In this chapter and the following three, we discuss numerical methods that have been used for direct numerical simulations of multiphase flow. Although direct numerical simulations, or DNS, mean slightly different things to different people, we shall use the term to refer to computations of complex unsteady flows where all continuum length and time scales are fully resolved. Thus, there are no modeling issues beyond the continuum hypothesis. The flow within each phase and the interactions between different phases at the interface between them are found by solving the governing conservation equations, using grids that are finer and time steps that are shorter than any physical length and time scale in the problem.

The detailed flow field produced by direct numerical simulations allows us to explore the mechanisms governing multiphase flows and to extract information not available in any other way. For a single bubble, drop, or particle, we can obtain integrated quantities such as lift and drag and explore how they are affected by free stream turbulence, the presence of walls, and the unsteadiness of the flow. In these situations it is possible to take advantage of the relatively simple geometry to obtain extremely accurate solutions over a wide range of operating conditions. The interactions of a few bubbles, drops, or particles is a more challenging computation, but can be carried out using relatively modest computational resources. Such simulations yield information about, for example, how bubbles collide or whether a pair of buoyant particles, rising freely through a quiescent liquid, orient themselves in a preferred way. Computations of one particle can be used to obtain information pertinent to modeling of dilute multiphase flows, and studies of a few particles allow us to assess the importance of rare collisions.

The previous chapter, with its direct simulation of the fluid flow and a modeling approach to the particle phase, may be seen as a transition between the methods for a fully resolved simulation described in the first part of this book and those for a coarse-grained description based on the averaging approach described in Chapter 8. We now turn to the latter, which in practice are the only methods able to deal with the complex flows encountered in most situations of practical interest such as fluidized beds, pipelines, energy generation, sediment transport, and others. This chapter and the next one are devoted to numerical methods for so-called two-fluid models in which the phases are treated as interpenetrating continua describing, e.g. a liquid and a gas, or a fluid and a suspended solid phase. These models can be extended to deal with more than two continua and, then, the denomination multifluid models might be more appropriate. For example, the commercial code OLGA (Bendiksen et al., 1991), widely used in the oil industry, recognizes three phases, all treated as interpenetrating continua: a continuous liquid, a gas, and a disperse liquid phase present as drops suspended in the gas phase. The more recent PeTra (Petroleum Transport, Larsen et al., 1997) also describes three phases: gas, oil, and water. Recent approaches to the description of complex boiling flows recognize four interpenetrating phases: a liquid phase present both as a continuum and as a dispersion of droplets, and a gas/vapor phase also present as a continuum and a dispersion of bubbles.

Brief history of the lattice Boltzmann method

In recent years, the lattice Boltzmann method (LBM) (Chen and Doolen, 1998; Succi, 2001) has become a popular numerical scheme for simulating fluid flows and modeling physics in fluids. The lattice Boltzmann method is based on a simplified mesoscopic equation, i.e. the discrete Boltzmann equation. By starting from mesoscopic modeling, instead of doing numerical discretizations of macroscopic continuum equations, the LBM can easily incorporate underlying physics into numerical solutions. By developing a simplified version of the kinetic equation, one avoids solving complicated kinetic equations such as the full Boltzmann equation. The kinetic feature of the LBM provides additional advantages of mesoscopic modeling, such as easy implementation of boundary conditions and fully parallel algorithms. This is arguably the major reason why the LBM has been quite successful in simulating multiphase flows. Furthermore, because of the availability of very fast and massively parallel computers, there is a current trend to use codes that can exploit the intrinsic features of parallelism. The LBM fulfills these requirements in a straightforward manner.

The lattice Boltzmann method was first proposed by McNamara and Zanetti in 1988 in an effort to reduce the statistical noise in lattice gas automaton (LGA) simulations (Rothman and Zaleski, 2004). The difference is that LBM uses real numbers to count particle population, while LGA only allows integers. The result has been phenomenal: the lattice Boltzmann model substantially reduced the statistic noise observed in LGA simulations.

This is a graduate-level textbook intended to serve as an introduction to computational approaches which have proven useful for problems arising in the broad area of multiphase flow. Each chapter contains references to the current literature and to recent developments on each specific topic, but the primary purpose of this work is to provide a solid basis on which to build both applications and research. For this reason, while the reader is expected to have had some exposure to graduate-level fluid mechanics and numerical methods, no extensive knowledge of these subjects is assumed. The treatment of each topic starts at a relatively elementary level and is developed so as to enable the reader to understand the current literature.

A large number of topics fall under the generic label of “computational multiphase flow,” ranging from fully resolved simulations based on first principles to approaches employing some sort of coarse-graining and averaged equations. The book is ideally divided into two parts reflecting this distinction. The first part (Chapters 2–5) deals with methods for the solution of the Navier-Stokes equations by finite difference and finite element methods, while the second part (Chapters 9–11) deals with various reduced descriptions, from point-particle models to two-fluid formulations and averaged equations. The two parts are separated by three more specialized chapters on the lattice Boltzmann method (Chapter 6), the boundary integral method for Stokes flow (Chapter 7), and on averaging and the formulation of averaged equation (Chapter 8).

