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VTT PUBLICATIONS 722
Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema,
Markku Kataja, Antti Koponen, Mikko Manninen &
Veikko Taivassalo
Multiphase Flow Dynamics
Theory and Numerics
VTT PUBLICATIONS 722
Multiphase Flow Dynamics
Theory and Numerics
Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema,
Markku Kataja, Antti Koponen,
Mikko Manninen & Veikko Taivassalo
ISBN 978-951-38-7365-3 (soft back ed.)
ISSN 1235-0621 (soft back ed.)
ISBN 978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp)
ISSN 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp)
Copyright © VTT 2009
JULKAISIJA – UTGIVARE – PUBLISHER
VTT, Vuorimiehentie 3, PL 1000, 02044 VTT
puh. vaihde 020 722 111, faksi 020 722 4374
VTT, Bergsmansvägen 3, PB 1000, 02044 VTT
tel. växel 020 722 111, fax 020 722 4374
VTT Technical Research Centre of Finland, Vuorimiehentie 3, P.O. Box 1000, FI-02044 VTT, Finland
phone internat. +358 20 722 111, fax + 358 20 722 4374
Edita Prima Oy, Helsinki 2009
3
Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen
& Veikko Taivassalo. Multiphase Flow Dynamics. Theory and Numerics [Monifaasivirtausten
dynamiikka. Teoriaa ja numeriikkaa]. Espoo 2009. VTT Publications 722. 113 p. + app. 4 p.
Keywords multiphase flows, volume averaging, ensemble averaging, mixture models, multifluid
finite volume method, multifluid finite element method, particle tracking, the lattice-
BGK model
Abstract
The purpose of this work is to review the present status of both theoretical and
numerical research of multiphase flow dynamics and to make the results of that
fundamental research more readily available for students and for those working
with practical problems involving multiphase flow. Flows that appear in many of
the common industrial processes are intrinsically multiphase flows – e.g. flows
of gas-particle suspensions, liquid-particle suspensions, and liquid-fiber suspen-
sions, as well as bubbly flows, liquid-liquid flows, and the flow through porous
medium. In the first part of this publication we give a comprehensive review of
the theory of multiphase flows accounting for several alternative approaches.
The second part is devoted to numerical methods for solving multiphase flow
equations.
4
Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen
& Veikko Taivassalo. Multiphase Flow Dynamics. Theory and Numerics [Monifaasivirtausten
dynamiikka. Teoriaa ja numeriikkaa]. Espoo 2009. VTT Publications 722. 113 s. + liitt. 4 s.
Avainsanat multiphase flows, volume averaging, ensemble averaging, mixture models, multifluid
finite volume method, multifluid finite element method, particle tracking, the lattice-
BGK model
Tiivistelmä
Työssä tarkastellaan monifaasivirtausten teoreettisen ja numeerisen tutkimuksen
nykytilaa, ja muodostetaan tuon perustutkimuksen tuloksista selkeä kokonaisuus
opiskelijoiden ja käytännön virtausongelmien kanssa työskentelevien käyttöön.
Monissa teollisissa prosesseissa esiintyvät virtaukset ovat olennaisesti moni-
faasivirtauksia – esimerkiksi kaasu-partikkeli-, neste-partikkeli- ja neste-kuitu-
suspensioiden virtaukset, sekä kuplavirtaukset, neste-neste-virtaukset ja virtaus
huokoisen aineen läpi. Julkaisun ensimmäisessä osassa tarkastellaan kattavasti
monifaasivirtausten teoriaa ja esitetään useita vaihtoehtoisia lähestymistapoja.
Toisessa osassa käydään läpi monifaasivirtauksia kuvaavien yhtälöiden numeeri-
sia ratkaisumenetelmiä.
