finite volume lattice boltzmann schemes in two and three dimensions simple and practical finite volume schemes for the lattice boltzmann equation are derived in two and three dimensions through the application of modern finite volume methods the schemes use a finite volume vortextype formulation based on quadrilateral elements in two dimensions and trilinear hexahedral elements in three dimensions it is shown that the schemes are applicable to domains with irregular boundaries of arbitrary shape in two and three dimensions s1063 651x 99 10809 2

numerical stability of explicit off lattice boltzmann schemes the off lattice boltzmann olb method consists of numerical schemes which are used to solve the discrete boltzmann equation unlike the commonly used lattice boltzmann method the spatial and time steps are uncoupled in the olb method in the currently proposed schemes which can be broadly classified into runge kutta based and characteristics based the size of the time step is limited due to numerical stability constraints in this work we systematically compare the numerical stability of the proposed schemes in terms of the maximum stable time step in line with the overall lb method we investigate the available schemes where the advection approximation is explicit and the collision approximation is either explicit or implicit the comparison is done by implementing these schemes on benchmark incompressible flow problems such as taylor vortex flow poiseuille flow and lid driven cavity flow it is found that the characteristics based olb schemes are numerically more stable than the runge kutta based schemes additionally we have observed that with respect to time step size the scheme proposed by bardow et al 2006 1 is the most numerically stable and computationally efficient scheme compared to similar schemes for the flow problems tested here

lattice boltzmann simulations on irregular grids introduction of the natrium library the lattice boltzmann method is a modern approach to simulate fluid flow in its original formulation it is restricted to regular grids second order discretizations and a unity cfl number this paper describes our new off lattice boltzmann solver natrium an extensible and parallel c code to perform lattice boltzmann simulations on irregular grids natrium also allows high order spatial discretizations and non unity cfl numbers to be used we demonstrate how these features can efficiently decrease the number of grid points required in a simulation and thus reduce the computational time compared to the standard lattice boltzmann method we detail the implementation of a recently proposed semi lagrangian lattice boltzmann method and prove its efficiency in comparisons to other state of the art off lattice boltzmann schemes

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

acceleration of steady state lattice boltzmann simulations on non uniform mesh using local time step method in this paper the local time step method is applied to the lattice boltzmann method lbm on non uniform grid generalized form of interpolation supplemented lbm gilbm for steady flow simulations the local time step method is known to be an effective technique to accelerate the solution to the converged steady state on non uniform grid first code validation for global time step method is performed by solving flow around a cylinder at low reynolds numbers our results show good agreement with previous studies two dimensional flow simulations around an airfoil were performed in order to validate the code using the local time step method and the results are compared with the global time step solutions the pressure distribution and the aerodynamic coefficients of global and local time step results are in good agreement they are also consistent with the previous studies at the same time cpu time required to obtain the steady state solutions using the local time step are reduced by 70 80 compared with that using global time step

singular forces and pointlike colloids in lattice boltzmann hydrodynamics we present an accurate method to include arbitrary singular distributions of forces in the lattice boltzmann formulation of hydrodynamics we validate our method with several examples involving stokeslet stresslet and rotlet singularities finding excellent agreement with analytical results a minimal model for sedimenting particles is presented using the method in the dilute limit this model has accuracy comparable to but computational efficiency much greater than algorithms that explicitly resolve the size of the particles

flow simulation around cambered airfoil by using conformal mapping and intermediate domain in lattice boltzmann method in this paper the developed interpolation lattice boltzmann method is used for simulation of unsteady fluid flow it combines the desirable features of the lattice boltzmann and the joukowski transformation methods this approach has capability to simulate flow around curved boundary geometries such as airfoils in a body fitted grid system simulation of unsteady flow around a cambered airfoil in a non uniform grid for the first time is considered to show the capability of this method for modeling of fluid flow around complex geometries and complicated long term periodic flow phenomena the developed solver is also coupled with a fast adaptive grid generator in addition the new approach retains all the advantages of the standard lattice boltzmann method the strouhal number the pressure the drag and the lift coefficients obtained from the simulations agree well with classical computational fluid dynamics simulations numerical studies for various test cases illustrate the strength of this new approach

