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The accuracy of moment-based boundary conditions for no slip walls in lattice Boltzmann simulations is examined numerically by using the dipole-wall collision benchmark test for both normal and oblique cases. In the normal case the dipole hits the wall perpendicularly while in the oblique case the dipole hits the wall at an angle of 30° to the horizontal. Boundary conditions are specified precisely at grid points by imposing constraints upon hydrodynamic moments only. These constraints are then translated into conditions for the unknown lattice Boltzmann distribution functions at boundaries. The two relaxation time (TRT) model is used with a judiciously chosen product of the two relaxation times. Stable results are achieved for higher Reynolds number up to 10,000 for the normal collision and up to 7500 for the oblique case. Excellent agreement with benchmark data is observed and the local boundary condition implementation is shown to be second order accurate.

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... Although the BB scheme is a simple and common method to apply the no-slip boundary condition, it introduces an additional error (a purely artificial numerical slip error) into the DUGKS. The NEBB scheme eliminates the error of the numerical slip, but the closure to find the unknowns at the wall is somewhat arbitrary [40]. There are only a few studies on boundary conditions for the DUGKS, which have not been studied thoroughly. ...

... Recently, the moment-based boundary condition has received increased interest and attention [40][41][42][43][44][45], which is based on the moments of the LBM [40] and has not yet been introduced to the DUGKS. Numerical simulations show that the moment-based boundary condition converges with second-order accuracy using dipole-wall collisions [40], natural convection in the square cavity [41], and lid-driven cavity flow [43][44][45]. ...

... Recently, the moment-based boundary condition has received increased interest and attention [40][41][42][43][44][45], which is based on the moments of the LBM [40] and has not yet been introduced to the DUGKS. Numerical simulations show that the moment-based boundary condition converges with second-order accuracy using dipole-wall collisions [40], natural convection in the square cavity [41], and lid-driven cavity flow [43][44][45]. ...

The boundary conditions are crucial for numerical methods. This study aims to contribute to this growing area of research by exploring boundary conditions for the discrete unified gas kinetic scheme (DUGKS). The importance and originality of this study are that it assesses and validates the novel schemes of the bounce back (BB), non-equilibrium bounce back (NEBB), and Moment-based boundary conditions for the DUGKS, which translate boundary conditions into constraints on the transformed distribution functions at a half time step based on the moment constraints. A theoretical assessment shows that both present NEBB and Moment-based schemes for the DUGKS can implement a no-slip condition at the wall boundary without slip error. The present schemes are validated by numerical simulations of Couette flow, Poiseuille flow, Lid-driven cavity flow, dipole–wall collision, and Rayleigh–Taylor instability. The present schemes of second-order accuracy are more accurate than the original schemes. Both present NEBB and Moment-based schemes are more accurate than the present BB scheme in most cases and have higher computational efficiency than the present BB scheme in the simulation of Couette flow at high Re. The present Moment-based scheme is more accurate than the present BB, NEBB schemes, and reference schemes in the simulation of Poiseuille flow and dipole–wall collision, compared to the analytical solution and reference data. Good agreement with reference data in the numerical simulation of Rayleigh–Taylor instability shows that they are also of use to the multiphase flow. The present Moment-based scheme is more competitive in boundary conditions for the DUGKS.

... The moment-method BC introduced by A. Hantsch et al. is an extension of the work by Noble et al. [30] and Bennett [3]. T. Reis et al. [55] then studied the accuracy of moment-method boundary conditions for no-slip walls using the dipole-wall collision benchmark. This method is a general methodology for imposing macroscopic boundary conditions in the LBM. ...

... As the example in Ref. [55] shows, for a stationary and noslip wall, by choosing three independent equations from the set of equations, unknown PDFs for left side boundary can be specified as in Eq. (A.4): ...

... As explained in Ref. [55], to apply the moment-method to convex corners and to simulate a no-slip wall, five equations are needed as there are five unknowns. The unknown PDFs for a typical D2Q9 lattice stencil are depicted in Fig. A.35. So, in this method, five equations from the existing general system of equations are chosen. ...

... Latt and Chopard (2007) validated their approach against findings in Clercx and Bruneau (2006), highlighting the accuracy of their approach. Interestingly, moment-based boundary conditions were tested against the dipole-wall collision for no-slip boundaries (Mohammed, Graham and Reis, 2018) and slip boundaries (Mohammed, Graham and Reis, 2020), showing very high numerical properties. More recently, De Rosis and Coreixas (2020) proposed a collision operator in the space of central moments (Geier, Greiner and Korvink (2006), Asinari (2008) , De Rosis and Enan (2021)) and tested it against the dipole-wall collision. ...

... More recently, De Rosis and Coreixas (2020) proposed a collision operator in the space of central moments (Geier, Greiner and Korvink (2006), Asinari (2008) , De Rosis and Enan (2021)) and tested it against the dipole-wall collision. Their results showed a very good agreement with findings in Clercx and Bruneau (2006) and Mohammed et al. (2018). Central-momentsbased collision operators were also adopted to perform accurate simulations of thermal flows (Hajabdollahi and Premnath (2018), Hajabdollahi, Premnath and Welch (2019)). ...

... In this section, we assess the accuracy of the proposed approach. First, we compare our findings against reference data by Clercx and Bruneau (2006) and Mohammed et al. (2018), where the collision of a dipole against a straight wall is considered. Then, the effectiveness of the combined LBM-IBM to estimate forces acting upon a moving immersed solid body is evaluated against the well-known benchmark problem proposed by Dütsch, Durst, Becker and Lienhart (1998), where a round cylinder undergoes harmonic horizontal oscillations. ...

In this paper, the flow physics generated by the collision of a vortex dipole that moves against a spinning round cylinder is investigated numerically. Fluid dynamics is predicted by a combined central-moments-based lattice Boltzmann-immersed boundary method. First, the model is validated against well established consolidated benchmark problems, showing very high accuracy properties. Then, results from a comprehensive numerical campaign are presented. A wide set of values of the Reynolds number (Re) is investigated, ranging from 10 to 1000. The cylinder is forced to spin around its centre with different angular velocities, which are obtained by varying the spinning number (Sp) between 0 (corresponding to the static case) and 0.75. The generation of secondary vortices as a consequence of the impact is elucidated and linked to the time evolution of the kinetic energy, enstrophy and hydrodynamic forces. Interestingly, we find that the flow physics changes drastically when Re ≥ 250, independently from the value of Sp. Through a closer look at the vorticity field, we find that the impact creates two primary-secondary structures and a second impingement takes place when Re ≥ 250. Interestingly, the normalised drag force (𝐶𝑑) is found to constantly fluctuates around a mean value. Oscillations are due to the vorticity created by the rotation of the cylinder and are more
emphasised as Sp grows. Specifically, 𝐶𝑑 can achieve marked negative values as a consequence of the velocity field created by the cylinder during its rotation.

... We further evaluate the numerical performance of the D3Q19-CM-LBM by examining the flow physics gener-ated by a dipole-wall collision 72,73 . Let us consider a square domain (x, y) ∈ [−1 : 1] 2 , enclosed by no-slip walls at each side. ...

