Computational Methods for Multi Phase Flow

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COMPUTATIONAL METHODS FOR MULTIPHASE FLOW

Predicting the behavior of multiphase flows is a problem of immense im-portance for both industrial and natural processes. Thanks to high-speed computers and advanced algorithms, it is starting to be possible to simulate such flows numerically. Researchers and students alike need to have a one-stop account of the area, and this book is that: it’s a comprehensive and self-contained graduate-level introduction to the computational modeling of multiphase flows. Each chapter is written by a recognized expert in the field and contains extensive references to current research. The books is orga-nized so that the chapters are fairly independent, to enable it to be used for a range of advanced courses. In the first part, a variety of different numer-ical methods for direct numernumer-ical simulations are described and illustrated with suitable examples. The second part is devoted to the numerical treat-ment of higher-level, averaged-equations models. No other book offers the simultaneous coverage of so many topics related to multiphase flow. It will be welcomed by researchers and graduate students in engineering, physics, and applied mathematics.

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COMPUTATIONAL METHODS FOR

MULTIPHASE FLOW

Edited by

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Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press

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First published in print format

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Preface

Computation has made theory more relevant

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 treat-ment of each topic starts at a relatively eletreat-mentary 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 mul-tiphase flow,” ranging from fully resolved simulations based on first prin-ciples to approaches employing some sort of coarse-graining and averaged equations. The book is ideally divided into two parts reflecting this distinc-tion. 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 descrip-tions, from point-particle models to two-fluid formulations and averaged equations. The two parts are separated by three more specialized chap-ters 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 (Chapter8).

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. Hope-fully this feature avoids the need for a sequential reading of the chapters, pos-sibly aside from some introductory material mostly presented in Chapter1. The objective of this work is to describe computational methods, rather

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than the physics of multiphase flow. With this aspect in mind, the primary criterion in the selection of specific examples has been their usefulness to illustrate the capabilities of an algorithm rather than the characteristics of particular flows.

The original idea for this book was conceived when we chaired the Study Group on Computational Physics in connection with the Workshop on Sci-entific Issues in Multiphase Flow. The workshop, chaired by Prof. T.J. Hanratty, was sponsored by the U.S. Department of Energy and held on the campus of the University of Illinois at Urbana-Champaign on May 7–9 2002; a summary of the findings has been published in the International Journal of Multiphase Flow, Vol. 29, pp. 1041–1116 (2003). As we started to col-lect material and to receive input form our colleagues, it became clearer and clearer that multiphase flow computation has become an activity with a major impact in industry and research. While efforts in this area go back at least five decades, the great improvement in hardware and software of the last few years has provided a significant impulse which, if anything, can be expected to only gain momentum in the coming years.

Most multiphase flows inherently involve a multiplicity of both temporal and spatial scales. Phenomena at the scale of single bubbles, drops, solid particles, capillary waves, and pores determine the behavior of large chem-ical reactors, energy production systems, oil extraction, and the global cli-mate itself. Our ability to see how the integration across all these scales comes about and what are its consequences is severely limited by this mind-boggling complexity. This is yet another area where computing offers a powerful tool for significant progress in our ability to understand and predict.

Basic understanding is achieved not only through the simulation of actual physical processes, but also with the aid of computational “experiments.” Multiphase flows are notorious for the difficulties in setting up fully con-trolled physical experiments. However, computationally, it is possible, for example, to include or not include gravity, account for the effects of a well-characterized surfactant, and others. It is now possible to routinely compute the behavior of relatively simple systems, such as the breakup of jets and the shape of bubbles. The next few years are likely to result in an explosion of results for such relatively simple systems where computations will help us gain a very complete picture of the relevant physics over a large range of parameters. A strong impulse to these activities will be imparted by effective computational methods for multiscale problems, which are rapidly developing.

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Preface ix description and closure models to account for the unresolved phenomena. The formulation of these closures will greatly beneﬁt from the detailed sim-ulation of the underlying microphysics. The situation is similar to single-phase turbulent ﬂows where, in the last two decades, simulations have played a major role, e.g. in developing large-eddy models.

It is in the examination of very complex, very large-scale systems, where it is necessary to follow the evolution of an enormous range of scales for a long time, that the major challenges and opportunities lie. Such simulations, in which it is possible to get access to the complete data and to control accu-rately every aspect of the system, will not only revolutionize our predictive capability, but also open up new opportunities for controlling the behavior of such systems.

It is our firm belief that today we stand at the threshold of exciting develop-ments in the understanding of multiphase flows for which computation will prove an essential element. All of us – authors and editors – sincerely hope that this book will contribute to further progress in this field.

Andrea Prosperetti Gretar Tryggvason

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The editors and the contributors to the present volume wish to acknowledge the help and support received by several individuals and organizations in connection with the preparation of this work.

• S. Balachandar’s research was supported by the ASCI Center for the Simulation of Advanced Rockets at the University of Illinois at Urbana-Champaign through the U.S. Department of Energy (subcontract number B341494).

• Jerzy Blawzdziewicz would like to acknowledge the support provided by NSF CAREER grant CTS-0348175.

• Howard H. Hu’s research was supported by NSF grant CTS-9873236 and by DARPA through a grant to the University of Pennsylvania. • M. Yousuff Hussaini would like to acknowledge NSF contract DMS

0108672, and the support and encouragement of Provost Lawrence G. Abele.

• Li-Shi Luo would like to acknowledge the support provided by NSF grant CTS-0500213.

• Sreekanth Pannala and Sankaran Sundaresan would like thank Tom O’Brien, Madhava Syamlal and the MFIX team. The contribution has been partly authored by a contractor of the U.S. Government under Contract No. DE-AC05-00OR22725. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.

• Andrea Prosperetti expresses his gratitude to Drs. Anthony J. Baratta, Cesare Frepoli, Yao-Shin Hwang, Raad Issa, John H. Mahaﬀy, Randi Moe, Christopher J. Murray, Fadel Moukalled, Sylvain Pigny,

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Acknowledgments xi Iztok Tiselj, and Vaughn E. Whisker. His work was supported by NSF grant CTS-0210044 and by DOE grant DE-FG02-99ER14966.

• Mark Sussman’s contribution was supported in part by the National Science Foundation under contract DMS 0108672

• Gretar Tryggvason would like to thank his graduate students and colla-borators who have contributed to his work on multiphase ﬂows. He would also like to acknowledge support by DOE grant DE-FG02-03ER46083, NSF grant CTS-0522581, as well as NASA projects NAG3-2535 and NNC05GA26G, during the preparation of this book.

