An interaction potential based lattice Boltzmann method with adaptive mesh refinement (AMR) for two-phase flow simulation

The lattice Boltzmann method (LBM) for two-phase flow simulation is often hindered by insufficient resolution at the interface. As a result, the LBM simulation of bubbles in bubbling flows is commonly limited to spherical or slightly deformed bubble shapes. In this study, the adaptive mesh refinement method for the LBM is developed to overcome such a problem. The approach for this new method is based on the improved interaction potential model, which is able to maintain grid-independent fluid properties in the two-fluid phases and at the interface. The LBM–AMR algorithm is described, especially concerning the LBM operation on a non-uniform mesh and the improved interaction potential model. Numerical simulations have been performed to validate the method in both single phase and multiphase flows. The 2D and 3D simulations of the buoyant rise of bubbles are conducted under various conditions. The agreement between the simulated bubble shape and velocity with experiments illustrates the capability of the LBM–AMR approach in predicting bubble dynamics even under the large bubble deformation conditions. Further, the LBM–AMR technique is capable of simulating a complex topology change of the interface. Integration of LBM with AMR can significantly improve the accuracy and reduce computation cost. The method developed in this study may appreciably enhance the capability of LBM in the simulation of complex multiphase flows under realistic conditions.     

Thermodynamic Foundations of Kinetic Theory and Lattice Boltzmann Models for Multiphase Flows

This paper demonstrates that thermodynamically consistent lattice Boltzmann models for single-component multiphase flows can be derived from a kinetic equation using both Enskog's theory for dense fluids and mean-field theory for long-range molecular interaction. The lattice Boltzmann models derived this way satisfy the correct mass, momentum, and energy conservation equations. All the thermodynamic variables in these LBM models are consistent. The strengths and weaknesses of previous lattice Boltzmann multiphase models are analyzed.     

自由上浮气泡运动特性的光滑粒子流体动力学模拟

基于虚功原理, 在Hu X Y等和Grenier N等的研究结果基础上推导了多相流光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)控制方程, 采用精度较高的黏性力和表面张力模型, 发展了一套适用于具有大密度比和大黏性比界面的多相流SPH方法. 首先, 通过施加人工位移修正, 适当背景压力和异相界面力, 使得计算全程粒子分布相对均匀, 改善了界面处的失稳现象, 防止了异相界面处粒子的非物理性穿透; 在此基础上, 利用方形流体团振荡模型对表面张力模型进行了验证, 数值结果与解析解甚为吻合; 然后采用上浮气泡经典数值算例对比研究了不同黏性力计算方法、不同核函数的适用性以及人工位移修正的效果; 最后, 对单个气泡的上浮、变形、撕裂以及垂向两个气泡的追赶、融合等现象进行了模拟, 初步揭示了气泡上浮过程中各种有趣物理现象的细节过程和动力学机理.     

相同尺度下气泡与复杂壁面的耦合特性研究

采用格子Boltzmann方法(LBM)建立了气液固三相耦合的动力学模型,研究了相同尺度下上浮气泡与复杂壁面的相互耦合作用.首先,基于黏性流体理论,通过构建一组格子Boltzmann(LB)方程来描述气液两相的运动,并以LB离散体积力的形式计入了黏性力、表面张力和重力.同时,采用LBM中的Half-way反弹模型与有限差分格式相结合的方式进行固壁边界的处理.然后,利用本文建立的模型,对不同特征尺寸比条件下,气泡与考虑边缘效应的平面固壁和曲面固壁的耦合特性进行了研究.研究发现固壁边界条件以及特征尺寸比对气泡的运动和拓扑结构的变化都具有明显的非线性影响.最后,研究了流体属性对气泡与复杂壁面耦合规律的影响.     