flow simulation around an airfoil by lattice boltzmann method on generalized coordinates flow simulations around an airfoil naca0012 using lattice boltzmann method lbm are performed in this paper originally lbm is an incompressible flow solver on orthogonal coordinates however in order to resolve the boundary layer of the airfoil accurately the algorithm is extended to generalized coordinates also suitable wall boundary condition is introduced for generalized coordinates first of all the present numerical method is validated at relatively small reynolds number re 500 the velocity profile of the boundary layer is compared in detail with the cfl3d results and consistent results are obtained second the present method is combined with the baldwin lomax turbulence model and results at relatively high reynolds number re 5x10 5 are shown the results are in good agreement with the experimental data with the same grid resolution the present method can solve high reynolds

a simplified finite volume lattice boltzmann method for simulations of fluid flows from laminar to turbulent regime part i numerical framework and its application to laminar flow simulation abstract in this paper an unstructured grid based finite volume lattice boltzmann method fvlbm that can be used for the simulation of incompressible laminar flows is presented and studied in detail this method is derived from a simple modification of the cell vertex unstructured grid based fvlbm proposed by stiebler et al 2006 compared with other complex flux reconstruction methods the present scheme has a low computational cost and can also achieve second order spatial accuracy furthermore depending on the use of the different temporal discretization schemes the temporal accuracy can be adjusted for both steady and unsteady flows besides some comparisons of the computational cost and accuracy with another fvlbm scheme are also presented meanwhile different boundary conditions are illustrated that are easy to implement on complex geometries to validate the present method four cases are carried out including a couette flow driven by one plate for an accuracy test flow in a square cavity flow around a single circular cylinder and a more complex flow around double circular cylinders numerical experiments show that the present scheme can simulate steady and unsteady flows at relatively high reynolds number with relatively few grid cells thus demonstrating the good capability of the present method

finite element lattice boltzmann simulations of free surface flow in a concentric cylinder the annular centrifugal contactor is a compact mixer centrifuge device designed for liquid liquid extraction operations in processes for recycling nuclear fuel the flow in the annulus of a centrifugal contactor is similar to the taylor vortex flow with superimposed axial flow the major differences are that the contactor has one end that is open to air and the free surface effects dominate the annular flow in this study we demonstrate the application of a finite element lattice boltzmann equation fe lbe method to the annular mixing geometry as found in a centrifugal contactor the computational results are validated with available experimental observations with the reynolds number re in the range of 1250

high order upwind compact finite difference lattice boltzmann method for viscous incompressible flows abstract in this work a high order upwind compact finite difference lattice boltzmann method ucdlbm is developed to efficiently solve viscous incompressible flow problems a fifth order upwind compact difference scheme is adopted to discretize the spatial derivatives of the lattice boltzmann equation and the third order total variation diminishing runge kutta scheme is utilized for the discretization of the temporal term compared to the existing central compact finite difference lattice boltzmann method cfdlbm the present ucdlbm can prevent non physical oscillations without filtering due to the natural dissipative property of upwind schemes three benchmark problems involving the taylor green vortex problem the doubly periodic shear layer flow problem and the lid driven square cavity flow problem are numerically solved to demonstrate the accuracy and efficiency of the present method numerical results computed are in good agreement with the analytical solution or other available numerical results and the present ucdlbm is less time consuming than the cfdlbm without degenerating the order of accuracy of the numerical solutions

a stable discretization of the lattice boltzmann equation for simulation of incompressible two phase flows at high density ratio a stable discretization of the lattice boltzmann equation lbe for non ideal gases is presented for simulation of incompressible two phase flows having high density and viscosity ratios the stiffness of the discretized forcing terms in lbe for non ideal gases is known to trigger severe numerical instability and restrict practical application of the lbe method use of a proper pressure updating scheme is also crucial to the stability of the lbe method because of non negligible pressure variation across the phase interface to deal with these issues we propose a stable discretization scheme which assumes the low mach number approximation and utilizes the stress and potential forms of the surface tension force the incompressible transformation and the consistent discretization of the intermolecular forcing terms the proposed stable discretization scheme is applied to simulate 1 d advection equation with a source term a stationary droplet droplet oscillation and droplet splashing and deposition on a thin film at a density ratio of 1000 with varying reynolds numbers the numerical solutions of stationary and oscillatory droplets agree well with analytic solutions including the laplace s law the time history of the spread factor of the liquid sheet emitted after the droplet impact also follows the known spreading power law