• Duan Z. Zhang would like to acknowledge many important discussions and physical insights oﬀered by Dr. F. H. Harlow. The Joint DoD/ DoE Munitions Technology Development Program provided the ﬁnancial support for this work.

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1 Introduction: A computational approach to

multiphase ﬂow

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 in-cluded 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, multi-phase flow denotes a subset of this very large class of problems. A pre-cise 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 in-trinsic 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. How-ever, the flow in a network of ducts, or wind blowing through a tree canopy, may be – and have been – studied as multiphase flow problems.

These examples point to a frequent feature of multiphase ﬂow systems, namely the complexity arising from the mutual interaction of many subsys-tems. But – as a counterexample to the extent that it may be regarded as

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‘simple’ – one may consider a single small bubble as an instance of multiphase ﬂow, particularly if the study focuses on features that would be relevant to an assembly of such entities.

The interaction among many entities, such as bubbles, drops, or particles immersed in the fluid, is not the only source of the complexity usually exhib-ited by multiphase flow phenomena. There may be many other components as well, such as the very physics of the problem (e.g. the advancing of a solid–liquid–gas contact line, or the transition between different gas–liquid flow regimes), the simultaneous occurring of phenomena spanning widely different scales (e.g. oil recovery, where the flow at the single pore level impacts the behavior of the entire reservoir), the presence of a disturbed interface (e.g. surface waves on a falling film, or large, highly deformable drops or bubbles), turbulence, and others.

This complexity strongly limits the usefulness of purely analytical meth-ods. For example, even for the ﬂow around bodies with a simple shape such as spheres, most analytical results are limited to very small or very large Reynolds numbers. The more common and interesting situation of inter-mediate Reynolds numbers can hardly be studied by these means. When two or more bodies interact, or the ambient ﬂow is not simple, the power of analytical methods is reduced further.

In a laboratory, it may even be difficult to set up a multiphase flow ex-periment with the necessary degree of control: the breakup of a drop in a turbulent flow or a precise characterization of the bubble or drop size dis-tribution may be examples of such situations. Furthermore, many of the experimental techniques developed for single-phase flow encounter severe difficulties in their extension to multiphase systems. For example, even at volume fractions of a few percent, a bubbly flow may be nearly opaque to op-tical radiation so that visualization becomes problematic. The clustering of suspended particles in a turbulent flow depends on small-scale details which it may be very difficult to resolve. Little information about atomization can be gained by local probes, while adequate seeding for visualization may be impossible.

In this situation, numerical simulation becomes an essential tool for the investigation of multiphase flow. In a limited number of cases, computa-tion can solve actual practical problems which lend themselves to direct numerical simulation (e.g. the flow in microfluidic devices), or for which suf-ficiently reliable mathematical models exist. But, more frequently, compu-tation is the only available tool to investigate crucial physical aspects of the situation of interest, for example the role of gravity, or surface tension, which can be set to arbitrary values unattainable with physical experimentation.

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Multiphase Flow 3 Furthermore, the complexity of multiphase ﬂows often requires reduced descriptions, for example by means of averaged equations, and the formu-lation of such reduced models can greatly beneﬁt from the insight provided by computational results.

The last decade has seen the development of powerful computational ca-pabilities which have marked a turning point in multiphase ﬂow research. In the chapters that follow, we will give an overview of many of these devel-opments on which future progress will undoubtedly be built.

1.1 Some typical multiphase ﬂows

Having given up on the idea of providing a definition, we may illustrate the scope of multiphase flow phenomena by means of some typical examples. Here we encounter an embarrassment of riches. In technology, electric power generation, sprays (e.g. in internal combustion engines), pipelines, catalytic oil cracking, the aeration of water bodies, fluidized beds, and distillation columns are all legitimate examples. As a matter of fact, it is estimated that over half of anything which is produced in a modern industrial soci-ety depends to some extent on a multiphase flow process. In Nature, one may cite sandstorms, sediment transport, the “white water” produced by breaking waves, geysers, volcanic eruptions, acquifiers, clouds, and rain. The number of items in these lists can easily be made arbitrarily large, but it may be more useful to consider with a minimum of detail a few representative situations.

A typical example of a multiphase flow of major industrial interest is a fluidized bed (see Section 10.4). Conceptually, this device consists of a vertical vessel containing a bed of particles, which may range in size from tens of microns to centimeters. A fluid (a liquid or, more frequently, a gas) is pumped through the porous bottom of the vessel and through the bed. As the flow velocity is increased, initially one observes an increasing pressure drop across the bed. However, when the pressure drop reaches a value close to the weight of the bed per unit area, the particles become suspended in the fluid stream and the bed is said to be fluidized. These systems are useful as they promote an intimate contact between the particles and the fluid which facilitates, e.g., the combustion of material with a low caloric content (such as low-grade coal, or even domestic garbage), the in situ absorption of the pollutants deriving from the combustion (e.g. limestone particles absorbing SO2), the action of a catalyst (e.g. in oil cracking), and others.

In order for the bed to fulﬁll these functions, it is desirable that it remain homogeneous, which is exceedingly diﬃcult to obtain. Indeed, under most

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conditions, one observes large volumes of fluid, called bubbles, which contain a much smaller concentration of particles than the average, and which rise through the bed venting at its surface. In the regime commonly called “channeling,” these particle-free fluid structures span the entire height of the bed. It is evident that both bubbling and channeling reduce the effectiveness of the system as they cause a large fraction of the fluid to leave the bed contacting only a limited number of particles. The transition from the state of uniform fluidization to the bubbling regime is thought to be the result of an instability which is still incompletely understood after several decades of study. The resulting uncertainty hampers both design tasks, such as scale-up, and performance, by requiring operation with conservative safety margins. Several different types of fluidized beds exist. Figure 1.1 shows a diagram of a circulating fluidized bed, so called because the particles are ejected from the top of the riser and then returned to the bed. The figure illustrates the wide variety of situations encountered in this system: the dense particle flow in the standpipe, the fast and dilute flow in the riser, the balance between centrifugal and gravitational forces in the cyclones, and wall effects.

