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Preface
Multi-phase flows exist in many natural and technological systems.
Owing to the many challenges that are being faced by many applied sci-
entists, practising engineers, undergraduate and graduate students or
researchers engaging in the subject of multi-phase flows, the advance-
ment of digital computers has apparently re-focused the reliance on com-
putational predictions to better understand and predict the multi-phase
phenomenon. With increasing prevalence, the feasibility of application of
computational models is certainly gaining significant traction. Modelling
multi-phase flows, in general, encompasses a wide spectrum of different
length scales, a broad range of engineering disciplines, and a multitude of
different computational approaches. In spite of the comprehensive treat-
ment in the subject of multi-phase flows, a single text that provides a sys-
tematic exposition of a physical model utilizing the computational fluid
dynamics concepts, techniques and models remains elusive. Therefore,
the authors truly understand the difficulties faced by multi-phase
modellers to gain sufficient knowledge in employing the necessary com-
putational techniques and models in their investigative studies. For the
uninitiated multi-phase modeller who is learning about the basic con-
cepts of computational fluid dynamics for the first time, the nitty-gritty
elements within this particular mathematically sophisticated discipline
can be rather difficult to master. For practical multi-phase flows, appro-
priate models in aptly handling the complexity associated with the state
of the different phases are, in general, required. Each of these models
requires some prior basic knowledge of which the multi-phase mod-
eller may not have sufficiently acquired during his/her course of study.
Therefore, the authors fully acknowledge the predicament of multi-phase
modellers in attempting to obtain sensible results for a range of flow
problems by simply concentrating on the application rather than under-
standing the basic theory behind the formulation and limits of applicabil-
ity of these models.
Because of the increasing importance on the use of computational
modelling of multi-phase flows, the aim of the present text is to try to
present the basic techniques used in the treatment of multi-phase flows.
The authors have tried to strike a balance by covering relevant materi-
als that may be regarded as ‘old-fashioned’ as well as ‘new-fashioned’.
xi
xii Preface
On the former, the older, tried-and-proven ideas form a wonderfully
intuitive and meaningful learning experience for the uninitiated multi-
phase modeller. On the latter, some extensive development on the com-
putational techniques pertaining specifically to handle different forms of
multi-phase flows has been presented in order to keep-up-to-date of the
current computational approaches that are currently being employed in
the subject of multi-phase flows.
Because of the complex nature of multi-phase flows, the authors have
adopted an unconventional style of presenting the sequence of materi-
als in this book which has proven to be rather effective in conveying the
treatment of different classifications of multi-phase flows. For each clas-
sification of multi-phase flows, every effort has been made to discuss the
materials in easy-to-understand chapters (Chapters 4–9). Each of these
chapters have been purposefully written to cover the relevant transport
equations, multi-phase models and numerical methods or techniques
that are required to solve the appropriate multi-phase flow. One feature
of these chapters is that the materials are targeted precisely for the treat-
ment of particular multi-phase flow under consideration. For a more
comprehensive review on the background theory and development espe-
cially on the governing equations and numerical methods, the reader can
refer to Chapter 2, the governing equations and boundary conditions,
and Chapter 3, solution methods for multi-phase flows, respectively,
which incidentally form the foundation chapters of this book where the
building blocks of Chapters 4–9 are built on top of it. More importantly,
the authors have provided an extensive range of worked examples in
elucidating the application of specific computational techniques to best
handle the different classifications of multi-phase flows. As such these
materials will go a long way to even satisfy the initiated multi-phase
modeller through an intuitive, physically oriented approach to multi-
phase modelling. The future trends in handling turbulent multi-phase
flows are subsequently described in Chapter 10. There, the authors aim
to present other sophisticated models to circumvent current approaches
to multi-phase modelling and point towards the possible realization in
resolving more complex multi-phase flows.
Dr Yeoh would like to acknowledge the untiring support and bound-
less tolerance of his wife, Natalie, and his older daughters, Genevieve and
Ellana, and the new arrival of his youngest daughter, Clarissa, for their
enormous understanding and unflinching encouragement during the
seemingly unending hours spent in preparing and writing.
