computational methods for multiphase flow

Furthermore, the complexity of multiphase flows often requires reduceddescriptions, for example by means of averaged equations, and the formu-lation of such reduced models can greatly ben

Predicting the behavior of multiphase flows is a problem of immense portance for both industrial and natural processes Thanks to high-speedcomputers and advanced algorithms, it is starting to be possible to simulatesuch 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 andself-contained graduate-level introduction to the computational modeling ofmultiphase flows Each chapter is written by a recognized expert in the fieldand 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

im-a rim-ange of im-advim-anced courses In the first pim-art, im-a vim-ariety of different ical methods for direct numerical simulations are described and illustratedwith suitable examples The second part is devoted to the numerical treat-ment of higher-level, averaged-equations models No other book offers thesimultaneous coverage of so many topics related to multiphase flow It will

numer-be welcomed by researchers and graduate students in engineering, physics,and applied mathematics

MULTIPHASE FLOW

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Preface page vii

1 Introduction: A computational approach to multiphase flow 1

A Prosperetti and G Tryggvason

2 Direct numerical simulations of finite Reynolds number flows 19

G Tryggvason and S Balachandar

3 Immersed boundary methods for fluid interfaces 37

G Tryggvason, M Sussman and M.Y Hussaini

S Balachandar

H Hu

6 Lattice Boltzmann models for multiphase flows 157

S Chen, X He and L.-S Luo

A Prosperetti, S Sundaresan, S Pannala and D.Z Zhang

A Prosperetti

v

Computation has made theory more relevant

This is a graduate-level textbook intended to serve as an introduction tocomputational approaches which have proven useful for problems arising inthe broad area of multiphase flow Each chapter contains references to thecurrent literature and to recent developments on each specific topic, but theprimary purpose of this work is to provide a solid basis on which to buildboth applications and research For this reason, while the reader is expected

to have had some exposure to graduate-level fluid mechanics and numericalmethods, no extensive knowledge of these subjects is assumed The treat-ment 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 tiphase flow,” ranging from fully resolved simulations based on first prin-ciples to approaches employing some sort of coarse-graining and averagedequations The book is ideally divided into two parts reflecting this distinc-tion The first part (Chapters 2 5) deals with methods for the solution ofthe 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 averagedequations The two parts are separated by three more specialized chap-ters on the lattice Boltzmann method (Chapter 6), the boundary integralmethod for Stokes flow (Chapter 7), and on averaging and the formulation

mul-of averaged equation (Chapter8)

This is a multi-author volume, but we have made an effort to unify thenotation 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

vii

than the physics of multiphase flow With this aspect in mind, the primarycriterion in the selection of specific examples has been their usefulness toillustrate the capabilities of an algorithm rather than the characteristics ofparticular flows.

The original idea for this book was conceived when we chaired the StudyGroup 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 thecampus 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 clearerand 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 thelast few years has provided a significant impulse which, if anything, can beexpected to only gain momentum in the coming years

Most multiphase flows inherently involve a multiplicity of both temporaland spatial scales Phenomena at the scale of single bubbles, drops, solidparticles, 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 scalescomes about and what are its consequences is severely limited by this mind-boggling complexity This is yet another area where computing offers apowerful tool for significant progress in our ability to understand andpredict

Basic understanding is achieved not only through the simulation of actualphysical 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, forexample, to include or not include gravity, account for the effects of a well-characterized surfactant, and others It is now possible to routinely computethe behavior of relatively simple systems, such as the breakup of jets andthe 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 byeffective computational methods for multiscale problems, which are rapidlydeveloping

At a practical, industrial level, simulation must rely on an averaged

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 longtime, that the major challenges and opportunities lie Such simulations, inwhich 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 predictivecapability, 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 ments in the understanding of multiphase flows for which computation willprove an essential element All of us – authors and editors – sincerely hopethat this book will contribute to further progress in this field

develop-Andrea ProsperettiGretar Tryggvason

The editors and the contributors to the present volume wish to acknowledgethe help and support received by several individuals and organizations inconnection 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 Champaign through the U.S Department of Energy (subcontract numberB341494)