Preface
This monograph was originally compiled within the project ”Dynamics of
Multiphase Flows” which was a part of the Finnish national Computational
Fluid Dynamics Technology Programme 1995–1999. The purpose of this
work is to review the present status of both theoretical and numerical re-
search of multiphase flow dynamics and to make the results of that funda-
mental research more readily available for students and for those working
with practical problems involving multiphase flow. Indeed, flows that ap-
pear in many of the common industrial processes are intrinsically multiphase
flows. For example, gas-particle suspensions or liquid-particle suspensions
appear in combustion processes, pneumatic conveyors, separators and in nu-
merous processes within chemical industry, while flows of liquid-fiber suspen-
sions are essential in paper and pulp industry. Bubbly flows may be found
in evaporators, cooling systems and cavitation processes, while liquid-liquid
flows frequently appear in oil extraction. A specific category of multiphase
flows is the flow through porous medium which is important in filtration and
precipitation processes and especially in numerous geophysical applications
within civil and petroleum engineering.
The advanced technology associated with these flows has great econom-
ical value. Nevertheless, our basic knowledge and understanding of these
processes is often quite limited as, in general, is our capability of solving
these flows. In this respect, the condition within multiphase flow problems
is very much different from the conventional single phase flows. At present,
relatively reliable models exist and versatile commercial computer programs
are available and capable of solving even large scale industrial problems of
single phase flows. Advanced commercial computer codes now include fea-
tures which also facilitate numerical simulation of multiphase flows. In most
cases, however, a realistic numerical solution of practical multiphase flows
requires, not only a powerful computer and an effective code, but deep un-
derstanding of the physical content and of the nature of the equations that
are being solved as well as of the underlying dynamics of the microscopic
processes that govern the observed behaviour of the flow.
In the first part of this monograph we give a comprehensive review of the
theory of multiphase flows accounting for several alternative approaches. We
also give general quidelines for solving the ’closure problem’, which involves,
5
Preface
e.g., characterising the interactions between different phases and thereby
deriving the final closed set of equations for the particular multiphase flow
under consideration. The second part is devoted to numerical methods for
solving those equations.
6
Contents
Abstract 3
Preface 5
1 Equations of multiphase flow 9
1.1 Introduction............................ 9
1.2 Volumeaveraging ........................ 12
1.2.1 Equations......................... 12
1.2.2 Constitutiverelations .................. 17
1.3 Ensembleaveraging........................ 23
1.4 Mixturemodels.......................... 27
1.5 Particletrackingmodels ..................... 32
1.5.1 Equationofmotionforasingleparticle ........ 33
1.5.2 Particledispersion .................... 36
1.6 Practicalclosureapproaches................... 45
1.6.1 Diluteliquid-particlesuspension ............ 45
1.6.2 Flowinaporousmedium ................ 48
1.6.3 Densegas-solidsuspensions ............... 51
1.6.4 Constitutive equations for the mixture model . . . . . 55
1.6.5 Dispersionmodels .................... 58
2Numericalmethods 64
2.1 Introduction............................ 64
2.2 MultifluidFiniteVolumeMethod................ 66
2.2.1 Generalcoordinates ................... 67
2.2.2 Discretizationofthebalanceequations......... 69
2.2.3 Rhie-Chowalgorithm .................. 75
2.2.4 Inter-phasecouplingalgorithms............. 77
2.2.5 Solutionofvolumefractionequations ......... 79
2.2.6 Pressure-velocitycoupling................ 80
2.2.7 Solutionalgorithm .................... 83
2.3 MultifluidFiniteElementMethod ............... 84
2.3.1 StabilizedFiniteElementMethod ........... 86
7
Tiivistelmä 4
2.3.2 Integrationandisoparametricmapping ........ 90
2.3.3 Solutionofthediscretizedsystem............ 91
2.4 Particletracking ......................... 93
2.4.1 Solution of the system of equation of motion . . . . . 94
2.4.2 Solutionofparticletrajectories ............. 95
2.4.3 Sourcetermcalculation ................. 96
2.4.4 Boundary condition . .................. 96
2.5 Mesoscopicsimulationmethods ................ 96
2.5.1 Thelattice-BGKmodel ................. 98
2.5.2 Boundary conditions . .................. 99
2.5.3 Liquid-particlesuspensions ...............100
2.5.4 Applicability of mesoscopic methods . . ........101
Bibliography
Appendix 1
8
104
1. Equations of multiphase
flow
1.1 Introduction
A multiphase fluid is composed of two or more distinct components or
’phases’ which themselves may be fluids or solids, and has the character-
istic properties of a fluid. Within the dicipline of multiphase flow dynamics
the present status is quite different from that of the single phase flows. The
theoretical background of the single phase flows is well established (the crux
of the theory being the Navier-Stokes equation) and apparently the only
outstanding practical problem that still remains unsolved is turbulence, or
perhaps more generally, problems associated with flow stability. While it is
rather straightforward to derive the equations of the conservation of mass,
momentum and energy for an arbitrary mixture, no general counterpart of
the Navier-Stokes equation for multiphase flows have been found. Using a
proper averaging procedure it is however quite possible to derive a set of
”equations of multiphase flows” which in principle correctly describes the
dynamics of any multiphase system and is subject only to very general as-
sumptions (see section 1.2 below). The drawback is that this set of equations
invariably includes more unknown variables than independent equations,
and can thus not be solved. In order to close this set of equations, addi-
tional system dependent costitutive relations and material laws are needed.
