A consistent hydrodynamic boundary condition for the lattice Boltzmann method

Transcription

1 A consistent hydrodynamic boundary condition for the lattice Boltzmann method David R. Noble Department of Mechanical and ndustrial Engineering, University of llinois, Urbana, llinois Shiyi Chen Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico John G. Georgiadis Department of Mechanical and ndustrial Engineering, University of llinois, Urbana, llinois Richard 0. Buckius Department of Mechanical and ndustrial Engineering, University of llinois, Urbana, llinois (Received 8 February 1994; accepted 23 September 1994) A hydrodynamic boundary condition is developed to replace the heuristic bounce-back boundary condition used in the majority of lattice Boltzmann simulations. This boundary condition is applied to the two-dimensional, steady flow of an incompressible fluid between two parallel plates. Poiseuille flow with stationary plates, and a constant pressure gradient is simulated to machine accuracy over the full range of relaxation times and pressure gradients. A second problem involves a moving upper plate and the injection of fluid normal to the plates. The bounce-back boundary condition is shown to be an inferior approach for simulating stationary walls, because it actually mimics boundaries that move with a speed that depends on the relaxation time. When using accurate hydrodynamic boundary conditions, the lattice Boltzmann method is shown to exhibit second-order accuracy American nstitute of Physics. 1. NTRODUCTON The lattice Boltzmann method (LBM) is a numerical scheme based on kinetic theory for modeling viscous flows.ls4 LBM has been used to model various problems, including multiphase flow,596 magnetohydrodynamics,7 8 and chemically reacting flows. n LBM, the state of the fluid is represented using a particle distribution that is analogous to the particle distribution function in kinetic theory. Despite the success of LBM in a number of physical situations, the rigorous modeling of general hydrodynamic boundary conditions (which is necessary for realistic flows) has remained a formidable obstacle. n general, a technique should be able to accommodate both Dirichlet and Neumann boundary conditions. For example, it might be desirable to impose a fixed velocity gradient, a special inlet profile, or a specified velocity at a solid wall. Up to this point, the bulk of LBM simulations have utilized heuristic techniques to model stationary walls. The task of modeling general Dirichlet boundary conditions is handled in this work by formulating a rigorous hydrodynamic boundary condition that is free from the errors that plague heuristic techniques. The most rational approach to impose a Dirichlet boundary condition in LBM requires answering this question: knowing the velocity of the fluid at a boundary, can an appropriate particle distribution be found such that imposing this distribution provides the prescribed velocity condition? Studies on this inverse problem of reproducing the particle distribution from a desired velocity distribution include the work by Rem and Somers, in which constraints on the particle distribution are developed in terms of the macroscopic velocity for lattice gas simulations. The recent work by Skordos focuses on this inverse problem in formulating initial conditions and boundary conditions. Ladd12 addresses this problem in the study of particulate suspensions. The majority of LBM studies, however, have used combinations of periodic conditions and stationary walls. The walls have been modeled using a heuristic technique, the bounce-back method, which originated in lattice gas simulations.13 Recent studies of this technique have analyzed the errors incorporated by the bounce-back boundary condition Work has also been done to improve the accuracy of this method for specific problems, but has not produced methods to implement boundaries accurately for the variety of conditions experienced in realistic flow~. ~, ~ An additional drawback is that the bounce-back technique cannot be simply extended to handle moving walls or mass injection through the walls. n this paper a method is presented to calculate the particle distribution at the boundaries from the velocity boundary conditions and the particle distributions of the neighboring fluid nodes of the boundary. To demonstrate the accuracy of this approach, the technique is applied to the twodimensional, steady flow of an