Multiphase Lattice Boltzmann Methods (eBook)

Theory and Application

Haibo Huang, Michael Sukop, Xiyun Lu (Autoren)

eBook Download: EPUB

2015
John Wiley & Sons (Verlag)
9781118971345 (ISBN)

Infos

Theory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference  on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the model most suitable for the problems they are interested in. The book is targeted at graduate students and researchers who plan to investigate multiphase flows using LBMs.

Throughout the text most of the popular multiphase LBMs are analyzed both theoretically and through numerical simulation. The authors present many of the mathematical derivations of the models in greater detail than is currently found in the existing literature. The approach to understanding and classifying the various models is principally based on simulation compared against analytical and observational results and discovery of undesirable terms in the derived macroscopic equations and sometimes their correction. A repository of FORTRAN codes for multiphase LBM models is also provided.

Haibo Huang is an Associate Professor in the University of Science and Technology of China. He was a Courtesy Associate Professor during his stays at Florida International University.
 
Michael C. Sukop is Professor of Hydrogeology at Florida International University in Miami and author of 'Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers'. His research emphasis is on flow and transport in porous media.
 
Xiyun Lu is a Professor of Fluid Mechanics in the University of Science and Technology of China. His research interests mainly include computational fluid dynamics, turbulence simulation and biomechanics.

Textauszug

Chapter 1
Introduction

Multiphase fluid phenomena and flows occur when two or more fluids that do not readily mix (such as air and water) share an interface. Multiphase fluid interactions are nearly ubiquitous in natural and industrial processes. Multiphase phenomena and flows can involve single component multiphase fluids, e.g., water and its own vapor, and multi-component multiphase fluids, e.g., oil/water. Some practical examples of multiphase fluid problems are the recovery and enhanced recovery of petroleum resources from reservoirs, non-aqueous phase liquid contamination of groundwater, soil water behavior, surface wetting phenomena, fuel cell operation, and the movement and evolution of clouds.

Computational fluid dynamics (CFD) has become very important in fluid flow studies. The Lattice Boltzmann method (LBM) has developed very quickly in the last two decades and has become a novel and powerful CFD tool – particularly for multiphase flows. The LBM has some major advantages compared to traditional CFD methods. First, it originates from Boltzmann's kinetic molecular dynamics – a more foundational level than normal continuum approaches. The LBM is able to recover the traditional macroscopic scale continuity and Navier–Stokes (N–S) equations, which are discretized and solved numerically in the common CFD methods. In the LBM, the more fundamental Boltzmann equation is directly discretized. Alternatively, the LBM can be viewed from its discrete-particle, more molecular-dynamics-like lattice gas origins. Second, in the LBM the pressure is usually related to the density through an ideal gas equation of state (for single-phase flow) or through a non-ideal van der Waals-like equation of state for some types of complex multiphase fluids. The pressure fields can be obtained directly once the density field is known. Hence, the Poisson equation – which can be computationally expensive – does not have to be solved in the LBM. The third advantage of the LBM is that the method is easy to parallelize due to the locality of much of the computation. Finally, no-slip boundary condition can be easily handled by simple bounce-back scheme.

The LBM has had great success in studies of single-phase flows, with commercial software known as POWERFLOW (Exa Corporation, https://www.exa.com/), based on the LBM, appearing about ten years ago. In contrast, multiphase LBMs are still undergoing development and there are many multiphase Lattice Boltzmann models available.

1.1 History of the Lattice Boltzmann method

LBMs trace their roots to cellular automata, which were originally conceived by Stanislaw Ulam and John von Neumann in the 1940s. Cellular automata consist of a discretization of space on which individual cells exist in a particular state (say 0 or 1), and update their state at each time step according to a rule that takes as input the states of some set of the cell's neighbors. Sukop and Thorne (2006) provide an introduction to cellular automata. Wolfram (1983, 2002) studied simple cellular automata systematically and inspired some of the earliest application to fluids, leading to the first paper to propose a lattice gas cellular automaton (LGCA) for the N–S equations (Frisch et al. 1986). The use of a triangular grid restored some of the symmetry required to properly simulate fluids. Rothman and Zaleski (1997), Wolf-Gladrow (2000), Succi (2001), and Sukop and Thorne (2006) all provide instructive information on this model and the extensions that appeared. All of the LGCA models suffer from inherent defects, however, in particular the lack of Galilean invariance for fast flows and statistical noise (Qian et al. 1992, Wolf-Gladrow 2000). These are explicit particle-based Boolean models that include the random fluctuations that one would expect at a molecular level of gas simulation and hence required extensive averaging to recover the smooth behavior expected at macroscopic scales.

A second major step towards the modern LBM was taken by McNamara and Zanetti (1988), who dispensed with the individual particles of the LGCAs and replaced them with an averaged but still directionally discrete distribution function. This completely eliminated the statistical noise of the LGCA. A major simplification was introduced by Qian et al. (1992): the collision matrix of Higuera et al. (1989) is replaced by a single relaxation time, leading to the Bhatnagar, Gross, and Krook (BGK) model. After that, the LBM developed very quickly. Sukop and Thorne (2006) showed that there were fewer than 20 papers on the topic in 1992; more than 600 were published in 2013.

