Lattice Boltzmann equation for axisymmetric thermal flows

A simple lattice Boltzmann equation (LBE) model for axisymmetric thermal flow is proposed in this paper. The flow field is solved by a quasi-two-dimensional nine-speed (D2Q9) LBE, while the temperature field is solved by another four-speed (D2Q4) LBE. The model is validated by a thermal flow in a pipe and some nontrivial thermal buoyancy-driven flows in vertical cylinders, including Rayleigh-Bénard convection, natural convection, and heat transfer of swirling flows. It is found that the numerical results agree excellently with analytical solution or other numerical results.     

The computation of strain rate tensor in multiple-relaxation-time lattice Boltzmann model

The multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is an important class of LB model with lots of advantages over the traditional single-relaxation-time (SRT) LB model. Generally, the computation of strain rate tensor is crucial for the MRT-LB simulations of some complex flows. At present, only two formulae are available to compute the strain rate tensor in the MRT LB model. One is to compute the strain rate tensor using the non-equilibrium parts of macroscopic moments (Yu formula). The other is to compute the strain rate tensor using the non-equilibrium parts of density distribution functions (Chai formula). The mathematical expressions of these two formulae are so different that we do not know which formula to choose for computing the strain rate tensor in the MRT LB model. To overcome this problem, this paper presents a theoretical study of the relationship between Chai and Yu formulae. The results show that the Yu formula can be deduced from the Chai formula, although they have their own advantages and disadvantages. In particular, the Yu formula is computationally more efficient, while the Chai formula is applicable to more lattice patterns of the MRT LB models. Furthermore, the derivation of the Yu formula in a particular lattice pattern from the Chai formula is more convenient than that proposed by Yu et al.     

基于Lattice Boltzmann模型的液-液混合流模拟

引入了一种二元Lattice Boltzmann Model（LBM）,实现丁两种液体组成的混合流的模拟．不同于其它的类似模型,它区分考虑了流体的粘性和扩散特性,可以很容易地模拟各种互溶或者不互溶的混合流现象．此外,由于LBM的运算大都是线性的局部运算,这使得它很容易在可编程图形处理器（Graphics Process Unit,GPU）上进行加速,从而进行实时模拟．给出了若干二元混合流的模拟结果．     

Multi-relaxation-time lattice Boltzmann model for axisymmetric flows

In this paper, a lattice-Boltzmann equation (LBE) with multi relaxation times (MRT) is presented for axisymmetric flows. The model is an extension of a recent model with single-relaxation-time [Guo et al., Phys. Rev. E 79, 046708 (2009)], which was developed based on the axisymmetric Boltzmann equation. Due to the use of the MRT collision model, the present model can achieve better numerical stability. The model is validated by some numerical tests including the Hagen-Poiseuille flow, the pulsatile Womersley flow, and the external flow over a sphere. Numerical results are in excellent agreement with analytical solutions or other available data, and the improvement in numerical stability is also confirmed.     

Optimization Strategies for the Entropic Lattice Boltzmann Method

The entropic formulation of the lattice Boltzmann method (LBM) features enhanced numerical stability due to its compliance with the Boltzmann H-theorem. This stability comes at the price of some computational overhead, associated with the need of adjusting the local relaxation time of the standard LBM in such a way as to secure compliance with the H-theorem. In this paper, we discuss a number of possible optimization strategies to reduce the computational overhead of entropic LBMs.     

Single bubble rising dynamics for moderate Reynolds number using Lattice Boltzmann Method

Dynamics of a single rising gas bubble is studied using a Lattice Boltzmann Method (LBM) based on the Cahn-Hilliard diffuse interface approach. The bubble rises due to gravitational force. However, deformation and velocity of the bubble depend on the balance of other forces produced by surface tension, inertia, and viscosity. Depending on the primary forces acting on the system, bubble dynamics can be classified into different regimes. These regimes are achieved computationally by systematically changing the values of Morton number (Mo) and Bond number (Bo) within the following ranges (1×10^-5<Mo<3×10⁴) and (1<Bo<1×10³). Terminal shape and Reynolds number (Re) are interactive quantities that depend on size of bubble, surface tension, viscosity, and density of surrounding fluid. Accurate simulation of terminal shape and Re for each regime could be satisfactorily predicted and simulated, since they are also functions of Mo and Bo. Results are compared with previous experimental results.     

Investigation of Lattice Boltzmann wetting boundary conditions for capillaries with irregular polygonal cross-section

We have investigated the performance of an alternative wetting boundary condition for complex geometries in a phase field Lattice Boltzmann scheme, which is an alternative to the commonly used formulation by Yeomans and coworkers. Though our boundary condition is much simpler in its implementation, all investigated schemes show proper droplet spreading behaviour following the Cox–Voinov law. Still, numerical artefacts like spurious velocities or chequer board effects in the pressure field can be significantly reduced by the use of a two-relaxation-time (TRT) scheme, likewise recent studies by the Yeomans group. The outstanding property of our implementation is the presence of an (artificial) thin wetting layer, which influences the relation between the saturation (_Sw$S_w$) and capillary pressure _pcap$p_{cap}$ in channels with irregular polygonal cross section. The _pcap(_Sw)$p_{cap}\left(S_w\right)$ relation from our simulation follows the shifted-Young–Laplace (sYL) law, showing that the physics of this wetting layer is similar to precursor films due to Van der Waals forces. With the knowledge of the thickness of the wetting layer, simulation results can be translated back to realistic pore configurations with thinner wetting layers.     

