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 higher-order 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.

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.

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.

A novel method is developed to simulate suspensions of deformable particles by coupling the lattice-Boltzmann method (LBM) for the fluid phase to a linear finiteelement analysis (FEA) describing particle deformation. The methodology addresses the need for an efficient method to simulate large numbers of three-dimensional and deformable particles at high volume fraction in order to capture suspension rheology, microstructure, and self-diffusion in a variety of applications. The robustness and accuracy of the LBM-FEA method is demonstrated by simulating an inflating thinwalled sphere, a deformable spherical capsule in shear flow, a settling sphere in a confined channel, two approaching spheres, spheres in shear flow, and red blood cell deformation in flow chambers. Additionally, simulations of suspensions of hundreds of biconcave red blood cells at 40 % volume fraction produce continuum-scale physics and accurately predict suspension viscosity and the shear-thinning behaviour of blood. Simulations of fluid-filled spherical capsules which have red-blood-cell membrane properties also display deformation-induced shear-thinning behaviour at 40 % volume fraction, although the suspension viscosity is significantly lower than blood.

No abstract

An investigation of margination dependence on hematocrit, platelet shape, and viscosity ratio of plasma to cytoplasm is presented. Whole blood is modeled as a suspension of deformable red blood cells (RBCs) and rigid platelets in a viscous liquid. The fluid phase is simulated using the lattice-Boltzmann method, the RBC membranes are modeled with a coarse-grained spectrin-link method, and the dynamics of rigid particles are updated using Newton's equations of motion for axisymmetric shapes. The results emphasize that an increase in hematocrit increases the rate of margination. The viscosity ratio between the interior cytoplasm and suspending fluid can considerably alter the rate of margination. The aspect ratio of surrogate platelet particles influences the rate of margination as well. Spherical particles tend to migrate more quickly than disks. Highly viscous or rigid RBCs slow down margination.

SUMMARYThe implementation of a spectrin-link (SL) red blood cell (RBC) membrane method coupled with a lattice-Boltzmann (LB) fluid solver is discussed. Details of the methodology are included along with subtleties associated with its integration into a massively parallel hybrid LB finite element (FE) suspension flow solver. A comparison of the computational performance of the coupled LB-SL method with that of the previously implemented LB-FE is given for an isolated RBC and for a dense suspension in HagenPoiseuille flow. Validating results for RBCs isolated in shear and parachuting in microvessel flow are also presented.

The basic dynamics of a prolate spheroidal particle suspended in shear flow is studied using lattice Boltzmann simulations. The spheroid motion is determined by the particle Reynolds number (${\mathit{Re}}_{p} $) and Stokes number ($\mathit{St}$), estimating the effects of fluid and particle inertia, respectively, compared with viscous forces on the particle. The particle Reynolds number is defined by ${\mathit{Re}}_{p} = 4G{a}^{2} / \nu $, where $G$ is the shear rate, $a$ is the length of the spheroid major semi-axis and $\nu $ is the kinematic viscosity. The Stokes number is defined as $\mathit{St}= \alpha \boldsymbol{\cdot} {\mathit{Re}}_{p} $, where $\alpha $ is the solid-to-fluid density ratio. Here, a neutrally buoyant prolate spheroidal particle ($\mathit{St}= {\mathit{Re}}_{p} $) of aspect ratio (major axis/minor axis) ${r}_{p} = 4$ is considered. The long-term rotational motion for different initial orientations and ${\mathit{Re}}_{p} $ is explained by the dominant inertial effect on the particle. The transitions between rotational states are subsequently studied in detail in terms of nonlinear dynamics. Fluid inertia is seen to cause several bifurcations typical for a nonlinear system with odd symmetry around a double zero eigenvalue. Particle inertia gives rise to centrifugal forces which drives the particle to rotate with the symmetry axis in the flow-gradient plane (tumbling). At high ${\mathit{Re}}_{p} $, the motion is constrained to this planar motion regardless of initial orientation. At a certain critical Reynolds number, ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $, a motionless (steady) state is created through an infinite-period saddle-node bifurcation and consequently the tumbling period near the transition is scaled as $\vert {\mathit{Re}}_{p} - {\mathit{Re}}_{c} {\vert }^{- 1/ 2} $. Analyses in this paper show that if a transition from tumbling to steady state occurs at ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $, then any parameter $\beta $ (e.g. confinement or particle spacing) that influences the value of ${\mathit{Re}}_{c} $, such that ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $ as $\beta = {\beta }_{c} $, will lead to a period that scales as $\vert \beta - {\beta }_{c} {\vert }^{- 1/ 2} $ and is independent of particle shape or any geometric aspect ratio in the flow.