It is evident that a system of this complexity is way beyond the reach of direct numerical simulation. Indeed, the mathematical models in use rely on averaged equations which, however, still suffer from several problems as will be explained in Chapters 8 and 10. Attempts to improve these equa-tions must rely on a good understanding of the flow through assemblies of particles or, at the very least, of the flow around a particle suspended in a fluid stream, possibly spatially non-uniform and temporally varying. Fur-thermore, interactions with the walls are important. These considerations are a powerful motivation for the development of numerical methods for the detailed simulation of particle–fluid flow. Some methods suitable for this purpose are described in Chapters4 and 5 of this book.

An important natural phenomenon involving ﬂuid–particle interactions is

sediment transport in rivers, coastal areas, and others. A signiﬁcant

differ-ence with the case of fluidized beds is that, in this case, gravity tends to act orthogonally to the mean flow. This circumstance greatly affects the balance of forces on the particles, increasing the importance of lift. This component of the hydrodynamic force on bodies of a general shape is still insufficiently understood and, again, the computational methods described in Chapters2–5are an effective tool for its investigation.

A bubble column is the gas–liquid analog of a ﬂuidized bed. The bubbles are introduced at the bottom of a liquid-ﬁlled column with the purpose of increasing the interfacial area available for a gas–liquid chemical reaction,

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Multiphase Flow 5

fluidizing

gas

riser

aeration gas

standpipe

Fig. 1.1. This figure shows schematically one of several different configurations of a circulating fluidized bed loop used in engineering practice. The particles flow downward through the aerated “standpipe,” and enter the bottom of a fast fluidized bed “riser.” The particles are centrifugally separated from the gas in a train of “cyclones.” In this diagram, the particles separated in the primary cyclone are returned to the standpipe while the fate of the particles removed from the secondary cyclone is not shown.

of aerating the liquid, or even to lift the liquid upward in lieu of a pump. Spatial inhomogeneities arise in systems of this type as well, and their effect can be magnified by the occurrence of coalescence which may produce very large gas bubbles occupying nearly the entire cross-section of the column and separated by so-called liquid “slugs.” The transition from a bubbly to a slug-flow regime is a typical phenomenon of gas–liquid slug-flows, of great practical importance but still poorly understood. Here, in addition to understanding how the bubbles arrange themselves in space, it is necessary to model the

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forces which cause coalescence and the coalescence process itself. These are evidently major challenges in free-surface ﬂows: Chapters 10 and 11

describe some computational methods capable of shedding light on such phenomena.

Another system in which coalescence plays a major role is in clouds and rain formation. Small water droplets fall very slowly and are easy prey to the convective motions of the atmosphere. For rain to fall, the drops need to grow to a sufficient size. Condensation is impeded by the slowness of vapor diffusion through the air to reach the drop surface. The only possible expla-nation of the observed short time scale for rain formation is the occurrence of coalescence. Simple random collisions caused by turbulence are very unlikely in dilute conditions. Rather, the process must rely on a subtler influence of turbulence which can be studied with the aid of an approximation in which the finite size of the droplets is (partially) disregarded. This approach to the study of turbulence–particle interaction is a powerful one described in Chapter9. This is another example in which a critical ingredient to improve modeling is a better understanding of fluctuating hydrodynamic forces on particle assemblies which can only be gained by computational means.

Other important gas–liquid flows occur in pipelines. Here free gas may exist because it is originally present at the inlet, as in many oil pipelines, but it may also be due to the ex-solution of gases originally dissolved in the liquid as the pressure along the pipeline falls. Depending on the liquid and gas flow rates and on the slope of the pipeline, one may observe a whole variety of flow regimes such as bubbly, stratified, wavy, slug, annular, and others. Each one of them reacts differently to an imposed pressure gradient. For example, in a stratified flow, a given pressure drop would produce a much larger flow rate of the gas phase than of the liquid phase, unlike a bubbly or slug-flow regime. In slug flow, solid surfaces such as pumps and tube walls are often subjected to large fluctuating forces which may cause dangerous vibration and fatigue. It is therefore of great practical importance to be able to predict which flow regime would occur in a given situation, the operational limits to remain in the desired regime, and how the system would react to transients such as start-ups and shut-downs. The experimental effort devoted to this subject has been very considerable, but progress has proven to be frustratingly slow and elusive. The computational methods described in Chapters3,10, and11are promising tools for a better understanding of these problems.

Even remaining at the level of the momentum coupling between the phases, all of the examples described so far are challenging enough that a com-plete understanding is not yet available. When energy coupling becomes

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Multiphase Flow 7 important, such as in combustion and boiling, the difficulties increase and, with them, the prospect of progress by computational means. Boiling is the premier process by which electric power is generated world-wide, and is considered to be a vital means of heat removal in the computers of the future and human activities in space. Yet, this is another instance of those pro-cesses which have been very reluctant to yield their secrets in spite of nearly a century of experimental and theoretical work. Vital questions such as nucleation site density, bubble–bubble interaction, and critical heat flux are still for the most part unanswered. For space applications, understanding the role of gravity is an absolute prerequisite but microgravity experimen-tation is costly and fraught with difficulties. Once again, compuexperimen-tation is a most attractive proposition. In this book, space constraints prevent us from getting very far into the treatment of nonadiabatic multiphase flow. A very brief treatment of energy coupling in the context of averaged equations is presented in Chapter11.

1.2 A guided tour

The book can be divided into two parts, arranged in order of increasing complexity of the systems for which the methods described can be used. The ﬁrst part, consisting of Chapters 2–7, describes methods suitable for the detailed solution of the Navier–Stokes equations for typical situations of interest in multiphase ﬂow. Chapter 8 introduces the concept of averaged equations, and methods for their solution take up the second part of the book, Chapters9to11.

In Chapter2we introduce the idea of direct numerical simulation of mul-tiphase flows, discussing the motivation behind such simulations and what to expect from the results. We also give a brief overview of the various numerical methods used for such simulations and present in some detail elementary techniques for the solution of the Navier–Stokes equations. In Chapter3, numerical methods for fluid–fluid simulations are discussed. The methods presented all rely on the use of a fixed Cartesian grid to solve the fluid equations, but the phase boundary is tracked in different ways, using either marker functions or connected marker particles. Computation of flows over stationary solid particles is discussed in Chapter 4. We first give an overview of methods based on the use of fixed Cartesian grids, along similar lines as the methods presented in Chapter 3, and then move on to meth-ods based on body-fitted grids. While less versatile, these latter methmeth-ods are capable of producing very accurate results for relatively high Reynolds number, thus providing essentially exact solutions that form the basis for

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the modeling of forces on single particles. Simulations of more complex solid-particle flows are introduced in Chapter 5, where several versions of finite element arbitrary Lagrangian–Eulerian methods, based on unstruc-tured tetrahedron grids that adapt to the particles as they move, are used to simulate several moving solid particles. One of the important applications of simulations of this type may be in formulating closures of the averaged quantities necessary for the modeling of multiphase flows in average terms. Chapter6introduces the lattice Boltzman method for multiphase flows and in Chapter7 we discuss boundary integral methods for Stokes flows of two immiscible fluids or solid particles in a viscous fluid. While restricted to a somewhat special class of flows, boundary integral methods can reduce the computational effort significantly and yield very accurate results.