Urbana-• 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 contributionhas been partly authored by a contractor of the U.S Government underContract No DE-AC05-00OR22725 Accordingly, the U.S Governmentretains a non-exclusive, royalty-free license to publish or reproduce thepublished 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,

x

Iztok Tiselj, and Vaughn E Whisker His work was supported by NSFgrant 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 wouldalso like to acknowledge support by DOE grant DE-FG02-03ER46083,NSF grant CTS-0522581, as well as NASA projects NAG3-2535 andNNC05GA26G, 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 financialsupport for this work

in-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 situationshould be considered as a multiphase flow problem depends more on thepoint 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 ofmultiphase flow, even though some of the physical processes responsible fortheir behavior may be quite similar to those affecting gas–liquid stratifiedflows, 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 ofinterest, 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

1

‘simple’ – one may consider a single small bubble as an instance of multiphaseflow, 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 particlesimmersed 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 asolid–liquid–gas contact line, or the transition between different gas–liquidflow regimes), the simultaneous occurring of phenomena spanning widelydifferent scales (e.g oil recovery, where the flow at the single pore levelimpacts the behavior of the entire reservoir), the presence of a disturbedinterface (e.g surface waves on a falling film, or large, highly deformabledrops or bubbles), turbulence, and others

This complexity strongly limits the usefulness of purely analytical ods For example, even for the flow around bodies with a simple shape such

meth-as spheres, most analytical results are limited to very small or very largeReynolds numbers The more common and interesting situation of inter-mediate Reynolds numbers can hardly be studied by these means Whentwo or more bodies interact, or the ambient flow is not simple, the power ofanalytical methods is reduced further

In a laboratory, it may even be difficult to set up a multiphase flow periment with the necessary degree of control: the breakup of a drop in aturbulent flow or a precise characterization of the bubble or drop size dis-tribution may be examples of such situations Furthermore, many of theexperimental techniques developed for single-phase flow encounter severedifficulties in their extension to multiphase systems For example, even atvolume fractions of a few percent, a bubbly flow may be nearly opaque to op-tical radiation so that visualization becomes problematic The clustering ofsuspended particles in a turbulent flow depends on small-scale details which

ex-it may be very difficult to resolve Lex-ittle information about atomization can

be gained by local probes, while adequate seeding for visualization may beimpossible

In this situation, numerical simulation becomes an essential tool for theinvestigation of multiphase flow In a limited number of cases, computa-tion can solve actual practical problems which lend themselves to directnumerical 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 thesituation of interest, for example the role of gravity, or surface tension, whichcan be set to arbitrary values unattainable with physical experimentation

Furthermore, the complexity of multiphase flows often requires reduceddescriptions, for example by means of averaged equations, and the formu-lation of such reduced models can greatly benefit from the insight provided

devel-1.1 Some typical multiphase flows

Having given up on the idea of providing a definition, we may illustrate thescope of multiphase flow phenomena by means of some typical examples.Here we encounter an embarrassment of riches In technology, electric powergeneration, sprays (e.g in internal combustion engines), pipelines, catalyticoil cracking, the aeration of water bodies, fluidized beds, and distillationcolumns are all legitimate examples As a matter of fact, it is estimatedthat over half of anything which is produced in a modern industrial soci-ety depends to some extent on a multiphase flow process In Nature, onemay cite sandstorms, sediment transport, the “white water” produced bybreaking waves, geysers, volcanic eruptions, acquifiers, clouds, and rain Thenumber 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 representativesituations

A typical example of a multiphase flow of major industrial interest is

a fluidized bed (see Section 10.4) Conceptually, this device consists of avertical vessel containing a bed of particles, which may range in size fromtens 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 Asthe flow velocity is increased, initially one observes an increasing pressuredrop 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 thefluid stream and the bed is said to be fluidized These systems are useful

as they promote an intimate contact between the particles and the fluidwhich 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 particlesabsorbing 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 remainhomogeneous, which is exceedingly difficult to obtain Indeed, under most

conditions, one observes large volumes of fluid, called bubbles, which contain

a much smaller concentration of particles than the average, and which risethrough the bed venting at its surface In the regime commonly called