Later Lallemand and Luo (2000) and Luo (1998) showed that the LBM can be derived from the continuous Boltzmann equation (Boltzmann 1964/1995). Hence, it can be considered as a special discretized form of the Boltzmann equation (Nourgaliev et al. 2003). From the Chapman–Enskog expansion (Wolf-Gladrow 2000), the governing continuity and N–S equations can be recovered from the LBM. Without solving Poisson's equation, the pressure field can be obtained directly from the density distributions.

Today, the use of LBM spans a broad variety of disciplines. For example, an overview of the LBM for material science and engineering can be found in Raabe (2004). Application of the LBM to biophysics can be found in Boyd et al. (2005) and Sun et al. (2003).

1.2 The Lattice Boltzmann method

The LBM can be derived from the BGK approximation of the Boltzmann equation (He and Luo 1997),

1.1

where is the single-particle distribution function in the phase space , and is the Maxwell–Boltzmann distribution function. x is the position vector, is the microscopic velocity, F(x,t) is a body force, and is the relaxation time, which determines the kinematic viscosity.

In the lattice BGK method, a discrete distribution function is introduced to represent the fluid. This distribution function satisfies the following Lattice Boltzmann equation (He and Luo 1997):

1.2

where is the density distribution function related to the discrete velocity direction i and is a relaxation time, which is related to the kinematic viscosity by , where is the sound speed. is the source term added into the standard Lattice Boltzmann equation. The equilibrium distribution function can be calculated as (Luo 1998)

1.3

In Eqs (1.2)) and (1.3)) the are the discrete velocities, as defined below, and s are weights, as given in Table 1.1. is the macroscopic density and u is the macroscopic velocity vector. Discrete velocity models are usually specified as DnQm, where n is the space dimension and m is the number of velocities. The popular 2D and 3D discrete velocity models are D2Q7, D2Q9, D3Q15, and D3Q19, which are shown in Figure 1.1.

Table 1.1 Overview of the weighting coefficients and sound speeds.

Model

D2Q7

(), (),

D2Q9

(), (), ()

D3Q15

(), (), ()

D3Q19

(), (), ()

Figure 1.1 Discrete velocity models (a) D2Q7, (b) D2Q9, (c) D3Q15, and (d) D3Q19.

For the D2Q7 model (Frisch et al. 1986), the discrete velocities are

For the D2Q9 model, the discrete velocities are given by (Qian et al. 1992)

For the D3Q15 model (Wolf-Gladrow 2000), the velocities are

For the D3Q19 model (Wolf-Gladrow 2000), they are

In the above equations, c is the lattice speed and is defined as . Here, we define 1 lattice unit () as 1 lu, 1 time step () as 1 ts, and 1 mass unit as 1 mu. There are other velocity models available, for example the D3Q27 model (He and Luo 1997), but we do not use them in simulations in this book.

In Eq. (1.3) s are weighting coefficients that can be derived theoretically (He and Luo 1997). can be derived from

1.4

where when , otherwise and we use the Einstein summation convention as detailed in the appendix to this chapter. Hence, or . As a detailed example, the computation of for the D2Q9 model is given in the following (calculation of each term from to is shown):

1.5

while the contribution from () is

1.6

In Eq. (1.3) is the density of the fluid, which can be obtained from:

1.7

This is simply the sum of the , revealing them as portions of the overall density associated with one of the discrete velocity directions. For , the macroscopic fluid velocity is given by

1.8

or in terms of the vector components of u as

1.9

which means the discrete velocities weighted by the directional densities. Application examples for viscous single-phase flow can be found in Yu et al. (2003), Dünweg and Ladd (2009), Aidun and Clausen (2010), and many others. A discussion on the H theorem in the context of the LBM can be found in Succi et al. (2002).

1.3 Multiphase LBM

Numerous macroscopic numerical methods have been developed for solving the two-phase N–S equations (Scardovelli and Zaleski 1999), such as the...

Produkt-Details

• Erscheint lt. Verlag: 11.6.2015
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Angewandte u. ökonomische Geologie • Angewandte u. ökonomische Geologie • Bauingenieur- u. Bauwesen • chemical engineering • Chemical Engineering Fundamentals • Chemische Verfahrenstechnik • Civil Engineering & Construction • earth sciences • Economic & Applied Geology • Environmental Engineering • geology, geophysics, geochemistry, applied geology, economic geology, petroleum science, mechanical geology, geological engineering, earth science, modeling, statistics, probability, multiphase LBMs, lattic boltzmann methods, FORTRAN codes • Geowissenschaften • Grundlagen der Chemischen Verfahrenstechnik • Umwelttechnik
• ISBN-13: 9781118971345 / 9781118971345