This is a multi-author volume, but we have made an effort to unify the notation and to include cross-referencing among the different chapters.

In the first chapters of this book we have seen methods suitable for a first-principles simulation of the interaction between a fluid and solid objects immersed in it. The associated computational burden is considerable and it is evident that those methods cannot handle large numbers of particles. In this chapter we develop an alternative approach which, while approximate, permits the simulation of thousands, or even millions, of particles immersed in a flow. The key feature which renders this possible is that the exchanges of momentum (and also possibly mass and energy) between the particle and the surrounding fluid are modeled, rather than directly resolved. This implies an approximate representation that is based on incorporating assumptions into the development of the mathematical model.

One of the most common approaches used today to model many particle-laden flows is based on the “point-particle approximation,” i.e. the treatment of individual particles as mathematical point sources of mass, momentum, and energy. This approximation requires an examination of the assumptions and limitations inherent to this approach, aspects that are given consideration in this chapter. Point-particle methods have relatively wide application and have proven a useful tool for modeling many complex systems, especially those comprised of a very large ensemble of particles. Details of the numerical aspects inherent to point-particle treatments are highlighted.

We start by putting point-particle methods into the context established earlier in this text and, in particular, in the previous chapter.

Boundary integral methods are powerful numerical techniques for solving multiphase hydrodynamic and aerodynamic problems in conditions where the Stokes or potential-flow approximations are applicable. Stokes flows correspond to the low Reynolds number limit, and potential flows to the high Reynolds number regime where fluid vorticity can be neglected. For both Stokes and potential flows, the velocity field in the system satisfies linear governing equations. The total flow can thus be represented as a superposition of flows produced by appropriate point sources and dipoles at the fluid interfaces.

In the boundary integral approach the flow equations are solved directly for the velocity field at the fluid interfaces, rather than in the bulk fluid. Thus, these methods are well suited for describing multiphase systems. Examples of systems for which boundary integral algorithms are especially useful include suspensions of rigid particles or deformable drops under Stokes-flow conditions. Applications of boundary integral methods in fluid dynamics, however, cover a broader range. At one end of this range are investigations of the hydrodynamic mobility of macromolecules; at the other end are calculations of the flow field around an airplane wing in a potential flow approximation. Here we will not address the potential flow case, limiting ourselves to Stokes flow.

Introduction

In the present chapter we discuss boundary integral methods for multiphase flows in the Stokes-flow regime. We review the governing differential equations, derive their integral form, and show how to use the resulting boundary integral equations to determine the motion of particles and drops. Specific issues that are relevant for the numerical implementation of these equations are also described.

The previous chapters have been devoted to methods capable of delivering “numerically exact” solutions of the Navier–Stokes equations as applied to various multiphase flow problems. In spite of their efficiency, these methods still require a substantial amount of computation even for relatively simple cases. It is therefore evident that the simulation of more complex flows approaching those encountered in most natural situations or technological contexts (sediment transport, fluidized beds, electric power generation, and many others) cannot be pursued by those means but must be based on a different approach. Furthermore, even if we did have detailed knowledge, e.g., of the motion of all the particles and of the interstitial fluid, most often, for practical purposes, we would be interested in quantities obtained by applying some sort of averaging to this immense amount of information. This observation suggests that it might be advantageous to attempt to formulate equations governing the time evolution of these averages directly. In this approach, rather than aiming at a detailed solution of the Navier–Stokes equations, we would be satisfied with a reduced description based on simplified mathematical models. While one may try to base such models on intuition, a more reliable way is perhaps to start from the exact equations and carry out a process of averaging which would filter out the inessential details retaining the basic physical processes which determine the behavior of the system.

Introduction

The issue of averaging in multiphase flow is a long-standing one with a history which stretches nearly as far back as for single-phase turbulence.

This book deals with multiphase flows, i.e. systems in which different fluid phases, or fluid and solid phases, are simultaneously present. The fluids may be different phases of the same substance, such as a liquid and its vapor, or different substances, such as a liquid and a permanent gas, or two liquids. In fluid-solid systems, the fluid may be a gas or a liquid, or gases, liquids, and solids may all coexist in the flow domain.

Without further specification, nearly all of fluid mechanics would be included in the previous paragraph. For example, a fluid flowing in a duct would be an instance of a fluid-solid system. The age-old problem of the fluid-dynamic force on a body (e.g. a leaf in the wind) would be another such instance, while the action of wind on ocean waves would be a situation involving a gas and a liquid.

In the sense in which the term is normally understood, however, multiphase flow denotes a subset of this very large class of problems. A precise definition is difficult to formulate as, often, whether a certain situation should be considered as a multiphase flow problem depends more on the point of view – or even the motivation – of the investigator than on its intrinsic nature. For example, wind waves would not fall under the purview of multiphase flow, even though some of the physical processes responsible for their behavior may be quite similar to those affecting gas–liquid stratified flows, e.g. in a pipe – a prime example of a multiphase system. The wall of a duct or a tree leaf may be considered as boundaries of the flow domain of interest, which would not qualify these as multiphase flow problems.