Lattice-Boltzmann discretization

These authors use the lattice-Boltzmann discretization method for solving the Navier-Stokes equations. In lattice-Boltzmann discretization, the motion of the fluid is represented by the motion and collision of discrete particles moving between the nodes of a fixed grid (the lattice). This discretization method, which has recently become more frequently used, but is still not standard, is thus completely different from the ones briefly discussed in Sect. 7.1.1. If the rules for the particle collisions and the morphology of the lattice upon which they move are set correctly, the dynamics of these particles will represent a numerical solution to the Navier-Stokes equations (Wolf-Gladrow, 2000 Chen and Doolen, 1998). 

Figure 7. Discrete particle methods for simulation of fluids. LB (lattice Boltzmann),...

Sectional and class methods for the solution of the collisional KE are generally called discrete-velocity methods (DVM). These methods are based on the simple idea of discretizing the velocity space into a grid constituted by a finite number of points. The existing methods are characterized by different grid structures (Aristov, 2001). For example, lattice Boltzmann methods discretize the velocity space into a regular cubic lattice with a constant lattice size (Li-Shi, 2000), whereas other methods employ different discretization schemes (Monaco Preziosi, 1990). By using a similar approach to that used with PBE, it is possible to define A,- as the number density of the particles with velocity and the discretized KE becomes... 

Li-Shi, L. 2000 Some recent results on discrete velocity models and ramifications for lattice Boltzmann equation. Computer Physics Communications 129, 63-74. 

Abe Takashi. 1997. Derivation of the lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation. Journal of Computational Physics. 131 (1). 

The lattice Boltzmann equation (LBE) is obtained as a dramatic simplification of the Boltzmann equation, (20.1) along with its associated equations, (20.2-20.4). In particular, it was discovered that the roots of the Gauss-Hermite quadrature used to exactly and numerically represent the moment integrals in (20.4), corresponds to a particular set of few discrete particle velocity directions c in the LBE [11]. Eurther-more, the continuous equihbrium distribution (20.3) is expanded in terms of the fluid velocity m as a polynomial, where the continuous particle velocity is replaced by the discrete particle velocity set ga obtained as discussed above, i.e., becomes f 1. After replacing the continuous distribution function / =f x, t) in (20.1) by the discrete distribution function/ =f x, (20.1) is integrated by considering... 

Lee, T. and C.-L. Lin. A Stable Discretization of the Lattice Boltzmann Equation for Simulation of Incompressible Two-Phase Flows at High Density Ratio. J. Comput. Phys. 206 16 (2005). 

The lattice Boltzmann method is a mesoscopic simulation method for complex fluid systems. The fluid is modeled as fictitious particles, and they propagate and coUide over a discrete lattice domain at discrete time steps. Macroscopic continuum equations can be obtained from this propagation-colhsion dynamics through a mathematical analysis. The particulate nature and local d3mamics also provide advantages for complex boundaries, multiphase/multicomponent flows, and parallel computation. 

The lattice Boltzmann method (LBM) is a relatively new simulation technique for complex fluid systems and has attracted great interests from researchers in computational physics and engineering. Unlike traditional computation fluid dynamics (CFD) methods to numerically solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy), LBM models the fluid as fictitious particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. Due to its particulate nature and local dynamics, LBM has several advantages over conventional CFD methods, especially in dealing with complex boundaries, incorporation of microscopic interactions, and parallel computation [1, 2]. 

In LBM, a fluid is modeled as fictitious particles moving in a lattice domain at discrete time steps. The major variable in LBM is the density distribution fi x, t), indicating the quantity of particles moving along the /-th lattice direction at position X and time t. The time evolution of density distributions is governed by the so-called lattice Boltzmann equation with a BGK collision term [1, 2] ... 

Lattice Boltzmann Method (LBM), Fig. 1 The discrete lattice velocities of a D2Q9 lattice structure Co = (0, 0) corresponds to the rest portion of the particles/o... 

LBM has been developed to simulate flows in microchannels based on kinetics equations and statistical physics. In LBM, the motion of the fluid is modeled by a lattice-Boltzmann equation for the distribution function of the fluid molecules. The discrete velocity Boltzmann equation corresponding to the Navier-Stokes equations can be written as... 

LBM was originally proposed by McNamara and Zanetti [3] to circumvent the limitations of statistical noise that plagued lattice gas automata (EGA). LBM is a simplified kinetic (mesoscopic) and discretized approximation of the continuous Boltzmarui equation. LBM is mesoscopic in nature because the particles are not directly related to the number of molecules like in DSMC or MD but representative of a collection of molecules. Hence, the computational cost is less demanding compared with DSMC and MD. Typical LBM consists of the lattice Boltzmann equation (LBE), lattice stmcture, transformation of lattice units to physical units, and boundary conditions. 

Lattice Boltzmann equation can be obtain through two ways, first is through of "cellular automaton" and second starting from Boltzmann equation, it was review previously, for carries out derivation of Boltzmann s lattice equation is necessary the space time discretization. Immediately presents brief description of second way, it shows by p>ace series. 

The Lattice Boltzmann Method (LBM), its simple form consist of discreet net (lattice), each place (node) is represented by unique distribution equation, which is defined by particle s velocity and is limited a discrete group of allowed velocities. During each discrete time step of the simulation, particles move, or hop, to the nearest lattice site along their direction of motion, where they "collide" with other particles that arrive at the same site. The outcome of the collision is determined by solving the kinetic (Boltzmann) equation for the new particle-distribution function at that site and the particle distribution function is updated (Chen Doolen, 1998 Wilke, 2003). Specifically, particle distribution function in each site f[(x,t), it is defined like a probability of find a particle with direction velocity. Each value of the index i specifies one of the allowed directions of motion (Chen et al., 1994 ThAurey, 2003). 

