A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

In this work,we proposed a diffuse-interface model for the dendritic growth with thermosolutal convection.In this model,the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface,which is described by the order parameter,and the diffuse-interface based governing equations for the dendritic growth are presented.To solve the model for the dendritic growth with thermosolutal convection,we also developed a diffuse-interface multirelaxation-time lattice Boltzmann(LB)method.In this method,the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction,and the treatment on the complex fluid-solid interface can be avoided.In addition,four LB models are designed for the phase-field equation,concentration equation,temperature equation and the Navier-Stokes equations in a unified framework.Finally,we performed some simulations of the dendritic growth to test the present diffuse-interface LB method,and found that the numerical results are in good agreements with some previous works.

Gas Flow Through Square Arrays of Circular Cylinders with Klinkenberg Effect:A Lattice Boltzmann Study

It is well known that,as non-continuum gas flows through microscale porous media,the gas permeability derived from Darcy law is larger than the absolute permeability,which is caused by the so-called Klinkenberg effect or slippage effect.In this paper,an effective definition of Knudsen number for gas flows through square arrays of circular cylinders and a local boundary condition for non-continuum gas flows are first proposed,and then the multi-relaxation-time lattice Boltzmann equation including discrete effects on boundary condition is used to investigate Klinkenberg effect on gas flow through circular cylinders in square arrays.Numerical results show that the celebrated Klinkenberg equation is only correct for low Knudsen number,and secondorder correction to Klinkenberg equation is necessary with the increase of Knudsen number.Finally,the present numerical results are also compared to some available results,and in general an agreement between them is observed.

Implementation of Multi-GPU Based Lattice Boltzmann Method for Flow Through Porous Media

The lattice Boltzmann method(LBM)can gain a great amount of performance benefit by taking advantage of graphics processing unit(GPU)computing,and thus,the GPU,ormulti-GPU based LBMcan be considered as a promising and competent candidate in the study of large-scale fluid flows.However,the multi-GPU based lattice Boltzmann algorithm has not been studied extensively,especially for simulations of flow in complex geometries.In this paper,through coupling with the message passing interface(MPI)technique,we present an implementation of multi-GPU based LBM for fluid flow through porous media as well as some optimization strategies based on the data structure and layout,which can apparently reduce memory access and completely hide the communication time consumption.Then the performance of the algorithm is tested on a one-node cluster equipped with four Tesla C1060 GPU cards where up to 1732 MFLUPS is achieved for the Poiseuille flow and a nearly linear speedup with the number of GPUs is also observed.

Parareal-Richardson Algorithm for Solving Nonlinear ODEs and PDEs

The parareal algorithm,proposed firstly by Lions et al.[J.L.Lions,Y.Maday,and G.Turinici,A”parareal”in time discretization of PDE’s,C.R.Acad.Sci.Paris Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to solve the timedependent problems parallel in time.This algorithm has received much interest from many researchers in the past years.We present in this paper a new variant of the parareal algorithm,which is derived by combining the original parareal algorithm and the Richardson extrapolation,for the numerical solution of the nonlinear ODEs and PDEs.Several nonlinear problems are tested to show the advantage of the new algorithm.The accuracy of the obtained numerical solution is compared with that of its original version(i.e.,the parareal algorithm based on the same numerical method).

Simulation of Power-Law Fluid Flows in Two-Dimensional Square Cavity Using Multi-Relaxation-Time Lattice Boltzmann Method

In this paper,the power-law fluid flows in a two-dimensional square cavity are investigated in detail with multi-relaxation-time lattice Boltzmann method(MRTLBM).The influence of the Reynolds number(Re)and the power-law index(n)on the vortex strength,vortex position and velocity distribution are extensively studied.In our numerical simulations,Re is varied from 100 to 10000,and n is ranged from 0.25 to 1.75,covering both cases of shear-thinning and shear-thickening.Compared with the Newtonian fluid,numerical results show that the flow structure and number of vortex of power-law fluid are not only dependent on the Reynolds number,but also related to power-law index.

A Pseudopotential Lattice Boltzmann Analysis for Multicomponent Flow

This paper presents a pseudopotential lattice Boltzmann analysis to show the deficiency of previous pseudopotential models,i.e.,inconsistency between equilibrium velocity and mixture velocity.To rectify this problem,there are two strategies:decoupling relaxation time and kinematic viscosity or introducing a system mixture relaxation time.Then,we constructed two modified models:a two-relaxationtime(TRT)scheme and a triple-relaxation-time(TriRT)scheme to decouple the relaxation time and kinematic viscosity.Meanwhile,inspired by the idea of a system mixture relaxation time,we developed three mixture models under different collision schemes,viz.mix-SRT,mix-TRT,and mix-TriRT models.Afterwards,we derived the advection-diffusion equation for the multicomponent system and derived the mutual diffusivity in a binarymixture.Finally,we conducted several numerical simulations to validate the analysis on these models.The numerical results show that these models can obtain smaller spurious currents than previous models and have a wider range for the accessible viscosity ratio with fourth-order isotropy.Compared to previous models,presentmodels avoid complex matrix operations and only fourth-order isotropy is required.The increased simplicity and higher computational efficiency of these models make them easy to apply to engineering and industrial applications.