格子Boltzmann方法求解Burgers方程

众所周知,格子方法（包括格子气和格子Ｂｏｌｔｚｍａｎｎ方法）在计算物理领域取得巨大进展。与之形成鲜明对比,格子方法的数学理论始终处于停滞前的状况。为求解Ｂｕｒｇｅｒｓ方程,一类带有ＢＧＫ模型格子方法被构造出来,经过变量替换,发现他们属于三层非性差分方法。使用极值原理,给出此类格式稳定性的严格证明,最后,从数值实验中可以看出,使用ＬＢＭ得到的结果,与经典二阶守恒差分方法的结果符合得非常好。     

A lattice Boltzmann method for immiscible multiphase flow simulations using the level set method

We consider the lattice Boltzmann method for immiscible multiphase flow simulations. Classical lattice Boltzmann methods for this problem, e.g. the colour gradient method or the free energy approach, can only be applied when density and viscosity ratios are small. Moreover, they use additional fields defined on the whole domain to describe the different phases and model phase separation by special interactions at each node. In contrast, our approach simulates the flow using a single field and separates the fluid phases by a free moving interface. The scheme is based on the lattice Boltzmann method and uses the level set method to compute the evolution of the interface. To couple the fluid phases, we develop new boundary conditions which realise the macroscopic jump conditions at the interface and incorporate surface tension in the lattice Boltzmann framework. Various simulations are presented to validate the numerical scheme, e.g. two-phase channel flows, the Young–Laplace law for a bubble and viscous fingering in a Hele-Shaw cell. The results show that the method is feasible over a wide range of density and viscosity differences.     

Cavitation Bubble Collapse near a Curved Wall by the Multiple-Relaxation-Time Shan-Chen Lattice Boltzmann Model

The cavitation bubble collapse near a cell can cause damage to the cell wall. This effect has received inereasing attention in biomedical supersonics. Based on the lattice Boltzmann method, a multiple-relaxation-time Shan-Chen model is built to study the cavitation bubble collapse. Using this model, the cavitation phenomena induced by density perturbation are simulated to obtain the coexistence densities at certain temperature and to demonstrate the Young-Laplace equation. Then, the cavitation bubble collapse near a curved rigid wall and the consequent high-speed jet towards the wall are simulated. Moreover, the influences of initial pressure difference and bubble-wall distance on the cavitation bubble collapse are investigated.     

Physical modeling of multiphase flow via lattice Boltzmann method: Numerical effects, equation of state and boundary conditions

The aims of the present paper are threefold. First, we further study the fast Fourier transform thermal lattice Boltzmann (FFT-TLB) model for van der Waals (VDW) fluids proposed in Phys. Rev. E, 2011, 84(4): 046715. We analyze the merits of the FFT approach over the traditional finite difference scheme and investigate the effects of smoothing factors on accuracy and stability in detail. Second, we incorporate the VDW equation of state with flexible parameters into the FFT-TLB model. As a result, the revised model may be used to handle multiphase flows with various critical densities and temperatures. Third, we design appropriate boundary conditions for systems with solid walls. These improvements, from the views of numerics and physics, significantly extend the application scope of the model in science and engineering.     

近复杂几何固壁空泡溃灭的伪势格子Boltzmann建模*

格子Boltzmann方法伪势多相模型具有高效性和复杂几何边界实施的简易性。该文采用改进作用力的伪势多相模型,通过优化参数实现最大程度的热力学一致性,进而提高模型的密度比和稳定性。分别从伪速度、网格不变性、Young-Laplace验证等方面研究了改进模型的性能。通过改进的模型模拟了复杂几何固壁附近空泡溃灭过程。分析了空化泡溃灭阶段的密度场、压力场和速度场演化过程,以及复杂几何固壁附近的空泡动力学特性。结果表明伪势格子Boltzmann方法在探索空泡溃灭和复杂几何固壁间的相互作用规律研究中是一种有效的工具。     

Landau–Pomeranchuk–Migdal effect in a quark–gluon plasma and the Boltzmann equation

We show how the Landau–Pomeranchuk–Migdal effect on photon production rates in a quark–gluon plasma can be derived via the usual Boltzmann equation. To do this, we first derive the electromagnetic polarization tensor using linear response theory, and then formulate the Boltzmann equation including the collisions mediated by soft gluon exchanges. We then identify the resulting expression for the production rate with that obtained by the field-theoretic formalism recently proposed by Arnold, Moore and Yaffe. To illustrate the LPM effect we solve the Boltzmann equation in the diffusion approximation.     