boundary conditions for the upwind finite difference lattice boltzmann model evidence of slip velocity in micro channel flow we conduct a systematic study of the effect of various boundary conditions bounce back and three versions of diffuse reflection for the two dimensional first order upwind finite difference lattice boltzmann model simulation of couette flow in a micro channel using the diffuse reflection boundary condition reveals the existence of a slip velocity that depends on the knudsen number e l l where l is the mean free path and l is the channel width for walls moving in opposite directions with speeds u w the slip velocity satisfies u s l i p 2 eu w a l l 1 2 e in the case of poiseuille flow in a micro channel the slip velocity is found to depend on the lattice spacing ds and knudsen number e to both first and second order the best results are obtained for diffuse reflection boundary conditions that allow thermal mixing at a wall located at half lattice spacing outside the boundary nodes

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

high order finite elements applied to the discrete boltzmann equation a discontinuous galerkin approach for solving the discrete boltzmann equation is presented allowing to compute approximate solutions for fluid flow problems based on a two dimensional high order finite element and an explicit euler time stepping scheme the d2q9 model is discretized and the results are compared to the exact solution of the navier stokes equation four numerical examples are considered including stationary and instationary problems with curved boundaries it is demonstrated that the proposed method allows to obtain the desired highly efficient exponential convergence copyright 2006 john wiley sons ltd

a new splitting scheme to the discrete boltzmann equation for non ideal gases on non uniform meshes we present a novel numerical procedure for solving the discrete boltzmann equations dbe on non uniform meshes our scheme is based on the strang splitting method where we seek to investigate two phase flow applications in this note we investigate the onset of parasitic currents which arise in many computational two phase algorithms to the best of our knowledge the results presented in this work show for the first time a spectral element discontinuous galerkin sedg discretization of a discrete boltzmann equation which successfully eliminates parasitic currents on non uniform meshes with the hope that this technique can be used for applications in complex geometries calculations are performed on non uniform mesh distributions by using high order spectral body fitting quadrilateral elements validation and verification of our work is carried out by comparing results against the classical 2d younglaplace law problem for a static drop

a multispeed discrete boltzmann model for transcritical 2d shallow water flows in this work a discrete boltzmann model for the solution of transcritical 2d shallow water flows is presented and validated in order to provide the model with transcritical capabilities a particular multispeed velocity set has been employed for the discretization of the boltzmann equation it is shown that this particular set naturally yields a simple and closed procedure to determine higher order equilibrium distribution functions needed to simulate transcritical flow the model is validated through several classical benchmarks and is proven to correctly and accurately simulate both 1d and 2d transitions between the two flow regimes

lattice boltzmann method on curvilinear coordinates system using an interpolation based strategy the lattice boltzmann method is extended to apply to general curvilinear coordinate systems as an example a cylindrical coordinate system is used to simulate two dimensional flow around a circular cylinder numerical simulations are carried out for impulsive initial conditions with reynolds numbers up to 104 the agreement of our results with previous computational and experimental results is satisfactory compared with previous lattice boltzmann simulations of the same problem our new approach greatly enhances the computational efficiency

implementation of a high order compact finite difference lattice boltzmann method in generalized curvilinear coordinates abstract in this work the implementation of a high order compact finite difference lattice boltzmann method cfdlbm is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non uniform grids the incompressible form of the discrete boltzmann equation with the bhatnagar gross krook bgk approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates herein the spatial derivatives in the resulting lattice boltzmann lb equation in the computational plane are discretized by using the fourth order compact finite difference scheme and the temporal term is discretized with the fourth order runge kutta scheme to provide an accurate and efficient incompressible flow solver a high order spectral type low pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions flow non linearities and grid non uniformity all boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates the accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems a sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are unsteady couette flow and steady flow in a 2 d cavity with non uniform grid and steady and unsteady flows over a circular cylinder and the naca0012 hydrofoil at different flow conditions results obtained for the above test cases are in good agreement with the existing numerical and experimental results the study shows the present solution methodology based on the implementation of the high order compact finite difference lattice boltzmann method cfdlbm in the generalized curvilinear coordinates is robust efficient and accurate for solving steady and unsteady incompressible flows over practical geometries