Professor Tu would like to express his deep gratitude to his wife, Xue, and
his son, Tian, who have provided their support in the writing of this text.
Special thanks are given to Geoff Smaldon, who first initiated the idea
of this book, and to Jonathan Simpson, who took over the project and
Preface xiii
provided much encouragement in the writing of this book. The authors
are especially grateful to Elsevier, the publisher, which has offered
immense assistance both in academic elucidation and professional skills
in the publication process.
To those that the authors have failed to mention but have been
involved in one way or another, the authors extend their deepest heart-
felt appreciation.
Guan Heng Yeoh
Jiyuan Tu
C H A P T E R
Introduction
. ClassifiCation and phenomenologiCal
disCussion
In the context of fluid mechanics, multi-phase flows can be taken as simply
any fluid flow system consisting of two or more distinct phases flowing
simultaneously in mixture, having some level of phase separation at a
scale well above the molecular level. Multi-phase flows can, in general,
exist in many different forms. Depending on combinations of phases,
two-phase flows can be classified according to the state of the different
phases: gas–solid flows or liquid–solid flows or gas–liquid flows.
Gas–solid flows, identified as gas–solid or gas–droplet flows, is con-
cerned with the motion of suspended solid or droplet in the gas phase.
Depending on the particle number density, such flows can be character-
ized as either being dilute or dense. When the particle number density is
relatively small, the influence of the gas flow is the dominant effect. Such
flows are referred to as dilute gas–particle flows which are governed pre-
dominantly by the surface and body forces acting on the particles. For
the special case of very dilute gas–particle flows, the solid particles act as
tracers which do not contribute in altering the gas flow. When the par-
ticle number density is taken to be sufficiently large, the motion of solid
particles is now controlled by particle–particle interactions. Categorically
known as dense gas–particle flows, collisions that exist between the solid
particles significantly influence the movement of these particles in the gas
phase. In bounded flow domains, the motion of solid particles following
impact on the boundary walls, which is different when compared to the
free flight of solid particles in air, is affected by the surface characteristics
and material properties. Gas–particle flows can be referred to as dispersed
flows in which the solid particles constitute the dispersed phase and the
gas constitutes the continuous phase.
Computational Techniques for Multi-Phase Flows © 22000190, Elsevier Ltd.
1. InTRoduCTIon
Liquid–solid flows consist of the transport of solid particles in liquid
flow. More appropriately termed as liquid–particle flows or slurry flows,
they can also be categorized as dispersed flows in which the liquid repre-
sents the continuous phase. In comparison to gas–particle flows, the liq-
uid and solid phases are mainly driven by and respond largely as one
to pressure gradients since the density ratio between phases is normally
low and the drag between phases is significantly high in such flows.
Of significant concern is the sedimentation behaviour of solid particles
within the liquid flow, which is strongly governed by the size of particles
of the dispersed phase and the flow conditions of the continuous phase.
Gas–liquid flows can, in principle, assume several different config-
urations. One example is the motion of bubbles in a liquid flow, while
another is the motion of liquid droplets in a gas. These two examples can
also be categorized as dispersed flows. For the first example, the liquid is
taken as the continuous phase and the bubbles are considered as discrete
constituents of the dispersed phase. For the second example, the gas,
however, is taken as the continuous phase and the droplets are now con-
sidered as the finite fluid particles of the dispersed phase. Since bubbles
or droplets are permitted to deform freely within the continuous phase,
they can take on different geometrical shapes: spherical, elliptical, dis-
torted, toroidal, cap and so on. In addition to dispersed flows, gas–liquid
flows often exhibit other complex interfacial structures, namely, separated
flows and mixed or transitional flows. Figure 1.1 summarizes the vari-
ous configurations that can be found for gas–liquid flows (Ishii & Hibiki,
2006). The transitional or mixed flows denote the transition between the
dispersed flows and separated flows, which is obviously characterized by
the presence of both of these flows. The change of interfacial structures
Flow transition
Dispersed flows Mixed or Transitional flows Separated flows
Bubbly Droplet Cap, Slug or Bubbly Droplet Bubbly droplet Film Annular
Churn-turbulent annular annular annular
Gas Liquid Gas Gas Gas core Gas core Liquid film in Liquid core
bubbles droplets pocket bubbles with with droplets gas and gas film
in liquid in liquid in liquid in liquid droplets and liquid Gas film in Gas core
film with and liquid film with gas gas and liquid
gas core film bubbles film
figuRe . Classification of gas–liquid flows (adapted from Ishii & Hibiki, 2006).