... In turn, they show a slight mismatch (up to 3%) with respect to the LB study by Mohammed et al. 72 . It should be noted that findings in Ref. 72 are closer to the reference ones by Clercx & Bruneau 73 . We address this behavior to the adoption in Ref. 72 of (i) a more accurate boundary condition and (ii) a lower Mach number. ...

... It should be noted that findings in Ref. 72 are closer to the reference ones by Clercx & Bruneau 73 . We address this behavior to the adoption in Ref. 72 of (i) a more accurate boundary condition and (ii) a lower Mach number. The time evolutions of the energy and enstrophy are reported in Figure 6. ...

In a recent work [A. De Rosis, R. Huang, and C. Coreixas, "Universal formulation of central-moments-based lattice Boltzmann method with external forcing for the simulation of multiphysics phenomena", Phys. Fluids 31, 117102 (2019)], a multiple-relaxation-time lattice Boltzmann method (LBM) has been proposed by means of the D3Q27 discretization, where the collision stage is performed in the space of central moments (CMs). These quantities relax towards an elegant Galilean invariant equilibrium, and can also include the effect of external accelerations. Here, we investigate the possibility to adopt a coarser lattice composed of 19 discrete velocities only. The consequences of such a choice are evaluated in terms of accuracy and stability through multiphysics benchmark problems based on single-, multi-phase and magnetohydrodynamics flow simulations. In the end, it is shown that the reduction from 27 to 19 discrete velocities have only little impact on the accuracy and stability of the CM-LBM for moderate Reynolds number flows in the weakly compressible regime.

... We further evaluate the numerical performance of the D3Q19-CM-LBM by examining the flow physics generated by a dipolewall collision. 70,71 Let us consider a square domain (x, y) ∈ [−1:1] 2 , enclosed by no-slip walls at each side. The velocity is initialized as In Table II, the values of the energy E and enstrophy Ψ at salient time instants are reported. ...

... The findings from the D3Q19-CM-LBM run are compared to the reference solution in Ref. 71 and to a recent LB effort. 70 One can immediately observe that the D3Q19-and the D3Q27-CM-LBMs produce identical results. In turn, they show a slight mismatch (up to 3%) with respect to the LB study by Mohammed et al. 70 It should be noted that the findings in Ref. 70 are closer to the reference ones by Clercx and Bruneau. ...

... 70 One can immediately observe that the D3Q19-and the D3Q27-CM-LBMs produce identical results. In turn, they show a slight mismatch (up to 3%) with respect to the LB study by Mohammed et al. 70 It should be noted that the findings in Ref. 70 are closer to the reference ones by Clercx and Bruneau. 71 We address this behavior to the adoption in Ref. 70 of (i) a more accurate boundary condition and (ii) a lower Mach number. ...

In a recent work [A. De Rosis, R. Huang, and C. Coreixas, ``Universal formulation of central-moments-based lattice Boltzmann method with external forcing for the simulation of multiphysics phenomena", Phys. Fluids 31, 117102 (2019)], a multiple-relaxation-time lattice Boltzmann method (LBM) has been proposed by means of the D3Q27 discretization, where the collision stage is performed in the space of central moments (CMs). These quantities relax towards an elegant Galilean invariant equilibrium, and can also include the effect of external accelerations. Here, we investigate the possibility to adopt a coarser lattice composed of 19 discrete velocities only. The consequences of such a choice are evaluated in terms of accuracy and stability through multiphysics benchmark problems based on single-, multi-phase and magnetohydrodynamics flow simulations. In the end, it is shown that the reduction from 27 to 19 discrete velocities have only little impact on the accuracy and stability of the CM-LBM for moderate Reynolds number flows in the weakly compressible regime.

... Combination of unknowns at east boundary ρ, ρu x , Π xxf3 +f 6 +f 7 ρu y , Π xy , Q xxyf6 −f 7 Π yy , Q xyy , S xxyyf6 +f 7 known to be a numerically favourable value 6,31,40 . Mass and momentum are conserved by collisions so can be obtained directly fromf j just as from f j ...

... Various grid resolutions N lb are used to examine the convergence of the TRT-LBE. The convergence study for this work is based on the previous study of dipole wall collision with noslip boundaries in Mohammed et al. 6 , see Table III. However, at large slip lengths for Re ≥ 5000 the simulations needed less refined grids to converge than for the no-slip case, because the wall velocity gradient is smaller for slip conditions than for no-slip and the near-wall flow is thus more easily resolved. ...

... The generation of these dipoles depends on the conditions of the wall. In the case of slip boundaries and compared with the no-slip case 6 , the number of dipole rebounds from the boundary decreases and fewer vortices are generated from the boundary. The dynamics of the collision with a slip wall are less complicated than for the no-slip collision. ...

We study the physics of flow due to the interaction between a viscous dipole and boundaries that permit slip. This includes partial and free slip, and interactions near corners. The problem is investigated by using a two relaxation time lattice Boltzmann equation with moment-based boundary conditions. Navier-slip conditions, which involve gradients of the velocity, are formulated and applied locally. The implementation of free-slip conditions with the moment-based approach is discussed. Collision angles of 0°, 30°, and 45° are investigated. Stable simulations are shown for Reynolds numbers between 625 and 10 000 and various slip lengths. Vorticity generation on the wall is shown to be affected by slip length, angle of incidence, and Reynolds number. An increase in wall slippage causes a reduction in the number of higher-order dipoles created. This leads to a decrease in the magnitude of the enstrophy peaks and reduces the dissipation of energy. The dissipation of the energy and its relation to the enstrophy are also investigated theoretically, confirming quantitatively how the presence of slip modifies this relation.

... Tao et al. (Shi et al., 2018a, 2018b proposed the immersed boundary-discrete unified gas kinetic scheme (IB-DUGKS) to successfully simulate the sedimentation of circular and elliptical particles with no-slip conditions in a fluid domain. It should be noted that they adopted an iterative forcing scheme (Luo et al., 2007) to avoid the phenomenon of unphysical streamline penetration rendering the collapse of mass conservation (Shu et al., 2007). While iterative forcing scheme is effective in correcting the slip error, it involves additional computational cost and implementation effort. ...

... To validate the numerical stability of the DUGKS at a high Reynolds number, we simulate the Couette flow at Re = 10 6 with coarse mesh (16 × 16). The top and bottom walls (the distance between the two walls is H) adopt the no-slip boundary condition (Mohammed et al., 2018). The top wall moves in the horizontal direction with a constant velocity of U 0 . ...

