a viscosity adaption method for lattice boltzmann simulations abstract in this work we consider the limited fitness for practical use of the lattice boltzmann method for non newtonian fluid flows several authors have shown that the lbm is capable of correctly simulating those fluids however due to stability reasons the modeled viscosity range has to be truncated the resulting viscosity boundaries are chosen arbitrarily because the correct simulation mach number for the physical problem is unknown a priori this easily leads to corrupt simulation results a viscosity adaption method vam is derived which drastically improves the applicability of lbm for non newtonian fluid flows by adaption of the modeled viscosity range to the actual physical problem this is done through tuning of the global mach number to the solution dependent shear rate we demonstrate that the vam can be used to accelerate lbm simulations and improve their accuracy for both steady state and transient cases

truncation errors and the rotational invariance of three dimensional lattice models in the lattice boltzmann method abstract the application of the lattice boltzmann method lbm in three dimensional isothermal hydrodynamic problems often adopts one of the following models d3q15 d3q19 or d3q27 although all of them retrieve consistent navier stokes dynamics in the continuum limit they are expected to behave differently at discrete level the present work addresses this issue by performing a lbm truncation error analysis as a conclusion it is theoretically demonstrated that differences among the aforementioned cubic lattices lie in the structure of their non linear truncation errors while reduced lattice schemes such as d3q15 and d3q19 introduce spurious angular dependencies through non linear truncation errors the complete three dimensional cubic lattice d3q27 is absent from such features this result justifies the superiority of the d3q27 lattice scheme to cope with the rotational invariance principle in three dimensional isothermal hydrodynamic problems particularly when convection is not negligible such a theoretical conclusion also finds support in numerical tests presented in this work a poiseuille duct flow and a weakly rotating duct flow

admissible mimo singular observation lqg designs abstract the widely known lqg design method for linear time invariant mimo systems may lead under certain circumstances to closed loop systems with improper sensitivity matrices such systems have no stability robustness and we refer to them as inadmissible ones this phenomenon is analyzed in the paper necessary and sufficient conditions guaranteeing admissibility of lqg designs both in the frequency and the time domains are presented and proven these conditions are based on the available information in the statement of the problem thus enabling a priori detection of inadmissibility in optimal systems the key tool in the analysis is the extension of results from polynomial matrices to rational matrices a suboptimal design aimed directly at avoiding inadmissibility and which involves the minimum necessary changes in the lqg cost function is suggested illustrative examples are used to demonstrate the main results

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

topology optimization in thermal fluid flow using the lattice boltzmann method this paper proposes a topology optimization method for thermal fluid flow problems using the lattice boltzmann method lbm the design sensitivities are derived based on the adjoint lattice boltzmann method albm whose basic idea is that the adjoint problem is first formulated using a continuous adjoint approach and the adjoint problem is then solved using the lbm in this paper the discrete velocity boltzmann equation in which only the particle velocities are discretized is introduced to the albm to deal with the various boundary conditions in the lbm the novel sensitivity analysis is applied in two flow channel topology optimization problems 1 a pressure drop minimization problem and 2 a heat exchange maximization problem several numerical examples are provided to confirm the utility of the proposed method

axisymmetric lattice boltzmann method a lattice boltzmann method is developed for incompressible axisymmetric flows both force and source or sink terms are incorporated into the lattice boltzmann equation in a natural way which is consistent in dimension with the lattice boltzmann equation the correct macroscopic equations for incompressible axisymmetric flows are recovered through the chapman enskog expansion it turns out that the added terms are nothing but the additional in the governing equations for the axisymmetric flows compared with the navier stokes equations resulting in a simple and efficient model this provides an additional unique advantage that the proposed scheme is naturally suitable for general axisymmetric flows involving more physical phenomena two numerical simulations have been presented to verify the method