1.2 TyPICAL PRACTICAL PRobLEms InvoLvIng muLTI-PHAsE fLows
occurs through the presence of bubble–bubble interactions due to coales-
cence and break-up and any prevailing phase change process.
Free surface flows, which are complicated by the presence of well-
defined interfaces, belong to the specific consideration of immiscible
liquid flows. Strictly speaking, such flows do not fall into the class of
two-phase flows. For practical purposes, they can be treated as two-phase
mixtures. In contrast to the above classifications of two-phase flows, free
surface flows, which comprise mainly of gas and liquid flows, are usually
treated with both phases considered as continuous. Similarly, the process
of freezing or solidification may also be judged as another special case
of a two-phase mixture. Here, the liquid and solid regions can be treated
separately and then coupled through appropriate kinematic and dynamic
conditions at the interface.
Of a more sophisticated form, three-phase gas–liquid–solid flows
are also encountered in a number of engineering applications of techni-
cal relevance. For this particular class of multi-phase flows, we consider
the solid particles and gas bubbles as being the discrete constituents of
the dispersed phase co-flowing with the continuous liquid phase. The
co-existence of three phases considerably complicates the fluid flow due to
an array of phenomena associated with particle–particle, bubble–bubble,
particle–bubble, particle–fluid and bubble–fluid interactions modifying
the flow physics.
. typiCal pRaCtiCal pRoblems involving
multi-phase flows
Many practical problems involving multi-phase flows are broadly fea-
tured in a range of modern technological industries as well as within our
body system and in the environment we live in. Examples of multi-phase
flows based upon the different classifications are listed below.
Gas–particle flows
l Natural sand storms, volcanoes, avalanches
l Biological aerosols, dust particles, smoke (finely soot particles), rain
droplets, mist formation
l Industrial pneumatic conveyers, dust collectors, fluidized beds, solid
propellant rockets, pulverized solid particles, spray drying, spray casting
Liquid–solid flows
l Natural sediment transport of sand in rivers and sea, soil erosion,
mud slides, debris flows, iceberg formation
l Biological blood flow
l Industrial slurry transportation, flotation, fluidized beds, water jet
cutting, sewage treatment plants
1. InTRoduCTIon
Gas–liquid flows
l Natural ocean waves
l Biological blood flow
l Industrial boiling water and pressurized water nuclear reactors,
chemical reactor desalination systems, boilers, heat exchangers,
internal combustion engines, liquid propellant rockets, fire sprinkler
suppression systems
Liquid–liquid flows
l Industrial emulsifiers, fuel-cell systems, micro-channel applications,
extraction systems
Gas–liquid–solid flows
l Industrial air lift pumps, fluidized beds, oil transportation
The examples listed above are intended only to provide the reader an
overview of the broad range of applications and types of multi-phase
flows that could be experienced in the many diverse natural, biological
and industrial systems. Needless to say, no list can be fully exhaustive
when multi-phase flows are involved. This clearly reflects the ubiquitous
challenges that exist when such flows are constantly being dealt with.
In all the above systems, the complex nature of multi-phase flows
in contrast to single-phase flows originates because of the existence of
dynamically changing interfaces, significant discontinuities of the fluid
properties and complicated flow field near the interface. When one or
both of the phases becomes turbulent, interactions between the turbulent
eddies and the interfacial structures and exchanges between individual
phases introduce additional complexities to the flow phenomena. Multi-
phase flow physics are also multi-scale in nature. It is therefore neces-
sary to account for the cascading effects of the various flow physics at
different scales: (i) large flow structures encompassing the different indi-
vidual phases within the fluid flow at the device scale, (ii) local structural
changes due to agglomeration/coalescence and breakage processes of
discrete constituents at the mesoscale and (iii) motion of discrete constitu-
ents within the continuum fluid at the micro-scale.