Considering the many forms of industrial multiphase flows, such as flow in
a fluidized bed, bubbly flow in nuclear reactors, gas-particle flow in com-
bustion reactors and fiber suspension flows within pulp and paper industry,
it seems virtually impossible to infer constitutive laws that would correctly
describe interactions and material properties of the various phases involved,
and that would be common even for these few systems. Furthermore, even
in a laminar flow of, e.g., liquid-particle suspensions, the presence of parti-
cles induces fluctuating motion of both particles and fluid. Analogously to
the Reynolds stresses that arise from time averaging the turbulent motion
of a single phase fluid, averaging over this ”pseudo-turbulent” motion in
multiphase systems leads to additional correlation terms that are unknown
9
1. Equations of multiphase flow
a priori. For genuinely turbulent multiphase flows, the dynamics of the
turbulence and the interaction between various phases are problems that
presumably will elude general and practical solution for decades to come.
A direct consequence of the complexity and diversity of these flows is
that the dicipline of multiphase fluid dynamics is and may long remain a
prominently experimental branch of fluid mechanics. Preliminary small scale
model testing followed by a trial and error stage with the full scale system is
still the only conceivable solution for many practical engineering problems
involving multiphase flows. Inferring the necessary constitutive relations
from measured data and verifying the final results are of vital importance
also within those approaches for which theoretical modeling and subsequent
numerical solution is considered feasible.
In general, a multiphase fluid may be a relatively homogeneous mixture
of its components, or it may be manifestly inhomogeneous in macroscopic
scales. While much of the general flow dynamics covered by the present
monograph can be applied to both types of multiphase fluids, we shall mainly
ignore here macroscopically inhomogeneous flows such as stratified flow and
plug flow of liquid and gas in a partially filled tube. In what follows we
thus restrict ourselves to flows of macroscopically homogeneous multiphase
fluids. In modeling such flows, several alternative approaches can be taken.
Perhaps the most frequently used method is to treat the multicomponent
mixture, e.g., a liquid-particle suspension, effectively as a single fluid with
rheological properties that may depend on local particle concentration. This
approach may be used in cases where the velocities of various phases are
nearly equal and when the effect of the interactions between the phases can
be adequately described by means of rheological variables such as viscosity.
The advantage of these ’homogeneous models’ is that numerical solution
may be attempted utilizing conventional single fluid algorithms and effec-
tive commercial programs. In some cases the method can be improved by
adding a separate particle tracking feature or an additional particle trans-
port mechanism superposed on the mean flow. Although various traditional
methods based on a single fluid approach may be sufficient for predicting
gross features of certain special cases of also multiphase flows, it has become
increasingly clear, that in numerous cases of practical interest, an adequate
description requires recognition of the underlying multiphase character of
the system.