Lattice Boltzmann simulation of lid-driven flow in trapezoidal cavities

In this paper, an incompressible lattice Bhatnagar–Gross–Krook (LBGK) model proposed by Guo et al. is used to simulate lid-driven flow in a two-dimensional isosceles trapezoidal cavity. Due to the complex boundary of the trapezoidal cavity, here the extrapolation scheme proposed by Guo et al. is used to treat curved boundary. In our numerical simulations, the effects of the Reynolds number (Re) and the top angle θ on the strength, center position and number of vortices in the isosceles trapezoidal cavities are studied. Re is varied from 100 to 15,000, and the top angle θ ranges from 50 to 90. Numerical results show that, as Re increases, the phenomena in the cavity become more and more complex, and the number of the vortexes increases. We also found that the vortex near the bottom wall breaks up into two smaller vortices as θ increases up to a critical value. Furthermore, as Re is increased, the flow in the cavity undergoes a complex transition (from steady to the periodic flow, and finally to the chaotic flow). At last, the scope of critical Re for flow transition from steady to periodic state, and from periodic to chaotic state is presented for different top angles θ.     

Fully Lagrangian and Lattice Boltzmann Methods for the Advection-Diffusion Equation

In the last decade or so, the Lattice–Boltzmann method (LBM) has achieved great success in computational fluid dynamics. The Fully–Lagrangian method (FLM) is the generalization of LBM for conservation systems. LBM can also be developed from FLM. In this paper a FL model and a LB model are developed for D-dimensional advection-diffusion equation. The LB model can be viewed as an improved version of the FL model. Numerical results of simulation of 1-dimensional advection-diffusion equation are presented. The numerical results are found to be in good agreement with the analytic solution.     

Multi relaxation time lattice Boltzmann simulations of deep lid driven cavity flows at different aspect ratios

In this paper, the multi relaxation time (MRT) lattice Boltzmann equation (LBE) was used to compute lid driven cavity flows at different Reynolds numbers (100–7500) and cavity aspect ratios (1–4 cavity width depth). Steady solutions were obtained for square cavity flows, however for deep cavity flows at 1.5 and 4 cavity width depth, unsteady solutions prevail at Re = 7500, where periodic flow exists manifested by the rapid changes of the shapes and locations of the corner vortices in strong contrast of the stationary primary vortex. The merger of the bottom corner vortices into a primary vortex and the reemergence of the corner vortices as the Reynolds number increases are more evident for the deep cavity flows. For the four cavity width depth cavity, four primary vortices were predicted by MRT model for Reynolds number beyond 1000, which were not predicted by previous single relaxation time (SRT) BGK LBE model, and this was verified by complementary Navier–Stokes simulations. Also, MRT model is more suitable for parallel computations than its BGK counterpart, due to the more intense local computations of the multi relaxation time procedure.     

Lattice Boltzmann method simulation of electroosmotic stirring in a microscale cavity

The suitable surface modification of microfluidic channels can enable a neutral electrolyte solution to develop an electric double layer (EDL). The ions contained within the EDL can be moved by applying an external electric field, inducing electroosmotic flows (EOFs) that results in associated stirring. This provides a solution for the rapid mixing required for many microfluidic applications. We have investigated EOFs generated by applying a steady electric field across a square cavity that has homogenous electric potentials along its walls. The flowfield is simulated using the lattice Boltzmann method. The extent of mixing is characterized for different electrode configurations and electric field strengths. We find that rapid mixing can be achieved by using this simple configuration which increases with increasing electric field strength. The mixing time for water-soluble organic molecules can be decreased by four orders of magnitude by suitable choice of wall zeta potential and electric field. We dedicate this paper to the memory of our colleagues Professors Kevin Granata and Liviu Librescu who fell tragically on April 16, 2007 while answering their call to serve higher education. They continue to inspire us. AM gratefully acknowledges support from Jadavpur University under the World Bank funded Technical Education Quality Improvement Programme of the Government of India and the hospitality of the Virginia Tech ESM Department where he conducted a portion of this work.     

格子波尔兹曼方法的医学图像同步去噪增强算法

格子波尔兹曼的理论基础为分子动理学和统计力学,具有算法简单高效,执行速度快,易于并行处理等优点,近年来成功地应用于图像处理领域.本文针对医学图像反差较低且包含噪声污染的特点,通过设计分段线性拉伸函数作为格子波尔兹曼方程的外力项,实现医学图像的同步去噪和反差增强.同时构建基于TV下降流的PDE反差增强模型与本文的方法进行对比,实验表明,本文的方法处理效果优于TV下降流对应的效果.     