Optimal low-dispersion low-dissipation LBM schemes for computational aeroacoustics

Lattice Boltmzann Methods (LBM) have been proved to be very effective methods for computational aeroacoustics (CAA), which have been used to capture the dynamics of weak acoustic fluctuations. In this paper, we propose a strategy to reduce the dispersive and disspative errors of the two-dimensional (2D) multi-relaxation-time lattice Boltzmann method (MRT-LBM). By presenting an effective algorithm, we obtain a uniform form of the linearized Navier–Stokes equations corresponding to the MRT-LBM in wave-number space. Using the matrix perturbation theory and the equivalent modified equation approach for finite difference methods, we propose a class of minimization problems to optimize the free-parameters in the MRT-LBM. We obtain this way a dispersion-relation-preserving LBM (DRP-LBM) to circumvent the minimized dispersion error of the MRT-LBM. The dissipation relation precision is also improved. And the stability of the MRT-LBM with the small bulk viscosity is guaranteed. Von Neuman analysis of the linearized MRT-LBM is performed to validate the optimized dispersion/dissipation relations considering monochromatic wave solutions. Meanwhile, dispersion and dissipation errors of the optimized MRT-LBM are quantitatively compared with the original MRT-LBM. Finally, some numerical simulations are carried out to assess the new optimized MRT-LBM schemes.     

A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows

A momentum exchange-based immersed boundary-lattice Boltzmann method is presented in this Letter for simulating incompressible viscous flows. This method combines the good features of the lattice Boltzmann method (LBM) and the immersed boundary method (IBM) by using two unrelated computational meshes, an Eulerian mesh for the flow domain and a Lagrangian mesh for the solid boundaries in the flow. In this method, the non-slip boundary condition is enforced by introducing a forcing term into the lattice Boltzmann equation (LBE). Unlike the conventional IBM using the penalty method with a user-defined parameter or the direct forcing scheme based on the Navier–Stokes (NS) equations, the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. Numerical examples show that the present method can provide very accurate numerical results.     

A fluctuating lattice Boltzmann scheme for the one-dimensional KPZ equation

Based on the well-known mapping between the Burgers equation with noise and the Kardar–Parisi–Zhang (KPZ) equation for fluctuating interfaces, we develop a fluctuating lattice Boltzmann (LB) scheme for growth phenomena, as described by the KPZ formalism. A very simple LB-KPZ scheme is demonstrated in 1+1 spacetime dimensions, and is shown to reproduce the scaling exponents characterizing the growth of one-dimensional fluctuating interfaces.     

偏晶合金液-液相分离的格子玻尔兹曼方法模拟

本文建立了一个模拟在弥散相液滴的扩散长大、碰撞凝并和Ostwald熟化等因素的作用下偏晶合金液-液相分离过程的二维格子玻尔兹曼方法 (lattice Boltzmann method, LBM) 模型.该模型结合了Shan-Chen的两相流模型和Qin的介观粒子相互作用势模型的优点,并在LB演化方程中引入了反映相变的源项.应用该模型模拟研究了偏晶合金液-液相分离过程中单液滴的生长、两液滴的合并和多液滴的生长规律.结果表明在两液相区中第二相单个液滴的生长是一个通过扩散从非平衡态到平衡态过渡的过程.两液滴合并 关键词： 偏晶合金 液-液相分离 格子玻尔兹曼方法     

基于树网格的格子Boltzmann方法以及曲线边界的处理

将格子玻尔兹曼方法(LBM)和BFECC技术相结合,在四叉树网格上实现,并用于二维不可压缩流动的求解.二维四叉树(三维八叉树)网格是一种非均匀网格,利用树结构进行数据存储,具有非常高的检索效率.BFECC是一种数值技巧,可以把求解输运方程的奇数阶格式通过误差修正提高一阶精度.基于四叉树网格的LBM,引入BFECC后,能有效减小输运过程中非均匀网格插值产生的误差,在总体上实现二阶精度,和经典的LBM方法相当.最后给出二维顶盖驱动方腔流动算例和二维圆柱绕流算例,从中可以看出格式的有效性.     