Many of the mixing simulations described in the previous section deal with the modeling of mass transfer between miscible fluids [33, 70-77]. These are the simulations which require a solution of the convection-difliision equation for the concentration fields. For the most part, the transport of a dilute species with a typical diSusion coeflEcient 10 m s between two miscible fluids with equal physical properties is simulated. It has already been mentioned that due to the discretization of the convection-diffusion equation and the typically small diffusion coefficients for liquids, these simulations are prone to numerical diffiision, which may result in an over-prediction of mass transfer efficiency. Using a lattice Boltzmann method, however, Sullivan et al. [77] successfully simulated not only the diffusion of a passive tracer but also that of an active tracer, whereby two miscible fluids of different viscosities are mixed. In particular, they used a coupled hydrodynamic/mass transfer model, which enabled the effects of the tracer concentration on the local viscosity to be taken into account. 

Let a generic site density p, represent either the (scalar) number density n, or the components of the (vector) momentum density /icn,v,. Similarly to lattice-Boltzmann (LB) methods, we evolve the system in time by alternating discrete eonveetive and dissipative timesteps. The physieal picture is that moleeular eages are eonveeted aeeording to the mean velocity field at a site, between diffusive hops of molecules from an occupied site (eage) to adjaeent empty sites (cages). 

Particle-based simulation techniques include atomistic MD and coarse-grained molecular dynamics (CG-MD). Accelerated dynamics methods, such as hyperdynamics and replica exchange molecular dynamics (REMD), are very promising for circumventing the timescale problem characteristic of atomistic simulations. Structure and dynamics at the mesoscale level can be described within the framework of coarse-grained particle-based models using such methods as stochastic dynamics (SD), dissipative particle dynamics (DPD), smoothed-particle hydrodynamics (SPH), lattice molecular dynamics (LMD), lattice Boltzmann method (IBM), multiparticle collision dynamics (MPCD), and event-driven molecular dynamics (EDMD), also referred to as collision-driven molecular dynamics or discrete molecular dynamics (DMD). 

The dynamics is obtained hy numerical solving a set of the coupled Boltzmann-BGK transport equations (d. eqn [35]) on a spatial lattice in discrete time steps with a discrete set of microscopic vdodties. At each time step, the prohahility density evolved hy each LB equation is adverted to nearest neighhoting lattice sites and modified by molecular collisions, which are local and conserve mass and momentum. As a result, a LB fluid is shown to obey the Navier-Stokes equation (in the limit of a small lattice spacing and small time step). For dilute polymer solutions, the method typically involves phenomenological coupling between the polymer chain and the flowing fluid in the form of a linear friction term based on an effective viscosity. 

Commercial codes, e.g. PowerFLOW, which use lattice-based approaches are available, and this particular code was used in the present work. Based on discrete forms of the kinetic theory equations, this code employs an approach that is an extension of lattice gas and lattice Boltzmann methods in which particles exist at discrete locations in space, and are allowed to move in given directions at particular speeds over discrete time intervals. The particles reside on a cubic lattice composed of voxels, and move from one voxel to another at each time step. Solid surfaces are accommodated through the use of surface elements, and arbitrary surface shapes can be represented. Particle advection, and particle-particle and particle-surface interactions, are all considered at a microscopic level to simulate fluid behaviour in a way which ensures conservation of mass, momentum and energy, and which recovers solutions of the continuum flow... 

Direct numerical simulations (DNS) At the most detailed level of description, the gas flow field is modeled at scales smaller than the size of the solid particles. The interaction of the gas phase with the solid phase is incorporated by imposing no-slip boundary conditions at the surface of the solid particles. This model thus allows one to measure the effective momentum exchange between the two phases, which is a key input in aU the higher scale models. Many different types of DNS models exist, such as the lattice Boltzmann model (Ladd, 1994 Ladd and Verberg, 2001) or immersed boundary techniques (Peskin (2002), UMmann (2005)). The goal of these simulations is to construct drag laws for dense gas—solid systems, which are used in the discrete particle type models. 

In this study, the Boltzmann equation is solved with the help of a single relaxation time collision operator approximated by the Bhatnagar-Gross-Krook (BGK) approach [1], Here, the relaxation of the distribution function to an equilibrium distribution is supposed to occur at a constant relaxation parameter r. The substitution of the continuous velocities in the Boltzmann equation by discrete ones leads to the discrete Boltzmann equation, where fai = fm(x, t). The number of available discrete velocity directions ai that connect the lattice nodes with each other depends on the applied model. In this work, the D3Q19 model is used which applies for a three-dimensional grid and provides 19 distinct propagation directions. Discretising time and space with At and Ax = At yields the Lattice-Boltzmann equation ... 

Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete eqnalions associated with a finite set of discrete points within the domain and specified by the user, directly or through an antomated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD). 

Hydrodynamic flow also can be incorporated via a discretized version of the Boltzmann equation. This approach is known as the lattice Boltzmann (LB) method. The idea is to solve the linearized Boltzmann equation on an underlying lattice to propagate molecular populations ( fictitious solvent particles), which define the density and velocity on each lattice site. Similar to SRD, the monomers are treated as hydrodynamic point sources in the continuous space. The flow velocity at the monomer positions follows from a linear interpolation between neighboring lattice sites. 

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