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

spectral element discontinuous galerkin lattice boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator in this paper a spectral element discontinuous galerkin sedg lattice boltzmann discretization and an exponential time marching scheme are used to study the flow field past two circular cylinders in tandem arrangement the basic idea is to discretize the streaming step of the lattice boltzmann equation by using the sedg method to get a system of ordinary differential equations odes whose exact solutions are expressed by using a large matrix exponential the approximate solution of the resulting odes are obtained from a projection method based on a krylov subspace approximation this approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension the exponential time integration scheme is useful especially when computations are carried out at high courant friedrichs lewy cfl numbers where most explicit time marching schemes are inaccurate simulations of flow were carried out for a circular cylinder at re 20 and for two circular cylinders in tandem at re 40 and a spacing of 2 5d where d is the diameter of the cylinders we compare our results with those from a fourth order runge kutta scheme that is restricted by the cfl number in addition important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme these results are compared with results from our simulation using the rk scheme and with existing benchmark results

chebyshev collocation spectral lattice boltzmann method for simulation of low speed flows in this study the chebyshev collocation spectral lattice boltzmann method ccslbm is developed and assessed for the computation of low speed flows both steady and unsteady flows are considered here the discrete boltzmann equation with the bhatnagar gross krook approximation based on the pressure distribution function is considered and the space discretization is performed by the chebyshev collocation spectral method to achieve a highly accurate flow solver to provide accurate unsteady solutions the time integration of the temporal term in the lattice boltzmann equation is made by the fourth order runge kutta scheme to achieve numerical stability and accuracy physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used an iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows the main advantage of using the ccslbm over other high order accurate lattice boltzmann method lbm based flow solvers is the decay of the error at exponential rather than at polynomial rates note also that the ccslbm applied does not need any numerical dissipation or filtering for the solution to be stable leading to highly accurate solutions three two dimensional 2d test cases are simulated herein that are a regularized cavity the taylor vortex problem and doubly periodic shear layers the results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement the computational efficiency of the proposed solution methodology based on the ccslbm is also examined by comparison with those of the standard streaming collision classical lbm and two finite difference lbm solvers the study indicates that the ccslbm provides more accurate and efficient solutions than these lbm solvers in terms of cpu and memory usage and an exponential convergence is achieved rather than polynomial rates the solution methodology proposed the ccslbm is also extended to three dimensions and a 3d regularized cavity is simulated the corresponding results are presented and validated indications are that the ccslbm developed and applied herein is robust efficient and accurate for computing 2d and 3d low speed flows note also that high accuracy solutions obtained by applying the ccslbm can be used as benchmark solutions for the assessment of other lbm based flow solvers

an eulerian description of the streaming process in the lattice boltzmann equation this paper presents a novel strategy for solving discrete boltzmann equation dbe for simulation of fluid flows this strategy splits the solution procedure into streaming and collision steps as in the lattice boltzmann equation lbe method the streaming step can then be carried out by solving pure linear advection equations in an eulerian framework this offers two significant advantages over previous methods first the relationship between the relaxation parameter and the discretization of the collision term developed from the lbe method is directly applicable to the dbe method the resulting dbe collision step remains local and poses no constraint on time step second decoupling of the advection step from the collision step facilitates implicit discretization of the advection equation on arbitrary meshes an implicit unstructured dbe method is constructed based on this strategy and is evaluated using several test cases of flow over a backward facing step lid driven cavity flow and flow past a circular cylinder the speedup of convergence for some cases improves by a factor of about 20

a new lower bound for online strip packing in this paper we consider the online strip packing problem in which a list of online rectangles has to be packed without overlap or rotation into a strip of width 1 and infinite length so as to minimize the required height of the packing we derive a new improved lower bound of 3 5 2 2 618 for the competitive ratio for this problem this result improves the best known lower bound of 2 589

some progress in lattice boltzmann method part i nonuniform mesh grids a new lattice boltzmann algorithm is proposed to simulate the navier stokes equation on arbitrary nonuniform mesh grids the new algorithm retains the advantages of the lattice boltzmann method parallel of algorithm ease of programming and ability to incorporate microscopic interactions a simulation of flow in a two dimensional symmetric channel with sudden expansion is carried out using the new algorithm on a nonuniform mesh the results of the simulation are in excellent agreement with previous experimental and numerical results

