Lattice-Boltzmann approach

One alternative to solving the equations of change of continuum mechanics for simulating droplet collisions is the lattice-Boltzmann approach [63-67]. This technique describes the liquid dynamics on the basis of the dynamics of particle motion, which represents the liquid dynamic behavior and is governed by the lattice-Boltzmann... 

A possible solution for simulating transport processes in larger domains can be the lattice Boltzmann approach see, for example, [11, 12]. The great advantage of this method is that it allows for a massive parallelization of the implementation, which permits computation on large domains. 

Original developments in this area stem from the work of Frisch et al. (1986) who employed the technique of lattice gas hydrodynamics in which the fluid is modelled as a cellular automaton and the flow represented by the motion of particles on a lattice. More numerically efficient variants of this method, such as the lattice Boltzmann approach (McNamara and Zanetti, 1988), were subsequently developed. 

Kattige and Rowley investigated the capsule-filling properties of lactose/poloxamer dispersions in hard gelatin capsules using rheological techniques [81,82]. Recently, Zhang and coworkers [83] developed an immersed boundary lattice Boltzmann approach to simulate deformable capsules in flows. 

Langevin simulations of time-dependent Ginzburg-Landau models have also been performed to study other dynamical aspects of amphiphilic systems [223,224]. An attractive alternative approach is that of the Lattice-Boltzmann models, which take proper account of the hydrodynamics of the system. They have been used recently to study quenches from a disordered phase in a lamellar phase [225,226]. 

The emphasis in this chapter is on the fruitful application of Large Eddy Simulations for reproducing the local and transient flow conditions in which these processes are carried out and on which their performance depends. In addition, examples are given of using Direct Numerical Simulations of flow and transport phenomena in small periodic boxes with the view to find out about relevant details of the local processes. Finally, substantial attention is paid throughout this chapter to the attractiveness and success of exploiting lattice-Boltzmann techniques for the more advanced CFD approaches. 

Lattice-Boltzmann is an inherently time-dependent approach. Using LB for steady flows, however, and letting the flow develop in time from some starting condition toward a steady-state is not a very good idea, since the LB time steps need to be small (compared to, e.g., FV time steps) in order to meet the incompressibility constraint. 

A demonstration of this approach has been reported to evaluate the ability of a lattice-Boltzmann code to predict both spatially resolved flow fields and MR propagators characterizing flow through random packings of spheres (model fixed beds) for flows defined by Peclet (Pe) and Reynolds numbers in the range 182 < Pc <3 50 and 0.4 < Re <0.77 (85). Excellent agreement was found between the numerical predictions and experimental measurements. Current interest in this field addresses the validation and development of numerical codes predicting flows at Reynolds numbers more appropriate to real catalytic reactors. 

The mesoscopic modeling approach consists of a stochastic reconstruction method for the generation of the CL and GDL microstructures, and a two-phase lattice Boltzmann method for studying liquid water transport and flooding phenomena in the reconstructed microstructures. 

The second contribution spans an even larger range of length and times scales. Two benchmark examples illustrate the design approach polymer electrolyte fuel cells and hard disk drive (HDD) systems. In the current HDDs, the read/write head flies about 6.5 nm above the surface via the air bearing design. Multi-scale modeling tools include quantum mechanical (i.e., density functional theory (DFT)), atomistic (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational fluid mechanics, and system optimization) levels. 

Lattice Boltzmann (LB) is a relatively new simulation technique and it represents an alternative numerical approach in the hydrodynamics of complex fluids. The LB method can be interpreted as an unusual finite-difference solution of the continuity... 

The most expensive part of a simulation of a system with explicit solvent is the computation of the long-range interactions because this scales as Consequently, a model that represents the solvent properties implicitly will considerably reduce the number of degrees of freedom of the system and thus also the computational cost. A variety of implicit water models has been developed for molecular simulations [56-60]. Explicit solvent can be replaced by a dipole-lattice model representation [60] or a continuum Poisson-Boltzmann approach [61], or less accurately, by a generalised Bom (GB) method [62] or semi-empirical model based on solvent accessible surface area [59]. Thermodynamic properties can often be well represented by such models, but dynamic properties suffer from the implicit representation. The molecular nature of the first hydration shell is important for some systems, and consequently, mixed models have been proposed, in which the solute is immersed in an explicit solvent sphere or shell surrounded by an implicit solvent continuum. A boundary potential is added that takes into account the influence of the van der Waals and the electrostatic interactions [63-67]. 

Only a few LES simulations have been reported describing the turbulent flow in single phase stirred tanks (e.g., [20, 77, 18]). The lattice-Boltzmann method is used in the more recent publications since this scheme is considered to be an efficient Navier-Stokes solver. Nevertheless, the computational requirements of these models are still prohibitive, therefore the application of this approach is restricted to academic research. No direct simulations of these vessels have been performed yet. 

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... 

The figure shows two slices through the center of a packed-sand core in identical position, and scanned after (a) the core has been drained slowly to a final capillary pressure of 490 cm using a multi-step approach (left), and (b) the core has been drained very quickly using just one large pressure step of 490 cm. Clearly, there are major differences in overall amount of water retained in the core and the water distribution is also very different. It is expected that these experiments will lead to quantitative interpretations of flow-rate dependent phenomena and, potentially, verification of Lattice Boltzmann numerical simulations of flow and transport in unsaturated porous media. 

As an illustration of the capabilities of 3D numerical approaches, in the next section a deeper insight in the fluid flow and mass transport in packed-bed membrane reactors is given with the help of a 3D reactor model, based on the lattice Boltzmann equation method. 

In the last decade, we have found a rapid development of the method known as the lattice Boltzmann equation (LBE) method. Because of its physical soundness and outstanding amenability to parallel processing, the LBE method has been successfully applied for the simulation of a variety of flow- and mass-transport situations, including flows through porous media, turbulence, advective diffusion, multiphase and reactive flows, to name but a few. In particular, an effective numerical approach to simulate advective-diffusion transport (the moment-... 

Inamuro, T., N. Konishi, and F. Ogino. A Galilean Invariant Model of the Lattice Boltzmann Method for Multiphase Huid Flows Using Free-Energy Approach. Comput. Phys. Commun. 129 32-45 (2000). 

Noncontinuous approach can be deterministic or stochastic. In deterministic approaches, such as the molecular dynamics (MD) method and the lattice Boltzmann method (LBM), the particle or molecule s trajectory, velocity, and intermolecular collision are calculated or simulated in a deterministic manner. In the stochastic approaches, such as the direct simulation Monte Carlo (DSMC) method, randomness is introduced into the solution variables. 

However, for Knudsen numbers higher than unity, as it could be the case in nanochaimels (gas flow in nanochannels) or in low-pressure flows in microchannels, the continuum approach is no longer valid, and molecular methods such as the Direct Simulation Monte Carlo - DSMC - (Monte Carlo method) or the Lattice Boltzmann methods (Lattice Boltzmann method) should be used. 

Complementarily, several numerical approaches have also been proposed either at the molecular scale, using molecular dynamics [17], or at larger mesoscopic scales using finite element methods, lattice-Boltzmann simulations, or phase-field... 

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