incompressible fluid between two infinite, parallel plates. Two different flow configurations are used to test the method. The first problem is that of Poiseuille flow with a fixed pressure gradient and stationary walls. Using this problem, the hydrodynamic condition presented here is compared to the heuristic bounce-back condition. The second problem involves a moving upper plate and mass injection normal to the plates. Grid function convergence tests are also performed using five mesh sizes. From these results, it is shown that LBM is truly a second-order scheme as long as the boundary conditions are accurately modeled. Phys. Fluids 7 (l), January /95/7(1)/203/7/$ American nstitute of Physics 203

2 . THEORY A. Lattice Boltzmann theory The lattice Boltzmann method employed in this study uses a hexagonal lattice (Frisch-Hasslacher-Pomeau model),17 in which each node has six nearest neighbors. A unit vector is assigned for each direction, 27r(i- 1) 27r(i- 1) 6, sin 6, i=1,2,..., 6. (1) The particle distribution, fi(x,t), is the probability of finding a particle at location x and time t that is moving in direction e;. Also included in the distribution is a rest particle contribution, fo(x,t). The primary variables, density, and velocity are found from this particle distribution according to the relations C fi*ei= PW C i fi=p* Using the pressure-corrected LBM proposed by Chen et al.,lg9 the Boltzmann equation is integrated in the form fi(x+ei,t+l)=fi(x,t)+~i[f(x,t)l. (4) The Boltzmann equation describes the evolution of the particle distribution function from two contributions: a streaming term and a collision term. The streaming term models the advection of the particle distribution from a reference node to its nearest neighbor in the direction of its velocity ei. The second term pi represents the local change in the particle distribution owing to particle collision. f the distribution is expanded about its equilibrium value, the collision operator becomes Wf aai(f q) ) = fimq) + - of, (fj-fl ) +0[(fi-fiq)21* J (5) At equilibrium, it can be noted that Qvq)=O. For simplicity, it is assumed that the distribution relaxes to equilibrium at a single rate, 7, where $j is the Kronecker delta function. Assuming that higher-order terms can be neglected, the collision term is written in its linearized form as flj(f )=-( l/t)(fi-fiq)* (7) The remaining step is to determine an equilibrium particle distribution, such that the continuum fluid equations are macroscopically recovered. The following distribution has been proposed by Chen et al.: (2) (3) P-do cq=b PD + a (ei.u)+p D 20,412 (ei-u)(ei-u) where flq is the equilibrium distribution of particles moving in direction i, yoq is the equilibrium distribution of rest particles, D is the dimension rank (2 for two-dimensions), b is the number of lattice directions (six for a hexagonal lattice), c is the lattice unit length (unity for two-dimensions), and do is the average rest particle number. By performing a Taylor expansion in time and space and using the Chapman-Enskog expansion, the correct form of the Navier-Stokes equations are recovered.18 Using this technique, the solution of the fluid equations is reduced to two major steps. First, in a collision step the distributions undergo relaxation toward equilibrium according to Eq. (7). Second, the particle distributions stream to their nearest neighbors. The next step in the formulation is to develop a method for imposing a fixed pressure gradient in the x direction. This gradient maintains the flow and is imposed by applying a force to each fluid node in the domain. Force is generally equivalent to a change of momentum. n a procedure that is equivalent to the forcing performed at random sites in LGA simulations, a uniform pressure gradient is imposed by adding a quantity, P, to forward directions, while subtracting this same quantity from backward directions. This does not alter the density at that node, but adds momentum in the positive x direction. Using this type of forcing, the integrated Boltzmann equation becomes where fi(x+ei,t+l>=fi(x,t)+ pi= + P, i= 1,2,6, pi -P, po=o. i=3,4,5, i [~q(x,t)-fi(x,t)l+pi, (9) (10) (11) Application of the force does not alter the density at that node, but adds momentum in the positive x direction: C pi*ei=4pt?,. 02) (13) n terms of P, the imposed pressure gradient during a time step At is dt= - 2 At=A(pu)=4P(e,l. (14) 204 Phys. Fluids, Vol. 7, No. 1, January 1995 Noble et a/.

