A novel incompressible finite-difference lattice Boltzmann equation for particle-laden flow

In this paper, we propose a novel incompressible finite-difference lattice Boltzmann Equation （FDLBE）. Because source terms that reflect the interaction between phases can be accurately described, the new model is suitable for simulating two-way coupling incompressible multiphase flow The 2-D particle-laden flow over a backward-facing step is chosen as a test case to validate the present method. Favorable results are obtained and the present scheme is shown to have good prospects in practical applications.

A Generalised Lattice Boltzmann Equation on Unstructured Grids

This paper presents a new finite-volume discretization of a generalised LatticeBoltzmann equation (LBE) on unstructured grids. This equation is the continuumLBE, with the addition of a second order time derivative term (memory), and is derivedfrom a second-order differential form of the semi-discrete Boltzmann equationin its implicit form. The new scheme, named unstructured lattice Boltzmann equationwith memory (ULBEM), can be advanced in time with a larger time-step than the previousunstructured LB formulations, and a theoretical demonstration of the improvedstability is provided. Taylor vortex simulations show that the viscosity is the same aswith standard ULBE and demonstrates that the new scheme improves both stabilityand accuracy. Model validation is also demonstrated by simulating backward-facingstep flow at low and moderate Reynolds numbers, as well as by comparing the reattachmentlength of the recirculating eddy behind the step against experimental andnumerical data available in literature.

Lattice Boltzmann Flux Solver:An Efficient Approach for Numerical Simulation of Fluid Flows

A lattice Boltzmann flux solver(LBFS)is presented for simulation of fluid flows.Like the conventional computational fluid dynamics(CFD)solvers,the new solver also applies the finite volume method to discretize the governing differential equations,but the numerical flux at the cell interface is not evaluated by the smooth function approximation or Riemann solvers.Instead,it is evaluated from local solution of lattice Boltzmann equation(LBE)at cell interface.Two versions of LBFS are presented in this paper.One is to locally apply one-dimensional compressible lattice Boltzmann(LB)model along the normal direction to the cell interface for simulation of compressible inviscid flows with shock waves.The other is to locally apply multi-dimensional LB model at cell interface for simulation of incompressible viscous and inviscid flows.The present solver removes the drawbacks of conventional lattice Boltzmann method(LBM)such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.Numerical examples show that the present solver can be well applied to simulate fluid flows with non-uniform mesh and curved boundary.

Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

A method combining the immersed boundary technique and a multi- relaxation-time （MRT） lattice Boltzmann flux solver （LBFS） is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations （LBEs）. The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution. The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.

Multi-relaxation-time lattice Boltzmann simulations of lid driven flows using graphics processing unit

Large eddy simulation （LES） using the Smagorinsky eddy viscosity model is added to the two-dimensional nine velocity components （D2Q9） lattice Boltzmann equation （LBE） with multi-relaxation-time （MRT） to simulate incompressible turbulent cavity flows with the Reynolds numbers up to 1 × 10^7. To improve the computation efficiency of LBM on the numerical simulations of turbulent flows, the massively parallel computing power from a graphic processing unit （GPU） with a computing unified device architecture （CUDA） is introduced into the MRT-LBE-LES model. The model performs well, compared with the results from others, with an increase of 76 times in computation efficiency. It appears that the higher the Reynolds numbers is, the smaller the Smagorinsky constant should be, if the lattice number is fixed. Also, for a selected high Reynolds number and a selected proper Smagorinsky constant, there is a minimum requirement for the lattice number so that the Smagorinsky eddy viscosity will not be excessively large.

Investigation of turbulence and skin friction modification in particle-laden channel flow using lattice Boltzmann method

Interaction between turbulence and particles is investigated in a channel flow. The fluid motion is calculated using direct numerical simulation(DNS) with a lattice Boltzmann(LB) method, and particles are tracked in a Lagrangian framework through the action of force imposed by the fluid. The particle diameter is smaller than the Kolmogorov length scale, and the point force is used to represent the feedback force of particles on the turbulence. The effects of particles on the turbulence and skin friction coefficient are examined with different particle inertias and mass loadings. Inertial particles suppress intensities of the spanwise and wall-normal components of velocity, and the Reynolds shear stress. It is also found that, relative to the reference particle-free flow,the overall mean skin-friction coefficient is reduced by particles. Changes of near wall turbulent structures such as longer and more regular streamwise low-speed streaks and less ejections and sweeps are the manifestation of drag reduction.