A ghost fluid,level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection

In this paper, we present the development of a sharp numerical scheme for multiphase electrohydrodynamic (EHD) flows for a high electric Reynolds number regime. The electric potential Poisson equation contains EHD interface boundary conditions, which are implemented using the ghost fluid method (GFM). The GFM is also used to solve the pressure Poisson equation. The methods detailed here are integrated with state-of-the-art interface transport techniques and coupled to a robust, high order fully conservative finite difference Navier–Stokes solver. Test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the EHD numerical scheme. The method is then applied to simulate a charged liquid kerosene jet.     

Chapman–Enskog analysis for finite-volume formulation of lattice Boltzmann equation

The classical Chapman–Enskog expansion is performed for the recently proposed finite-volume formulation of lattice Boltzmann equation (LBE) method [D.V. Patil, K.N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys. 228 (2009) 5262–5279]. First, a modified partial differential equation is derived from a numerical approximation of the discrete Boltzmann equation. Then, the multi-scale, small parameter expansion is followed to recover the continuity and the Navier–Stokes (NS) equations with additional error terms. The expression for apparent value of the kinematic viscosity is derived for finite-volume formulation under certain assumptions. The attenuation of a shear wave, Taylor–Green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation.     

Quantum theory of irreversibility

A generalization of the Gibbs–von Neumann entropy is proposed based on the quantum BBGKY (Bogolyubov–Born–Green–Kirkwood–Yvon) hierarchy as the non-equilibrium entropy for an N$N$-body system. By using a generalization of the Liouville–von Neumann equation describing the evolution of a density superoperator, the entropy production for an isolated system is calculated, being non-zero in general. The existence of a non-zero entropy production allows us, following the procedure of non-equilibrium thermodynamics to introduce a master matrix for which a microscopic expression is obtained. After this, as a test of our theory the quantum Boltzmann equation is derived in terms of a transition superoperator related to this master matrix.     

凝固前沿和气泡相互作用的大密度比格子玻尔兹曼方法模拟

建立了二维双组分两相流的大密度比格子玻尔兹曼方法 (lattice Boltzmann method, LBM)模型. 该模型基于改进的Shan-Chen伪势多相流LBM模型, 结合采用不同时间步长的方法, 实现密度比达800以上的气液两相流模拟. 为了对模型进行验证, 模拟了在不同气液相互作用系数和密度比条件下气泡内外压力差与其半径之间的关系, 其结果满足Laplace定律. 将所建立的大密度比LBM与介观尺度的元胞自动机(cellular automaton, CA)和有限差分法(FDM)相耦合, 用LBM模拟气液两相流, 用CA方法模拟固相生长, 用有限差分法模拟温度场, 采用LBM-CA-FDM耦合模型对定向凝固过程中凝固前沿的气泡与液-固界面之间的相互作用进行模拟研究. 结果表明, 绝热气泡的存在影响了温度场分布, 使得凝固前沿接近气泡时, 液-固界面凸起, 在不同的固相生长速度条件下, 出现凝固前沿淹没气泡或气泡脱离凝固前沿的不同情况, 模拟结果与实验结果符合良好. 关键词： 格子玻尔兹曼方法 元胞自动机 凝固 气泡     

FFT-LB Modeling of Thermal Liquid-Vapor System

We present an improved lattice Boltzmann(LB) model for thermal liquid-vapor system.In the new model,the Windowed Fast Fourier Transform(WFFT) and its inverse are used to calculate both the convection term and the external force term of the LB equation.By adopting the WFFT scheme,Gibbs oscillations can be damped effectively in unsmooth regions while high resolution feature of the spectral method can be retained in smooth regions.As a result,spatial discretization errors are dramatically decreased,conservation of the total energy is much better preserved,and the spurious velocities near the liquid-vapor interface are significantly reduced.The high resolution,together with the low complexity of the WFFT approach,endows the proposed method with considerable potential for studying a wide class of problems in the field of multiphase flows.