Chapters 8–11 constitute the second part of the book and deal with sit-uations for which the direct solution of the Navier–Stokes eqsit-uations would require excessive computational resources and the use of reduced descrip-tions becomes necessary. The basis for these descripdescrip-tions is some form of averaging applied to the exact microscopic laws and, accordingly, the first chapter of this group outlines the averaging procedure and illustrates how the various reduced descriptions in the literature and in the later chap-ters are rooted in it. A useful approximate treatment of disperse flows – primarily particles suspended in a gas – is based on the use of point-particle models, which are considered in Chapter 9. In these models, the fluid mo-mentum equation is augmented by point forces which represent the effect of the particles, while the particle trajectories are calculated in a Lagrangian fashion by adopting simple parameterizations of the fluid-dynamic forces. The fluid component of the model, therefore, looks very much like the ordi-nary Navier–Stokes equations, and it can be treated by the same methods developed for single-phase computational fluid dynamics. At present, this is the only well-developed reduced-description approach capable of incorporat-ing the direct numerical simulation of turbulence, and efforts are currently under way to apply to it the ideas and methods of large-eddy simulation.

The point-particle model is only valid when the particle concentration is so low that particle–particle interactions can be neglected, and the particles are smaller than the smallest flow length scale, e.g. in turbulent flow, the Kolmogorv scale. Therefore, while useful, the range of applicability of the approach is rather limited. The following two chapters deal with models based on a different philosophy of broader applicability, that of

interpene-trating continua. In the underlying conceptual picture it is supposed that

the various phases are simultaneously present in each volume element in proportions which vary with time and position. Each phase is described by

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Multiphase Flow 9 a continuity, momentum, and energy equation, all of which contain terms describing the exchange of mass, momentum, and energy among the phases. Numerically, models of this type pose special challenges due to the nearly omnipresent instabilities of the equations, the constraint that the volume fractions occupied by each phase necessarily lie between 0 and 1, and many others.

In principle, the interpenetrating-continua modeling approach is very broadly applicable to a large variety of situations. A model suitable for one application, for example stratified flow in a pipeline, differs from that applicable to a different one, for example, pneumatic transport, mostly in the way in which the interphase interaction terms are specified. It turns out that, for computational purposes, most of these specific models share a very similar structure. A case in point is the vast majority of multiphase flow models adopted in commercial codes. Two broad classes of numerical methods are available. In the first one, referred to as the segregated approach and described in Chapter 10, the various balance equations are solved se-quentially in an iterative fashion starting from an equation for the pressure. The general idea is derived from the well-known SIM P LE method of single-phase computational fluid mechanics. The other class of methods, described in Chapter11, adopts a more coupled approach to the solution of the equa-tions and is suitable for faster transients with stronger interacequa-tions among the phases.

1.3 Governing equations and boundary conditions

In view of the prominent role played by the incompressible single-phase Navier–Stokes equations throughout this book, it is useful to summarize them here. It is assumed that the reader has a background in ﬂuid mechan-ics and, therefore, no attempt at a derivation or an in-depth discussion will be made. Our main purpose is to set down the notation used in later chap-ters and to remind the reader of some fundamental dimensionless quantities which will be frequently encountered.

If ρ(x, t) and u(x, t) denote the ﬂuid density and velocity ﬁelds at position

x and time t, the equation of continuity is

∂ρ

∂t +∇∇∇ · (ρu) = 0. (1.1)

For incompressible ﬂows this equation reduces to

∇∇

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This latter equation embodies the fact that each ﬂuid particle conserves its volume as it moves in the ﬂow.

In conservation form, the momentum equation is ∂

∂t(ρu) +∇∇ · (ρuu) = ∇∇ ∇ · σσσ + ρf,∇ (1.3) in which f is an external force per unit volume acting on the ﬂuid. Very often, the force f will be the acceleration of gravity g. However, as in Chapter 9, one may think of very small suspended particles as exerting point forces which can also be described by the ﬁeld f. The stress tensor σσσ may be

decomposed into a pressure p and viscous part τττ : σσ

σ =−pI + τττ, (1.4)

in which I is the identity two-tensor. In most of the applications that follow, we will be dealing with Newtonian ﬂuids, for which the viscous part of the stress tensor is given by

τττ = 2µe, e = 1

2

∇∇∇u + ∇∇∇uT_, _(1.5)

in which µ is the coeﬃcient of (dynamic) viscosity, e the rate-of-strain tensor, and the superscript T denotes the transpose; in component form:

eij = 1 2 ∂ui ∂x_j + ∂uj ∂x_i , (1.6)

in which x = (x1, x2, x3). With (1.5), (1.3) takes the familiar form of the

Navier–Stokes momentum equation for a Newtonian, constant-properties ﬂuid: ∂u ∂t +∇∇∇ · (uu) = − 1 ρ∇∇∇p + ν∇∇∇ 2_{u + f ,} _(1.7)

in which ν = µ/ρ is the kinematic viscosity. Because of (1.2), this equation may be written in non-conservation form as

∂u

∂t + (u· ∇∇) u = −∇

1

ρ∇∇∇p + ν∇

2_{u + f ,} _(1.8)

where the notation implies that the i-th component of the second term is given by [(u· ∇∇) u]∇ _i = 3 j=1 u_j∂ui ∂x_j. (1.9)

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Multiphase Flow 11 the reduced or modiﬁed pressure, i.e. the pressure in excess of the hydrostatic contribution,

pr = p + ρU (1.10)

in terms of which (1.8) becomes

∂u

∂t + (u· ∇∇∇) u = −

1

ρ∇∇p∇

r_{+ ν}_∇2_u. _(1.11)

In particular, for the gravitational force,U = −ρg · x.

