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 (hyper-viscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which 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 2D 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 wave-length 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.

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Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability

With its roots in kinetic theory and the cellular automaton concept, the lattice-Boltzmann (LB) equation can be used to obtain continuum flow quantities from simple and local update rules based on particle interactions. The simplicity of formulation and its versatility explain the rapid expansion of the LB method to applications in complex and multiscale flows. We review many significant developments over the past decade with specific examples. Some of the most active developments include the entropic LB method and the application of the LB method to turbulent flow, multiphase flow, and deformable particle and fiber suspensions. Hybrid methods based on the combination of the Eulerian lattice with a Lagrangian grid system for the simulation of moving deformable boundaries show promise for more efficient applications to a broader class of problems. We also discuss higherorder boundary conditions and the simulation of microchannel flow with finite Knudsen number. Additionally, the remarkable scalability of the LB method for parallel processing is shown with examples. Teraflop simulations with the LB method are routine, and there is no doubt that this method will be one of the first candidates for petaflop computational fluid dynamics in the near future.

Lattice-Boltzmann Method for Complex Flows

Discrete particle simulation of particulate systems: Theoretical developments

This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,(69, 70) taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.

Lattice-Boltzmann Simulations of Particle-Fluid Suspensions

Viscous flow computations with the method of lattice Boltzmann equation

A lattice-Boltzmann method has been developed to simulate suspensions of both spherical and non-spherical particles in finite-Reynolds-number flows. The results for sedimentation of a single elliptical particle are shown to be in excellent agreement with the results of Huang, Hu & Joseph (1998) who used a finite-element method. Sedimentation of two-dimensional circular and rectangular particles in a two-dimensional channel and three-dimensional spherical particles in a tube with square cross-section is simulated. Computational results are consistent with experimentally observed phenomena, such as drafting, kissing and tumbling.

Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows

This paper reports the results of lattice Boltzmann simulations of the rotation behaviour of neutrally buoyant spheroidal particles in a three-dimensional Couette flow. We find several distinctive states depending on the Reynolds number range and particle shape. As the Reynolds number increases, rotation may change from one state to another. For a prolate spheroid, two rotation transitions are found. In the low Reynolds number range 0 <R < R 1 ≈ 205, the prolate spheroid rotates around its minor axis, which is parallel to the vorticity vector of the flow. The rate of rotation is a periodic function of time. In the intermediate Reynolds number range R1 <R< R2 ≈ 345, the prolate spheroid precesses about the vorticity direction with a nutational motion. The angular velocities are periodic functions of time. The mean nutation angle between the major axis and the vorticity increases monotonically as the Reynolds number increases. In the high Reynolds number range R2 <R <467, the prolate spheroid rotates with a constant rate around its major axis, which is parallel to the vorticity. For an oblate spheroid, only one rotation transition is observed. In the lower Reynolds number range 0 <R<R � 1 ≈ 220, the oblate spheroid finally spins with a constant rate around its minor axis (the symmetric axis of the revolution), which is parallel to the vorticity vector. In the higher Reynolds number range 220 ≈ R � 1 <R <467, the oblate spheroid still spins with a constant rate around its minor axis but there is a finite inclination angle between the minor axis and the vorticity vector. This angle increases as the Reynolds number increases.

Rotational and orientational behaviour of three-dimensional spheroidal particles in Couette flows

Dynamic behavior of collision of elastic spheres in viscous fluids

The simulations of elliptical particles in a pressure driven flow are performed using a lattice Boltzmann (LB) method. Effects of multi-particle interaction on the lateral migration and orientation of both neutrally and non-neutrally buoyant particles are investigated. Low and itermediate solid concentrations in terms of area fraction f

 a
 =13, 25, and 40% are included in these simulations.

Lateral Migration and Orientation of Elliptical Particles in Poiseuille Flows

The rotational states of three-dimensional nonspherical particles including cylindrical and disk-shaped, as well as prolate and oblate, ellipsoidal in a Couette flow are studied by using a lattice Boltzmann simulation. We discover that rotation of a nonspherical particle exhibits several different states, depending on the ranges of the particle Reynolds numbers and the geometric shape of the particle. As the Reynolds number increases, the rotation transits from one state to another state.

Dewei Qi

Papers

Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows

Rotational and orientational behaviour of three-dimensional spheroidal particles in Couette flows

Dynamic behavior of collision of elastic spheres in viscous fluids

Lateral Migration and Orientation of Elliptical Particles in Poiseuille Flows

Transitions in rotations of a nonspherical particle in a three-dimensional moderate Reynolds number Couette flow