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

a hybrid wavelet based adaptive immersed boundary finite difference lattice boltzmann method for two dimensional fluid structure interaction abstract a second generation wavelet based adaptive finite difference lattice boltzmann method fd lbm is developed in this paper in this approach the adaptive wavelet collocation method awcm is firstly to the best of our knowledge incorporated into the fd lbm according to the grid refinement criterion based on the wavelet amplitudes of density distribution functions an adaptive sparse grid is generated by the omission and addition of collocation points on the sparse grid the finite differences are used to approximate the derivatives to eliminate the special treatments in using the fd based derivative approximation near boundaries the immersed boundary method ibm is also introduced into fd lbm by using the adaptive technique the adaptive code requires much less grid points as compared to the uniform mesh code as a consequence the computational efficiency can be improved to justify the proposed method a series of test cases including fixed boundary cases and moving boundary cases are invested a good agreement between the present results and the data in previous literatures is obtained which demonstrates the accuracy and effectiveness of the present awcm ib lbm

an immersed boundary method for fluids using the xfem and the hydrodynamic boltzmann transport equation abstract this paper presents a stabilized finite element formulation of the hydrodynamic boltzmann transport equation hbte to predict nearly incompressible fluid flow the hbte is discretized with hermite polynomials in the velocity variable and a streamline upwind petrov galerkin formulation is used to discretize the spatial variable a nonlinear stabilization scheme is presented from which a simple linear stabilization scheme is constructed in contrast to the navier stokes ns equations the hbte is a first order equation and allows for conveniently enforcing dirichlet conditions along immersed boundaries a simple and efficient formulation for enforcing dirichlet boundary conditions is presented and its accuracy is studied for immersed boundaries captured by the extended finite element method xfem numerical experiments indicate that both the linear and non linear stabilization methods are sufficiently accurate and stable but the linear formulation reduces the computational cost significantly the accuracy of enforcing boundary conditions is satisfactory and shows second order convergence as the mesh is refined augmenting the boundary condition formulation with a penalty term increases the accuracy of enforcing the boundary condition constraints but may degrade the accuracy of the global solution comparisons with results of a single relaxation time lattice boltzmann method show that the proposed finite element method features greater robustness and lesser dependence of the computational costs on the level of mesh refinement

initial and boundary conditions for the lattice boltzmann method an alternative approach of implementing initial and boundary conditions for the lattice boltzmann method is presented the basic idea is to calculate the lattice boltzmann populations at a boundary node from the fluid variables that are specified at this node using the gradients of the fluid velocity the numerical performance of the lattice boltzmann method is tested on several problems with exact solutions and is also compared to an explicit finite difference projection method the discretization error of the lattice boltzmann method decreases quadratically with finer resolution both in space and in time the roundoff error of the lattice boltzmann method creates problems unless double precision arithmetic is used

derivation of the lattice boltzmann method by means of the discrete ordinate method for the boltzmann equation in this note we show that the lattice boltzmann bgk method in common use can be derived from the boltzmann equation itself according to the present derivation it is apparent that the equation thus derived can be reduced to the navier stokes equation at a limit of small knudsen number since the equation thus derived is the boltzmann equation itself hence the present derivation method gives us a simple and flexible recipe to construct the lattice boltzmann method and makes it easy to construct an extended lattice boltzmann method in various ways 15 refs 1 fig