“channeling,” these particle-free fluid structures span the entire height of thebed 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 bedcontacting 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 asscale-up, and performance, by requiring operation with conservative safetymargins 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 figureillustrates the wide variety of situations encountered in this system: thedense particle flow in the standpipe, the fast and dilute flow in the riser, thebalance between centrifugal and gravitational forces in the cyclones, andwall effects

It is evident that a system of this complexity is way beyond the reach ofdirect numerical simulation Indeed, the mathematical models in use rely

on averaged equations which, however, still suffer from several problems aswill be explained in Chapters 8 and 10 Attempts to improve these equa-tions must rely on a good understanding of the flow through assemblies ofparticles or, at the very least, of the flow around a particle suspended in afluid stream, possibly spatially non-uniform and temporally varying Fur-thermore, interactions with the walls are important These considerationsare a powerful motivation for the development of numerical methods for thedetailed simulation of particle–fluid flow Some methods suitable for thispurpose 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 toact orthogonally to the mean flow This circumstance greatly affects thebalance of forces on the particles, increasing the importance of lift Thiscomponent of the hydrodynamic force on bodies of a general shape is stillinsufficiently 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 ofincreasing the interfacial area available for a gas–liquid chemical reaction,

fluidizing gas

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 effectcan be magnified by the occurrence of coalescence which may produce verylarge gas bubbles occupying nearly the entire cross-section of the column andseparated by so-called liquid “slugs.” The transition from a bubbly to a slug-flow regime is a typical phenomenon of gas–liquid flows, of great practicalimportance but still poorly understood Here, in addition to understandinghow the bubbles arrange themselves in space, it is necessary to model the

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

describe some computational methods capable of shedding light on suchphenomena

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 tothe convective motions of the atmosphere For rain to fall, the drops need togrow to a sufficient size Condensation is impeded by the slowness of vapordiffusion 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 ofcoalescence Simple random collisions caused by turbulence are very unlikely

in dilute conditions Rather, the process must rely on a subtler influence ofturbulence which can be studied with the aid of an approximation in whichthe finite size of the droplets is (partially) disregarded This approach tothe study of turbulence–particle interaction is a powerful one described inChapter9 This is another example in which a critical ingredient to improvemodeling is a better understanding of fluctuating hydrodynamic forces onparticle 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 inthe liquid as the pressure along the pipeline falls Depending on the liquidand gas flow rates and on the slope of the pipeline, one may observe awhole variety of flow regimes such as bubbly, stratified, wavy, slug, annular,and others Each one of them reacts differently to an imposed pressuregradient For example, in a stratified flow, a given pressure drop wouldproduce 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 aspumps and tube walls are often subjected to large fluctuating forces whichmay cause dangerous vibration and fatigue It is therefore of great practicalimportance to be able to predict which flow regime would occur in a givensituation, the operational limits to remain in the desired regime, and howthe system would react to transients such as start-ups and shut-downs Theexperimental effort devoted to this subject has been very considerable, butprogress has proven to be frustratingly slow and elusive The computationalmethods described in Chapters3,10, and11are promising tools for a betterunderstanding 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

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 isconsidered to be a vital means of heat removal in the computers of the futureand 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 asnucleation site density, bubble–bubble interaction, and critical heat flux arestill for the most part unanswered For space applications, understandingthe role of gravity is an absolute prerequisite but microgravity experimen-tation is costly and fraught with difficulties Once again, computation is amost attractive proposition In this book, space constraints prevent us fromgetting very far into the treatment of nonadiabatic multiphase flow A verybrief treatment of energy coupling in the context of averaged equations ispresented in Chapter11

