Lattice Boltzmann Methods (LBM) emerged over twenty years ago as a new way of analyzing physical systems, mostly of the fluid flow type. While historically LBM evolved from lattice gas automata models, it is now better interpreted as a systematic approximation to the Boltzmann kinetic theory. Unlike a real microscopic many-body system, LBM represents a fluid system as a simple dynamical model residing in some special discrete phase space. The governing equation resulting from such formulation is called lattice Boltzmann equation. Although an LBM model is a drastic simplification to the corresponding realistic fluid system at the micro-dynamical level, correct physical properties at the macroscopic level can be shown to be recovered with proper formulation. This fact allows using LBM as an alternative computational fluid dynamics (CFD) approach for studying realistic physical phenomena. The fundamental difference between LBM and conventional CFD is that the former is based on a microscopic (or perhaps “mesoscopic”) level description via Boltzmann - like kinetic equation, instead of solving hydrodynamic equations such as the Navier-Stokes equation. This difference has enabled a number of key advantages of LBM over the conventional approach. In addition to the familiar benefits such as being an attractive computational method for efficient and robust fluid simulation, LBM, due to its underlying kinetic theory basis also opens a way to simulate a wider variety of hydrodynamic phenomena for a broader range of physical regimes than the conventional computational methods. Many of such problems known to be extremely difficult or impossible to treat by the conventional techniques can be now approached.
Besides the basic scientific interest associated with LBM, this new field has a significant potential in real world applications. With the rapid progress in the information and computer technologies, computer aided engineering (CAE) has now become the leading trend in modern industrial engineering processes. The computer - based process allows a virtual platform for studying physical problems often impossible in real experiment, thus enabling deeper understanding of the underlying phenomena. Consequently, the new process greatly enhances and empowers innovation. In this context, LBM is not only offering great opportunities in the fluid dynamics area but is already making substantial impact in the CAE - based real world engineering process.
In the recent years LBM has become a very active and fast growing research field, as evident from the richness of topics covered in this eBook. This method now is not only standing on a much more solid theoretical foundation but also has a substantially expanded domain of applications. Comparing to the early years when LBM only described the simple Navier-Stokes fluids, it is now used to handle a wide variety of complex fluids and flows, as well as physical systems beyond fluid dynamics. As indicated in the title, this eBook covers a set of important extensions in LBM ranging from reacting flows to transport phenomena to fluid-solid interactions. Such problems are of central importance in many real world applications. LBM models of these physical systems open a way for deeper understanding of their essential physical properties. In addition to these extensions, the eBook also includes some of the state of the art theoretical underpinnings in LBM fundamentals and formulations. This eBook edited by Professor Matthias Ehrhardt will certainly be useful for a large audience of scholars.
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This is the third volume of a new eBook series that is devoted to very recent research trends in computational physics. Hereby, it focus on the computational perspectives of current physical challenges, publishing new numerical techniques for the solution of mathematical equations including chapters describing certain real-world applications concisely. The goal of this series is to emphasize especially approaches that are of interdisciplinary nature. The scientific topics in the fields of modeling, numerical methods and practical applications include e.g. the coupling between free and porous media flow, coupling of flow and transport models, coupling of atmospheric and ground water models, etc.
This volume contains 9 chapters devoted to mathematical analysis of different issues related to tailor-made Lattice Boltzmann Methods (LBMs), advanced numerical techniques for physicochemical flows and fluid structure interaction and practical applications to real world problems. This eBook consists of 9 invited chapters that are structured in the five parts introduction, regularization, asymptotic analysis and lifting of Lattice Boltzmann Methods, reactive flow and physicochemical transport, Lattice Boltzmann Methods for fluid-structure interaction (FSI) and finally practical applications.
In the first part we first present a troughout introduction into the general topic and review the known derivation of the Lattice Boltzmann Method from the Boltzmann equation using finite differences and approximate the collision term with a single relaxation time model (BGK) Also, a multi-relaxation-time model is shortly presented. By applying a multiscale expansion (Chapman-Enskog), the solution of the numerical method is verified as a meaningful approximation of the solution of the Navier-Stokes equations. Next, the LBM is extended to handle coupled problems, like the movement of objects in a flow, the coupling to heat transport and the coupling of electric circuits with power dissipation (as heat) and heat transport.