Genuine models for multiphase flows have been developed mainly follow-
ing two different approaches. Within the ’Eulerian approach’ all phases are
treated formally as fluids which obey normal one phase equations of motion
in the unobservable ’mesoscopic’ level (e.g, in the size scale of suspended par-
ticles) — with appropriate boundary conditions specified at phase bound-
aries. The macroscopic flow equations are derived from these mesoscopic
equations using an averaging procedure of some kind. This averaging pro-
cedure can be carried out in several alternative ways such as time averaging
10
1. Equations of multiphase flow
[Ish75, Dre83], volume averaging [Ish75, Dre83, Soo90, Dre71, DS71, Nig79]
and ensemble averaging [Ish75, Dre83, Buy71, Hwa89, JL90]. Various combi-
nations of these basic methods can also been considered [Ish75]. Irrespective
of the method used, the averaging procedure leads to equations of the same
generic form, namely the form of the original phasial equations with a few
extra terms. These extra terms include the interactions (change of mass,
momentum etc. ) at phase boundaries and terms analogous to the ordinary
Reynold’s stresses in the turbulent single phase flow equations. Each averag-
ing procedure may however provide a slightly different view in the physical
interpretation of these additional terms and, consequently, may suggest dif-
ferent approach for solving the closure problem that is invariably associated
with the solution of these equations. The manner in which the various pos-
sible interaction mechanisms are naturally divided between these additional
terms, may also depend on the averaging procedure being used.
The advantage of the Eulerian method is its generality: in principle it can
be applied to any multiphase system, irrespective of the number and nature
of the phases. A drawback of the straightforward Eulerian approach is that it
often leads to a very complicated set of flow equations and closure relations.
In some cases, however, it is possible to use a simplified formulation of the
full Eulerian approach, namely ’mixture model’ (or ’algebraic slip model’).
The mixture model may be applicable, e.g., for a relatively homogeneous
suspension of one or more species of dispersed phase that closely follow
the motion of the continuous carrier fluid. For such a system the mixture
model includes the continuity equation and the momentum equations for
the mixture, and the continuity equations for each dispersed phase. The
slip velocities between the continuous phase and the dispersed phases are
inferred from approximate algebraic balance equations. This reduces the
computational effort considerably, especially when several dispersed phases
are considered.
Another common approach is the so called ’Lagrangian method’ which
is mainly restricted to particulate suspensions. Within that approach only
the fluid phase is treated as continuous while the motion of the discontin-
uous particulate phase is obtained by integrating the equation of motion
of individual particles along their trajectories. (In practical applications a
”particle” may represent a single physical particle or a group of particles.)
In this chapter we review the theoretical basis of multiphase flow dy-
namics. We first derive the basic equations of multiphase flows within the
Eulerian approach using both volume averaging and ensemble averaging,
and discuss the guiding principles for solving the closure problem within the
Eulerian scheme. We then consider the basic formalism and applicability of
the mixture model and the Lagrangian multiphase model. Finally, we give
a few practical examples of possible closure relations.
11
1. Equations of multiphase flow
1.2 Volume averaging
1.2.1 Equations
In this section we shall derive the ’equations of multiphase flow’ using the
volume averaging method. To this end, we first define appropriate volume
averaged dynamic flow quantities and then derive the required flow equa-
tions for those variables by averaging the corresponding phasial equations
[Ish75, Dre83, Soo90, Dre71, DS71, Nig79]. While ensemble averaging may
appear as the most elegant approach from the theoretical point of view,
volume averaging provides perhaps the most intuitive and straightforward
interpretation of the dynamic quantities and interaction terms involved. Vol-
ume averaging also illustrates the potential problems and intricacies that are
common to all averaging methods within Eulerian approach.
Volume averaging is based on the assumption that a length scale Lc
exists such that lLcL, where Lis the ’macroscopic’ length scale of
the system and lis a length scale that we shall call ’mesoscopic’ in what
follows. The mesoscopic length scale is associated with the distribution of
the various phases within the mixture. (The ’microscopic’ length scale would
then be the molecular scale.)
Vα
Vβ
Vγ
dAα
Aα
γ
ˆ
nα
uα
uA
pα
ρα
τα
Phase α
Phase β
Phase γ
V=Vα+Vβ+Vγ
Figure 1.1: Averaging volume Vincluding three phases α,β, and γ.