1.2 A guided tour

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

In Chapter2we introduce the idea of direct numerical simulation of tiphase flows, discussing the motivation behind such simulations and what

mul-to expect from the results We also give a brief overview of the variousnumerical methods used for such simulations and present in some detailelementary techniques for the solution of the Navier–Stokes equations InChapter3, numerical methods for fluid–fluid simulations are discussed Themethods presented all rely on the use of a fixed Cartesian grid to solve thefluid equations, but the phase boundary is tracked in different ways, usingeither marker functions or connected marker particles Computation of flowsover stationary solid particles is discussed in Chapter 4 We first give anoverview of methods based on the use of fixed Cartesian grids, along similarlines as the methods presented in Chapter 3, and then move on to meth-ods based on body-fitted grids While less versatile, these latter methodsare capable of producing very accurate results for relatively high Reynoldsnumber, thus providing essentially exact solutions that form the basis for

the modeling of forces on single particles Simulations of more complexsolid-particle flows are introduced in Chapter 5, where several versions offinite 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 averagedquantities 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 twoimmiscible fluids or solid particles in a viscous fluid While restricted to asomewhat special class of flows, boundary integral methods can reduce thecomputational effort significantly and yield very accurate results

Chapters 8 11 constitute the second part of the book and deal with uations for which the direct solution of the Navier–Stokes equations wouldrequire excessive computational resources and the use of reduced descrip-tions becomes necessary The basis for these descriptions is some form ofaveraging applied to the exact microscopic laws and, accordingly, the firstchapter of this group outlines the averaging procedure and illustrates howthe 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-particlemodels, which are considered in Chapter 9 In these models, the fluid mo-mentum equation is augmented by point forces which represent the effect ofthe particles, while the particle trajectories are calculated in a Lagrangianfashion 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 methodsdeveloped for single-phase computational fluid dynamics At present, this isthe only well-developed reduced-description approach capable of incorporat-ing the direct numerical simulation of turbulence, and efforts are currentlyunder 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

sit-so low that particle–particle interactions can be neglected, and the particlesare smaller than the smallest flow length scale, e.g in turbulent flow, theKolmogorv scale Therefore, while useful, the range of applicability of theapproach 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 inproportions which vary with time and position Each phase is described by

a continuity, momentum, and energy equation, all of which contain termsdescribing the exchange of mass, momentum, and energy among the phases.Numerically, models of this type pose special challenges due to the nearlyomnipresent instabilities of the equations, the constraint that the volumefractions occupied by each phase necessarily lie between 0 and 1, and manyothers.

In principle, the interpenetrating-continua modeling approach is verybroadly applicable to a large variety of situations A model suitable forone application, for example stratified flow in a pipeline, differs from thatapplicable to a different one, for example, pneumatic transport, mostly inthe way in which the interphase interaction terms are specified It turnsout that, for computational purposes, most of these specific models share avery similar structure A case in point is the vast majority of multiphaseflow 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 quentially in an iterative fashion starting from an equation for the pressure

se-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 tions and is suitable for faster transients with stronger interactions amongthe phases

equa-1.3 Governing equations and boundary conditions

In view of the prominent role played by the incompressible single-phaseNavier–Stokes equations throughout this book, it is useful to summarizethem 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 ters and to remind the reader of some fundamental dimensionless quantitieswhich will be frequently encountered

chap-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

This latter equation embodies the fact that each fluid particle conserves itsvolume 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 σ σ σ may be

decomposed into a pressure p and viscous part ττ τ :

σ

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 thestress tensor is given by

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:

e ij = 12

the reduced or modified pressure, i.e the pressure in excess of the hydrostatic

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

We have already noted at the beginning of this chapter that multiphaseflows are often characterized by the presence of interfaces When there is amass 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 isrepresented as

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

a Taylor series expansion,

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)

At solid surfaces, for viscous flow, one usually imposes the no-slip condition,

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

(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 impermeablesurface,

The tangential velocity of a fluid interface can only be unambiguosly 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 purelygeometric interface, the tangential velocity is meaningless as a mapping ofthe interface on itself cannot have physical consequences For example, inthe case of an expanding sphere such as a bubble, a rotation around thefixed center cannot have quantitative effects In the case of two fluids sep-arated by a purely geometric interface, the velocity field of each fluid mustindividually satisfy (1.17) but, rather than (1.18), the proper condition isone of continuity of the tangential velocity:

It is interesting to note that, while both (1.18) and (1.20) are essentiallyphenomenological relations, in the case of inviscid fluids with a constantsurface 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 theinterface:

When the interface separates a liquid from a gas or a vapor, the dynamicaleffects 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 notthe tangential condition (1.20)