We have already noted at the beginning of this chapter that multiphase ﬂows are often characterized by the presence of interfaces. When there is a mass ﬂux ˙m across (part of) the boundary S separating two phases 1 and 2

as, for example, in the presence of phase change at a liquid–vapor interface, conservation of mass requires that

˙

m≡ ρ2(u2− w) · n = ρ1(u1− w) · n (1.12)

where n is the unit normal and w· n the normal velocity of the interface itself. An expression for this quantity is readily found if the interface is represented as

S(x, t) = 0. (1.13)

Indeed, at time t + dt, we will have S(x + wdt, t + dt) = 0 from which, after a Taylor series expansion,

∂S

∂t + w· ∇∇S = 0∇ on S = 0. (1.14)

But the unit normal, directed from the region where S < 0 to that where

S > 0, is given by n = ∇∇S∇ |∇∇S|∇ , (1.15) so that n· w = − 1 |∇∇S|∇ ∂S ∂t. (1.16)

If S = 0 denotes an impermeable surface, as in the case of a solid wall, ˙m = 0

so that n· u = n · w. In this case, by (1.12), (1.16) becomes the so-called

kinematic boundary condition: ∂S

∂t + u· ∇∇S = 0∇ on S = 0. (1.17)

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which requires the tangential velocity of the ﬂuid to match that of the boundary:

n× (u − w) = 0 on S = 0. (1.18)

(It is well known that there are situations, such as contact line motion, where this relation does not reﬂect the correct physics. Several more or less ad hoc models to treat these cases exist, but a “standard” one has yet to emerge.) Upon combining (1.14) and (1.18) one simply ﬁnds, for an impermeable surface,

u = w on S = 0. (1.19)

The tangential velocity of a fluid interface can only be unambiguosly de-fined when the interface points carry some attribute other than their geomet-ric location in space, such as the concentration of a surfactant1_{. For a purely} geometric interface, the tangential velocity is meaningless as a mapping of the interface on itself cannot have physical consequences. For example, in the case of an expanding sphere such as a bubble, a rotation around the fixed center cannot have quantitative effects. In the case of two fluids sep-arated by a purely geometric interface, the velocity field of each fluid must individually satisfy (1.17) but, rather than (1.18), the proper condition is one of continuity of the tangential velocity:

n× (u1− u2) = 0. (1.20)

It is interesting to note that, while both (1.18) and (1.20) are essentially phenomenological relations, in the case of inviscid ﬂuids with a constant surface tension (1.20) is actually a consequence of the conservation of tan-gential momentum provided ˙m= 0. When ˙m = 0, the combination of (1.17) for each ﬂuid and (1.20) renders the entire velocity continuous across the interface:

u1 = u2 on S = 0. (1.21)

When the interface separates a liquid from a gas or a vapor, the dynamical eﬀects of the latter can often be modeled in terms of pressure alone, neglect-ing viscosity. In this case, only the normal condition (1.17) applies, but not the tangential condition (1.20).

For solid boundaries with a prescribed velocity, the condition (1.19), possi-bly augmented by suitable conditions at infinity and at the initial instant, is sufficient to find a well-defined solution to the Navier–Stokes equations (1.2) 1 _{In spite of its simplicity, the interface model described here is often adequate for many}

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Multiphase Flow 13 and (1.7) or (1.8). For a free surface, a further condition is required to determine the motion of the surface itself. This condition arises from a momentum balance across the interface which stipulates that the jump in the surface tractions t = σσσ· n, combined with the momentum ﬂuxes, be

balanced by the action of surface tension:

(σσσ2− σσσ1)· n − ˙m (u2− u1) =−∇∇ · [(I − nn) γ] = − (I − nn) · ∇∇ ∇∇γ + γκn,

(1.22) where γ is the surface tension coeﬃcient and

κ =∇∇∇ · n, (1.23)

the local mean curvature of the surface. It will be recognized thatI − nn is the projector on the plane tangent to the interface. The signs in Eq. (1.22) are correct provided S is defined so that S > 0 in fluid 2 and S < 0 in fluid 1. In practice, it is more convenient to decompose this condition into its normal and tangential parts. The former is

−p2+ p1+ n· (τττ2− τττ1)· n − ˙m (u2− u1)· n = γκ (1.24)

while the tangential component is, by (1.18),

n× (τττ2− τττ1)· n = − (I − nn) · ∇∇∇γ. (1.25)

If, in place of p, the reduced pressure pr defined in (1.10) is used, the right-hand sides of (1.22) and (1.24) acquire an additional contribution necessary to cancel the difference between the potentialsU in the two fluids; for ex-ample, (1.24) becomes

−pr

2+ pr1+ n· (τττ2− τττ1)· n = γκ + ρ1U1− ρ2U2. (1.26)

Let us now consider a rigid body of mass mb, inertia tensor Jb, volume

Vband surface Sbimmersed in the ﬂuid. According to the laws of dynamics,

the motion of such a body is governed by an equation specifying the rate of change of the linear momentum

d

dt(mbv) = F

h_{+ F}e_{+ m}

bg, (1.27)

and of the angular momentum

d

dt(Jb· ΩΩ) = LΩ

h_{+ L}e_. _(1.28)

Here v is the velocity of the body center of mass, ΩΩΩ the angular veloc-ity about the center of mass, and F and L denote forces and couples,

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respectively; the superscripts “h” and “e” distinguish between forces and couples of hydrodynamic and other, external, origin. The former are given by

Fh = Sb σ σ σ· n dSb, Lh= Sb x× [σσσ · n] dSb, (1.29) where x is measured from the center of mass and the unit normal n is directed outward from the body. When the fluid stress in Fh is expressed in terms of the ordinary pressure p, the buoyancy force arises as part of the hydrodynamic force. Sometimes it may be more useful to express the fluid stress in terms of the reduced pressure pr defined in (1.10). In the case of gravity, U = −ρg · x and (1.27) takes the form

d

dt(mbv) = F

h

r + Fe+ (mb− ρVb) g. (1.30)

The position X of the center of mass and the orientation of the body (for example, the three Euler angles), ΘΘ, depend on time according to theΘ kinematic relations dX dt = v, dΘΘΘ dt = ΩΩ,Ω (1.31) respectively.