a spectral element discontinuous galerkin lattice boltzmann method for nearly incompressible flows we present a spectral element discontinuous galerkin lattice boltzmann method for solving nearly incompressible flows decoupling the collision step from the streaming step offers numerical stability at high reynolds numbers in the streaming step we employ high order spectral element discontinuous galerkin discretizations using a tensor product basis of one dimensional lagrange interpolation polynomials based on gauss lobatto legendre grids our scheme is cost effective with a fully diagonal mass matrix advancing time integration with the fourth order runge kutta method we present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous galerkin approach we show convergence studies for couette flows and demonstrate two benchmark cases with lid driven cavity flows for re 400 5000 and flows around an impulsively started cylinder for re 550 9500 computational results are compared with those of other theoretical and computational work that used a multigrid method a vortex method and a spectral element model

finite volume versus streaming based lattice boltzmann algorithm for fluid dynamics simulations a one to one accuracy and performance study a finite volume fv discretization method for the lattice boltzmann lb equation which combines high accuracy with limited computational cost is presented in order to assess the performance of the fv method we carry out a systematic comparison focused on accuracy and computational performances with the standard streaming lattice boltzmann equation algorithm in particular we aim at clarifying whether and in which conditions the proposed algorithm and more generally any fv algorithm can be taken as the method of choice in fluid dynamics lb simulations for this reason the comparative analysis is further extended to the case of realistic flows in particular thermally driven flows in turbulent conditions we report the successful simulation of high rayleigh number convective flow performed by a lattice boltzmann fv based algorithm with wall grid refinement

evaluation of the finite element lattice boltzmann method for binary fluid flows abstract in contrast to the commonly used lattice boltzmann method off lattice boltzmann methods decouple the velocity discretization from the underlying spatial grid thereby potentially increasing the geometric flexibility of the method the current work combines characteristic based integration of the streaming step with the free energy based multiphase model by lee and lin 2005 this allows for simulation time steps more than an order of magnitude larger than the relaxation time unlike previous work by wardle and lee 2013 that integrated intermolecular forcing terms in the advection term the current scheme applies collision and forcing terms locally for a simpler finite element formulation a series of thorough benchmark studies reveal that this does not compromise stability and that the scheme is able to accurately simulate flows at large density and viscosity contrasts

on the finite difference based lattice boltzmann method in curvilinear coordinates the lattice boltzmann method is a microscopic based approach for solving the fluid flow problems at the macroscopic scales the presently popular method uses regularly spaced lattices and cannot handle curved boundaries with desirable flexibility to circumvent such difficulties a finite difference based lattice boltzmann method fdlbm in curvilinear coordinates is explored using body fitted coordinates with non uniform grids several test cases including the impulsively started cylindrical couette flow steady state cylindrical couette flow steady flow over flat plates and steady flow over a circular cylinder are used to examine various issues related to the fdlbm the effect of boundary conditions for the distribution functions on the solution the merits between second order central difference and upwind schemes for advection terms and the effect of the reynolds number are investigated favorable results are obtained using fdlbm in curvilinear coordinates indicating that the method is potentially capable of solving finite reynolds number flow problems in complex geometries

regular article an accurate curved boundary treatment in the lattice boltzmann method the lattice boltzmann equation lbe is an alternative kinetic method capable of solving hydrodynamics for various systems major advantages of the method are due to the fact that the solution for the particle distribution functions is explicit easy to implement and natural to parallelize because the method often uses uniform regular cartesian lattices in space curved boundaries are often approximated by a series of stairs that leads to reduction in computational accuracy in this work a second order accurate treatment of the boundary condition in the lbe method is developed for a curved boundary the proposed treatment of the curved boundaries is an improvement of a scheme due to o filippova and d hanel 1998 j comput phys 147 219 the proposed treatment for curved boundaries is tested against several flow problems 2 d channel flows with constant and oscillating pressure gradients for which analytic solutions are known flow due to an impulsively started wall lid driven square cavity flow and uniform flow over a column of circular cylinders the second order accuracy is observed with a solid boundary arbitrarily placed between lattice nodes the proposed boundary condition has well behaved stability characteristics when the relaxation time is close to 1 2 the zero limit of viscosity the improvement can make a substantial contribution toward simulating practical fluid flow problems using the lattice boltzmann method