3 This method of imposing a pressure gradient is incorporated into the LBM procedure by adding a forcing step following the collision and before the streaming step. B. Heuristic boundary conditions The typical method of modeling stationary walls in LBM simulations is to use the bounce-back boundary condition. This technique has its roots in the lattice gas formulation, and was selected in order to maintain the proper momentum balance for the discrete particles used in these simulations. Using this technique, particles that stream into the walls bounce back and exit the wall in the direction from which they came. This method was carried over to LBM simulations, even though the representation in terms of discrete particles has been abandoned. The bounce-back procedure was shown to be a first-order approximation for a no-slip wall lying halfway between the wall nodes and the first set of fluid nodes outside the wa11. 4 Examination of second-order terms has shown that the location of zero momentum is displaced from this halfway point.r5 n these studies, an effective wall location was found by determining the point at which the velocity vanishes. An alternate interpretation of these findings is that, to first order, the velocity vanishes halfway between the wall nodes and the first fluid nodes outside the wall. Higher-order effects may produce a nonzero velocity at this location. An effective wall velocity is then defined as the velocity at this point. Using this interpretation, the ability of the bounce-back boundary condition to model a stationary wall can be examined. C. Hydrodynamic boundary conditions Rather than developing a technique that maintains a discrete particle momentum balance, the hydrodynamic approach seeks to maintain a specified velocity profile on the boundaries. During each time step in the LBM procedure, the particle distribution at each node is modified by collision, forcing, and streaming. The goal of the hydrodynamic approach is to prescribe this process in such a fashion that the desired velocity conditions are satisfied at the end of the time step. A detailed look at the LBM algorithm reveals how the particle distributions should be prescribed to provide the desired velocity conditions. Figure 1 depicts the vicinity of a LBM boundary node, and is helpful for this discussion. The upper nodes in this figure are termed interior, or fluid nodes (denoted by subscript f ). The lower nodes, in the gray section, are termed wall nodes (denoted by subscript w). The nodes on the boundary between the fluid mass and the wall mass are the boundary nodes (denoted by subscript b). Thus, the neighbors of a boundary node are classified in three groups: xb -ef,b, neighbors of the boundary node that lie within the fluid; xb-e&b, neighbors of the boundary node that lie on the boundary; and Xb-%-+b, neighbors of the boundary node that lie within the wall. The six lattice directions have been classified in three groups where f -+ b denotes directions from neighboring fluid nodes Fluid Nodes. f Boundary Nodes. b FG. 1. The neighbors of a boundary node using the hexagonal (FHP) lattice. to the boundary node of interest, b-+b denotes directions from neighboring boundary nodes to the boundary node of interest, and w--t b denotes directions from neighboring wall nodes to the boundary node of interest. The velocity and density at a new time step are found from dw+l)=c fi(x,t+l), (15) p(x,t+ l)u(x,t+ 1)=x f,(x,t+ l).ei. (16) The particle distribution at a new time step can be expressed in terms of the particle distributions of the neighbors at the previous time step using Eq. (lo), which can be restated as fjx,t+l)=fl (x-ej,t)+pi, (17) where fl is used to denote the particle distribution following the collision step: r (x-ei,t)=fi(x-ei,t)+ -fi(x-ei,t)l. Thus, the velocity and density are i [fiq(x-ei,t) (18) (19) p(x,t+l)u(x,t+l)=c [r'(x-ei,t)+pi]*c!,. (20) These expressions can be simplified using Eqs. (12)-(13). This yields f(x,t+l)=c [f3=-i,~)l, Phys. Fluids, Vol. 7, No. 1, January 1995 Noble et al. 205