geometry models of porous media based on voronoi tessellations and their porosity permeability relations in this paper we present methods that directly model the random structure of porous media using voronoi tessellations three basic structures were generated and they correspond to porous medium geometries with intersecting fractures granular interconnected tubes tubular and fibers fibrous fluid flow through these models was solved by a massively parallelized lattice boltzmann code we established the porosity permeability relations for these basic geometry models it is found that for granular and tubular geometries the specific surface area is a critical structural parameter that can bring their porosity permeability relations together under a unified kozeny carman equation a connected fracture network superimposed on the basic voronoi structure increases the dimensionless permeability relative to the kozeny carman equation isolated large pores vugs on the other hand decreases the dimensionless permeability relative to the kozeny carman equation the kozeny carman equation however cannot distinguish a heterogeneous structure with an embedded partially penetrating fracture the porosity permeability relation for fibrous geometries in general agrees with those established for simple cubic body centered cubic and face centered cubic models in the dilute limit however the dependence on the solid fraction is weaker in voronoi geometries indicating weaker hydrodynamic interactions among randomly interconnected fibers than those in the idealized models

lattice boltzmann accuracy in pore scale flow simulation we investigate the possibility of using nominally second order accurate techniques for resolving flow about solid boundaries as a means of improving accuracy and reducing grid resolution requirements in pore scale simulations an lbgk method is used to calculate flow in several geometries of increasing complexity using a first order accurate and two nominally second order accurate methods for no slip boundaries the geometries include uniform flow past an isolated sphere quadratic flow past a sphere near a wall flow through a bcc array of spheres and through a randomly packed bed of spheres the packed bed flows are also used to compare hydrodynamic dispersion results the results confirm second order accurate behavior where navier stokes flows are clearly developed however 3d pore scale simulations involve a trade off between resolution of the flow and the number of pore spaces and there is a resolution threshold below which certain flow features such as recirculation are not resolved we conjecture that most simulations will tend to operate near this threshold because of the competing demands for resolution and statistical accuracy we consider local flow features and the velocity distribution in addition to hydraulic permeability and drag to provide a fuller understanding of accuracy near this threshold

alternative kinetic theory based lattice boltzmann model for incompressible axisymmetric flows abstract based on the axisymmetric boltzmann equation an incompressible lattice boltzmann model for axisymmetric flows is proposed within the framework of the kinetic theory based model developed by guo et al 2009 while retaining the advantages of the guo et al 2009 model in terms of the solid physics basis and simple source terms involving no gradient calculations the present model further improves the numerical stability and reduces compressibility errors and computational requirements armed with the assumption that the fluid density is a constant and thus the fluid pressure has no direct relation with the density the incompressibility conditions are realized by applying the hermite expansion then the present model employs a novel way of calculating the fluid pressure which is derived from the modified second order moment equation additionally based on the regularized lattice bgk rlbgk model an extra relaxation parameter pertaining to the ghost mode is introduced to enhance the numerical stability of the present model the accuracy and applicability of the present model are verified by both the chapman enskog theoretical analysis and numerical validations it is demonstrated via well acknowledged test cases that the present model is accurate and reliable for incompressible axisymmetric flows and is able to effectively reduce the compressibility errors vis a vis the guo et al 2009

implementation aspects of 3d lattice bgk in many realistic fluid dynamical simulations the specification of the boundary conditions the error sources and the number of time steps to reach a steady state are important practical considerations in this paper we study these issues in the case of the lattice bgk model the objective is to present a comprehensive overview of some pitfalls and shortcomings of the lattice bgk method and to introduce some new ideas useful in practical simulations we begin with an evaluation of the widely used bounce back boundary condition in staircase geometries by simulating flow in an inclined tube it is shown that the bounce back scheme is first order accurate in space when the location of the non slip wall is assumed to be at the boundary nodes moreover for a specific inclination angle of 45 degrees the scheme is found to be second order accurate when the location of the non slip velocity is fitted halfway between the last fluid nodes and the first solid nodes the error as a function of the relaxation parameter is in that case qualitatively similar to that of flat walls next a comparison of simulations of fluid flow by means of pressure boundaries and by means of body force is presented a good agreement between these two boundary conditions has been found in the creeping flow regime for higher reynolds numbers differences have been found that are probably caused by problems associated with the pressure boundaries furthermore two widely used 3d models namelyd3q15andd3q19 are analysed it is shown that thed3q15model may induce artificial checkerboard invariants due to the connectivity of the lattice finally a new iterative method which significantly reduces the saturation time is presented and validated on different benchmark problems