a characteristic galerkin method for discrete boltzmann equation abstract the characteristic galerkin finite element method for the discrete boltzmann equation is presented to simulate fluid flows in complex geometries the inherent geometric flexibility of the finite element method permits the easy use of simple cartesian variables on unstructured meshes and the mesh clustering near large gradients the characteristic galerkin procedure with appropriate boundary condition results in accurate solutions with little numerical diffusion several test cases are conducted including unsteady couette flows lid driven cavity flows and steady flow past a circular cylinder on unstructured meshes the numerical results are in good agreement with previous analytical if applicable numerical and experimental results

lattice boltzmann method for 3 d flows with curved boundary in this work we investigate two issues that are important to computational efficiency and reliability in fluid dynamic applications of the lattice boltzmann equation lbe 1 computational stability and accuracy of different lattice boltzmann models and 2 the treatment of the boundary conditions on curved solid boundaries and their 3 d implementations three athermal 3 d lbe models q15d3 q19d3 and q27d3 are studied and compared in terms of efficiency accuracy and robustness the boundary treatment recently developed by filippova and hanel 1998 j comp phys 147 219 and mei et al 1999 j comp phys 155 307 in 2 d is extended to and implemented for 3 d the convergence stability and computational efficiency of the 3 d lbe models with the boundary treatment for curved boundaries were tested in simulations of four 3 d flows 1 fully developed flows in a square duct 2 flow in a 3 d lid driven cavity 3 fully developed flows in a circular pipe and 4 a uniform flow over a sphere we found that while the 15 velocity 3 d q15d3 model is more prone to numerical instability and the q27d3 is more computationally intensive the q19d3 model provides a balance between computational reliability and efficiency through numerical simulations we demonstrated that the boundary treatment for 3 d arbitrary curved geometry has second order accuracy and possesses satisfactory stability characteristics

derivation and analysis of lattice boltzmann schemes for the linearized euler equations we derive lattice boltzmann lbm schemes to solve the linearized euler equations in 1d 2d and 3d with the future goal of coupling them to an lbm scheme for navier stokes equations and a finite volume scheme for linearized euler equations the derivation uses the analytical maxwellian in a bgk model in this way we are able to obtain second order schemes in addition we perform an l 2 stability analysis numerical results validate the approach

least squares finite element scheme for the lattice boltzmann method on an unstructured mesh a numerical model of the lattice boltzmann method lbm utilizing least squares finite element method in space and the crank nicolson method in time is developed this method is able to solve fluid flow in domains that contain complex or irregular geometric boundaries by using the flexibility and numerical stability of a finite element method while employing accurate least squares optimization fourth order accuracy in space and second order accuracy in time are derived for a pure advection equation on a uniform mesh while high stability is implied from a von neumann linearized stability analysis implemented on unstructured mesh through an innovative element by element approach the proposed method requires fewer grid points and less memory compared to traditional lbm accurate numerical results are presented through two dimensional incompressible poiseuille flow couette flow and flow past a circular cylinder finally the proposed method is applied to estimate the permeability of a randomly generated porous media which further demonstrates its inherent geometric flexibility

analyticity properties and power law estimates of functions in percolation theory we consider percolation on the sites of a graphg e g a regulard dimensional lattice all sites ofg are occupied vacant with probabilityp respectively q 1 p independently of each other w denotes the cluster of occupied sites containing a fixed site which will usually be taken to be the origin andw the cardinality ofw the percolation probability is the probability that w i e p pp w some critical values ofp ph andpt are defined respectively as the smallest value ofp for which p 0 and for which the expectation of w is infinite formally ph inf p p 0 andpt inf p ep w we show for fairly general graphsgthat ifp pt thenpp w n decreases exponentially inn for the special casesg g0 the simple quadratic lattice andg1 the graph which corresponds to bond percolation on 2 we obtain upper and lower bounds for p of the formc p ph and bounds forep w of the formc p ph we also investigate smoothness properties of p ep number of clusters per site ep w 1 w 1 this function was introduced by sykes and essam who assumed that has exactly one singularity namely atp ph for the graphsg0 andg1 i e site or bond percolation on 2 we show that p is analytic atp ph and has two continuous derivatives atp ph the emphasis is on rigorous proofs