4 p(x,t+l)u(x,t+ 1)=x [~ (X-ei,t)].ei+4Pe,. 1 (22) The above expressions show the dependence of the velocity and density at the end of a time step on the particle distributions of the neighbors following the collision step. For a boundary node, some of the neighbors lie on the boundary, some within the fluid, and some within the wall. The components of the distribution that come from the neighboring fluid nodes and boundary nodes are known from the particle distributions at those locations. The components that come from the neighboring wall nodes are unknown, however, since the particle distribution is not calculated within the wall. For the scenario diagrammed in Fig. 1, the fluid nodes (2 and 3) contribute two components of the particle distribution (fs and f6). Neighboring boundary nodes (4 and 1) also contribute two components (ft and f4). The wall nodes (5 and 6) contribute two components cf2 and f3), but these latter contributions are unknown. t is therefore useful to separate the contributions of the neighboring fluid nodes and boundary nodes from those contributed by the wall nodes. With this distinction, the expressions for the velocity and density become dxb?+ )= 2 fl (xb-ei,f) + c t? (Xb-ei7t), (23) prior to the forcing step, such that the hydrodynamic boundary conditions will be satisfied at the end of the time step. t is more useful, however, to prescribe the components of the particle distribution at the boundary at the end of the time step that have come from the wall nodes. The constraints that govern these components can be expressed in terms of the velocity and density along with the components that have come from neighboring boundary and fluid nodes. This can be shown by writing Eqs. (15)-(16) for a boundary node in the form dxb,t+l)= c fi(xb,t+l)+ 2 fi(xb,t+l), p(x,t+ l)u(xb,t+ )= c fi(xb,t+ l) ej i=w-b (27) + C fkxb,t+l)%, (28) where the particle distribution at the boundary node at the end of the time step has been separated into two types of contributions: components that come from wall nodes and components that come from fluid or boundary nodes. Solving for the contributions from the wall nodes produces 2 fi(xb,t+l)=p(xb,t+l)- c fi(xb,t+l)v i=f,b-+b (29) p(xb,t+l)u(xb,t+l)= 2 r (xb-ei,t)-ei + c ~ (xb-ei,t) ei+4pe,. i=w-+b (24) For the boundary nodes, however, the velocity at the end of the time step is known. Therefore, this boundary condition is used to write an equation to prescribe the neighbor contributions from the wall nodes, such that the desired velocity and density are obtained at the end of the time step. The resulting equations are c flu(xb-ei,t)=d Xb,t+ )- C r (Xb-ei,t), (25) C i=w-+b ~"(Xb-ei,t).ei=P(Xb,t+l)U(Xb,t+l) -i=z -+ b f? (Xb-ei,t)*ei -4Pe,. (26) This form of the boundary condition prescribes the particle distributions within the wall following the collision step and - C fi(x,t+l).ej. (30) t is noted that the components that come from fluid or boundary nodes are computed using the normal LBM evolution equation [Eq. (lo)]: -fi(xb-ei,t)]+pi,. (31) A procedure for applying the hydrodynamic boundary condition in terms of the particle distributions now emerges. First, the LBM procedure advances normally, producing the contributions from the nodes lying within the fluid and those lying on the boundary. Then these contributions are subtracted from the known velocity and density to give the contribution of the nodes lying within the wall. From these contributions, the wall node particle distributions are calculated. By applying this boundary condition to the scenario shown in Fig. 1, the specifics of handling a boundary that runs along the lattice are demonstrated. n this example, Eqs. (29)-(30) yield, for the new time, t+l, fz+f3=p-(fo+flff4+f5+f6h (32) f2-f3=w-p f1-2 f4-f5+fd, (33) f2+f3=(2mw+(f5+fd. (34) 206 Phys. Fluids, Vol. 7, No. 1, January 1995 Noble et al.