To begin with, we consider a representative averaging volume V∼L3
c
which contains distinct domains of each phase such that V=PαVαwhere
Vαis the volume occupied by phase αwithin V(see Fig. 1.1). We as-
sume that for each phase αthe usual fluid mechanical equations for mass,
momentum and energy conservation are valid at any interior point of Vα,
12
1. Equations of multiphase flow
namely
∂
∂t ρα+∇ · (ραuα) = 0 (1.1)
∂
∂t (ραuα) + ∇ · (ραuαuα) = −∇pα+∇ · τα+Fα(1.2)
∂
∂t (ραEα) + ∇ · (ραuαEα) = (1.3)
−∇ · (uαpα) + ∇ · (uα·τα) + uα·Fα−∇·Jqα +JEα .
Here,
ρα= density of pure phase α
uα= flow velocity
pα= pressure
τα= deviatoric stress tensor
Eα= total energy per unit mass
Fα= external force density
Jqα = heat flux into phase α
JEα = heat source density.
Eqns. (1.1) through (1.3) are assumed to be valid both for laminar and for
turbulent flow. These equations are valid even if one of the phases is not
actually a fluid but consists, e.g., of solid particles suspended in a fluid. In
that case the stress tensor ταcontains viscous stresses for the fluid and elastic
deviatoric stresses for the particles. However, the concept of ’pressure’ may
not always be very useful for a solid material. In such cases it may be
preferable to use the total stress tensor σα=−pα11 + ταinstead, whence
−∇pα+∇ · τα=∇ · σα.
Similarly to single phase flows, the energy equation (1.3) is necessary only
in the presence of heat transfer. For simplicity, we shall from now on neglect
the energy equation and consider only mass and momentum equations. For
derivation of the energy equation for multiphase flows, see Refs. [Soo90] and
[Hwa89].
Eqns. (1.1) and (1.2) for phase αare subject to the following boundary
conditions at the interface Aαγ between phase αand any other phase γ
inside volume V(see Fig. 1.2) [Soo90].
ρα(uα−uA)·ˆnα+ργ(uγ−uA)·ˆnγ= 0 (1.4)
ραuα(uα−uA)·ˆnα+ργuγ(uγ−uA)·ˆnγ= (1.5)
(−pα11 + τα)·ˆnα+ (−pγ11 + τγ)·ˆnγ− ∇Aσαγ +2σαγ
|RA|ˆ
RA,
where
ˆnα= unit outward normal vector of phase α
13
1. Equations of multiphase flow
uA
ˆ
nγ
ˆ
nα
α
γ
dA
Figure 1.2: A portion of the interface between phases αand γ.
uA= velocity of the interface
ˆ
RA=RA/|RA|
RA= inerface curvature radius vector
σαγ = interface surface tension
∇A=∇ − ˆ
RA· ∇ = surface gradient operator
11 = second rank unit tensor
The interface Aα=SγAαγ may, however, have a very complicated shape
which depends on time and which actually should be solved simultaneously
with the flow equations. Therefore, it is usually not possible to apply the
boundary conditions (1.4) and (1.5) and to solve the mesoscopic equations
(1.1) and (1.2) in the usual manner. This is the basic reason why we have
to resort to averaged equations, in general.
For any quantity qα(scalar, vector or tensor) defined in phase αwe
define the following averages [Ish75, Hwa89]
hqαi=1
VZVα
qαdV (1.6)
˜qα=1
VαZVα
qαdV =1
φαhqαi(1.7)
¯qα=RVαραqαdV
RVαραdV =hραqαi
φα˜ρα
,(1.8)
where
φα=Vα/V. (1.9)
is the volume fraction of phase αand is subject to the constraint that
X
α
φα= 1.(1.10)
14
1. Equations of multiphase flow
The quantities defined by Eqns. (1.6), (1.7) and (1.8) are called the partial
average, the intrinsic or phasic average and the Favr´e or mass weighted
average of qα, respectively. At this point we leave until later the decision of
which particular average of each flow quantity we should choose to appear
as the final dynamic quantity of the averaged theory.