Boundary treatment for the lattice Boltzmann method using adaptive relaxation times

In this paper, we propose a new boundary treatment with almost second-order accuracy that does not require neighboring lattice information. In order to achieve improved accuracy for the boundary lattices, we used adaptive relaxation times reflecting boundary length scales that were unequal to the length scale of the internal fluid region lattices. Since the boundary treatment using adaptive relaxation times at the boundaries was formulated without information about the neighboring lattices, it could be easily applied to complex geometries. Numerical results using the proposed boundary treatment showed almost second-order accuracy for two-dimensional and three-dimensional problems without using information from neighboring lattices, unlike interpolation or extrapolation methods.     

Lattice Boltzmann schemes for quantum applications

We review the basic ideas behind the quantum lattice Boltzmann equation (LBE), and present a few thoughts on the possible use of such an equation for simulating quantum many-body problems on both (parallel) electronic and quantum computers.     

Lattice Boltzmann based PDE solver on the GPU

In this paper, we propose a hardware-accelerated PDE (partial differential equation) solver based on the lattice Boltzmann model (LBM). The LBM is initially designed to solve fluid dynamics by constructing simplified microscopic kinetic models. As an explicit numerical scheme with only local operations, it has the advantage of being easy to implement and especially suitable for graphics hardware (GPU) acceleration. Beyond the Navier–Stokes equation of fluid mechanics, a typical LBM can be modified to solve the parabolic diffusion equation, which is further used to solve the elliptic Laplace and Poisson equations with a diffusion process. These PDEs are widely used in modeling and manipulating images, surfaces and volumetric data sets. Therefore, the LBM scheme can be used as an GPU-based numerical solver to provide a fast and convenient alternative to traditional implicit iterative solvers. We apply this method to several examples in volume smoothing, surface fairing and image editing, achieving outstanding performance on contemporary graphics hardware. It has the great potential to be used as a general GPU computing framework for efficiently solving PDEs in image processing, computer graphics and visualization.     

Godunov-type upwind flux schemes of the two-dimensional finite volume discrete Boltzmann method

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major conclusions can be made. First, the Godunov-type schemes display higher spatial accuracy than the non-Godunov ones as the result of a more advanced treatment of the advection. Second, the PL and PP schemes are much more accurate than the PC scheme for velocity solutions. Third, there exists a threshold spatial resolution below which the PL scheme is better than the PP scheme and above which the PP scheme becomes more accurate. Fourth, besides increasing spatial resolution, increasing temporal resolution can also improve the accuracy in space for the PL and PP schemes.     

General regularized boundary condition for multi-speed lattice Boltzmann models

The lattice Boltzmann method is nowadays a common tool for solving computational fluid dynamics problems. One of the difficulties of this numerical approach is the treatment of the boundaries, because of the lack of physical intuition for the behavior of the density distribution functions close to the walls. A massive effort has been made by the scientific community to find appropriate solutions for boundaries. In this paper we present a completely generic way of treating a Dirichlet boundary for two- and three-dimensional flat walls, edges or corners, for weakly compressible flows, applicable for any lattice topology. The proposed algorithm is shown to be second-order accurate and could also be extended for compressible and thermal flows.     

Lattice Boltzmann simulations of the motion induced by variable surface tension

Methods for implementing variable surface tension in the multiphase Lattice Boltzmann model with the color model and Shan-Chen scheme are tested by analyzing the models’ abilities to reproduce a theoretical result by Levich and Kuznetzov. If the surface tension around a droplet is asymmetrical, the droplet moves towards the side where the surface tension is lower. The droplet’s velocity is proportional to the surface tension gradient, the droplet’s radius, and the inverse of the viscosity. The model is tested to determine whether the simulated droplets move in the manner predicted by theory. Although the discreteness of the underlying lattice causes a spurious oscillation to the velocity, the numerical results concerning the average velocity show a good correspondence between theory and the model in regards to the surface tension gradient and droplet size. The color model also produces good simulations in the scenarios with different viscosities, while the diffusive properties and unknown relationships between the parameters and surface tension in the Shan-Chen model make the numerical results of that model more dubious, even though several of the results are qualitatively in agreement.     

Simulation of incompressible two-phase flows with large density differences employing lattice Boltzmann and level set methods

A hybrid lattice Boltzmann and level set method (LBLSM) for two-phase immiscible fluids with large density differences is proposed. The lattice Boltzmann method is used for calculating the velocities, the interface is captured by the level set function and the surface tension force is replaced by an equivalent force field. The method can be applied to simulate two-phase fluid flows with the density ratio up to 1000. In case of zero or known pressure gradient the method is completely explicit. In order to validate the method, several examples are solved and the results are in agreement with analytical or experimental results.