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Lattice Boltzmann Method for Modeling Convective Heat Transfer in Porous Media
Published in Yasser Mahmoudi, Kamel Hooman, Kambiz Vafai, Convective Heat Transfer in Porous Media, 2019
Gholamreza Imani, Kamel Hooman
The continuous BTE with the Bhatnagar–Gross–Krook (BGK) collision operator (Bhatnagar et al. 1954) describes the evolution of f(r,ξ,t) as (Chapman et al. 1990): () ∂f(r,ξ,t)∂t+ξ.∇f(r,ξ,t)=−f(r,ξ,t)−feq(r,ξ,t)γ
Lattice Boltzmann Method and Its Applications in Microfluidics
Published in Sushanta K. Mitra, Suman Chakraborty, Fabrication, Implementation, and Applications, 2016
Historically, LBM originated from the lattice gas automata (LGA), which can be considered as a simplified, fictitious version of MD in which space, time, and particle velocities are all discrete. Each lattice node is connected to its neighbors by, for example, six lattice velocities in a hexagonal Frisch–Hasslacher–Pomeau (FHP) model [3, 4]. There can be either one or zero particle at a lattice node moving along a lattice direction, according to a set of collision rules. Good collision rules should conserve the particle number (mass), momentum, and energy before and after the collision. In spite of its many successful applications, LGA suffers from several native defects, including the lack of Galilean invariance, the presence of statistical noise, and the exponential complexity for three-dimensional lattices [3]. The main motivation for the transition from LGA to LBM was the desire to remove statistical noise by replacing the Boolean particle number in a lattice direction with its ensemble average, the so-called density distribution function. Accompanying with this replacement, the discrete collision rules also have to be modified as a continuous function—the collision operator. In the LBM development, an important simplification is the approximation of the collision operator with the Bhatnagar–Gross–Krook (BGK) relaxation term. This lattice BGK model makes simulations more efficient and allows flexibility of the transport coefficients. Through a Chapman–Enskog analysis, one can recover the governing continuity and Navier–Stokes equations from the LBM algorithm [2]. In addition, the pressure field is also directly available from the density distributions, and hence there is no additional Poisson equation to be solved as in the traditional CFD methods. It has been shown that the LBM scheme can also be interpreted as a special discretized form of the continuous Boltzmann equation [5–7].
Electrons in Semiconductors
Published in Hualin Zhan, Graphene-Electrolyte Interfaces, 2020
In contrast to the example shown in appendix A, where (∂f∂t)c= 0, the collision term (∂f∂t)c must be non-zero when studying the charge transport in solid materials. This is because charge transport processes are often collisional and the distribution function changes in the presence of an external perturbation, such as an electric field. A commonly used model for the collision term is the BGK (Bhatnagar-Gross-Krook) operator (∂f∂t)c=-f-f0τ, where f = f0 + f1. f0 and f1 represent the equilibrium term and the perturbation term of the distribution function, respectively. This is a coarse approximation of the collision process stating that the change of the distribution function is linear and happens instantly after a duration (relaxation time τ). As the collision process is simply described using the relaxation time, Eq. 2.3 is also called the relaxation time approximation.
Probing the Rayleigh–Benard convection phase change mechanism of low-melting-point metal via lattice Boltzmann method
Published in Numerical Heat Transfer, Part A: Applications, 2018
Double distribution function thermal lattice Boltzmann model was first proposed by Shan [34], it was widely used to simulate the conjugate thermal flow problems. In this model, density distribution function is solved to obtain the flow field and temperature distribution function is calculated to get the temperature field. In general, a single-relaxation time collision model, namely, the Bhatnagar–Gross–Krook (BGK) approximate collision model, is used in LBM, for the reason that this collision model is easy to understand and implement. However, there exists numerical instability for this model especially when the dimensionless relaxation time approaches to 0.5. Multirelaxation time algorithm can greatly improve the numerical stability, and some double multirelaxation time models have been developed for solid–liquid phase change problems in porous media [353637] and/or with internal fins [36, 38].
Mesoscopic-based finite volume solutions for waterhammer flows
Published in Journal of Hydraulic Research, 2019
Sara Mesgari Sohani, Mohamed Salah Ghidaoui
The collision operator term in Eq. (1) prevents us from obtaining an exact solution of the Boltzmann equation due to the operator's complexity. Therefore, approximate forms of this operator have been proposed over the last century (Cercignani, 1988). Among these approximate methods, the model proposed by Bhatnagar–Gross–Krook in 1954 ( the BGK model) is the most popular in fluid applications because of its simplicity and accuracy. The one-dimensional BGK Boltzmann equation without external forces, then, is: where τ is the average time interval between successive particle collisions. Moreover, the fundamental laws of classical physics dictate that the mass, momentum, and energy of the particles are collision invariant (Vincenti & Kruger, 1965). Therefore: The above relation is often called the compatibility condition. The collision term in the BGK model causes the equilibrium velocity-distribution function (g) to deviate from the non-equilibrium velocity-distribution function (f). This makes the BGK model capable of capturing both equilibrium and non-equilibrium gas flow accurately and robustly (Vincenti & Kruger, 1965; Xu, 1998). A bridge between the BGK Boltzmann equation on the one side and Navier–Stokes or Euler equations on the other side is established using the Chapman–Enskog expansion solution (Kogan, 1967). Throughout the established connection, the collision term τ needs to be a function of the local macroscopic flow variables as follows (Xu, 1998): where μ is the dynamic viscosity; P is the fluid pressure.
Numerical investigation of transient responses of triangular fins having linear and power law property variation under step changes in base temperature and base heat flux using lattice Boltzmann method
Published in Numerical Heat Transfer, Part A: Applications, 2021
Abhishek Sahu, Shubhankar Bhowmick
Here, distribution function fistream through a discretized space with velocity ci toward its neighboring lattice, where the collision occurs. Boltzmann collision operator Ω required very heavy and lengthy calculation, different collision operator are available to model the collision process, the most commonly used one is Bhatnagar–Gross–Krook (BGK) collision operator [37] due to simple linear operation. The LBM along with the BGK collision operator and its discretized form [24, 25] are given as in Eqs. (12) and (13), respectively