For solid boundaries with a prescribed velocity, the condition (1.19), bly augmented by suitable conditions at infinity and at the initial instant, issufficient to find a well-defined solution to the Navier–Stokes equations (1.2)

possi-1 In spite of its simplicity, the interface model described here is often adequate for many

applica-tions Much more sophisticated models exist as described, for example, in Edwards et al (1991 ).

and (1.7) or (1.8) For a free surface, a further condition is required todetermine the motion of the surface itself This condition arises from amomentum balance across the interface which stipulates that the jump in

the surface tractions t = σ σ · n, combined with the momentum fluxes, be

balanced by the action of surface tension:

(σ σ2− σσσ1)· n − ˙m (u2 − u1) =−∇ ∇ · [(I − nn) γ] = − (I − nn) · ∇ ∇ ∇ ∇γ + γκn,

(1.22)

where γ is the surface tension coefficient and

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 itsnormal 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),

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

Vb and 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 ofchange of the linear momentum

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,

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

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 ofgravity, 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 theΘkinematic relations

1.4 Some dimensionless groups

The use of dimensional analysis and dimensionless groups is a well-establishedpractice 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, mensionless ratios of the other scales to L A similar role may be played

di-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

a representative value of f Then the continuity equation remains formally

Re = ρLU

LU

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

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

On the other hand, when the flow is dominated by viscosity, the proper

pressure scale is ∆P = µU/L and the equation becomes

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

which, together with (1.34), are known as the Stokes equations.

Additional dimensionless groups arise from the boundary conditions Inthe 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

where κ ∗ = Lκ and the Weber number, expressing the ratio of inertial and

surface-tension-induced pressures, is defined by

W e = ρLU

2

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

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 √

where the capillary number, expressing the ratio of viscous to capillary

stresses, is defined by

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

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

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 ofthe hydrodynamic force|Fh| may be estimated in terms of a drag coefficient

Cd defined by

Cd = F

h 1

2ρAU2 r

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

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 thehydrodynamic force to find

τb∼ L

CdUr

ρb

ρ ∼ ρb ρ

Cd  1/Reb and we have

to different people, we shall use the term to refer to computations of plex unsteady flows where all continuum length and time scales are fullyresolved Thus, there are no modeling issues beyond the continuum hy-pothesis The flow within each phase and the interactions between differentphases at the interface between them are found by solving the governingconservation equations, using grids that are finer and time steps that areshorter than any physical length and time scale in the problem.

com-The detailed flow field produced by direct numerical simulations allows

us to explore the mechanisms governing multiphase flows and to extractinformation not available in any other way For a single bubble, drop, orparticle, 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 takeadvantage of the relatively simple geometry to obtain extremely accuratesolutions over a wide range of operating conditions The interactions of afew bubbles, drops, or particles is a more challenging computation, but can

be carried out using relatively modest computational resources Such lations yield information about, for example, how bubbles collide or whether

simu-a psimu-air of buoysimu-ant psimu-articles, rising freely through simu-a quiescent liquid, orientthemselves in a preferred way Computations of one particle can be used

to obtain information pertinent to modeling of dilute multiphase flows, andstudies 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 teracting particles that holds the greatest promise Such simulations canyield data for not only the collective lift and drag of dense systems, but

in-19

also about how the particles are distributed and what impact the formation

of structures and clusters has on the overall behavior of the flow Mostindustrial size systems, such as fluidized bed reactors or bubble columns,will remain out of reach of direct numerical simulations for the foreseeablefuture (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 solved by several hundred grid points in each spatial direction If we assumethat a single bubble can be adequately resolved by 25 grid points (sufficientfor clean bubbles at relatively modest Reynolds numbers), that the bubblesare, on the average, one bubble diameter apart (a void fraction of slightlyover 6%), and that we have a uniform grid with 10003 grid points, then wewould be able to accommodate 8000 bubbles High Reynolds numbers andsolid particles or drops generally require higher resolution Furthermore, thenumber of bubbles that we can simulate on a given grid obviously dependsstrongly on the void fraction It is clear, however, that DNS has opened

re-up completely new possibilities in the studies of multiphase flows which wehave 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 thedevelopment of numerical methods Advecting the phase boundary posesunique challenges and we will give a brief overview of such methods below,followed by a more detailed description in the next few chapters In mostcases, however, it is also necessary to solve the governing equations for thefluid flow For body-fitted and unstructured grids, these are exactly thesame as for flows without moving interfaces For the “one-fluid” approachintroduced in Chapter 3, we need to deal with density and viscosity fieldsthat 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 solvethe fluid equations After we briefly review the different ways of computingmultiphase flows, we will therefore outline in this chapter a relatively simplemethod to compute single-phase flows using a regular structured grid