To improve the endurance and speed of an autonomous underwater vehicle (AUV), we consider adopting the slip over a solid-liquid interface, which has the effect of drag reduction. However, the simulation of motion of an AUV with the slip remains a challenge due to the fluid-structure interaction and multiscale problems involving the moving boundary. To solve these problems, the present work proposes a novel and competitive mesoscopic method. Firstly, the Discrete Unified Gas Kinetic Scheme (DUGKS) coupled with the immersed boundary method is improved by correcting the boundary force, avoiding any additional computational time or implementation effort. Its effectiveness and accuracy are validated by two numerical tests involving the moving boundary. Furthermore, a fluid-solid interaction force is introduced to this method to model the slip condition due to its easy and effective implementation. We perform the experiment of motion of an AUV model in water to further validate the proposed method, which shows that it can predict acceptable results. Then, the method is applied to study the effect of the slip on the motion of the AUV model. The greater slip induced by the fluid-solid interaction force has a greater effect on reducing the drag and improving the velocity. It is indicated that increasing the fluid-solid interaction force can increase the amplitude and decrease the oscillation frequency of the drag curve. Overall, the slip can improve the endurance and speed of the AUV model.

... The density of each fluid is constant and the flow is unidirectional so that u = (u x (y), 0). The no-slip condition is imposed with the moment-based method and the constraints u x = u y = 0 and Π xx = P on the plates [49,50]. A brief overview of the moment-based method is given in Appendix B. Note that P on the plate is computed from the velocity conditions and known distributions, and the conditions on the moments have to be translated into conditions on "barred" functions [43]. ...

... We used a computational domain of size 257×1025 so that the grid spacing was ∆x = 1/256 and we set the timestep according to ∆t/∆x = 0.04 (meaning that U = 0.04 in lattice units). No-slip conditions were applied using the moment-based approach [50] and no flux conditions on the phase field [51]. Figure 9 plots on the top row the density at different moments in (nondimensional) time, T , when there is no surface tension (σ = 0). ...

A multiphase lattice Boltzmann model is constructed to numerically solve the one-fluid flow equations for immiscible fluids. The method features one solver for the macroscopic pressure and momentum and another for a scalar field that captures and sharpens the interface. The surface tension is set a priori and independently of other parameters. The interface capillary tensor is embedded within the moments of the lattice Boltzmann equation so that its divergence is captured locally. The algorithm is simple and can compute flows with large density and viscosity ratios while maintaining distributed but narrow interfaces. The model is validated against analytical solutions and benchmark simulations.

... Alternatively, a novel technique for implementing boundary conditions in the lattice Boltzmann method was proposed by Bennett [29] and used to impose the Navier-Maxwell conditions precisely for slip flow in microchannels by Reis and Dellar [16]. This methodology does not produce any artificial slip and can be used with any lattice Boltzmann collision operator [30][31][32]. It can also be used to impose boundary conditions that are consistent with the deviatoric stress and has been shown to predict non-Navier-Stokes behaviour [28,33], but until now this has not been used to compute the slip-flow regime. ...

... where we have used equation (14) noting that the flow is steady (∂ t = 0), unidirectional (u = u(y)), and the flow velocity is given by equation (31). The computed solution with the Navier-Stokes stress condition Γ xx = 0 that has often been used with the moment-based approach [16,30,38,39,41] is also shown. ...

A lattice Boltzmann method with moment-based boundary conditions is used to compute flow in the slip regime. Navier-Maxwell slip conditions and Burnett-order stress conditions that are consistent with the discrete velocity Boltzmann equation are imposed locally on stationary and moving boundaries. Micro-Couette and micro-lid-driven cavity flows are studied numerically at Knudsen and Mach numbers of the order O(10−1). The Couette results for velocity and the deviatoric stress at second order in Knudsen number are in excellent agreement with analytical solutions, and the cavity results are in excellent agreement with existing data. The algorithm is shown to compute nonequilibrium effects in the pressure that are in very good agreement with DSMC simulations of the Boltzmann equation but not captured by the Navier-Stokes equations.

... This approach, like all other on-node boundary conditions, is not easily extended to complicated geometries where boundaries are not aligned with grid points. It does, however, provide a general methodology for imposing hydrodynamic conditions (on velocity, pressure, or stress) locally and precisely at grid points and has ben shown to be very accurate [33]. The moment-based method has already been employed to implement diffusive slip in binary gases [34], Navier-Maxwell slip [35,36], contact angles in multiphase flows [37], and Dirichlet and von-Neumam conditions for natural convection [38]. ...

... Previous applications of the moment method determine the tangential momentum flux at a boundary by imposing ∂ x u x = 0. Moreover, it was always implicitly assumed that the deviatoric stress, T, is given by Newtonian constitutive equation [32][33][34][35][37][38][39]. A closer inspection shows that the stress obtained from the D2Q9 discrete velocity Boltzmann equation includes a non-zero Burnett contribution at second-order in relaxation time that reassembles a non-objective viscoelastic constitutive equation [12]. ...

Stress boundary conditions for the lattice Boltzmann equation that are consistent to Burnett order are proposed and imposed using a moment-based method. The accuracy of the method with complicated spatially-dependent boundary conditions for stress and velocity is investigated using the regularized lid-driven cavity flow. The complete set of boundary conditions, which involve gradients evaluated at the boundaries, are implemented locally. A recently-derived collision operator with modified equilibria and velocity-dependent collision rates to reduce the defect in Galilean invariance is also investigated. Numerical results are in excellent agreement with existing benchmark data and exhibit second-order convergence. The lattice Boltzmann stress field is studied and shown to depart significantly from the Newtonian viscous stress when the ratio of Mach to Reynolds numbers is not negligibly small.

... This appears counterintuitive and does not support findings achieved for non-conductive fluids, where it was found that the dipole-wall collision comes earlier for higher values of the Reynolds number. 6,13,23 The reason should be found in the presence of the magnetic field. As stated above, its role becomes stronger as Re grows and its braking effect on the flow field increases as well. ...

We investigate numerically the flow physics generated by the collision of a vortex against a wall in an electrically conductive fluid. Governing magnetohydrodynamic equations are solved by the lattice Boltzmann method. Our findings demonstrate that the presence of a magnetic field modifies significantly the vortex dynamics. Specifically, it exerts a braking effect on the vortex that increases with the magnetic Prandtl number. Our results are linked to the transfer of energy between the velocity and the magnetic fields, as well as to the evolution of their enstrophies.

... Either transient or steady-state, the recurrence equations linearly inter-relate the central-difference, equilibrium and non-equilibrium, operators to non-equilibrium solution. The recurrence equations are especially attractive for being intuitive, dimension-and lattice-transparent with the directional collisions, like the anisotropic-ADE L-basis [25] or its particular sub-class, the two-relaxation-time TRT collision [24,71,53,36]; the TRT is suitable for both hydrodynamic and advection-diffusion problems, [26]. The form of the recurrence solutions proves [42] that any steady-state TRT bulk solution is controlled by the free-tunable product of its two relaxation eigenfunctions, Λ = Λ − Λ + . ...