three time integration methods for incompressible flows with discontinuous galerkin boltzmann method abstract this paper presents three time integration methods for incompressible flows with finite element method in solving the lattice bgk boltzmann equation the space discretization is performed using nodal discontinuous galerkin method which employs unstructured meshes with triangular elements and high order approximation degrees the time discretization is performed using three different kinds of time integration methods namely direct decoupling and splitting from the storage cost temporal accuracy numerical stability and time consumption we systematically compare three time integration methods then benchmark fluid flow simulations are performed to highlight efficient time integration methods numerical results are in good agreement with others or exact solutions

lattice boltzmann method on composite grids a composite block structured lattice boltzmann method is proposed for the simulation of two dimensional incompressible fluid flows the grid structure is composed of a coarse base grid and one or several fine grid s the former covers the entire physical domain the latter are placed at regions where local grid refinement is desirable the simulation is first carried out on the base grid level at a smaller relaxation time allowing a rapid propagation of boundary information throughout the entire domain thus large scale flow features can be resolved efficiently at a relatively low cost at a later time fine grid variables are initiated the dependent variables on both grid levels are then advanced in time simultaneously with the fine grid boundary conditions obtained from the base grid solution at the grid interface as a demonstration the lid driven cavity flow is selected for study the results show good agreement with benchmark numerical data and those calculated from the finite volume u 2 mathrm rans model the proposed method is able to produce accurate solutions on fine grids with a considerable saving in cpu time

finite volume lattice boltzmann method we present a finite volume formulation for the lattice boltzmann method fvlbm based on standard bilinear quadrilateral elements in two dimensions the accuracy of this scheme is demonstrated by comparing the velocity field with the analytical solution of the navier stokes equations for time dependent rotating couette flow and taylor vortex flow to demonstrate the flexibility of the scheme we have also simulated a modified rotating couette flow where the inner cylinder has an elliptical shape the results agree with those obtained from the traditional marker and cell method the fvlbm scheme is applicable to arbitrarily shaped two dimensional regions and thus the range of applicability of the lattice boltzmann method has been significantly extended

performance evaluation of the general characteristics based off lattice boltzmann scheme and dugks for low speed continuum flows the general characteristics based off lattice boltzmann scheme proposed by bardow et al 1 hereafter bardow s scheme and the discrete unified gas kinetic scheme dugks 2 are two methods that successfully overcome the time step restriction by the collision time which is commonly seen in many other kinetic schemes in this work we first perform a theoretical analysis of the two schemes in the finite volume framework by comparing their numerical flux evaluations it is found that the effect of collision term is considered in the evaluation of the cell interface distribution function in both schemes which explains why they can overcome the time step restriction and can give accurate results even as the time step is much larger than the collision time the difference between the two schemes lies in the treatment of the integral of the collision term when evaluating the cell interface distribution function in which bardow s scheme uses the rectangular rule while dugks uses the trapezoidal rule the performance of the two schemes i e accuracy stability and efficiency are then compared by simulating several two dimensional flows including the unsteady taylorgreen vortex flow the steady lid driven cavity flow and the laminar boundary layer problem it is observed that dugks can give more accurate results than bardow s scheme with a same mesh size furthermore the numerical stability of bardow s scheme decreases as the courantfriedrichslewy cfl number approaches to 1 while the stability of dugks is not affected by the cfl number apparently as long as cfl 1 it is also observed that dugks is twice as expensive as the bardow s scheme with the same mesh size

viscosity of finite difference lattice boltzmann models two dimensional finite difference lattice boltzmann models for single component fluids are discussed and the corresponding macroscopic equations for mass and momentum conservation are derived by performing a chapman enskog expansion in order to recover the correct mass equation characteristic based finite difference schemes should be associated with the forward euler scheme for the time derivative while the space centered and second order upwind schemes should be associated to second order schemes for the time derivative in the incompressible limit the characteristic based schemes lead to spurious numerical contributions to the apparent value of the kinematic viscosity in addition to the physical value that enters the navier stokes equation formulae for these spurious numerical viscosities are in agreement with results of simulations for the decay of shear waves

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100