convergence of lattice boltzmann methods for stokes flows in periodic and bounded domains combining an asymptotic analysis of the lattice boltzmann method m junk z yang asymptotic analysis of lattice boltzmann boundary conditions j stat phys 121 2005 3 35 with the stability estimate presented in m junk w a yong weighted l 2 stability of the lattice boltzmann equation preprint we are able to prove some strict convergence results the proof applies to the lattice boltzmann method with linear collision operator both in the case of periodic domains and bounded domains if the dirichlet boundary condition is realized with the bounce back rule

analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model in this paper we analytically solve the velocity of the lattice boltzmann bgk equation lbgk for several simple flows the analysis provides a framework to theoretically analyze various boundary conditions in particular the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition we find that the slip velocity is zero as long as f e 0 at boundaries no matter what combination of distributions is chosen the schemes proposed by nobleet al and by inamuroet al yield the correct zeroslip velocity while some other schemes such as the bounce back scheme and the equilibrium distribution scheme would inevitably generate a nonzero slip velocity the bounce back scheme with the wall located halfway between a flow node and a bounce back node is also studied for the simple flows considered and is shown to produce results of second order accuracy the momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries to be specific the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids

an alternative lattice boltzmann model for three dimensional incompressible flow abstract in this work an alternative lattice boltzmann lb model for three dimensional 3d incompressible flow is proposed the equilibrium distribution function edf of the present model is directly derived in accordance with the incompressibility conditions by applying the hermite expansion moreover an alternative formula for pressure computation is designed from the second order moment of the distribution function the present 3d lb model inherits the advantageous features of guo s lb model the density is a constant the fluid pressure is independent of density and the navier stokes n s equations for incompressible flow can be derived two benchmark tests flow over a backward facing step and the lid driven cavity flow are applied to validate the present model accurate results for these tests are obtained with the present model and further comparisons with the previous lb models the standard lb model the he luo model and guo s lb model demonstrate that the present model provides better accuracy in the region of high deviatoric stress and such advantage is further enhanced by using the d3q27 lattice

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

an alternative implementation of the kinetic theory based axisymmetric lattice boltzmann model abstract the semi implicit centered scheme as an approximated implementation of the trapezium rule based scheme has proved to be capable of avoiding the discrete lattice effects in our recent forcing scheme analysis for the axisymmetric lb method and is applied here for an alternative implementation of the kinetic theory based axisymmetric model by guo et al in particular the external force terms and the first order part of the source terms accounting for the axisymmetric effect are discretized with the semi implicit centered scheme while the rest are represented with the trapezium rule the proposed axisymmetric lb scheme has the advantage of simpler calculations of the macroscopic variables since the zeroth and the first order moments of the distribution function are independent of the complicated source terms chapman enskog analysis in the appendix demonstrates that axisymmetric navier stokes n s equations are recovered from the proposed model and the error terms caused by the semi implicit centered scheme do not affect the overall accuracy of the present model moreover the accuracy and applicability of the proposed axisymmetric lb model are verified by the adopted numerical tests and the complicated macroscopic variables computations in the guo et al model and its derivative models are avoided by the present model