The scalar field and the non-equilibrium solutions of the linear advection-diffusion d2Q9 Lattice Boltzmann (LBM) two-relaxation-times (TRT) scheme are constructed analytically. The scheme copes with an infinite number of suitable, second-order accurate, equilibrium weights. Here, the simplest, translation-invariant geometry with an implicitly located, straight or diagonal, grid-aligned interface (boundary) is addressed. We show that these two interface (boundary) orientations are accommodated with the help of two distinctive, anisotropic, discrete-exponential algebraic solution components, referred to as the A-layer and the B-layer. Being unpredicted by the perturbative analysis, such as the Chapman-Enskog, asymptotic or truncation, their solution is derived symbolically from the TRT recurrence equations, subject to the local mass conservation solvability and effective closure conditions. When the interface (boundary) is “diagonal”, the A-layer perturbs the simplest physical solutions, like the piece-wise linear, polynomial or exponential scalar field, rendering the macroscopic solution weight-dependent and delaying its convergence to the first order; the A-layer base depends upon the weights, free relaxation parameter Λ and physical numbers. In contrast, the B-layer, invisible to the scalar field, typically accommodates the non-equilibrium discrepancy between the normal and diagonal directions on the “straight” interface (boundary); the B-layer base is fixed by Λ alone. The A-layer and B-layer may coexist and degrade the physical solution gradient and its convergence. Only the D2Q5 model is free from all these effects in the straight and diagonal orientations, while the diagonally-rotated D2Q5 model is unsuitable because of the “checkerboard” effect. These spurious corrections are not the Knudsen layers, but they present the LBM response for any-order bulk mismatch with the implicit or explicit interface (boundary) treatment; the A-layer and B-layer bring them in evidence and provide excellent benchmarks for their attenuation through interface-conjugate or adaptive refinement techniques. Our approach extends to any lattice, linear collision, source term, heterogeneity and LBM problem class.

To explore the anisotropic slip on hydrophobic surfaces, a new anisotropic slip boundary condition is proposed for three-dimensional simulations of liquid microflows using the lattice Boltzmann method with adjustable streamwise/spanwise slip length. The proposed boundary condition is derived based on the moment method, which is no longer limited to the assumption of the unidirectional steady flow. Numerical tests validated the effectiveness of the proposed method. Compared with the bounce-back and specular reflection scheme, the proposed method is more accurate and stable for capturing velocity profiles. The proposed method was applied to explore the effects of anisotropic slip on three-dimensional micro-lid-driven cavity flow. The numerical simulation results showed that anisotropic slip has a greater influence on the flow than pure streamwise/spanwise slip, and streamwise slip plays a more important role in influencing the flow than spanwise slip. The findings may hold significance for efficient development of microfluidic systems and micro-devices.

We introduce the steady-state two-relaxation-time (TRT) Lattice Boltzmann method. Owing to the symmetry argument, the bulk system and the closure equations are all expressed in terms of the equilibrium and non-equilibrium unknowns with the half discrete velocity set. The local mass-conservation solvability condition is adjusted to match the stationary, but also the quasi-stationary, solutions of the standard TRT solver. Additionally, the developed compact, boundary and interface-conjugate, multi-reflection (MR) concept preserves the efficient directional bulk structure and shares its parametrization properties. The method is exemplified in grid-inclined stratified slabs for two-phase Stokes flow and the linear advection-diffusion equation featuring the discontinuous coefficients and sources. The piece-wise parabolic benchmark solutions are matched exactly with the novel Dirichlet, pressure-stress, Neumann flux and Robin MR schemes. The popular, anti-bounce-back and shape-fitted Dirichlet continuity schemes are improved in the presence of both interface-parallel and perpendicular advection velocity fields. The steady-state method brings numerous advantages: it skips transient numerical instability, overpasses severe von Neumann parameter range limitations, tolerates high physical contrasts and arbitrary MR coefficients. The method is promising for faster computation of Stokes/Brinkman/Darcy linear flows in heterogeneous soil, but also heat and mass transfer problems governed by an accurate boundary and interface treatment.

This work addresses the Dirichlet boundary condition for momentum in the lattice Boltzmann method (LBM), with focus on the steady-state Stokes flow modelling inside non-trivial shaped ducts. For this task, we revisit a local and highly accurate boundary scheme, called the local second-order boundary (LSOB) method. This work reformulates the LSOB within the two-relaxation-time (TRT) framework, which achieves a more standardized and easy to use algorithm due to the pivotal parametrization TRT properties. The LSOB explicitly reconstructs the unknown boundary populations in the form of a Chapman–Enskog expansion, where not only first- but also second-order momentum derivatives are locally extracted with the TRT symmetry argument, through a simple local linear algebra procedure, with no need to compute their non-local finite-difference approximations. Here, two LSOB strategies are considered to realize the wall boundary condition, the original one called Lwall and a novel one Lnode, which operate with the wall and node variables, roughly speaking. These two approaches are worked out for both plane and curved walls, including the corners. Their performance is assessed against well-established LBM boundary schemes such as the bounce-back, the local second-order accurate CLI scheme and two different parabolic multi-reflection (MR) schemes. They are all evaluated for 3D duct flows with rectangular, triangular, circular and annular cross-sections, mimicking the geometrical challenges of real porous structures. Numerical tests confirm that LSOB competes with the parabolic MR accuracy in this problem class, requiring only a single node to operate.
This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

We analytically solve the {two dimensional, nine-velocity,} lattice Boltzmann model in planar channel flow and determine its deviatoric stress tensor. The shear component of its stress takes the expected Navier-Stokes form but the tangential component contains second order in Knudsen number contributions that one finds in solutions to the Burnett equations. Boundary conditions that neglect this Burnett contribution cause spurious grid-scale oscillations in the computed stress field within the computational domain. A moment-based boundary condition which considers the non--zero deviatoric stress is analysed and shown to completely eliminate the spurious oscillations seen in solutions using other boundary conditions. The analysis offers an explanation of previously reported optimal relaxation times in terms of the recurrence relation for the tangential stress and gives them an interpretation in terms of compact finite difference schemes.

In this paper, moment‐based boundary conditions for the lattice Boltzmann method are extended to three dimensions. Boundary conditions for velocity and pressure are explicitly derived for straight on‐grid boundaries for the D3Q19 lattice. The method is compared against the bounce‐back scheme using both single and two relaxation time collision schemes. The method is verified using classical benchmark test cases. The results show very good agreement with the data found in the literature. It is confirmed from the results that the derived moment‐based boundary scheme is of second order accuracy in grid spacing and does not produce numerical slip, and therefore offers a transparent way of accurately prescribing velocity and pressure boundaries that are aligned with grid points in 3D.

The accuracy of the Moment Method for imposing no-slip boundary conditions in the lattice Boltzmann algorithm is investigated numerically using lid-driven cavity flow. Boundary conditions are imposed directly upon the hydrodynamic moments of the lattice Boltzmann equations, rather than the distribution functions, to ensure the constraints are satisfied precisely at grid points. Both single and multiple relaxation time models are applied. The results are in excellent agreement with data obtained from state-of-the-art numerical methods and are shown to converge with second order accuracy in grid spacing.

We study a multiple relaxation time lattice Boltzmann model for natural convection with moment-based boundary conditions. The unknown primary variables of the algorithm at a boundary are found by imposing conditions directly upon hydrodynamic moments, which are then translated into conditions for the discrete velocity distribution functions. The method is formulated so that it is consistent with the second order implementation of the discrete velocity Boltzmann equations for fluid flow and temperature. Natural convection in square cavities is studied for Rayleigh numbers ranging from 10^3 to 10^8. An excellent agreement with benchmark data is observed and the flow fields are shown to converge with second order accuracy.