1.4 Some dimensionless groups

The use of dimensional analysis and dimensionless groups is a well-established practice in ordinary fluid dynamics and it is no less useful in multiphase flow. Each problem will have one or more characteristic length scales such as par-ticle size, duct diameter, and others. The spatial scale of each problem can therefore be represented by a characteristic length L and, possibly, di-mensionless ratios of the other scales to L. A similar role may be played by an intrinsic time scale τ due, for example, to an imposed time depen-dence of the flow or a force oscillating with a prescribed frequency, and by a velocity scale U . We introduce dimensionless variables x_∗, t_∗, and u_∗ by writing x = Lx_∗, t = τ t_∗, u = U u_∗. (1.32) Furthermore, we let ∇∇ ∇p = ∆P L ∇∇∇∗p∗, f = f f∗ (1.33)

where ∇∇∇_∗ denotes the gradient operator with respect to the dimension-less coordinate x_∗, ∆P is an appropriate pressure-diﬀerence scale, and f

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Multiphase Flow 15 a representative value of f. Then the continuity equation remains formally unaltered,

∇∇

∇∗· u∗ = 0, (1.34)

while the momentum equation (1.8) becomes 1 Sl ∂u_∗ ∂t_∗ + (u∗· ∇∇∇∗) u∗ =− ∆p ρU2∇∇∇∗p∗+ 1 Re∇ 2 ∗u∗+f L U2f∗. (1.35)

Here we have introduced the Strouhal number Sl, deﬁned by

Sl = U τ

L , (1.36)

which expresses the ratio of the intrinsic time scale τ to the convective time scale L/U . When no external or imposed time scale is present, τ = L/U and Sl = 1. The Reynolds number Re is deﬁned by

Re = ρLU

µ =

LU

ν , (1.37)

and, in addition to its usual meaning of the ratio of inertial to viscous forces, can be interpreted as the ratio of the viscous diﬀusion time L2_{/ν to}

the convective time scale L/U . When the force f is gravity, f = g =|g| and the group

F r = U

2

gL (1.38)

is known as the Froude number.

The appropriate pressure-difference scale depends on the situation. When fluid inertia is important, pressure differences scale proportionally to ρU2 so that we may take ∆P = ρU2 to find

1 Sl ∂u_∗ ∂t_∗ + (u∗· ∇∇∇∗) u∗ =−∇∇∇∗p∗+ 1 Re∇ 2 ∗u∗+f L_U2f∗. (1.39)

Frequently Sl = 1 and this equation becomes

∂u_∗ ∂t_∗ + (u∗· ∇∇∇∗) u∗ =−∇∇∇∗p∗+ 1 Re∇ 2 ∗u∗+f L U2f∗. (1.40)

On the other hand, when the ﬂow is dominated by viscosity, the proper pressure scale is ∆P = µU/L and the equation becomes

1 Sl ∂u_∗ ∂t_∗ + (u∗· ∇∇∇∗) u∗ =− 1 Re∇∇∇∗p∗+ 1 Re∇ 2 ∗u∗+ f L U2f∗. (1.41)

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then, the left-hand side of this equation is negligible; in dimensional form, what remains is

−∇∇∇p + µ∇2_{u + ρf = 0,} _(1.42) which, together with (1.34), are known as the Stokes equations.

Additional dimensionless groups arise from the boundary conditions. In the case of inertia-dominated pressure scaling, the normal stress condition (1.24) leads to

−p∗2+ p_∗1+ 1

Ren· (τττ∗2− τττ∗1)· n =

1

W eκ∗ (1.43)

where κ_∗ = Lκ and the Weber number, expressing the ratio of inertial and surface-tension-induced pressures, is deﬁned by

W e = ρLU

2

γ . (1.44)

In some cases, the characteristic velocity is governed by buoyancy, which leads to the estimate U ∼ (|ρ − ρ|/ρ)gL. A typical case is the rise of large gas bubbles (density ρ) in a free liquid or in a liquid-ﬁlled tube. In these cases, equation (1.44) becomes

Eo = Bo = |ρ − ρ

_|gL2

γ , (1.45)

a combination known as the E¨otv¨os number or Bond number. When ρ ρ, Eo is simply written as

Eo = ρgL

2

γ . (1.46)

The Morton number, deﬁned by

M o = gµ

4

ργ3, (1.47)

is often useful as, for ﬁxed g, it only depends on the liquid properties. If the Reynolds number is expressed in terms of the characteristic velocity √gL,

one immediately veriﬁes that M o = (Eo3/Re4). The Reynolds number constructed with the velocity(|ρ − ρ|/ρ)gL is the Galilei number

Ga =

gρ|ρ − ρ|L3

µ . (1.48)

In the opposite case of viscosity-dominated pressure scaling, the normal-stress condition (1.24) becomes

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Multiphase Flow 17 where the capillary number, expressing the ratio of viscous to capillary stresses, is deﬁned by

Ca = µU

γ . (1.50)

For small-scale phenomena dominated by surface tension and viscosity, the characteristic time due to the ﬂow, L/U , is of the order of ρL3_{/γ, while} the intrinsic time scale is the diﬀusion time L2_{/ν. In this case the inverse} of the Strouhal number (1.36) is known as the Ohnesorge number

Oh = √µ

ργL. (1.51)

An important dimensionless parameter governing the dynamics of a par-ticle in a flow is the Stokes number defined as the ratio of the characteristic time of the particle response to the flow to that of the flow itself:

St = τb

τ . (1.52)

This ratio can be estimated as follows. Let Ur denote the characteristic

particle–ﬂuid relative velocity and A its projected area on a plane normal to the relative velocity. When inertia is important, the order of magnitude of the hydrodynamic force|Fh| may be estimated in terms of a drag coeﬃcient

Cd deﬁned by Cd = Fh 1 2ρAUr2 , (1.53)

In problems where the scale of the relative velocity is determined by a bal-ance between the hydrodynamic and gravity forces, Ur may be estimated as

Ur∼ 1 Cd ρb ρ − 1 Lg (1.54)

where ρb is the density of the body and L = Vb/A is a characteristic body

length deﬁned in terms of the body volumeVb. The characteristic relaxation

time of the body velocity in the ﬂow, τb, may be determined by balancing

the left-hand side of the body momentum equation, ρbVbUr/τb, with the

hydrodynamic force to ﬁnd τb∼ L CdUr ρb ρ ∼ ρb ρ L Cd|ρb/ρ− 1| g . (1.55)

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Cd 1/Reb and we have Ur∼ ρb ρ − 1 L_ν2g, from which τb∼ ρb ρ L2 ν (1.56) so that St∼ ρb ρ L2 ντ. (1.57)