a mass conserving boundary condition for the lattice boltzmann equation method in lattice boltzmann lb simulations the widely used wall boundary conditions bcs proposed by filipova and hanel fh and mei luo and shyy mls result in constant mass leakage in certain circumstances in this paper we have analyzed the source of the leakage based on this analysis we propose a second order accurate mass conserving wall bc in our bc the distribution function at a wall node is decomposed into its equilibrium and non equilibrium parts the mass conservation is guaranteed by enforcing a mass conserving rule in the construction of the fictitious equilibrium distribution part we have shown through several benchmark test problems involving steady and unsteady flows that our new bc not only eliminates the constant mass leakage but also has many other advantages over the fh and mls bcs

the reliability of a stochastically complex dynamical system abstract the first passage problem is an important issue in the theory of dynamical systems it determines whether the system is safe and reliable or destroyed therefore taking a suitable force to control first passage failure is very significant for such systems in this paper we study the reliability of a stochastically complex dynamical system by using the stochastic averaging method the reliability results show that the reliability function is monotonically decreasing with respect to time and that the reliability of the system is relatively worse with external and parametric white noise excitations a bang bang feedback control is used to improve the reliability for the original system numerical results display that even a small control force can greatly enhance the reliability of the original system

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

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

analytic solutions of linearized lattice boltzmann equation for simple flows a general procedure to obtain analytic solutions of the linearized lattice boltzmann equation for simple flows is developed as examples the solutions for the poiseuille and the plane couette flows in two dimensional space are obtained and studied in detail the solutions not only have a component which is the solution of the navier stokes equation they also include a kinetic component which cannot be obtained by the navier stokes equation the kinetic component of the solutions is due to the finite mean free path effect comparison between the analytic results and the numerical results of lattice gas simulations is made and they are found to be in accurate agreement

rotational invariance in the three dimensional lattice boltzmann method is dependent on the choice of lattice the lattice boltzmann method has recently gained popularity as a tool for simulating complex fluid flows it uses discrete sets of velocity vectors or lattices to create a reduced model of the molecular dynamics of a continuum fluid while several lattices are believed to behave isotropically there are reports of qualitatively incorrect results however thus far the reason as to why a lack of isotropy occurs is not known based on the hypothesis that lower order lattices may not display rotational invariance this study tests the isotropy of the d3q15 d3q19 and d3q27 lattices by performing simulations at intermediate reynolds numbers 50 500 and low knudsen number 250 the d3q15 and d3q19 lattices produced different results depending on the plane of the lattice with which the flow was aligned lattice planes with fewer than six velocity vectors consistently produced results which were qualitatively different from the planes with six or more velocity vectors these errors were not observed at re 50 or when a d3q27 lattice was used they appeared to be independent of grid density collision operator and ma this suggests that the lattices which contain these planes are not fully isotropic and therefore do not properly replicate the behavior of a real fluid in this particular situation notably downstream from the expansion predictions made using these models in more complex geometries may therefore be affected by the orientation of the lattice when using lbm in cfd simulation including validation this study highlights the need for caution to ensure that the solution obtained is independent of the lattice orientation throughout the domain

one point boundary condition for the lattice boltzmann method we propose a correction to the bounce back boundary condition for lattice boltzmann algorithms which improves the accuracy of pressure from zero to first order and the accuracy of velocity from first to second order compared to interpolation based corrections our method has the advantage of being completely local in fact methods using interpolation face difficulties at boundary points where not enough neighboring nodes are available we show that a combination with our method offers a natural solution to this problem