Irrespective of the nature of the modeled conservation laws, we establish first the microscopic interface continuity conditions for Lattice Boltzmann (LB) multiple-relaxation time, link-wise collision operators with discontinuous components (equilibrium functions and/or relaxation parameters). Effective macroscopic continuity conditions are derived for a planar implicit interface between two immiscible fluids, described by the simple two phase hydrodynamic model, and for an implicit interface boundary between two heterogeneous and anisotropic, variably saturated soils, described by Richard's equation. Comparing the effective macroscopic conditions to the physical ones, we show that the range of the accessible parameters is restricted, e.g. a variation of fluid densities or a heterogeneity of the anisotropic soil properties. When the interface is explicitly tracked, the interface collision components are derived from the leading order continuity conditions. Among particular interface solutions, a harmonic mean value is found to be an exact LB solution, both for the interface kinematic viscosity and for the interface vertical hydraulic conductivity function. We construct simple problems with the explicit and implicit interfaces, matched exactly by the LB hydrodynamic and/or advection-diffusion schemes with the aid of special solutions for free collision parameters.

Consistent formulations of 2D and 3D pressure and velocity boundary conditions along both the stationary and non-stationary plane wall and corner for lattice Boltzmann simulations are proposed. The unknown distribution functions are made function of local known distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definitions of density and momentum. This boundary condition can be easily implemented on the wall and corner boundary using the same formulation. Discrete macroscopic equation is also derived for steady fully developed channel flow to assess the effect of the boundary condition on the solutions, where the resulting second order accurate central difference equation predicts continuous distribution across the boundary provided the boundary unknown distribution functions satisfy the macroscopic quantity. Three different local known distribution functions are experimented to assess both this observation and the applicability of the present formulation, and are scrutinized by calculating two-dimensional Couette-Poiseuille flow, Couette flow with wall injection and suction, lid-driven square cavity flow, and three-dimensional square duct flow. Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known distribution functions is as expected negligible, which are consistent with the results from the derived discrete macroscopic equation.

We propose a lattice Boltzmann approach for simulating contact angle phenomena in multiphase fluid systems. Boundary conditions for partially-wetted walls are introduced using the moment method. The algorithm with our boundary conditions allows for a maximum density ratio of 200,000 for neutral wetting. The achievable density ratio decreases as the contact angle departs from 90°, but remains of the order O(10^2 ) for all but extreme contact angles. In all simulations an excellent agreement between the simulated and nominal contact angles is observed.

A discrete model based on the Boltzmann equation with a body force and a single relaxation time collision model is derived for simulations of nonideal-gas flow. The interparticle interaction is treated using a mean-field approximation. The Boltzmann equation is discretized in a way that preserves the derivation of the hydrodynamic equations from the Boltzmann equation, using either the Chapman-Enskog method or the Grad 13-moment method. The previously proposed nonideal-gas lattice Boltzmann equation model can be analyzed with rigor.

Laboratory experiments and numerical simulations of oscillating spin-up in a square tank have been conducted to investigate the production of small-scale vorticity near the no-slip sidewalls of the container and the formation and subsequent decay of wall-generated quasi-two-dimensional vortices. The flow is made quasi-two-dimensional by a steady background rotation, and a small sinusoidal perturbation to the background rotation leads to the periodic formation of eddies in the corners of the tank by the roll-up of vorticity generated along the sidewalls. When the oscillation period is greater than the time scale required to advect a full-grown corner vortex to approximately halfway along the sidewall, dipole structures are observed to form. These dipoles migrate away from the walls, and the interior of the tank is continually filled with new vortices. The average size of these vortices appears to be largely controlled by the initial formation mechanism. Their vorticity decays from interactions with other stronger vortices that strip off filaments of vorticity, and by Ekman pumping at the bottom of the tank. Subsequent interactions between the weaker ‘old’ vortices and the ‘young’ vortices result in the straining, and finally the destruction, of older vortices. This inhibits the formation of large-scale vortices with diameters comparable to the size of the container.

Accurate numerical simulations of vortex dipoles impinging on flat boundaries have revealed interesting new features. In the case of free-slip boundaries the dipole does not rebound from the wall. In the case of nonslip walls rebounding occurs and complex interactions of secondary and tertiary vortices appear. The numerical simulation of the first dipole rebound from the wall agrees with experimental visualizations. Numerical experiments extending in time beyond the real experiments show multiple rebounding. Each rebound is associated with the detachment of a secondary vorticity layer from the wall, these layers merge, and at a value of Reynolds number Re = 1600, form a new dipole. This dipole has sufficient circulation to induce on itself a motion in the opposite direction to the motion of the initial dipole.

We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels. Rarefied effects lead to non-zero cross-channel velocities, and nonlinear variations in the pressure along the channel. Both effects are absent in flows driven by uniform body forces. We obtain second-order accuracy for the two components of velocity and the pressure relative to asymptotic solutions of the compressible Navier–Stokes equations with slip boundary conditions. Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier–Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. Our moment formalism is also valuable for analysing the existing boundary conditions. It reveals the origin of numerical slip in the bounce-back and other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself.

We present in detail a theoretical framework for representing hydrodynamic systems through a systematic discretization of the Boltzmann kinetic equation. The work is an extension of a previously proposed formulation. Conventional lattice Boltzmann models can be shown to be directly derivable from this systematic approach. Furthermore, we provide here a clear and rigorous procedure for obtaining higher-order approximations to the continuum Boltzmann equation. The resulting macroscopic moment equations at each level of the systematic discretization give rise to the Navier–Stokes hydrodynamics and those beyond. In addition, theoretical indications to the order of accuracy requirements are given for each discrete approximation, for thermohydrodynamic systems, and for fluid systems involving long-range interactions. All these are important for complex and micro-scale flows and are missing in the conventional Navier–Stokes order descriptions. The resulting discrete Boltzmann models are based on a kinetic representation of the fluid dynamics, hence the drawbacks in conventional higher-order hydrodynamic formulations can be avoided.

The lattice Boltzmann equation (LBE) is directly derived from the Boltzmann equation by discretization in both time and phase space. A procedure to systematically derive discrete velocity models is presented. A LBE algorithm with arbitrary mesh grids is proposed and a numerical simulation of the backward-facing step is conducted. The numerical result agrees well with experimental and previous numerical results. Various improvements on the LBE models are discussed, and an explanation of the instability of the existing LBE thermal models is also provided.

In this paper, the lattice Boltzmann equation is directly derived from the Boltzmann equation. It is shown that the lattice Boltzmann equation is a special discretized form of the Boltzmann equation. Various approximations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail. A general procedure to derive the lattice Boltzmann model from the continuous Boltzmann equation is demonstrated explicitly. The lattice Boltzmann models derived include the two-dimensional 6-bit, 7-bit, and 9-bit, and three-dimensional 27-bit models.