In particular, for a sphere of radius a, τb= 2ρba2/(9ρν) and one ﬁnds St = 2ρba

2

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2 Direct numerical simulations of ﬁnite Reynolds

number ﬂows

In this chapter and the following three, we discuss numerical methods that have been used for direct numerical simulations of multiphase flow. Al-though direct numerical simulations, or DNS, mean slightly different things to different people, we shall use the term to refer to computations of com-plex unsteady flows where all continuum length and time scales are fully resolved. Thus, there are no modeling issues beyond the continuum hy-pothesis. 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 ex-plore 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 simu-lations 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. It is, however, the possibility of conducting DNS of thousands of freely in-teracting particles that holds the greatest promise. Such simulations can yield data for not only the collective lift and drag of dense systems, but

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also about how the particles are distributed and what impact the formation of structures and clusters has on the overall behavior of the flow. Most industrial size systems, such as fluidized bed reactors or bubble columns, will remain out of reach of direct numerical simulations for the foreseeable future (and even if they were possible, DNS is unlikely to be used for routine design). However, the size of systems that can be studied is growing rapidly. It is realistic today to conduct DNS of fully three-dimensional systems re-solved by several hundred grid points in each spatial direction. If we assume that a single bubble can be adequately resolved by 25 grid points (sufficient for clean bubbles at relatively modest Reynolds numbers), that the bubbles are, on the average, one bubble diameter apart (a void fraction of slightly over 6%), and that we have a uniform grid with 10003 grid points, then we would be able to accommodate 8000 bubbles. High Reynolds numbers and solid particles or drops generally require higher resolution. Furthermore, the number of bubbles that we can simulate on a given grid obviously depends strongly on the void fraction. It is clear, however, that DNS has opened up completely new possibilities in the studies of multiphase flows which we have only started to explore.

In addition to relying on explosive growth in available computer power, progress in DNS of multiphase flows has also been made possible by the development of numerical methods. Advecting the phase boundary poses unique challenges and we will give a brief overview of such methods below, followed by a more detailed description in the next few chapters. In most cases, however, it is also necessary to solve the governing equations for the fluid flow. For body-fitted and unstructured grids, these are exactly the same as for flows without moving interfaces. For the “one-fluid” approach introduced in Chapter 3, we need to deal with density and viscosity fields that change abruptly across the interface and singular forces at the interface, but otherwise the computations are the same as for single-phase flow. Meth-ods developed for single-phase flows can therefore generally be used to solve the fluid equations. After we briefly review the different ways of computing multiphase flows, we will therefore outline in this chapter a relatively simple method to compute single-phase flows using a regular structured grid.

2.1 Overview

Many methods have been developed for direct numerical simulations of mul-tiphase flows. The oldest approach is to use one stationary, structured grid for the whole computational domain and to identify the different fluids by markers or a marker function. The equations expressing conservation of

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DNS of Finite Re Flows 21 mass, momentum and energy hold, of course, for any fluid, even when density and viscosity change abruptly and the main challenge in this ap-proach is to accurately advect the phase boundary and to compute terms concentrated at the interface, such as surface tension. In the marker-and-cell (MAC) method of Harlow and collaborators at Los Alamos (Harlow and Welch,1965) each fluid is represented by marker points distributed over the region that it occupies. Although the MAC method was used to produce some spectacular results, the distributed marker particles were not partic-ularly good at representing fluid interfaces. The Los Alamos group thus replaced the markers by a marker function that is a constant in each fluid and is advected by a scheme specifically written for a function that changes abruptly from one cell to the next. In one dimension this is particularly straightforward and one simply has to ensure that each cell fills completely before the marker function is advected into the next cell. Extended to two and three dimensions, this approach results in the volume-of-fluid (VOF) method.

The use of a single structured grid leads to relatively simple as well as efficient methods, but early difficulties experienced with the volume-of-fluid method have given rise to several other methods to advect a marker func-tion. These include the level-set method, originally introduced by Osher and Sethian (1988) but first used for multiphase flow simulations by Sussman, Smereka, and Osher (1994), the CIP method of Yabe and collaborators (Takewaki, Nishiguchi and Yabe,1985; Takewaki and Yabe,1987), and the phase field method used by Jacqmin (1997). Instead of advecting a marker function and inferring the location of the interface from its gradient, it is also possible to mark the interface using points moving with the flow and reconstruct a marker function from the interface location. Surface mark-ers have been used extensively for boundary integral methods for potential flows and Stokes flows, but their first use in Navier–Stokes computations was by Daly (1969a,b) who used them to calculate surface tension effects with the MAC method. The use of marker points was further advanced by the introduction of the immersed boundary method by Peskin (1977), who used connected marker points to follow the motion of elastic boundaries im-mersed in homogeneous fluids, and by Unverdi and Tryggvason (1992) who used connected marker points to advect the boundary between two different fluids and to compute surface tension from the geometry of the interface. Methods based on using a single structured grid, identifying the interface either by a marker function or connected marker points, are discussed in some detail in Chapter3 of this book.

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on stationary grids is their simplicity and efficiency. Since the interface is, however, represented on the grid as a rapid change in the material proper-ties, their formal accuracy is generally limited to first order. Furthermore, the difficulty that the early implementations of the “one-fluid” approach experienced, inspired several attempts to develop methods where the grid lines were aligned with the interface. These attempts fall, loosely, into three categories. Body-fitted grids, where a structured grid is deformed in such a way that the interface always coincides with a grid line; unstructured grids where the fluid is resolved by elements or control volumes that move with it in such a way that the interface coincides with the edge of an element; and what has most recently become known as sharp interface methods, where a regular structured grid is used but something special is done at the interface to allow it to stay sharp.

Body-fitted grids that conform to the phase boundaries greatly simplify the specification of the interaction of the phases across the interface. Fur-thermore, numerical methods on curvilinear grids are well developed and a high level of accuracy can be maintained both within the different phases and along their interfaces. Such grids were introduced by Hirt, Cook, and Butler (1970) for free surface flows, but their use by Ryskin and Leal (1983,1984) to find the steady state shape of axisymmetric buoyant bubbles brought their utility to the attention of the wider fluid dynamics community. Although body-fitted curvilinear grids hold enormous potential for obtaining accurate solutions for relatively simple systems such as one or two spherical parti-cles, generally their use is prohibitively complex as the number of particles increases. These methods are briefly discussed in Chapter 4. Unstructured grids, consisting usually of triangular (in two-dimensions) and tetrahedral (in three-dimensions) shaped elements offer extreme flexibility, both because it is possible to align grid lines to complex boundaries and also because it is possible to use different resolution in different parts of the computational domain. Early applications include simulations of the breakup of drops by Fritts, Fyre, and Oran (1983) but more recently unstructured moving grids have been used for simulations of multiphase particulate systems, as dis-cussed in Chapter5. Since body-fitted grids are usually limited to relatively simple geometries and methods based on unstructured grids are complex to implement and computationally expensive, several authors have sought to combine the advantages of the single-fluid approach and methods based on a more accurate representation of the interface. This approach was pio-neered by Glimm and collaborators many years ago (Glimm, 1982; Glimm and McBryan, 1985) but has recently re-emerged in methods that can be referred to collectively as “sharp interface” methods. In these methods the