A recurring theme throughout the study of multi-phase flows is the
requirement to model and predict the detailed behaviour of such flows
and the phenomena that they continue to manifest. Because of the lim-
ited scaling that can be realized in laboratory models, a reliable computa-
tional model, which should be thoroughly verified and validated through
the availability of experimental data, is essential for the confident extrap-
olation to the scale of the actual multi-phase flow system. At the current
availability of computational resources, it is becoming ever more possi-
ble to solve directly the transport equations governing the conservation
of mass, momentum and energy for each of the phases and to compute
1.3 ComPuTATIonAL fLuId dynAmICs As A REsEARCH TooL
every detail of the multi-phase flow, the motion of all the fluid around
and inside every particle, bubble and drop, and the position of every
interface. Nevertheless, such comprehensive treatment is still restricted
to turbulent flows of low Reynolds number and dynamics of a limited
amount of individual particles, bubbles and drops. On the other hand,
there is still an apparent need to resort to a macroscopic formulation
based on proper averaging which yields a multi-phase flow continuum
formulation, thereby effectively eliminating the interfacial discontinuities,
for large-scale multi-phase flow systems that are highly turbulent. Here,
the gross features are only predicted in which realistic models encom-
passing the turbulent effects for momentum and energy as well as for
interfacial exchanges for mass, momentum and energy transfer provide
the necessary closure to the conceptual modelling of multi-phase flows.
Owing to the demands of progressively accurate predictions for a great
number of important systems of interest, computational fluid dynamics
is becoming a powerful tool to be employed to resolve a range of multi-
phase flows either for pure/applied research or industrial applications.
The effective use of computational techniques to aptly handle different
classifications of multi-phase flows is exemplified in the next two sections.
. Computational fluid dynamiCs as a
ReseaRCh tool foR multi-phase flows
Computational fluid dynamics can be employed to reveal details of
peculiar flow physics that otherwise could not be visualized by experi-
ments, or clarify particular accentuating mechanisms that are consistently
being manifested in complex multi-phase flows. A range of pertinent multi-
phase studies based upon the judicious use of direct numerical simulation
or large eddy simulation is described below. Such an approach usually con-
tains very detailed information, producing an accurate realization about the
flow encapsulating a broad range of length and time scales. Because of the
wealth of information that can be attained, direct numerical simulation or
large eddy simulation is normally adopted as a research tool to effectively
provide a qualitative understanding of the flow physics and to possibly con-
struct a quantitative model, allowing other, similar, flows to be computed.
Consider the first example of stratified flows occurring in the atmo-
sphere–ocean interface. As depicted by the schematic drawing in Figure 1.2,
the transport processes at gas–liquid interfaces are controlled by a liquid
layer above the bottom wall which flows underneath the interface, while
the gas phase above can either be stationary or flowing along or against
the direction of the liquid flow.
The nature of the complex turbulence structures in the liquid flow
in the absence of significant wind stress has been studied by Pan and
1. InTRoduCTIon
Gas flow or no flow
Gas-liquid interface
Liquid flow
Wall
figuRe . Classification of gas–liquid flows (after Banerjee, 1994).
Upwelling
Vortex
figuRe . Visualization of free surface structures looking down at the free surface
from the top. Dark areas indicate the upwellings emanating from the bottom boundary
(after Banerjee, 1994).
Banerjee (1995). Direct numerical simulation, which entails all fluid
motions that are contained in the flow to be fully resolved, is performed
to understand the flow physics and the natural occurrence of the turbu-
lent structures. Assuming the interface to be a flat slip surface, interesting
turbulent structures are observed through the computations to be gen-
erated at the bottom boundary which subsequently impinge on the free
surface causing upwellings and associated vortical structures; these pre-
dictions have been found to correspond very closely to the visualization
of free structures as depicted in Figure 1.3. These vortices, being attached
to the free surface, are found to be very long lived and essentially quasi-
two-dimensional in nature such as illustrated in Figure 1.4. Direct numer-
ical simulation also indicates that these vortices as they approach towards
each other form a pair of counter-rotating vortices. These vortices then
merge, if they are of like sign and given a sufficient length of time, or
counter-rotate if they are unlike sign.