An comprehensive o verview of the lattice gas automata LGA and lattice Boltzmann equation LBE is presented in this article. The mathematical foundation of the LGA and LBE methods are discussed in detail. The connections between the LGA and LBE methods and other kinetic methods are pointed out. Future development of the LGA and LBE method concerning issues in the areas of hardware, modeling, and applications are also discussed.

Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by
non-linear (anisotropic) advection-diffusion equations, there are very few analytical results on their stability, even for
the isotropic linear equation. In this paper, the optimal two-relaxation-time (OTRT) model is defined, along with necessary and sufficient (easy to use) von Neumann stability conditions for a very general
anisotropic advection-diffusion equilibrium, in one to three dimensions, with or without numerical diffusion. Quite remarkably,
the OTRT stability bounds are the same for any Peclet number and they are defined by the adjustable equilibrium parameters.
Such optimal stability is reached owing to the free (“kinetic”) relaxation parameter. Furthermore, the sufficient stability bounds tolerate negative
equilibrium functions (the distribution divided by the local mass), often labeled as “unphysical”. We prove that the non-negativity
condition is (i) a sufficient stability condition of the TRT model with any eigenvalues for the pure diffusion equation, (ii) a sufficient stability condition of its OTRT and BGK/SRT sub-classes, for any linear anisotropic advection-diffusion equation, and (iii) unnecessarily more restrictive for any Peclet number than the optimal sufficient conditions. Adequate choices
of the two relaxation rates and the free-tunable equilibrium parameters make the OTRT sub-class more efficient than the BGK
one, at least in the advection-dominant regime, and allow larger time steps than known criteria of the forward time central
finite-difference schemes (FTCS/MFTCS) for both, advection and diffusion dominant regimes.
KeywordsLattice Boltzmann equation-Advection-diffusion equation-Necessary and sufficient stability conditions-Von Neumann stability analysis-Two-relaxation-time model-Forward time finite-difference schemes-BGK

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 fe=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.

Boundary conditions of lattice Boltzmann method to simulate flows embedded with a solid object is proposed. The closest nodes adjacent to the boundary in the fluid domain are used as boundary nodes of the flow domain. The fluid velocity of the boundary node is obtained by linear interpolation between the velocities of the solid object and the second fluid node further away. Then, distribution functions originating from the solid domain at the boundary nodes are modified using known distribution functions and correctors to satisfy the momentum. This boundary condition is an extended form of a method proposed by Hou et al. [C.F. Hou, C. Chang, C.A. Lin, Consistent boundary conditions for 2D and 3D Lattice Boltzmann simulations (submitted for publication)] for plane wall and regular geometry. The technique is examined by simulating decaying vortex, transient flow induced by an abruptly rotating ring and flow over an asymmetrically placed cylinder. Numerical simulations indicate that this method is second order accurate, and all the numerical results are compatible with the benchmark solutions.

Ph.D. Kurt A. Wiesenfeld

The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyperviscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters that optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some two-dimensional LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long-wavelength limit (wave vector k=0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.

The role of no-slip boundaries as an enstrophy source in two-dimensional (2D) flows has been investigated for high Reynolds numbers. Numerical simulations of normal and oblique dipole-wall collisions are performed to investigate the dissipation of the kinetic energy E(t), and the evolution of the enstrophy Omega(t) and the palinstrophy P(t). It is shown for large Reynolds numbers that dE(t)/dt=-2Omega(t)/Re proportional, variant 1/sqrt[Re] instead of the familiar relation dE(t)/dt proportional, variant 1/Re as found for 2D unbounded flows.

A new lattice Boltzmann (LB) model is introduced, based on a regularization of the pre-collision distribution functions in terms of the local density, velocity, and momentum flux tensor. The model dramatically improves the precision and numerical stability for the simulation of fluid flows by LB methods. This claim is supported by simulation results of some 2D and 3D flows.

In this paper, for the first time a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for the $n$-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlinear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard sphere systems, describing the time evolution of pair correlations. As a quantitative test we calculate equal time correlation functions in equilibrium for two models that violate semi-detailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid type model on a triangular lattice. The numerical predictions agree very well with computer simulations. Comment: 31 pages LaTeX, 12 uuencoded tar-compressed Encapsulated PostScript figures (`psfig' macro), hardcopies available on request, 78kb + 52kb

This thesis describes the development of a Lattice Boltzmann (LB) model for a binary gas mixture. Specifically, channel flow driven by a density gradient with diffusion slip occurring at the wall is studied in depth.
The first part of this thesis sets the foundation for the multi-component model used in the subsequent chapters. Commonly used single component LB methods use a non-physical equation of state, in which the relationship between pressure and density varies according to the scaling used. This is fundamentally unsuitable for extension to multi-component systems containing gases of differing molecular masses that are modelled with the ideal gas equation of state. Also, existing methods for implementing boundary conditions are unsuitable for extending to novel boundary conditions, such as diffusion slip. Therefore, a new single component LB derivation and a new method for implementing boundary conditions are developed, and validated against Poiseuille flow. However, including a physical equation of state reduces stability and time accuracy, leading to longer computational times, compared with 'incompressible' LB methods. The new method of analysing LB boundary conditions successfully explains observations from other commonly used schemes, such as the slip velocity associated with 'bounce-back'.
The new model developed for multi-component gases avoids the pitfalls of some other LB models, a single computational grid is shared by all the species and the diffusivity is independent of the viscosity. The Navier-Stokes equation for the mixture and the Stefan-Maxwell diffusion equation are both recovered by the model. However, the species momentum equations are not recovered correctly and this can lead to instability. Diffusion slip, the non-zero velocity of a gas mixture at a wall parallel to a concentration gradient, is successfully modelled and validated against a simple one-dimensional model for channel flow. To increase the accuracy of the scheme a second order numerical implementation is needed. This can be achieved using a variable transformation method which does not result in an increase in computational time.
Simulations were carried out on hydrogen and water diffusion through a narrow channel, with varying total pressure and concentration gradients. For a given value of the species mass flux ratio, the total pressure gradient was dependent on the species concentration gradients. These results may be applicable to fuel cells where the species mass flux ratio is determined by a chemical reaction and the species have opposing velocities. In this case the total pressure gradient is low and the cross-channel average mass flux of hydrogen is independent of the channel width.
Finally, solutions for a binary Stefan tube problem were investigated, in which the boundary at one end of a channel is permeable to hydrogen but not water. The water has no total mass flux along the channel but circulates due to the slip velocity at the wall. The cross-channel average mass flux of the hydrogen along the channel increases with larger channel widths. A fuel cell using a mixture of gases, one being inert, will experience similar circulation phenomena and, importantly, the width of the pores will affect performance.
This thesis essentially proves the viability of LB models to simulate multi-component gases with diffusion slip boundaries, and identifies the many areas in which improvements could be made.

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.