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DNS of Finite Re Flows 23 fluid domain is resolved by a structured grid, but the interface treatment is improved by, for example, introducing special difference formulas that incorporate the jump across the interface (Lee and LeVeque, 2003), using “ghost points” across the interface (Fedkiw et al.,1999), or restructuring the control volumes next to the interface so that the face of the control volume is aligned with the interface (Udaykumar et al., 1997). While promising, for the most part these methods have yet to prove that they introduce fun-damentally new capabilities and that the extra complication justifies the increased accuracy. We will briefly discuss “sharp interface methods” for simulations of the motion of fluid interfaces in Chapter 3 and in slightly more detail for fluid–solid interactions in Chapter4.

2.2 Integrating the Navier–Stokes equations in time

For a large class of multiphase flow problems, including most of the systems discussed in this book, the flow speeds are relatively low and it is appropriate to treat the flow as incompressible. The unique role played by the pressure for incompressible flows, where it is not a thermodynamic variable, but takes on whatever value is needed to enforce a divergence-free velocity field, requires us to pay careful attention to the order in which the equations are solved. There is, in particular, no explicit equation for the pressure and therefore such an equation has to be found as a part of the solution process. The standard way to integrate the Navier–Stokes equations is by the so called “projection method,” introduced by Chorin (1968) and Yanenko (1971). In this approach, the velocity is first advanced without accounting for the pressure, resulting in a field that is in general not divergence-free. The pressure necessary to make the velocity field divergence-free is then found and the velocity field corrected by adding the pressure gradient.

We shall first work out the details for a simple first-order explicit time integration scheme and then see how it can be modified to generate a higher order scheme. To integrate equations (1.2) and (1.7) (or 1.8) in time, we write un+1− un ∆t + Ah(u n_{) =}₋1 ρ∇hp + νDh(u n_{) + f}n b (2.1) ∇h· un+1= 0. (2.2)

The superscript n denotes the variable at the beginning of a time step of length ∆t and n + 1 denotes the new value at the end of the step. Ah is a

numerical approximation to the advection term, Dh is a numerical

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other force acting on the ﬂuid. ∇h means a numerical approximation to the divergence or the gradient operator.

In the projection method the momentum equation is split into two parts by introducing a temporary velocity u∗ such that un+1− un= un+1− u∗+ u∗− un. The first part is a predictor step, where the temporary velocity field is found by ignoring the effect of the pressure:

u∗− un

∆t =−Ah(u

n_{) + νD}

h(un) + fbn. (2.3) In the second step – the projection step – the pressure gradient is added to yield the ﬁnal velocity at the new time step:

un+1− u∗

∆t =−

1 ρ∇hp

n+1_. _(2.4)

Adding the two equations yields exactly equation (2.1).

To ﬁnd the pressure, we use equation (2.2) to eliminate un+1 from equa-tion (2.4), resulting in Poisson’s equation:

1 ρ∇ 2 hpn+1= 1 ∆t∇h· u ∗ _(2.5)

since the density ρ is constant. Once the pressure has been found, equa-tion (2.4) is used to ﬁnd the projected velocity at time step n + 1. We note that we do not assume that∇h· un= 0. Usually, the velocity ﬁeld at time

step n is not exactly divergence-free but we strive to make the divergence of the new velocity ﬁeld, at n + 1, zero.

As the algorithm described above is completely explicit, it is subject to rel-atively stringent time-step limitations. If we use standard centered second-order approximations for the spatial derivatives, as done below, stability analysis considering only the viscous terms requires the step size ∆t to be bounded by

∆t < C_νh

2

ν (2.6)

where C_ν = 1/4 and 1/6 for two- and three-dimensional ﬂows, respectively, and h is the grid spacing. The advection scheme is unstable by itself, but it is stabilized by viscosity if the step size is limited by

∆t < 2ν

q2, (2.7)

where q2 = u· u. More sophisticated methods for the advection terms, which are stable in the absence of viscosity and can therefore also be used to

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DNS of Finite Re Flows 25 integrate the Euler equations in time, are generally subject to the Courant– Friedrichs–Lewy (CFL) condition1. For one-dimensional ﬂow,

∆t < h

(|u|). (2.8)

Many advection schemes are implemented by splitting, where the flow is sequentially advected in each coordinate direction. In these cases the one-dimensional CFL condition applies separately to each step. For fully mul-tidimensional schemes, however, the stability analysis results in further re-duction of the size of the time step. General discussions of the stability of different schemes and the resulting maximum time step can be found in standard textbooks, such as Hirsch (1988), Wesseling (2001), or Ferziger and Perić (2002). In an unsteady flow, the CFL condition on the time step is usually not very severe, since accuracy requires the time step to be sufficiently small to resolve all relevant time scales. The limitation due to the viscous diffusion, equation (2.6), can be more stringent, particularly for slow flow, and the viscous terms are frequently treated implicitly, as dis-cussed below. For problems where additional physics must be accounted for, other stability restrictions may apply. When surface tension is impor-tant, it is generally found, for example, that it is necessary to limit the time step in such a way that a capillary wave travels less than a grid space in one time step.

The simple explicit forward-in-time algorithm described above is only ﬁrst-order accurate. For most problems it is desirable to employ at least a second-order accurate time integration method. In such methods the non-linear advection terms can usually be treated explicitly, but the viscous terms are often handled implicitly, for both accuracy and stability. If we use a second-order Adams–Bashforth scheme for the advection terms and a second-order Crank–Nicholson scheme for the viscous term, the predictor step is (Wesseling,2001) u∗− un ∆t =− 3 2Ah(u n_{) +}1 2Ah(u n−1_{) +}ν 2 Dh(un) + Dh(u∗) , (2.9)

and the correction step is

un+1− u∗

∆t =−∇hφ

n+1_. _(2.10)

1 _{Lewy is often spelled Levy.} _{This is incorrect.} _{Hans Lewy (1904–1988) was a well-known}