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### Citations

329 |
A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems”.
- Bhatnagar, Gross, et al.
- 1954
(Show Context)
Citation Context ...structed independently to better suit numerics [21]. The simplest lattice Boltzmann equation (LBE) is the lattice Bhatnagar-Gross-Krook (BGK) equation, based on a single-relaxation-time approximation =-=[1]-=-. Due to its extreme simplicity, the lattice BGK (LBGK) equation [29, 4] has become the most popular lattice Boltzmann model in spite of its well known de ciencies. The multiple-relaxation-time (MRT) ... |

190 |
Lattice BGK models for Navier-Stokes equation:
- Qian, D’Humières, et al.
- 1992
(Show Context)
Citation Context ...ttice Boltzmann equation (LBE) is the lattice Bhatnagar-Gross-Krook (BGK) equation, based on a single-relaxation-time approximation [1]. Due to its extreme simplicity, the lattice BGK (LBGK) equation =-=[29, 4]-=- has become the most popular lattice Boltzmann model in spite of its well known de ciencies. The multiple-relaxation-time (MRT) lattice Boltzmann equation was also developed at the same time [7]. The ... |

139 |
Numerical simulations of particulate suspensions via a discretized Boltzmann equation.
- Ladd
- 1994
(Show Context)
Citation Context ...en the kinematic and bulk viscosities. The MRT lattice Boltzmann equation has been persistently pursued, and much progress has been made. Successes include: formulation of optimal boundary conditions =-=[10, 22]-=-, interface conditions in multi-phase ows [11] and free surfaces [12], thermal [26] and viscoelastic models [13, 14], models with reduced lattice symmetries [8, 2], and improvement ofnumerical stabili... |

138 |
Recovery of the NavierStokes Equations Using a Lattice-Gas Boltzmann Method,” Phys.
- Chen, Chen, et al.
- 1992
(Show Context)
Citation Context ...ttice Boltzmann equation (LBE) is the lattice Bhatnagar-Gross-Krook (BGK) equation, based on a single-relaxation-time approximation [1]. Due to its extreme simplicity, the lattice BGK (LBGK) equation =-=[29, 4]-=- has become the most popular lattice Boltzmann model in spite of its well known de ciencies. The multiple-relaxation-time (MRT) lattice Boltzmann equation was also developed at the same time [7]. The ... |

119 |
Theory Of The Lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, And Stability," Physical Review E,
- Lallemand, Luo
- 2000
(Show Context)
Citation Context ...nterface conditions in multi-phase ows [11] and free surfaces [12], thermal [26] and viscoelastic models [13, 14], models with reduced lattice symmetries [8, 2], and improvement ofnumerical stability =-=[23]-=-. It should be stressed that most of the above results cannot be obtained with the LBGK models. Applying optimization techniques in coding, the computational e ciency of the RLBE method [7] can be fai... |

87 |
A priori derivation of the lattice Boltzmann equation,”
- He, Luo
- 1997
(Show Context)
Citation Context ...9 model). 2. Multiple-relaxation-time lattice Boltzmann equation. Although it can be shown that the lattice Boltzmann equation is a nite di erence form of the linearized continuous Boltzmann equation =-=[17,18]-=-, we present RLBE as a self-contained mathematical object representing a dynamical system with a number of moments in discrete space and time. The general RLBE model has three components. The nite rst... |

67 |
Boltzmann approach to lattice gas simulations,"
- Higuera, Jeminez
- 1989
(Show Context)
Citation Context ...BE in 3D, D3Q15 and D3Q19 models, 3D diagonal lid driven Subject classi cation. Fluid Mechanics 1. Introduction. The relaxation lattice Boltzmann equation (RLBE) was introduced by Higuera and Jimenez =-=[20]-=-toovercome some drawbacks of lattice gas automata (LGA) such as large statistical noise, limited range of physical parameters, non-Galilean invariance, and implementation di culty in three dimensions.... |

59 |
Momentum transfer of a boltzmann-lattice fluid with boundaries”.
- Bouzidi, Firdaouss, et al.
- 2001
(Show Context)
Citation Context ...ect; and the momentum transfer in the direction perpendicular to the moving lid is signi cantly weakened. The correct boundary conditions consistent with the bounce-back boundary conditions should be =-=[24, 25,3]-=-: e f = f +2w 0 UBC c 2 s e = f , 2w 0 UBC c 2 s ; (4.2) where f is the distribution function of e ,e . Nevertheless, the implementation prescribed by equation (4.1) does help to enhance the stability... |

34 |
An Introduction to the Theory of the Boltzmann Equation”,
- Harris
- 1971
(Show Context)
Citation Context ...i; e1xi; :::; eNxi) T , for i 2f1; 2; :::; Dg in D dimensions (e.g., fjexi; jeyi; jezig in three dimensions). The set fj ig is analogous to the Hermite tensor polynomials in continuous velocity space =-=[15, 16]-=-. It should be stressed that the orthogonal functions de ned on a nite set of discrete velocities fe g has some degeneracies which do not exist in the Hermite tensor polynomials in continuous space. O... |

