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

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

Viscous flow computations with the method of lattice Boltzmann equation

An efficient and robust computational method, based on the lattice-Boltzmann method, is presented for analysis of impermeable solid particle(s) suspended in fluid with inertia. In contrast to previous lattice-Boltzmann approaches, the present method can be used for any solid-to-fluid density ratio. The details of the numerical technique and implementation of the boundary conditions are presented. The accuracy and robustness of the method is demonstrated by simulating the flow over a circular cylinder in a two-dimensional channel, a circular cylinder in simple shear flow, sedimentation of a circular cylinder in a two-dimensional channel, and sedimentation of a sphere in a three-dimensional channel. With a solid-to-fluid density ratio close to one, new results from two-dimensional and three-dimensional computational analysis of dynamics of an ellipse and an ellipsoid in a simple shear flow, as well as two-dimensional and three-dimensional results for sedimenting ellipses and prolate spheroids, are presented.

Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation

The lattice Boltzmann equation method: theoretical interpretation, numerics and implications

The effect of inertia on the dynamics of a solid particle (a circular cylinder, an elliptical cylinder, and an ellipsoid) suspended in shear flow is studied by solving the discrete Boltzmann equation. At small Reynolds number, when inertia is negligible, the behaviour of the particle is in good agreement with the creeping flow solution showing periodic orbits. For an elliptical cylinder or an ellipsoid, the results show that by increasing the Reynolds number, the period of rotation increases, and eventually becomes infinitely large at a critical Reynolds number, Rec. At Reynolds numbers above Rec, the particle becomes stationary in a steady-state flow. It is found that the transition from a time-periodic to a steady state is through a saddle-node bifurcation, and, consequently, the period of oscillation near this transition is proportional to [mid ]p−pc[mid ]−1/2, where p is any parameter in the flow, such as the Reynolds number or the density ratio, which leads to this transition at p = pc. This universal scaling law is presented along with the physics of the transition and the effect of the inertia and the solid-to-fluid density ratio on the dynamics. It is conjectured that this transition and the scaling law are independent of the particle shape (excluding body of revolution) or the shear profile.

The dynamics and scaling law for particles suspended in shear flow with inertia

Computational methods based on the solution of the lattice-Boltzmann equation have been demonstrated to be effective for modeling a variety of fluid flow systems including direct simulation of particles suspended in fluid. Applications to suspended particles, however, have been limited to cases where the gap width between solid particles is much larger than the size of the lattice unit. The present extension of the method removes this limitation and improves the accuracy of the results even when two solid surfaces are near contact. With this extension, the forces on two moving solid particles, suspended in a fluid and almost in contact with each other, are calculated. Results are compared with classical lubrication theory. The accuracy and robustness of this computational method are demonstrated with several test problems.

Extension of the lattice-boltzmann method for direct simulation of suspended particles near contact

The dynamics and interaction of two circular cylinders settling in an infinitely long narrow channel (width equal to four times the cylinder diameter) is explained by direct computational analysis The results show that at relatively low Reynolds numbers (based on the average particle velocity and diameter), the particles undergo complex transitions to reach a low-dimensional chaotic state represented by a strange attractor As the Reynolds number increases, the initial periodic state goes through a turning point and a subcritical transition to another periodic branch Further increase in the Reynolds number results in a cascade of period-doubling bifurcations to a chaotic state represented by a low dimensional chaotic attractor The entire sequence of transitions takes place in a relatively narrow range of Reynolds number between 2 and 6 The physical reason for the period-doubling transitions is explained based on the interaction of the particles with each other and the channel walls The particles undergo near contact interactions as settling in the channel To accurately capture the dynamics, the computational method requires accurate resolution of the particle interactions The computational results are obtained with our lattice-Boltzmann method developed for suspended particles near contact The results show that even in the most simple and ideal multiparticle sedimentation, the system undergoes transition to complex dynamics at relatively low Reynolds number

Dynamics of particle sedimentation in a vertical channel: Period-doubling bifurcation and chaotic state

The clustering characteristic of purely hydrodynamically interacting particles suspended in pressure-driven flow in a circular cylinder is studied using direct numerical simulation based on the solution of the lattice-Boltzmann equation. We find a universal scaling relation for the cluster size distribution in the subcritical regime for all of the cases considered in this study. This scaling relation is independent of particle shape and concentration.

Cluster size distribution and scaling for spherical particles and red blood cells in pressure-driven flows at small Reynolds number.

the fundamental effect of inertia on particle motion, pattiWe have developed an efficient computational method cle interaction and microstructure, and effects on the bulk for hydrodynamic analysis of solid particle(s)suspended in properties, we need to solve the full momentum equation fluid. After a brief outline of the numerical technique, we to analyze the dynamics of individual particle suspended in demonstrate the accuracy and robustness of our method the fluid. Progress has been made towards this goal by the by comparing the results with (1) the asymptotic solutions recent finite element simulations of two-dimensional partiat small particle Reynolds number, Rep _ 1, for particle cles suspended in fluid by Feng, Hu, and Joseph (1994 a,b); sedimentation and particles suspended in shear flows, (2) Huang, Feng, and Joseph (1994), and Feng and Joseph 2-D Finite Element simulations of single and multi-particle (1995). These results provide considerable information sedimentation at finite Rep, and (3)experimental results, on the particlemotion and interaction at finite particle We present new results from 3-D computational analysis of Reynolds number. Their finite element simulation is based solid particle(s) sedimentation. The emphasis of this paper on solution of the Navier-Stokes equation for the fluid dois on demonstration of the accuracy, robustness, and commain using a commercial software, POLYFLOW. The proputational efficiency of our approach. Some new 2-D and cedure involves computation of the net force and torque on 3-D results for sedimenting ellipses and prolate spheroids the solid particle; explicit translation and rotation of the will also be presented. The results indicate that the sediparticle, remeshing, and projection of the solution from mentation dynamics of prolate spheroid objects in a chanthe previous time step onto the remeshed computational nel consists of a series of time-periodic equilibrium states domain at every time step (Hu et.al., 1992). Up to date, with marginal steady states in between, only 2-D solutions have been reported due to the excessive computational time required by this approach. Analysis of 3-D problems involving transport of solid particles require

https://smartech.gatech.edu/bitstream/1853/1945/1/tps-640.pdf

E-Jiang Ding

Papers

The dynamics and scaling law for particles suspended in shear flow with inertia

Extension of the lattice-boltzmann method for direct simulation of suspended particles near contact

Dynamics of particle sedimentation in a vertical channel: Period-doubling bifurcation and chaotic state

Cluster size distribution and scaling for spherical particles and red blood cells in pressure-driven flows at small Reynolds number.

Dynamic simulation of particles suspended in fluid