5 Now, two potential difficulties become apparent, First, there is three equations constraining the two unknowns, f2 and f3, Second, Eqs. (32) and (34) both constrain the quantity f2+ f3. Both issues are easily handled by considering the density to be an unknown quantity. This makes Eqs. (32)- (34) a straightforward system of three equations and three unknowns. This system of equations is solved to give the density at the boundary and the unknown components of the particle distribution. While the result given in Eqs. (29)-(30) is completely general and may be applied to any geometry, the example and results presented here all involve boundaries that run along the lattice. n order to handle other geometries, the boundary condition must be applied, and the resulting system of equations must be examined. This is the scope of future work. The fact that the density at the boundary becomes a computed quantity for flat walls is an important result. Since the density is related to the pressure through the isothermal equation of state, it is seen that, for flat walls, the boundary condition proposed here requires only that the two velocity components be prescribed. The pressure boundary condition is supplied by the algorithm. The need for such a feature in a boundary condition implementation is noted by Skordos. n practical simulations, the pressure is often an unknown quantity, and hence it is beneficial that the pressure is computed using the information provided by the neighbors and the known velocity boundary condition.. RESULTS A channel flow problem is considered with two different sets of boundary conditions. The first problem is Poiseuille flow with stationary upper and lower walls and a constant pressure gradient. n the second problem, the upper plate is moving at a constant velocity. n addition, a constant normal flow of fluid is injected through the bottom plate and is withdrawn at the same rate from the upper plate. This models a fluid being sheared between two porous plates through which an identical fluid is being injected perpendicular to the direction of shearing. This problem tests the ability of the boundary condition to model a wall condition with nonzero horizontal and vertical velocity components, u and U, respectively. n each problem the incompressible, steady, fully developed solution is sought. The x-momentum equation that govems both of these problems becomes au ap a2u u -=--+vy. ay ax ay (35) The advection term is zero for Poiseuille flow, and the pressure gradient term is zero for the porous plate problem. An analytical solution is easily obtained for each case. A. Poiseuille flow-stationary wall boundary condition Poiseuille flow is a simple problem, which serves as a useful benchmark. The analytical velocity profile is expressed as a parabola centered around the axis of the channel (with walls at y= -L, y= +L): where u=u,,(l-y2/l2), ap L2 *nax= --- ax 2pv (36) One unique feature of this solution is that it is represented exactly on a uniform mesh with a second-order finitedifference scheme. The truncation error expansion for the viscous term, ET, using a second-order-accurate central difference scheme, is 1 a4u E~=g~by~+&$Ay +--. (38) Since the derivatives in the truncation error are identically zero, Poiseuille flow is exactly represented using such a scheme. n this study, the proposed boundary condition is applied for all values of dpldx and Q- examined. The nondimensional velocity profile obtained by LBM matches that given by the analytical solution to machine accuracy. Regardless of the pressure gradient or the viscosity, LBM exactly predicts the parabolic profile. This is true, despite the fact that only 11 nodes were used to span the channel. This provides evidence of, and strongly suggests, that LBM is a second-orderaccurate scheme (at least) when the boundary conditions are handled correctly. The exact order of the method is obtained through a convergence test in Sec. B. Owing to the stationary walls involved in simple Poiseuille flow, it would be expected that the bounce-back boundary condition should be adequate for modeling the fluid boundaries. This is examined over a range of viscosity and pressure gradients. The simulation results are compared to the analytical solution for a channel having a width corresponding to walls that lie halfway between the wall nodes and the first fluid nodes outside the wall. The velocity profile within the channel again fits a perfect parabola, that is, a single parabola passes though all the points to within machine accuracy. The parabola differs, however, from the exact solution, in that the velocity does not exactly vanish at the walls. This result is shown in Fig. 2. l%o conclusions are made by careful consideration of these results. Since the anticipated parabolic profile is achieved throughout the domain, the error that is experienced with higher