30 | Theory of the lattice Boltzmann method: Lattice Boltzmann Models for Nonideal gases”, Physical Review E,
- Luo
- 2000
(Show Context)
Citation Context ...ect; and the momentum transfer in the direction perpendicular to the moving lid is signi cantly weakened. The correct boundary conditions consistent with the bounce-back boundary conditions should be =-=[24, 25,3]-=-: e f = f +2w 0 UBC c 2 s e = f , 2w 0 UBC c 2 s ; (4.2) where f is the distribution function of e ,e . Nevertheless, the implementation prescribed by equation (4.1) does help to enhance the stability... |

18 |
Lattice gas-dynamics with enhanced collisions, Europhys
- Higuera, Succi, et al.
- 1989
(Show Context)
Citation Context ...nderlying LGA models. It was soon realized that the connection to the LGA model could be abandoned and the equilibria and collision matrices could be constructed independently to better suit numerics =-=[21]-=-. The simplest lattice Boltzmann equation (LBE) is the lattice Bhatnagar-Gross-Krook (BGK) equation, based on a single-relaxation-time approximation [1]. Due to its extreme simplicity, the lattice BGK... |

16 |
Stabilization of thermal lattice Boltzmann models
- McNamara, Garcia, et al.
- 1995
(Show Context)
Citation Context ...istently pursued, and much progress has been made. Successes include: formulation of optimal boundary conditions [10, 22], interface conditions in multi-phase ows [11] and free surfaces [12], thermal =-=[26]-=- and viscoelastic models [13, 14], models with reduced lattice symmetries [8, 2], and improvement ofnumerical stability [23]. It should be stressed that most of the above results cannot be obtained wi... |

15 |
Lattice Boltzmann Equation on a Two-Dimensional Rectangular Grid
- Bouzidi, d´Humieres, et al.
(Show Context)
Citation Context ...ion of optimal boundary conditions [10, 22], interface conditions in multi-phase ows [11] and free surfaces [12], thermal [26] and viscoelastic models [13, 14], models with reduced lattice symmetries =-=[8, 2]-=-, and improvement ofnumerical stability [23]. It should be stressed that most of the above results cannot be obtained with the LBGK models. Applying optimization techniques in coding, the computationa... |

15 |
A Knudsen layer theory for lattice gases
- Cornubert, d'Humieres, et al.
- 1991
(Show Context)
Citation Context ...ntation prescribed by equation (4.1) does help to enhance the stability of the D3Q15 model. The `node' bounce-back boundary conditions are applied to the rest ve walls for no-slip boundary conditions =-=[6]-=-. The `node' bounce-back boundary conditions di er from the `link' bounce-back boundary conditions by a one-step delay in time but otherwise they are the same. This one-time-step delay seems to e ecti... |

15 |
Thirteen-velocity three-dimensional lattice Boltzmann model
- d’Humieres, Bouzidi, et al.
- 2001
(Show Context)
Citation Context ...ion of optimal boundary conditions [10, 22], interface conditions in multi-phase ows [11] and free surfaces [12], thermal [26] and viscoelastic models [13, 14], models with reduced lattice symmetries =-=[8, 2]-=-, and improvement ofnumerical stability [23]. It should be stressed that most of the above results cannot be obtained with the LBGK models. Applying optimization techniques in coding, the computationa... |

6 |
A free surface lattice-Boltzmann method for modelling the filling of expanding cavities by Bingham Fluids
- Ginzburg, Steiner
- 2002
(Show Context)
Citation Context ... has been persistently pursued, and much progress has been made. Successes include: formulation of optimal boundary conditions [10, 22], interface conditions in multi-phase ows [11] and free surfaces =-=[12]-=-, thermal [26] and viscoelastic models [13, 14], models with reduced lattice symmetries [8, 2], and improvement ofnumerical stability [23]. It should be stressed that most of the above results cannot ... |

4 |
Boundary ow condition analysis for the three-dimensional lattice Boltzmann model, J.Phys
- Ginzbourg, Alder
- 1994
(Show Context)
Citation Context ...en the kinematic and bulk viscosities. The MRT lattice Boltzmann equation has been persistently pursued, and much progress has been made. Successes include: formulation of optimal boundary conditions =-=[10, 22]-=-, interface conditions in multi-phase ows [11] and free surfaces [12], thermal [26] and viscoelastic models [13, 14], models with reduced lattice symmetries [8, 2], and improvement ofnumerical stabili... |

4 |
A lattice Boltzmann model for viscoelasticity, Int
- Giraud, d'Humieres, et al.
- 1997
(Show Context)
Citation Context ...ogress has been made. Successes include: formulation of optimal boundary conditions [10, 22], interface conditions in multi-phase ows [11] and free surfaces [12], thermal [26] and viscoelastic models =-=[13, 14]-=-, models with reduced lattice symmetries [8, 2], and improvement ofnumerical stability [23]. It should be stressed that most of the above results cannot be obtained with the LBGK models. Applying opti... |

