献结果一致, 验证了本文数值方法的可行性和准确性. 研究结果表明: 随着瑞利数和普朗特数的增大, 方腔内对流传热作用逐渐增强, 边界处形成明显的边界层; 考虑黏性热耗散和压缩功的模型对流作用相对增强, 黏性热耗散和压缩功对自然对流的影响在微尺度流动过程中不能忽略.     

Multiple-relaxation-time lattice Boltzmann model for compressible fluids

We present an energy-conserving multiple-relaxation-time finite difference lattice Boltzmann model for compressible flows. The collision step is first calculated in the moment space and then mapped back to the velocity space. The moment space and corresponding transformation matrix are constructed according to the group representation theory. Equilibria of the nonconserved moments are chosen according to the need of recovering compressible Navier-Stokes equations through the Chapman-Enskog expansion. Numerical experiments showed that compressible flows with strong shocks can be well simulated by the present model. The new model works for both low and high speeds compressible flows. It contains more physical information and has better numerical stability and accuracy than its single-relaxation-time version.     

三维方腔温盐双扩散的格子Boltzmann方法数值模拟

研究了温盐双扩散系统的多组分格子Boltzmann方法.通过对二维方腔的温盐双扩散系统的数值模拟，检验了方法的可行性及有效性，所得到的结果与差分法结果符合良好，继而将此方法推广到三维，建立了三维温盐双扩散系统的格子Boltzmann方法，对三维方腔双扩散问题进行了模拟和分析，并与差分法模拟的结果进行了比较，结果令人满意.最后，分析了格子Boltzmann方法在模拟双扩散对流问题时存在的局限性. 关键词： 格子Boltzmann方法 温盐双扩散 Boussinesq近似 数值模拟     

On the statistical dynamics of the Lorenz model

We use the theory of stochastic differential equations with rapidly fluctuating coefficients to study the statistical dynamics of the Lorenz model in the turbulent region. On the assumption that the system is ergodic we are able to calculate self-consistently several basic statistical quantities in terms of the parameters of the model. Our results are in good agreement with numerical computations.Supported financially by the Summer Study Program in Geophysical Fluid Dynamics at the Woods Hole Oceanographic Institution.     

A lattice Boltzmann method for thermal nonideal fluids

A Lattice Boltzmann Method for van der Waals fluids with variable temperature is described. Thermo-hydrodynamic equations are correctly reproduced at second order of a Chapman-Enskog expansion. The method is applied to study initial stages of phase separation of a fluid quenched by contact with colder walls. Thermal equilibration is favoured by pressure waves which propagate with the sound velocity.     

A lattice Boltzmann model for coupled diffusion

Diffusion coupling between different chemical components can have significant effects on the distribution of chemical species and can affect the physico-chemical properties of their supporting medium. The coupling can arise from local electric charge conservation for ions or from bound components forming compounds. We present a new lattice Boltzmann model to account for the diffusive coupling between different chemical species. In this model each coupling is added as an extra relaxation term in the collision operator. The model is tested on a simple diffusion problem with two coupled components and is in excellent agreement with the results obtained through a finite difference method. Our model is observed to be numerically very stable and unconditional stability is shown for a class of diffusion matrices. We further develop the model to account for advection and show an example of application to flow in porous media in two dimensions and an example of convection due to salinity differences. We show that our model with advection loses the unconditional stability, but offers a straight-forward approach to complicated two-dimensional advection and coupled diffusion problems.     

A lattice Boltzmann model with an amending function forsimulating nonlinear partial differential equations

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form $u_t+\alpha uu_{xx}+\beta u^n u_x+\gamma u_{xxx}+\xi u_{xxxx}=0$. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman--Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.