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

an investigation into curved and moving boundary treatments in the lattice boltzmann method curved boundary treatments provide a means of improving the computational accuracy of the conventional stair shaped approximation used in lattice boltzmann lb simulations furthermore curved boundary treatments can be extended to the modeling of moving boundary problems simply by adding a momentum term to the bounced distribution functions at the solid surface this study commences by investigating three conventional interpolating treatments for curved boundaries in lb problems namely the filippova and hanel fh model o filippova d hanel grid refinement for lattice bgk models j comput phys 147 1998 219 228 bouzidi s model m bouzidi m firdaouss p lallemand momentum transfer of a boltzmann lattice fluid with boundaries phys fluids 13 11 2001 3452 3459 and yu s model d yu r mei w shyy a unified boundary treatment in lattice boltzmann method aiaa 2003 0953 new york 2003 previous investigations have indicated that the interpolations would break the mass conservation at the boundaries since the inaccuracy in evaluation of the momentum transfer at boundary leads to a net mass flux based on this reason a concept of the interpolation free treatment for modeling the curved and moving boundary conditions is proposed to overcome the drawback of these interpolation based curved boundary treatments in present study two interpolation free models are then proposed namely on site interpolation free osif and composite interpolation free cpif models these proposed models are initially applied to simulate the flow in the channels containing a stationary square block positioned at various locations along the longitudinal axis the simulations results are then compared with those obtained using the three conventional interpolating treatments the interpolation free models are then applied to the case of moving boundary problems in which a square block and a cylindrical block respectively move with a constant speed along a channel containing stationary flows to test the galilean effect of the proposed cpif model a couette flow past the stationary square cylinder block with the moving top bottom walls is simulated overall the numerical results show that the proposed interpolation free curved treatment models significantly improve the accuracy of the mass flux computation near the solid surface and thus enhance the accuracy of the momentum interaction at the moving boundaries

local second order boundary methods for lattice boltzmann models a new way to implement solid obstacles in lattice boltzmann models is presented the unknown populations at the boundary nodes are derived from the locally known populations with the help of a second order chapman enskog expansion and dirichlet boundary conditions with a given momentum steady flows near a flat wall arbitrarily inclined with respect to the lattice links are then obtained with a third order error in particular couette and poiseuille flows are exactly recovered without the knudsen layers produced for inclined walls by the bounce back condition

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

on the cubic velocity deviations in lattice boltzmann methods the macroscopic equations derived from the lattice boltzmann equation are not exactly the navier stokes equations here the cubic deviation terms and the methods proposed to eliminate them are studied the most popular two and three dimensional models d2q9 d3q15 d3q19 d3q27 are considered in the paper it is demonstrated that the compensation methods provide only partial elimination of the deviations for these models it is also shown that the compensation of qian and zhou 1998 europhys lett 42 359 using the compensation parameter k 1 in a d2q9 or d3q27 model can eliminate all the cross terms perfectly but the deviation terms x u3x y u3y and z u3z still survive the compensation

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

galilean invariance study on different lattice boltzmann fluid solid interface approaches for vortex induced vibrations abstract the present work compares fluid solid interface approaches for the lattice boltzmann method lbm to study vortex induced vibrations viv two classes of fluid solid interface approaches namely the partially saturated methods psm noble and torczynski 1998 holdych 2003 krause et al 2017 trunk et al 2018 and moving boundary methods mbm bouzidi et al 2001 lallemand and luo 2003 yu et al 2003 ginzburg et al 2008 and filippova and hanel 1998 mei et al 1999 2000 are investigated first the galilean invariance of aerodynamic coefficients obtained by each scheme is examined the bhatnagar gross krook bgk lbm is utilized to simulate an eccentrically positioned cylinder in a transient couette flow in addition various refill methods for mbm and volume approximation techniques for psm are tested besides different error norms and a grid independence study of each method the galilean invariance violating frequencies are studied as well these error calculations are used to choose a representative for each class to simulate vortex shedding the viv test case describes a transverse oscillation of a cylinder in a freestream at a reynolds number of 100 both free and forced cylinder oscillations are examined to study known phenomena as lock in and lock out zones the results of mbm and psm are in good agreement to literature values and prove the suitability for viv simulations