In order to derive the governing equations for the averaged quantities
defined above, we wish to apply averaging to equations (1.1) and (1.2). To
this end, we notice that the following rules apply to the partial averages
(and to the other two averages),
hf+gi=hfi+hgi(1.11)
hhfigi=hfihgi(1.12)
hCi=Cfor constant C. (1.13)
It is also rather straightforward to show that the following rules hold for
partial averages of various derivatives of qα[Soo90],
h∇qαi=∇hqαi+1
VZAα
qαˆnαdA (1.14)
h∇ · qαi=∇ · hqαi+1
VZAα
qα·ˆnαdA (1.15)
h∂
∂t qαi=∂
∂t hqαi − 1
VZAα
qαuA·ˆnαdA. (1.16)
For later purposes, it is also useful to define the phase indicator function Θα
such that
Θα(r, t) = 1,r∈phase αat time t
0,otherwise.(1.17)
Using Eqns. (1.6) and (1.14) with qα= Θα, it is straightforward to see that
hΘαi=φα,(1.18)
and that 1
VZAα
ˆnαdA =−∇φα.(1.19)
Applying partial averaging on both sides of Eqns. (1.1) and (1.2) and using
Eqns. (1.11)-(1.16) the following equations are obtained
∂
∂t hραi+∇ · hραuαi= Γα(1.20)
∂
∂t hραuαi+∇ · hραuαuαi=−∇hpαi+∇ · hταi+hFαi
+Mα,(1.21)
15
1. Equations of multiphase flow
where the so called ’transfer integrals’ Γαand Mαare defined by
Γα=−1
VZAα
ρα(uα−uA)·ˆnαdA (1.22)
Mα=1
VZAα
(−pα11 + τα)·ˆnαdA
−1
VZAα
ραuα(uα−uA)·ˆnαdA. (1.23)
The flow equations as given by Eqns. (1.20) and (1.21) are not yet in
a closed form amenable for solution. Firstly, the properties of each pure
phase are not specified at this point. Secondly, the transfer integrals (1.22)
and (1.23), which include the interactions (mass and momentum transfer)
between phases, are still given in terms of integrals of the original mesoscopic
quantities over the unknown phase boundaries. The additional constitutive
relations, which are required to specify the material properties and to relate
the transfer integrals with the proper averaged quantities, are discussed in
more detail below. Thirdly, averages of various products of original variables
that appear on the left side of the equations are independent of each other.
Even if all the necessary constitutive relations are assumed to be known,
we still have more independent variables than equations for each phase.
In order to reduce the number of independent variables, we must express
averages of these products in terms of products of suitable averages. This
can be done in several alternative ways which may lead to slightly different
results. Here we shall use Favr´e averaging for velocity and, depending on
which is more convenient, either partial or intrinsic averaging for density
and pressure. Defining the velocity fluctuation δuαby
uα=¯
uα+δuα,(1.24)
it is easy to see that the averages of products that appear in Eqns. (1.20)
and (1.21) can be written as
hραuαi=hραi¯
uα=φα˜ρα¯
uα(1.25)
hραuαuαi=hραi¯
uα¯
uα+hραδuαδuαi(1.26)
=φα˜ρα¯
uα¯
uα+hραδuαδuαi
The averaged equations now acquire the form
∂
∂t (φα˜ρα) + ∇ · (φα˜ρα¯
uα) = Γα(1.27)
∂
∂t (φα˜ρα¯
uα) + ∇ · (φα˜ρα¯
uα¯
uα) =
−∇(φα˜pα) + ∇ · hταi+φα˜
Fα+Mα+∇ · hτδαi,(1.28)
16
1. Equations of multiphase flow
where
hτδαi=−hραδuαδuαi.(1.29)
This tensor is sometimes called a pseudo-turbulent stress tensor since it is
analogous to the usual Reynolds stress tensor of turbulent one phase flow.
Notice however, that tensor hτδαiis defined here as a volume average instead
of a time average as the usual Reynolds stress. It also contains momentum
fluxes that arise both from the turbulent fluctuations of the mesoscopic flow
and from the fluctuations of the velocity of phase αdue to the presence of
other phases. Consequently, tensor hτδαidoes not necessarily vanish even if
the mesoscopic flow is laminar.
Integrating the mesoscopic boundary conditions (1.4) and (1.5) over the
interphase Aαγ, summing over αand γand using definitions (1.22) and
(1.23), we find that
X
α
Γα= 0 (1.30)
X
α
Mα=−1
2VX
α,γ
α6=γZAαγ
(−∇Aσαγ +2σαγ
|RA|ˆ
RA)dA. (1.31)
Eqn. (1.30) ensures conservation of the total mass of the mixture, while
the right side of Eqn. (1.31) gives rise to surface effects such as ’capillary’
pressure differences between various phases.