values of r (the relaxation time) does not represent the breakdown of the LBM scheme, but rather the inability of the bounce-back boundary condition to accurately model hydrodynamic boundary conditions. n previous studies, departure from the expected results has been observed for high 7.l This departure has been attributed to the violation of the requirement that the mean-free path, which depends on 7, be much smaller than the lattice size of the system in order to recover the macroscopic hydrodynamic properties. At this moment, it is not clear, from a fundamental physics point of view, why the current hydrodynamic boundary condition gives correct results for all 7, regardless of the system size. The second conclusion drawn from these results is that the effectiveness of the bounce-back boundary condition to Phys. Fluids, Vol. 7, No. 1, January 1995 Noble et al. 207

6 L O Amlyiical, Resent Boundary Condition, all T U ~0.52, Bounce-back Bounday Condition + ~1.0, Bounce-back Boundary Condition + +=2.0, Bounce-back Boundary Condition t - %=3.0, Bounce-back Boundary Condition - e10.0, Bounce-back - ~50.0, Bounce-back z L LBM, Re= u :,,, / u v FG. 2. Poiseuille flow velocity profiles using the bounce-back boundary condition and the rigorous hydrodynamic boundary conditions developed here. For all values of T and the axial pressure gradient, the analytical and LBM solutions using the boundary condition method presented here fall on a single line. Since the points for the simulations using the hydrodynamic boundary condition agree to machine accuracy with the analytical solution, the points have been left off for clarity. The other curves result from LBM simulations using the bounce-back boundary condition for various values of 7. model a stationary wall varies with 7. The comparison of the profile obtained using the bounce-back condition with the analytical result for Poiseuille flow reveals that the bounceback condition is actually modeling walls which are moving with a velocity that depends on the fluid viscosity (which enters in the definition of 7). n general, the effective velocity of a wall modeled by the bounce-back condition may be a function of other parameters, in addition to the viscosity. For Poiseuille flow, however, this artificial movement of the walls is solely a function of the viscosity when using a single relaxation lattice Boltzmann scheme. Figure 3 shows the effective wall velocity, as compared to the maximum velocity FG. 4. Analytical and LBM velocity profiles for the porous plate problem using the rigorous hydrodynamic boundary conditions developed here. n these simulations, the numerical values for the parameters are U=O.1/2.1, L=2O(vm), v=f. (at the center of the channel) over the full range of 7: t is seen that the bounce-back boundary condition effectively models moving walls with velocity very nearly zero for small values of 7. For large r, the wall velocity approaches that of the center of the channel, and thus the velocity profile varies greatly from the analytical profile for stationary walls, as observed in Fig. 3. This error does not have to be tolerated, however, since the method proposed here is clearly successful in obtaining the exact analytical solution over the full range of viscosity values. B. Normal plate velocity problem The porous plate problem poses the challenge of imposing a more general Dirichlet condition at the boundaries in which both velocity components are nonzero. The analytical solution for the problem is given by ;=( e(yryy;l), (39) ' ' FG. 3. The effective wall velocity as a function of the relaxation time using the bounce-back boundary condition. This result is for Poiseuille flow, and, in general, the effective wall velocity may be a function of other flow parameters. T where U is the velocity of the upper plate, and the Reynolds number, Re, is based on the velocity of the injected fluid, uo, and the width of the channel, L. Figure 4 shows the LBM results plotted along with the analytical solution for various values of transverse velocity. For the case of zero transverse mass flow, LBM predicts the linear profile, as expected. As the transverse mass flow increases, the velocity profile becomes skewed, and the LBM and analytical results continue to compare well. This demonstrates the ability of these techniques for modeling various Dirichlet boundary conditions. The porous plate problem provides an independent measure of the order of the LBM scheme. To determine the convergence rate in space, simulations are performed on grids with 10, 20, 40, 80, and 160 lattice units in the y direction for the case of Re=4.4. The errors are compared by examining the difference between the exact solution and the simulation results at each y location that is common to all the grids. The relative error for a location is given by 208 Phys. Fluids, Vol. 7, No. 1, January 1995 Noble et al.