In the second part consisting of Chapters 2–5, on regularization, asymptotic analysis and lifting of Lattice Boltzmann Methods, the authors first describe how regularization of LBMs can be achieved by dampening systems using controlled numerical dissipation. Where effective, each of the proposed techniques corresponds to an additional injection of dissipation compared with the standard LBGK model. Using some standard 1D and 2D benchmarks including the shock tube and lid driven cavity, the authors show effectiveness and accuracy penalties of the proposed methods.
In Chapter 3 the authors present discrete-velocity models and LBMs for solving convection- radiation effects in thermal fluid flows. The discrete-velocity equations are derived from the continuous Boltzmann equation with appropriate scaling suitable for incompressible flows. The radiative heat flux in the energy equation is obtained using the discreteordinates solution of the radiative transfer equation. Numerical results for several test examples on coupled convection-radiation flows in two dimensional dimensions show that the developed models are competitive tools for convection-radiation problems.
In Chapter 4 the authors perform a detailed asymptotic analysis of different numerical schemes for the interaction between an incompressible fluid and a rigid structure within a Lattice Boltzmann (LB) framework. Hereby they focus on moving boundary LB schemes and on force computation through a momentum exchange algorithm (MEA). Again, all theoretical results are supported by several numerical experiments, considering both adhoc designed benchmarks, to validate accuracy properties of the schemes, and more general test setups.
In Chapter 5 the authors present an overview of the various lifting strategies for LBMs which is useful in coupled LBM and PDE models, where one part of the domain is described by a PDE while another part is modeled by an LBM. At the interface between these models the lifting operator provides the correct boundary conditions for the LBM domain. The authors discuss the accuracy, computational cost and convergence rate of analytical and numerical lifting procedures.
This topic is a good bridge to the third part on reactive flow and physicochemical transport, consisting of Chapters 6–7. The author of Chapter 6 explains the recent progress of the LBMs applied to diffusion-reaction processes for chemical and environmental applications, in particular to physicochemical processes that take place in environmental systems, such as aquatic systems, porous media, sediment, soils and biofilm layer on inert substrate. Furthermore, the role of Michaelis-Menten boundary condition at a consuming interface will be investigated. These results have been obtained by using a new computer program MHEDYN developed by the author to compute metal fluxes at planar consuming surfaces in multiligand, chemically heterogeneous environmental systems.
In Chapter 7 the authors present a LBM for modeling coupled fluid flow, solute transport, and chemical reaction at a fundamental scale where the flow is governed by continuum fluid equations. This numerical model accounts for multiple processes, including fluid flow, diffusion and advection of species, ion-exchange and mineral precipitation/ dissolution reactions, as well as the evolution of pore geometry due to dissolution/ precipitation. Homogeneous reactions are described either kinetically or through local equilibrium mass action relations. Heterogeneous reactions are incorporated into the LBM through boundary conditions imposed at the mineral surface.
In the fourth part on LBMs for FSI, the authors investigate the validity and efficiency of the coupling of the LBM with finite element methods (FEMs) as well as rigid body approaches to model FSI. The results using the fluid solver VirtualFluids on two- and three-dimensional benchmark configurations show that an explicit coupling scheme is able to produce accurate results which agree with reference solutions very well. Furthermore, the coupling to a rigid body dynamics engine (PhysicsEngine-pe) leads to the possiblity to compute FSI problems with a huge number of particles which is the basis for numerical simulations in geothermic drilling.
Finally, Chapter 9 entitled LBM for MILD Oxy-fuel Combustion Research: A Potential Powerful Tool Responding to the Man-made Global Warming considers a hot topic with socio-economic relevance. It deals with the numerical modeling and simulation of multicomponent combustion under MILD oxy-fuel operation by the LBM to deepen our understanding on this novel combustion technology. The main objective of this final chapter is to demonstrate the opportunities to build a new modeling framework to accelerate scientific discovery in this topic by utilizing unique features of the LBM and indicate the challenges that we should overcome since until recently the LBM could not be employed in such research field due to combustion’s inherent complexity.
We would like to thank Dr. Hudong Chen (Exa Corporation) for writing the foreword and for providing the figures for the title page (generated with Exa PowerFLOW/PowerVIZ) and Bentham Science Publishers, particularly the Manager Ms. Sana Mokarram for support and efforts.
Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Lehrstuhl für Angewandte Mathematik und Numerische Mathematik
Gaußstrasse 20, 42119 Wuppertal
List of Contributors
Bergische Universität Wuppertal