This thesis describes the development of a Lattice Boltzmann (LB) model for a binary gas mixture. Specifically, channel flow driven by a density gradient with diffusion slip occurring at the wall is studied in depth.
The first part of this thesis sets the foundation for the multi-component model used in the subsequent chapters. Commonly used single component LB methods use a non-physical equation of state, in which the relationship between pressure and density varies according to the scaling used. This is fundamentally unsuitable for extension to multi-component systems containing gases of differing molecular masses that are modelled with the ideal gas equation of state. Also, existing methods for implementing boundary conditions are unsuitable for extending to novel boundary conditions, such as diffusion slip. Therefore, a new single component LB derivation and a new method for implementing boundary conditions are developed, and validated against Poiseuille flow. However, including a physical equation of state reduces stability and time accuracy, leading to longer computational times, compared with 'incompressible' LB methods. The new method of analysing LB boundary conditions successfully explains observations from other commonly used schemes, such as the slip velocity associated with 'bounce-back'.
The new model developed for multi-component gases avoids the pitfalls of some other LB models, a single computational grid is shared by all the species and the diffusivity is independent of the viscosity. The Navier-Stokes equation for the mixture and the Stefan-Maxwell diffusion equation are both recovered by the model. However, the species momentum equations are not recovered correctly and this can lead to instability. Diffusion slip, the non-zero velocity of a gas mixture at a wall parallel to a concentration gradient, is successfully modelled and validated against a simple one-dimensional model for channel flow. To increase the accuracy of the scheme a second order numerical implementation is needed. This can be achieved using a variable transformation method which does not result in an increase in computational time.
Simulations were carried out on hydrogen and water diffusion through a narrow channel, with varying total pressure and concentration gradients. For a given value of the species mass flux ratio, the total pressure gradient was dependent on the species concentration gradients. These results may be applicable to fuel cells where the species mass flux ratio is determined by a chemical reaction and the species have opposing velocities. In this case the total pressure gradient is low and the cross-channel average mass flux of hydrogen is independent of the channel width.
Finally, solutions for a binary Stefan tube problem were investigated, in which the boundary at one end of a channel is permeable to hydrogen but not water. The water has no total mass flux along the channel but circulates due to the slip velocity at the wall. The cross-channel average mass flux of the hydrogen along the channel increases with larger channel widths. A fuel cell using a mixture of gases, one being inert, will experience similar circulation phenomena and, importantly, the width of the pores will affect performance.
This thesis essentially proves the viability of LB models to simulate multi-component gases with diffusion slip boundaries, and identifies the many areas in which improvements could be made.

We present a moment-based approach for implementing boundary conditions in a lattice Boltzmann formulation of magnetohydrodynamics. Hydrodynamic quantities are represented using a discrete set of distribution functions that evolve according to a cut-down form of Boltzmann’s equation from continuum kinetic theory. Electromagnetic quantities are represented using a set of vector-valued distribution functions. The nonlinear partial differential equations of magnetohydrodynamics are thus replaced by two constant-coefficient hyperbolic systems in which all nonlinearities are confined to algebraic source terms. Further discretising these systems in space and time leads to efficient and readily parallelisable algorithms. However, the widely used bounce-back boundary conditions place no-slip boundaries approximately half-way between grid points, with the precise position being a function of the viscosity and resistivity. Like most lattice Boltzmann boundary conditions, bounce-back is inspired by a discrete analogue of the diffuse and specular reflecting boundary conditions from continuum kinetic theory. Our alternative approach using moments imposes no-slip boundary conditions precisely at grid points, as demonstrated using simulations of Hartmann flow between two parallel planes.

Introduction Initial and Boundary Conditions for Lattice Boltzmann Method Improved Lattice Boltzmann Models Sample Applications of LBE for Isothermal Flows LBE for Low Speed Flows with Heat Transfer LBE for Compressible Flows LBE for Multiphase and Multi-component Flows LBE for Microscale Gas Flows Other Applications of LBE.

The collision of a dipolar vortex with a sliding wall is investigated numerically. Previous studies have shown that perpendicular vortex collisions with fixed walls may lead to a rebound of the primary dipole during which the symmetry of the vorticity field with respect to the dipole's axis is preserved. However, a wall sliding tangentially breaks this symmetry, leading to distinctive flow regimes for different wall speeds. The conditions for which the transition between these two regimes occur are studied, both numerically and analytically, in terms of the wall speed and the Reynolds number of the dipolar flow.

When the Lattice Boltzmann Method (LBM) is used for simulating continuum fluid flow, the discrete mass distribution must satisfy imposed constraints for density and momentum along the boundaries of the lattice. These constraints uniquely determine the three‐dimensional (3‐D) mass distribution for boundary nodes only when the number of external (inward‐pointing) lattice links does not exceed four. We propose supplementary rules for computing the boundary distribution where the number of external links does exceed four, which is the case for all except simple rectangular lattices. Results obtained with 3‐D body‐centered‐cubic lattices are presented for Poiseuille flow, porous‐plate Couette flow, pipe flow, and rectangular duct flow. The accuracy of the two‐dimensional (2‐D) Poiseuille and Couette flows persists even when the mean free path between collisions is large, but that of the 3‐D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3‐D low‐Reynolds‐number flow.

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.

A numerical experiment is presented that is taken from the recent literature. It has been devised as a benchmark case to test the quality of boundary conditions in numerical solvers for computational fluid dynamics. In this experiment, a two-dimensional system of two counter-rotating vortexes is brought into collision with a no-slip wall and rebounds from it. In the present paper, the benchmark is run with a lattice-Boltzmann numerical solver. Astonishingly accurate results are obtained with a straightforward boundary condition known under the name of bounce-back. This sample problem is also used to discuss techniques for the setup of an initial condition in the lattice Boltzmann method.

A study of the dynamics of a discrete two-dimensional system of classical particles is presented. In this model, dynamics and computations may be done exactly, by definition. The equilibrium state is investigated and the Navier-Stokes hydrodynamical equations are derived. Two hydrodynamical modes exist in the model: the sound waves and a kind of vorticity diffusion. In the Navier-Stokes equations one obtains a transport coefficient which is given by a Green-Kubo formula. The related time correlation function has been calculated in a numerical simulation up to a time of the order of 50 mean free flights. After a short time of exponential decay this time correlation behaves like t-S, the exponent being compared to theoretical predictions.

▪ Abstract We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.

We extend lattice Boltzmann (LB) methods to advection and anisotropic-dispersion equations (AADE). LB methods are advocated for the exactness of their conservation laws, the handling of different length and time scales for flow/transport problems, their locality and extreme simplicity. Their extension to anisotropic collision operators (L-model) and anisotropic equilibrium distributions (E-model) allows to apply them to generic diffusion forms. The AADE in a conventional form can be solved by the L-model. Based on a link-type collision operator, the L-model specifies the coefficients of the symmetric diffusion tensor as linear combination of its eigenvalue functions. For any type of collision operator, the E-model constructs the coefficients of the transformed diffusion tensors from linear combinations of the relevant equilibrium projections. The model is able to eliminate the second order tensor of its numerical diffusion. Both models rely on mass conserving equilibrium functions and may enhance the accuracy and stability of the isotropic convection–diffusion LB models.The link basis is introduced as an alternative to a polynomial collision basis. They coincide for one particular eigenvalue configuration, the two-relaxation-time (TRT) collision operator, suitable for both mass and momentum conservation laws. TRT operator is equivalent to the BGK collision in simplicity but the additional collision freedom relates it to multiple-relaxation-times (MRT) models. “Optimal convection” and “optimal diffusion” eigenvalue solutions for the TRT E-model allow to remove next order corrections to AADE. Numerical results confirm the Chapman–Enskog and dispersion analysis.