Lattice Boltzmann Study of Flow and Temperature Structures of Non-Isothermal Laminar Impinging Streams

Previous works on impinging streams mainly focused on the structures of flow field,but paid less attention to the structures of temperature field,which are very important in practical applications.In this paper,the influences of the Reynolds number(Re)and Prandtl number(Pr)on the structures of flow and temperature fields of non-isothermal laminar impinging streams are both studied numerically with the lattice Boltzmann method,and two cases with and without buoyancy effect are considered.Numerical results show that the structures are quite different in these cases.Moreover,in the case with buoyancy effect,some new deflection and periodic structures are found,and their independence on the outlet boundary condition is also verified.These findings may help to understand the flow and temperature structures of non-isothermal impinging streams further.

Evaluation of Three Lattice Boltzmann Models for Particulate Flows

A comparative study is conducted to evaluate three types of lattice Boltzmann equation(LBE)models for fluid flows with finite-sized particles,including the lattice Bhatnagar-Gross-Krook(BGK)model,the model proposed by Ladd[Ladd AJC,J.Fluid Mech.,271,285-310(1994);Ladd AJC,J.Fluid Mech.,271,311-339(1994)],and the multiple-relaxation-time(MRT)model.The sedimentation of a circular particle in a two-dimensional infinite channel under gravity is used as the first test problem.The numerical results of the three LBE schemes are compared with the theoretical results and existing data.It is found that all of the three LBE schemes yield reasonable results in general,although the BGK scheme and Ladd’s scheme give some deviations in some cases.Our results also show that the MRT scheme can achieve a better numerical stability than the other two schemes.Regarding the computational efficiency,it is found that the BGK scheme is the most superior one,while the other two schemes are nearly identical.We also observe that the MRT scheme can unequivocally reduce the viscosity dependence of the wall correction factor in the simulations,which reveals the superior robustness of the MRT scheme.The superiority of the MRT scheme over the other two schemes is also confirmed by the simulation of the sedimentation of an elliptical particle.

Non-Newtonian Effect on Hemodynamic Characteristics of Blood Flow in Stented Cerebral Aneurysm

Stent placement is considered as a promising and minimally invasive technique to prevent rupture of aneurysm and favor coagulation mechanism inside the aneurysm.Many scholars study the effect of the stent on blood flow in cerebral aneurysm by numerical simulations,and usually regard blood as the Newtonian fluid,blood,however,is a kind of non-Newtonian fluid in practice.The main purpose of the present paper is to investigate the effect of non-Newtonian behavior on the hemodynamic characteristics of blood flow in stented cerebral aneurysm with lattice Boltzmann method.The Casson model is used to describe the blood non-Newtonian character,which is one of the most popular models in depicting blood fluid.In particular,hemodynamic characteristics derived with Newtonian and non-Newtonian models are studied,and compared in detail.The results show that the non-Newtonian effect gives a great influence on hemodynamic characteristics of blood flow in stented cerebral aneurysm,especially in small necked ones.

Numerical Study on the Tortuosity of Porous Media via Lattice Boltzmann Method

In this paper,we simulate the pressure driven fluid flow at the pore scale level through 2-D porous media,which is composed of different curved channels via the lattice Boltzmann method.With this direct simulation,the relation between the tortuosity and the permeability is examined.The numerical results are in good agreement with the existing theory.

Coupled MRT Lattice Boltzmann Study of Electrokinetic Mixing of Power-Law Fluids in Microchannels with Heterogeneous Surface Potential

The electrokinetic mixing,as a powerful technique in microfluidic devices,is widely used in many applications.In this study,a more general dynamic model,which consists of Poisson equation,Nernst-Planck equation and Navier-Stokes equations,is used to describe the electrokinetic mixing of non-Newtonian fluids in microchannels.Furthermore,a coupled multiple-relaxation-time(MRT)lattice Boltzmann(LB)framework is developed to solve this complicated multi-physics transport phenomenon.In numerical simulations,we mainly consider the effects of the arrangement of nonuniform surface potentials and the power-law index on the mixing efficiency and the volumetric flow rate.Numerical results show that the mixing efficiency and the volumetric flow rate of shear-thinning fluids are higher than that of shear-thickening fluids under the same condition.It is also shown that for both types of fluids,there should be a balance between the mixing and liquid transport in electrokinetic microfluidics.

Lattice Boltzmann Method for Thermocapillary Flows

In this paper,we apply a recently proposed thermal axisymmetric lattice Boltzmann model to the thermocapillary driven flow in a cylindrical container.The temperature profiles and isothermal lines at the free surface with Prandtl(Pr)number fixed at 0.01 and Marangoni(Ma)number varying from 10 to 500 are measured and compared with the previous numerical results.In addition,we also give the numerical results for different Ma numbers at Pr=1.0.It is shown that present results greed well with those reported in previous studies.

Lattice BGK Model for Incompressible Axisymmetric Flows

In this paper,a lattice Boltzmann BGK(LBGK)model is proposed for simulating incompressible axisymmetric flows.Unlike other existing axisymmetric lattice Boltzmann models,the present LBGK model can eliminate the compressible effects only with the smallMach number limit.Furthermore the source terms of themodel are simple and contain no velocity gradients.Through the Chapman-Enskog expansion,the macroscopic equations for incompressible axisymmetric flows can be exactly recovered fromthe present LBGKmodel.Numerical simulations of theHagen-Poiseuille flow,the pulsatile Womersley flow,the flow over a sphere,and the swirling flow in a closed cylindrical cavity are performed.The results agree well with the analytic solutions and the existing numerical or experimental data reported in some previous studies.

Fast algorithms for Walsh transform in bit-reversed sequency order

Bit_reversed sequency order or bit_reversed Walsh order (M) is presented. Walsh functions in this order can be processed easily, and there is a simple relation between sequency order and bit_reversed sequency order. 8 fast algorithms for discrete Walsh transform in bit_reversed sequency order are given.