boundary conditions for thermal lattice boltzmann equation method we propose a thermal boundary condition treatment based on the bounce back idea and interpolation of the distribution functions for both the dirichlet and neumann normal derivative conditions in the thermal lattice boltzmann equation tlbe method the coefficients for the distribution functions involved are determined to satisfy the dirichlet or neumann condition with second order accuracy for the dirichlet condition there is an adjustable parameter in the treatment and three particular schemes are selected for demonstration while for the neumann condition the second order accurate scheme is unique when applied to inclined or curved boundaries the dirichlet condition treatment can be directly used while the neumann condition given in the normal direction of the boundary should be converted into derivative conditions in the discrete velocity directions of the tlbe model a spatially coupled formula relating the boundary temperature boundary normal heat flux and the distribution functions near the boundary is thus derived for the neumann problems on curved boundaries the applicability and accuracy of the present boundary treatments are examined with several numerical tests for which analytical solutions are available including the 2 dimensional 2 d steady state channel flow the 1 d transient heat conduction in an inclined semi infinite solid the 2 d steady state and transient heat conduction inside a circle and the 3 d steady state circular pipe flow while the dirichlet condition treatment leads to second order accuracy for the temperature field it only gives a first order accurate boundary heat flux because of the irregularity of the cuts by the curved boundary with the lattices with the neumann condition on the curved boundary the accuracy for the temperature field obtained is first order when the tangential temperature gradient on the boundary is decoupled second order convergence of the temperature field can be obtained with neumann conditions

optimal relaxation collisions for lattice boltzmann methods the optimal relaxation time of about 0 8090 has been proposed to balance the efficiency stability and accuracy at a given lattice size of numerical simulations with lattice boltzmann methods the optimal lattice size for a desired reynolds number can be refined by reducing the mach number for incompressible flows the functioned polylogarithm polynomials are defined and used to express the lattice boltzmann equations at different time scales and to analyze the impact of relaxation times and lattice sizes on truncation errors smaller truncation errors can be achieved when relaxation times are greater than 0 5 and less than 1 0 the steady state lid driven cavity flow was chosen to validate the code of lattice boltzmann procedures the applications of the optimal relaxation parameters numerically balance the stability efficiency and accuracy through hartmann flow the optimal relaxation time can also be used to select the initial lattice size for the channel flow over a square cylinder with a given mach number

quartic parameters for acoustic applications of lattice boltzmann scheme using the taylor expansion method we show that it is possible to improve the lattice boltzmann method for acoustic applications we derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy the corresponding discrete equations are solved with the help of symbolic manipulation the solutions are obtained in the case of d3q27 lattice boltzmann scheme various numerical tests support the coherence of this approach

an immersed boundary method based on the lattice boltzmann approach in three dimensions with application the immersed boundary ib method originated by peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid the navier stokes n s equations in the ib method are usually solved using numerical methods such as fft and projection methods here in our work the n s equations are solved by an alternative approach the lattice boltzmann method lbm compared to many conventional n s solvers the lbm can be easier to implement and more convenient for modeling additional physics in a problem this alternative approach adds extra versatility to the immersed boundary method in this paper we discuss the use of a 3d lattice boltzmann model d3q19 within the ib method we use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3d channel and determine a drag scaling law for the sheet our main conclusions are 1 the hybrid method is convergent with first order accuracy which is consistent with the immersed boundary method in general 2 the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow

theory of the lattice boltzmann equation symmetry properties of discrete velocity sets the lattice boltzmann equation replaces continuous particle velocity space by a finite set the velocity distribution function then varies over a finite dimensional vector space instead of over an infinite dimensional function space the number of linearly independent moments of the distribution function in a lattice boltzmann model cannot exceed the number of velocities finite dimensionality therefore necessarily induces linear dependences among the moments that do not exist in a continuous theory given a finite velocity set it is important to know which moments are free of these dependences elementary group theory is applied to the solution of this problem it is found that decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group makes it straightforward to uncover linear dependences among the moments the construction of some standard two and three dimensional models is reviewed from this viewpoint and procedures for constructing higher dimensional models are suggested