7 FG. 5. Local error and error norms corresponding to porous plate solutions with 10, 20, 40, 80, and 160 lattice units spanning the width of the domain for Re=4.4. The slope, m, of the convergence is in the range 2.001<m for each error. AY (40) where U* is the simulation result and u is the exact solution. The errors are greatest near the moving upper wall (where the gradients are higher) and smallest near the lower wall. These local errors along with two error norms are plotted logarithmically in Fig. 5. The error norms are found according to the formulas (41) (42) where the sums are taken over the grid locations that are common to all the grids. The slope, m, of the convergence is in the range 2.001<m<2.03 for each error. Clearly, LBM is a second-order scheme when boundaries are accurately modeled. V. CONCLUSONS n this work, the bounce-back boundary condition has been shown to be a deficient method of modeling walls in lattice Boltzmann method (LBM) simulations. A method is proposed that successfully models a variety of Dirichlet boundary conditions, including both horizontal and vertical fluid motion at the walls. This method involves calculating the particle distributions contributed by nodes just inside the wall, such that proper hydrodynamic boundary conditions are maintained while executing the regular LBM procedure. This method is successfully used to simulate parallel plate flow problems. The first problem, Poiseuille flow, displays the inadequacy of the bounce-back boundary condition adapted from lattice gas studies. Rather than modeling a stationary wall, the bounce-back condition models a wall with an effective velocity that depends on the fluid viscosity. n the second problem, the ability to handle various Dirichlet boundary conditions is shown by modeling a fluid being sheared between two plates while fluid is being injected through the lower plate and extracted from the upper plate. The spatial convergence rate is measured by using grids with 10, 20, 40, 80, and 160 lattice units spanning the domain. LBM is shown to be a second-order scheme for modeling viscous fluid flow when the boundary conditions are accurately modeled. ACKNOWLEDGMENTS This research is supported by a National Science Foundation Graduate Fellowship, NCSA Grant No. CBT N, NSF Grant No. CTS , and the Richard W. Kritzer Foundation. The manuscript has been greatly improved following the suggestions of an anonymous reviewer. G. McNamara and G. Zanetti, Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett. 61, 2332 (1988). F. Higuera and J. Jimenez, Lattice gas dynamics with enhanced collisions, Europhys. Lett. 9, 663 (1989). H. Chen, S. Chen, and W. H. Matthaeus, Recovery of the Navier-Stokes equations using a lattice Boltzmann method, Phys. Rev. A 45, R5339 (1991). 4Y. H. Qian, D. d Humikres, and P. Lallemand, Lattice BGK models for the Navier-Stokes equation, Europhys. Lett. 17, 479 (1992). 5A. K. Gunstensen, D. H. Rothman, and S. Zaleski, Lattice Boltzmann model of immiscible fluids, Phys. Rev. A 43, 4320 (1991). 6D. Gnmau, S. Chen, and K. Eggert, A lattice Boltzmann model for multiphase fluid flows, Phys. Fluids A 5, 2557 (1993). S. Y. Chen, H. D. Chen, D. Martinez, and W. Matthaeus, Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys. Rev. Lett. 67, 3776 (1991). sd. Martinez, S. Chen, and W. H. Mattaeus, Lattice Boltzmann magnetohydrodynamics, submitted to Phys. Plasmas. 9S. P. Dawson, S. Chen, and G. Doolen, Lattice Boltzmann computations for reaction-diffusion equations, J. Chem. Phys. 98, 1514 (1993). OP. C. Rem and J. A. Somers, Cellular automata on a transputer network, in Discrete Kinetic Theory, Lattice Gas Dynamics and Foundations of Hydrodynamics, edited by R. Monaco (World Scientific, Singapore, 1989), p P. A. Skordos, nitial and boundary conditions for the lattice Boltzmann method, Phys. Rev. E 48, 4823 (1993). A. J. C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part. Theoretical foundation, J. Fluid Mech. 271, 285 (1994). 13D. d Humibres and P. Lallemand, Numerical simulations of hydrodynamics with lattice gas automata in two dimensions, Complex Syst. 1, 599 (1987). 14R. Comubert, D. d Humieres, and D. Levermore, A Knudsen layer theory for lattice gases, Physica D 47, 241 (1991). 1. Ginzbourg and P. M. Adler, Boundary flow condition analysis for the three-dimensional lattice Boltzmann model, J. Phys. France 4, 191 (1994). r6d. P. Ziegler, Boundary conditions for lattice Boltzmann simulations, J. Stat. Phys. 71, 1171 (1993). U Frisch, B. Hasslacher, and Y. Pomeau, Lattice-gas automata for the Navier-Stokes equation, Phys. Rev. Len. 56, 1505 (1986). t8s. Chen 9 Z. Wang, X. Shari,, and G. D. Doolen, Lattice Boltzmann computational fluid dynamics in three dimensions, J. Stat. Phys. 68, 379 (1992). r9l P Kadanoff, G. R. McNamara, and G. Zanetti, A Poiseuille viscometer for lattice gas automata, Complex Syst. 1, 791 (1987). Phys. Fluids, Vol. 7, No. 1, January 1995 Noble et al. 209