2.1 Overview

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

mul-mass, momentum and energy hold, of course, for any fluid, even whendensity and viscosity change abruptly and the main challenge in this ap-proach is to accurately advect the phase boundary and to compute termsconcentrated at the interface, such as surface tension In the marker-and-cell(MAC) method of Harlow and collaborators at Los Alamos (Harlow andWelch,1965) each fluid is represented by marker points distributed over theregion that it occupies Although the MAC method was used to producesome spectacular results, the distributed marker particles were not partic-ularly good at representing fluid interfaces The Los Alamos group thusreplaced the markers by a marker function that is a constant in each fluidand is advected by a scheme specifically written for a function that changesabruptly from one cell to the next In one dimension this is particularlystraightforward and one simply has to ensure that each cell fills completelybefore the marker function is advected into the next cell Extended to twoand 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 asefficient methods, but early difficulties experienced with the volume-of-fluidmethod have given rise to several other methods to advect a marker func-tion These include the level-set method, originally introduced by Osher andSethian (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 thephase field method used by Jacqmin (1997) Instead of advecting a markerfunction and inferring the location of the interface from its gradient, it isalso possible to mark the interface using points moving with the flow andreconstruct a marker function from the interface location Surface mark-ers have been used extensively for boundary integral methods for potentialflows and Stokes flows, but their first use in Navier–Stokes computationswas by Daly (1969a,b) who used them to calculate surface tension effectswith the MAC method The use of marker points was further advanced bythe introduction of the immersed boundary method by Peskin (1977), whoused connected marker points to follow the motion of elastic boundaries im-mersed in homogeneous fluids, and by Unverdi and Tryggvason (1992) whoused connected marker points to advect the boundary between two differentfluids and to compute surface tension from the geometry of the interface.Methods based on using a single structured grid, identifying the interfaceeither by a marker function or connected marker points, are discussed insome detail in Chapter3 of this book

The attraction of methods based on the use the “one-fluid” formulation

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” approachexperienced, inspired several attempts to develop methods where the gridlines were aligned with the interface These attempts fall, loosely, into threecategories Body-fitted grids, where a structured grid is deformed in such away that the interface always coincides with a grid line; unstructured gridswhere 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; andwhat has most recently become known as sharp interface methods, where aregular 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 simplifythe specification of the interaction of the phases across the interface Fur-thermore, numerical methods on curvilinear grids are well developed and ahigh level of accuracy can be maintained both within the different phases andalong 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) tofind the steady state shape of axisymmetric buoyant bubbles brought theirutility to the attention of the wider fluid dynamics community Althoughbody-fitted curvilinear grids hold enormous potential for obtaining accuratesolutions for relatively simple systems such as one or two spherical parti-cles, generally their use is prohibitively complex as the number of particlesincreases These methods are briefly discussed in Chapter 4 Unstructuredgrids, 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 computationaldomain Early applications include simulations of the breakup of drops byFritts, Fyre, and Oran (1983) but more recently unstructured moving gridshave been used for simulations of multiphase particulate systems, as dis-cussed in Chapter5 Since body-fitted grids are usually limited to relativelysimple 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 neered by Glimm and collaborators many years ago (Glimm, 1982; Glimmand McBryan, 1985) but has recently re-emerged in methods that can bereferred to collectively as “sharp interface” methods In these methods the

pio-fluid domain is resolved by a structured grid, but the interface treatment

is improved by, for example, introducing special difference formulas thatincorporate the jump across the interface (Lee and LeVeque, 2003), using