Eqns. (1.27) and (1.28) together with constraints (1.10), (1.30) and
(1.31) are the most general averaged equations of multiphase flow (with
no heat transfer), which can be derived without reference to the particular
properties of the system (other than the general continuum assumptions).
The basic dynamical variables of the averaged theory can be taken to
be the three components of the mass-averaged velocities ¯
uαand the volume
fractions φα(or, alternatively, the averaged densities hραi). Provided that
all the other variables and terms that appear in Eqns. (1.27) and (1.28)
can be related to these basic variables using definitions (1.6) through (1.8),
constraints (1.10), (1.30) and (1.31) and constitutive relations, we thus have
a closed set of four unknown variables and four independent equations for
each phase α.
1.2.2 Constitutive relations
Eqns. (1.27) and (1.28) are, in principle, exact equations for the averaged
quantities. So far, they do not contain much information about the dynamics
of the particular system to be described. That information must be provided
by a set of system dependent constitutive relations which specify the material
properties of each phase, the interactions between different phases and the
(pseudo)turbulent stresses of each phase in the presence of other phases.
17
1. Equations of multiphase flow
These relations finally render the set of equations in a closed form where
solution is feasible.
At this point we do not attempt to elaborate in detail the possible strate-
gies for attaining the constitutive relations in specific cases, but simply state
the basic principles that should be followed in inferring such relations. The
unknown terms that appear in the averaged equations (1.27) and (1.28),
such as the transfer integrals and stress terms that still contain mesoscopic
quantities, should be replaced by new terms which
•depend only on the averaged dynamic quantities (and their deriva-
tives),
•have the same physical content, tensorial form and dimension as the
original terms,
•have the same symmetry properties as the original terms (isotropy,
frame indifference etc.),
•include the effects of all the physical processes or mechanisms that are
considered to be important in the system to be described.
Typically, constitutive relations are given in a form where these new terms
include free parameters which are supposed to be determined experimentally.
For more detailed discussion of the constitutive relations and constitutive
principles, see Refs. [DALJ90, Dre83, DLJ79, Dre76, Hwa89, HS89, HS91,
BS78, Buy92a, Buy92b].
In some cases constitutive laws can readily be derived from the proper-
ties of the mixture, or from the properties of the pure phase. For example,
the incompressibility of the pure phase αimplies the constitutive relation
˜ρα=constant. Similarly, the equation of state pα=Cρα, where C=constant
for the pure phase, implies ˜pα=C˜ρα. In most cases, however, the consti-
tutive relations must be either extracted from experiments, derived analyt-
ically under suitable simplifying assumptions, or postulated.
Including a given physical mechanism in the model by imposing proper
constitutive relations is not always straightforward even if adequate ex-
perimental and theoretical information is available. In particular, making
specific assumptions concerning one of the unknown quantities may induce
constraints on other terms. For example, the transfer integrals Γαand Mα
contain the effect of exchange of mass and momentum between the phases.
According to Eqn. (1.22), the quantity Γαgives the rate of mass transfer per
unit volume through the phase boundary Aαinto phase αfrom the other
phases. In a reactive mixture, where phase αis changed into phase γ, the
mass transfer term Γαmight be given in terms of the experimental rate of
the chemical reaction α→γ, correlated to the volume fractions φαand φγ,
and to the temperature of the mixture T. Similarly, the quantity Mαgives
the rate of momentum transfer per unit volume into phase αthrough the
18
1. Equations of multiphase flow
phase boundary Aα. The second integral on the right side of Eqn. (1.23)
contains the transfer of momentum carried by the mass exchanged between
phases. It is obvious that this part of the momentum transfer integral Mα
must be consistently correlated with the mass transfer integral Γα. Simi-
larly, the first integral on the right side of Eqn. (1.23) contains the change
of momentum of phase αdue to stresses imposed on the phase boundary
by the other phases. Physically, this term contains forces such as buoyancy
which may be correlated to average pressures and gradients of volume frac-
tions, and viscous drag which might be correlated to volume fractions and
average velocity differences. For instance in a liquid-particle suspension,
the average stress inside solid particles depends on the hydrodynamic forces
acting on the surface of the particles. The choice of, e.g., drag force correla-
tion between fluid and particles should therefore influence the choice of the
stress correlation for the particulate phase. While this particular problem
can be solved exactly for some idealized cases [DALJ90], there seems to be
no general solution available.