Solving generalized lattice Boltzmann model for 3-D cavity flows using CUDA-GPU

The generalized lattice Boltzmann equation(GLBE),with the addition of the standard Smagorinsky subgrid-stress(SGS) model,has been proved that it is more suitable for simulating high Reynolds number turbulent flows when compared with the lattice BGK Boltzmann equation(LBGK).However,the computing efficiency of lattice Boltzmann method(LBM) is too low to make it for practical applications,unless using a massive parallel computing clusters facility.In this study,the massive parallel computing power from an inexpensive graphic processor unit(GPU) and a typical personal computer has been developed for improving the computing efficiency,more than 100 times.This developed three-dimensional(3-D) GLBE-SGS model,with the D3Q19 scheme for simplifying collision and streaming courses,has been successfully used to study 3-D rectangular cavity flows with Reynolds number up to 10000.

Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

A lattice Boltzmann flux solver(LBFS)is presented in this work for simulation of incompressible viscous and inviscid flows.The new solver is based on Chapman-Enskog expansion analysis,which is the bridge to link Navier-Stokes(N-S)equations and lattice Boltzmann equation(LBE).The macroscopic differential equations are discretized by the finite volume method,where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers.The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.LBFS is validated by its application to simulate the viscous decaying vortex flow,the driven cavity flow,the viscous flow past a circular cylinder,and the inviscid flow past a circular cylinder.The obtained numerical results compare very well with available data in the literature,which show that LBFS has the second order of accuracy in space,and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.

Two-Relaxation-Time Lattice Boltzmann Scheme:About Parametrization,Velocity,Pressure and Mixed Boundary Conditions

We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamicequations with variable source terms based on equivalent equilibriumfunctions. A special parametrization of the free relaxation parameter is derived. Itcontrols, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopicsteady solution and governs the spatial discretization of transient flows. Inthis framework, the multi-reflection approach [16, 18] is generalized and extended forDirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions.We propose second and third-order accurate boundary schemes and adapt themfor corners. The boundary schemes are analyzed for exactness of the parametrization,uniqueness of their steady solutions, support of staggered invariants and for the effectiveaccuracy in case of time dependent boundary conditions and transient flow.When the boundary scheme obeys the parametrization properly, the derived permeabilityvalues become independent of the selected viscosity for any porous structureand can be computed efficiently. The linear interpolations [5, 46] are improved withrespect to this property.

Multiple-relaxation-time lattice Boltzmann model for binary mixtures of nonideal fluids based on the Enskog kinetic theory

In this paper, a lattice Boltzmann equation （LBE） model with multiple-relaxation-time （MRT） colli- sion operator is developed based on the Enskog theory for isothermal nonideal mixtures, which is an extension of the previous single relaxation time （SRT） LBE model （Guo and Zhao in Phys Rev E 68：035302, 2003）. The present MRT-LBE model overcomes some inherent defects of the original SRT-LBE model such as the fixed Schmidt num- ber and limited viscosity ratio. It is also interestingly shown that the widely used Shan-Chen （SC） model, which is constructed heuristically based on the pseudo-potential concept, can also be regarded as a special case of the present model, and thus putting a solid foundation for this well-accepted multiphase LBE model. A series of nu- merical simulations, including the static droplet and lay- ered co-current flow, are conducted to test the applicability of the present model for immiscible fluids with different Schmidt numbers and large viscosity ratio, which may be difficult for the original SRT-LBE model and the SC model.

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.