“ghost points” across the interface (Fedkiw et al.,1999), or restructuring thecontrol 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 theincreased accuracy We will briefly discuss “sharp interface methods” forsimulations of the motion of fluid interfaces in Chapter 3 and in slightlymore 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 systemsdiscussed 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 pressurefor incompressible flows, where it is not a thermodynamic variable, buttakes on whatever value is needed to enforce a divergence-free velocity field,requires us to pay careful attention to the order in which the equationsare solved There is, in particular, no explicit equation for the pressureand therefore such an equation has to be found as a part of the solutionprocess The standard way to integrate the Navier–Stokes equations is bythe so called “projection method,” introduced by Chorin (1968) and Yanenko(1971) In this approach, the velocity is first advanced without accountingfor the pressure, resulting in a field that is in general not divergence-free.The pressure necessary to make the velocity field divergence-free is thenfound and the velocity field corrected by adding the pressure gradient

We shall first work out the details for a simple first-order explicit timeintegration scheme and then see how it can be modified to generate a higherorder scheme To integrate equations (1.2) and (1.7) (or 1.8) in time, wewrite

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

approx-imation to the diffusion term, and fb is a numerical approximation to any

other force acting on the fluid ∇h means a numerical approximation to thedivergence 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 − u n= un+1 − u ∗+

u∗ − u n The first part is a predictor step, where the temporary velocityfield is found by ignoring the effect of the pressure:

Adding the two equations yields exactly equation (2.1)

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

equa-1

ρ ∇2

hp n+1= 1

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 · u n= 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 atively stringent time-step limitations If we use standard centered second-order approximations for the spatial derivatives, as done below, stability

rel-analysis considering only the viscous terms requires the step size ∆t to be

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

integrate the Euler equations in time, are generally subject to the Courant–Friedrichs–Lewy (CFL) condition1 For one-dimensional flow,

∆t < h

Many advection schemes are implemented by splitting, where the flow issequentially 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 instandard textbooks, such as Hirsch (1988), Wesseling (2001), or Ferzigerand Peri´c (2002) In an unsteady flow, the CFL condition on the timestep is usually not very severe, since accuracy requires the time step to besufficiently small to resolve all relevant time scales The limitation due tothe viscous diffusion, equation (2.6), can be more stringent, particularly forslow flow, and the viscous terms are frequently treated implicitly, as dis-cussed below For problems where additional physics must be accountedfor, other stability restrictions may apply When surface tension is impor-tant, it is generally found, for example, that it is necessary to limit the timestep in such a way that a capillary wave travels less than a grid space in onetime step

The simple explicit forward-in-time algorithm described above is onlyfirst-order accurate For most problems it is desirable to employ at least

a second-order accurate time integration method In such methods the linear advection terms can usually be treated explicitly, but the viscousterms are often handled implicitly, for both accuracy and stability If weuse a second-order Adams–Bashforth scheme for the advection terms and

non-a second-order Crnon-ank–Nicholson scheme for the viscous term, the predictorstep is (Wesseling,2001)

Here, φ is not exactly equal to the pressure, since the viscous term is not

computed at the new time level but at the intermediate step It is easilyseen that

−∇φ n+1 =−∇p n+1+ν

2 Dh(u

n+1)− Dh(u∗)

The intermediate velocity u∗ does not satisfy the divergence-free condition,

and a Poisson equation for the pseudo-pressure is obtained as before from

Eq (2.10) as

∇2

hφ n+1= ∇h · u ∗

This multidimensional Poisson’s equation must be solved before the final

velocity, un+1, can be obtained Since the viscous terms are treated itly, we must rearrange equation (2.9) to yield a Helmholtz equation for the

to (2.13), is usually not very stringent

The above two-step formulation of the time-splitting scheme is not unique;another variant is presented in Chapter9 We refer the reader to standardtextbooks, such as Ferziger and Peri´c (2002) and Wesseling (2001) for furtherdiscussions

2.3 Spatial discretization

Just as there are many possible time integration schemes, the spatial cretization of the Navier–Stokes equations – where continuous variables arereplaced by discrete representation of the fields and derivatives are ap-proximated by relations between the discrete values – can be accomplished

dis-in many ways Here we use the finite-volume method and discretize the