3 |
Generalized lattice-Boltzmann equation, inRare ed Gas Dynamics: Theory and Simulations
- d'Humieres
- 1992
(Show Context)
Citation Context ...n [29, 4] has become the most popular lattice Boltzmann model in spite of its well known de ciencies. The multiple-relaxation-time (MRT) lattice Boltzmann equation was also developed at the same time =-=[7]-=-. The MRT lattice Boltzmann equation (also referred to as the generalized lattice Boltzmann equation (GLBE) or the moment method) overcomes some obvious defects of the LBGK model, such as xed Prandtl ... |

2 |
Conditions aux limites des modeles cinetiques discrets: couche de Knudsen et obstacles, " Ph.D.l thesis (Ecole Normale Superieure
- Cornubert
- 1991
(Show Context)
Citation Context ...by a one-step delay in time but otherwise they are the same. This one-time-step delay seems to e ectively reduce oscillations caused by the parity invariance and thus enhances the numerical stability =-=[5]-=-. As the e ective width of the system is approximately 50 lattice units, the Reynolds number Re = 50UBC= was set by varying the viscosity . We computed the lower bounds of the viscosity for this parti... |

2 |
Principles of the kinetic theory of gases, in Encyclopedia of Physics, edited by
- Grad
- 1958
(Show Context)
Citation Context ...i; e1xi; :::; eNxi) T , for i 2f1; 2; :::; Dg in D dimensions (e.g., fjexi; jeyi; jezig in three dimensions). The set fj ig is analogous to the Hermite tensor polynomials in continuous velocity space =-=[15, 16]-=-. It should be stressed that the orthogonal functions de ned on a nite set of discrete velocities fe g has some degeneracies which do not exist in the Hermite tensor polynomials in continuous space. O... |

2 | Boltzmann model for the incompressible Navier-Stokes equation - Lattice - 1997 |

2 |
Uni ed Theory of the lattice Boltzmann models for nonideal gases
- Luo
- 1998
(Show Context)
Citation Context ...ect; and the momentum transfer in the direction perpendicular to the moving lid is signi cantly weakened. The correct boundary conditions consistent with the bounce-back boundary conditions should be =-=[24, 25,3]-=-: e f = f +2w 0 UBC c 2 s e = f , 2w 0 UBC c 2 s ; (4.2) where f is the distribution function of e ,e . Nevertheless, the implementation prescribed by equation (4.1) does help to enhance the stability... |

2 |
Lattice Boltzmann method for 3-d ows with curved boundary
- Mei, Shyy, et al.
(Show Context)
Citation Context ...e accurate than that obtained by using the D3Q15 RLBE model with di erent boundary conditions. This con rms the previous observation that the D3Q19 LBGK model is more stable than the D3Q15 LBGK model =-=[27]-=-. A further comparative study of these two RLBE models is left for future work. 5. Conclusions. In this paper we provide a synopsis of the multiple-relaxation-time LBE in three dimensions and demonstr... |

1 |
Introduction of upwind and free boundary into lattice Boltzmann method, in Discrete Modelling and Discrete Algorithms in Continuum Mechanics, edited by Th. Sonar and I. Thomas 12
- Ginzburg
- 2001
(Show Context)
Citation Context ...tors of M, fh jg, are orthogonal to each other but they are not normalized, i.e. h j i = k k k k . Note also that, with di erent ordering and normalization, the basis vectors fj kig given by Ginzburg =-=[9]-=- are the same as the ones given here, except j 1i and j 2i which are replaced by an orthogonal linear combination. This would make a di erence only when s1 6= s2. The 4th, 6th, and 8th row vectors of ... |

1 |
Surface tension models with di erent viscosities, Transport
- Ginzbourg, Alder
- 1995
(Show Context)
Citation Context ...tice Boltzmann equation has been persistently pursued, and much progress has been made. Successes include: formulation of optimal boundary conditions [10, 22], interface conditions in multi-phase ows =-=[11]-=- and free surfaces [12], thermal [26] and viscoelastic models [13, 14], models with reduced lattice symmetries [8, 2], and improvement ofnumerical stability [23]. It should be stressed that most of th... |

1 |
High-incidence 3-D lid-driven cavity ow
- Povitsky
(Show Context)
Citation Context ...ies of parities 0 and 1 does not have this checkerboard invariance. 4. Simulations. In order to demonstrate the enhanced stability of the RLBE approach, we simulated a diagonally lid-driven cavity ow =-=[28]-=- with a ow con guration shown in gure 1. The mesh is uniform and of size 52 3 . The boundary condition (BC) at the top plane (at y =1)isUBC = ,( p 2; 0; p 2)=20, so that UBC = kUBCk =0:1 in lattice un... |

1 |
The hydrodynamics of lattice gas automata, Phys
- Zanetti
- 1989
(Show Context)
Citation Context ...nce is the existence of spurious invariants that may lead to some undesirable artifacts in simulations, especially near boundaries. One such invariant is the staggered invariant in LGA and LBE models =-=[30]-=-. The D3Q15 model has also another special invariant not found in most LBE models: the parity (ri) ofavector ri =(xi; yi; zi) de ned on a three-dimensional cubic lattice by (ri) =(xi + yi + zi) (mod 2... |