A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

In this paper,a simplified lattice Boltzmann method(SLBM)without evolution of the distribution function is developed for simulating incompressible viscous flows.This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes(N-S)equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis.In SLBM,the equilibrium distribution function is calculated from the macroscopic variables,while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions.Therefore,SLBM tracks the evolution of the macroscopic variables rather than the distribution function.As a result,lower virtual memories are required and physical boundary conditions could be directly implemented.Through numerical test at high Reynolds number,the method shows very nice performance in numerical stability.An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space.More benchmark tests,including the Couette flow,the Poiseuille flow as well as the 2D lid-driven cavity flow,are conducted to further validate the present method;and the simulation results are in good agreement with available data in literatures.

Study of Simple Hydrodynamic Solutions with the Two-Relaxation-Times Lattice Boltzmann Scheme

For simple hydrodynamic solutions, where the pressure and the velocity arepolynomial functions of the coordinates, exact microscopic solutions are constructedfor the two-relaxation-time (TRT) Lattice Boltzmann model with variable forcing andsupported by exact boundary schemes. We show how simple numerical and analyticalsolutions can be interrelated for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) multi-reflection (MR) type schemes. Special care is taken toadapt themfor corners, to examine the uniqueness of the obtained steady solutions andstaggered invariants, to validate their exact parametrization by the non-dimensionalhydrodynamic and a “kinetic” (collision) number. We also present an inlet/outlet“constant mass flux” condition. We show, both analytically and numerically, that thekinetic boundary schemes may result in the appearance of Knudsen layers which arebeyond the methodology of the Chapman-Enskog analysis. Time dependent Dirichletboundary conditions are investigated for pulsatile flow driven by an oscillating pressuredrop or forcing. Analytical approximations are constructed in order to extend thepulsatile solution for compressible regimes.

Simulations of Boiling Systems Using a Lattice Boltzmann Method

We report about a numerical algorithm based on the lattice Boltzmann method and its applications for simulations of turbulent convection in multi-phase flows.We discuss the issue of’latent heat’definition using a thermodynamically consistent pseudo-potential on the lattice.We present results of numerical simulations in 3D with and without boiling,showing the distribution of pressure,density and temperature fluctuations inside a convective cell.

Comparision of DdQm models in image optical flow analysis based on LBM

The optical flow analysis of the image sequence based on the formal lattice Boltzmann equation, with different DdQm models, is discussed in this paper. The Mgorithm is based on the lattice Boltzmann method （LBM）, which is used in computational fluid dynamics theory for the simulation of fluid dynamics. At first, a generalized approximation to the formal lattice Boltzmann equation is discussed. Then the effects of different DdQm models on the results of the optical flow estimation are compared with each other, while calculating the movement vectors of pixels in the image sequence. The experimental results show that the higher dimension DdQm models, e.g., D3Q15 are more effective than those lower dimension ones.

Lattice Boltzmann Finite Volume Formulation with Improved Stability

The most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especiallywhen there is a need for high resolutions near the body or thewalls.Among the recent advances in lattice Boltzmann research to handle complex geometries,a particularly remarkable option is represented by changing the solution procedure from the original"stream and collide"to a finite volume technique.However,most of the presented schemes have stability problems.This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme.To enhance stability,upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice.The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers:boundary layer,mixing layer and plane jet flows.The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data.Results in self-similar coordinates are also investigated and show that the timeaveraged statistics for velocity and vorticity express self-similarity at low Reynolds numbers.Furthermore,the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent.The agreement of the numerical results with previous results is satisfactory.

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.

A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

The objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation.The proposed method is valid for both gas and liquid flows.A discrete flux scheme(DFS)is used to derive the governing equations for two distribution functions;one for mass and another for thermal energy.These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions.The zero-order moment equation of the mass distribution function is used to recover the continuity equation,while the first-order moment equation recovers the linear momentum equation.The recovered equations are correct to the first order of the Knudsen number(Kn);thus,satisfying the continuum assumption.Similarly,the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation.For aerodynamic flows,it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation(LBE)with a BGK-type model and a specified equation of state.Thus formulated,the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows.Examples of classical aeroacoustics,compressible flow with shocks,incompressible isothermal and non-isothermal Couette flows,stratified flow in a cavity,and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS.Very good to excellent agreement with known analytical and/or numerical solutions is obtained;thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.