Recent experiments on a freely evolving dipolar vortex in a homogeneous shallow fluid layer have clearly shown the importance of vertical secondary flows on top of the primary horizontal motion. The present contribution focuses on the interaction of such a dipolar vortex with a sidewall. Accurate measurements of the three velocity components in a single horizontal plane have been performed using the Stereoscopic Particle Image Velocimetry (SPIV) technique. The experimental results, supported by numerical simulations, indicate that the complex vertical structure of a shallow-layer dipole becomes even more complex during the collision process. The observed growth of the kinetic energy associated with enhanced vertical motion pinpoints the strong discrepancies between vortex-wall interactions in shallow fluid layers and in purely two-dimensional wall-bounded turbulence.

Lattice Boltzmann equations using multiple relaxation times are intended to be more stable than those using a single relaxation time. The additional relaxation times may be adjusted to suppress non-hydrodynamic modes that do not appear directly in the continuum equations, but may contribute to instabilities on the grid scale. If these relaxation times are fixed in lattice units, as in previous work, solutions computed on a given lattice are found to diverge in the incompressible (small Mach number) limit. This non-existence of an incompressible limit is analysed for an inclined one dimensional jet. An incompressible limit does exist if the non-hydrodynamic relaxation times are not fixed, but scaled by the Mach number in the same way as the hydrodynamic relaxation time that determines the viscosity.

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.

The effect of no-slip walls on the evolution of coherent, vortical structures in two-dimensional flows is studied by numerical calculations. The calculations are based on an accurate and efficient spectral scheme which has been developed for the solution of the 2D Navier-Stokes equations in the vorticity-stream function representation for bounded geometries. Fundamental processes connected to vorticity detachment from the boundary layers caused by the proximity of vortical structures are described. These processes include enstrophy enhancement of the main flow during bursting events, and pinning down of vortex dipoles by “vortex shielding”.

Benchmark results are reported of two separate sets of numerical experiments on the collision of a dipole with a no-slip boundary at several Reynolds numbers. One set of numerical simulations is performed with a finite differences code while the other set concerns simulations conducted with a Chebyshev pseudospectral code. Well-defined initial and boundary conditions are used and the accuracy and convergence of the numerical solutions have been investigated by inspection of several global quantities like the total kinetic energy, the enstrophy and the total angular momentum of the flow, and the vorticity distribution and vorticity flux at the no-slip boundaries. It is found that the collision of the dipole with the no-slip wall and the subsequent flow evolution is dramatically influenced by small-scale vorticity produced during and after the collision process. The trajectories of several coherent vortices are tracked during the simulation and show that in particular underresolved high-amplitude vorticity patches near the no-slip walls are potentially responsible for deteriorating accuracy of the computations in the course of time. Our numerical simulations clearly indicate that it is extremely difficult to obtain mode- or grid-convergence for this seemingly rather simple two-dimensional vortex–wall interaction problem.

We prove for generic steady solutions of the Lattice Boltzmann (LB) models that the variation of the numerical errors is set by specific combinations (called “magic numbers”) of the relaxation rates associated with the symmetric and anti-symmetric collision moments. Given the governing dimensionless physical parameters, such as the Reynolds or Peclet numbers, and the geometry of the computational mesh, the numerical errors remain the same for any change of the transport coefficients only when the “free” (“kinetic”) anti-symmetric rates and the boundary rules are chosen properly. The single-relaxation-time (BGK) model has no free collision rate and yields viscosity dependent errors with any boundary scheme for hydrodynamic problems. The simplest and most efficient collision operator for invariant errors is the two-relaxation-times (TRT) model. As an example, this model is able to compute viscosity independent permeabilities for any porous structure.These properties are derived from steady recurrence equations, obtained through linear combinations of the LB evolution equations, in which the equilibrium and non-equilibrium components are directly interconnected via finite-difference link-wise central operators. The explicit dependency of the non-equilibrium solution on the relaxation rates is then obtained. This allows us, first, to confirm the governing role of the “magic” combinations for steady solutions of the Stokes equation, second, to extend this property to steady solutions of the Navier–Stokes and anisotropic advection–diffusion equations, third, to develop a parametrization analysis of the microscopic and macroscopic closure relations prescribed via link-wise boundary schemes.

We study the velocity boundary condition for curved boundaries in the lattice Boltzmann equation (LBE). We propose a LBE boundary condition for moving boundaries by combination of the "bounce-back" scheme and spatial interpolations of first or second order. The proposed boundary condition is a simple, robust, efficient, and accurate scheme. Second-order accuracy of the boundary condition is demonstrated for two cases: (1) time-dependent two-dimensional circular Couette flow and (2) two-dimensional steady flow past a periodic array of circular cylinders (flow through the porous media of cylinders). For the former case, the lattice Boltzmann solution is compared with the analytic solution of the NavierâStokes equation. For the latter case, the lattice Boltzmann solution is compared with a finite-element solution of the NavierâStokes equation. The lattice Boltzmann solutions for both flows agree very well with the solutions of the NavierâStokes equations. We also analyze the torque due to the momentum transfer between the fluid and the boundary for two initial conditions: (a) impulsively started cylinder and the fluid at rest, and (b) uniformly rotating fluid and the cylinder at rest. \copyright2001 American Institute of Physics.

This contribution proposes an alternative lattice Boltzmann grid refinement algorithm that overcomes the drawbacks that plague existing approaches. We demonstrate that this algorithm is accurate and applicable for all values of the relaxation time. We also show that this algorithm can significantly speed up the flow settlement process. By using a hierarchy of grid levels, the stationary regime can be approached up to a thousand times faster than with a single grid resolution.

Pressure (density) and velocity boundary conditions inside a flow domain are studied for 2-D and 3-D lattice Boltzmann BGK models (LBGK) and a new method to specify these conditions are proposed. These conditions are constructed in consistency of the wall boundary condition based on an idea of bounceback of non-equilibrium distribution. When these conditions are used together with the improved incompressible LBGK model by Zou et al., the simulation results recover the analytical solution of the plane Poiseuille flow driven by pressure (density) difference with machine accuracy. Since the half-way wall bounceback boundary condition is very easy to implement and was shown theoretically to give second-order accuracy for the 2-D Poiseuille flow with forcing, it is used with pressure (density) inlet/outlet conditions proposed in this paper and in Chen et al. to study the 2-D Poiseuille flow and the 3-D square duct flow. The numerical results are approximately second-order accurate. The magnitude of the error of the half-way wall bounceback is comparable with that using some other published boundary conditions. Besides, the bounceback condition has a much better stability behavior than that of other boundary conditions. Comment: 18 pages, one figure