Perhaps the most intricate term which is to be correlated to the aver-
aged quantities through constitutive relations is the tensor hτδα igiven by
Eqn. (1.29). It contains the momentum transfer inside phase αwhich arises
from the genuine turbulence of phase αand from the velocity fluctuations
due to presence of other phases, and which are present also in the case
that the flow is laminar in the mesoscopic scale. Moreover, the truly tur-
bulent fluctuations of phase αmay be substantially modulated by the other
phases. Bearing in mind the intricacies that are encountered in modeling
turbulence in single phase flows, it is evident that inferring realistic consti-
tutive relations for tensor hτδαiremains as a considerable challenge. It may,
however, be attempted, e.g., for fluid-particle suspensions by generalising
the corresponding models for single phase flows, such as turbulence energy
dissipation models, large-eddy simulations or direct numerical simulations.
A recent review on the topic is given by Crowe, Troutt and Chung in Ref.
[CTC96].
In the remaining part of this section we shall shortly discuss a few par-
ticular cases where additional simplifying assumptions can be made, namely
liquid-particle suspension, bubbly flow and multifluid flow. These examples
emphasize further the circumstance that no general set of equations exists
that, as such, would be valid and readily solvable for an arbitrary multiphase
flow, or even for an arbitrary two-phase flow. Instead, the flow equations
appropriate for each particular system should be derived separately starting
from the general (but unclosed) set of equations given in section 1.2 and
utilizing all the specific assumptions and approximations that are plausible
for that system (or class of systems). We notice, however, that the assump-
tions made here concerning, e.g., bubbly flow may not be generally valid for
all such flows. Nor are they the only possible extra assumptions that can
be made, but should be taken merely as examples of the kind of hypotheses
19
1. Equations of multiphase flow
that are reasonable owing to the nature of that category of systems. De-
tailed examples of more complete closure relations will be given in section
1.6.
In chapter 2 (see section 2.5) we shall briefly discuss novel numerical
methods that can be used to infer constitutive relations by means of direct
numerical simulation in a mesoscopic level.
Liquid-particle suspension
Consider a binary system of solid particles suspended in a Newtonian liquid.
We denote the continuous fluid phase by subscript f and the dispersed par-
ticle phase by subscript d. We assume that both phases are incompressible,
that the suspension is non-reactive, i.e., there is no mass transfer between
the two phases, and that surface tension between solid and liquid is negligi-
ble. Both the densities ˜ρfand ˜ρdare thus constants, and
Γf= Γd= 0 (1.32)
Mf+Md= 0.(1.33)
The mutual momentum transfer integral can now be written as
M≡Mf=−Md=1
VZAf
(−pf11 + τf)·ˆnfdA
=−1
VZA
(−pf11 + τf)·ˆn dA, (1.34)
where A=Af=Adand ˆn =ˆnd=−ˆnf. Introducing the fluid pressure
fluctuation by δpf=pf−˜pfand using Eqn. (1.19), the momentum transfer
integral can be cast in the form
M= ˜pf∇φ+D,(1.35)
where
D=−1
VZA
(−δpf11 + τf)·ˆn dA, (1.36)
and φ=φf. The averaged flow equations can now be written in the final
form as
∂
∂t φ+∇ · (φ¯
uf) = 0 (1.37)
∂
∂t (1 −φ) + ∇ · ((1 −φ)¯
ud) = 0 (1.38)
˜ρf[∂
∂t (φ¯
uf) + ∇ · (φ¯
uf¯
uf)] = −φ∇˜pf+∇ · hτfi+φ˜
Ff
+D+∇ · hτδfi(1.39)
˜ρd[∂
∂t ((1 −φ)¯
ud) + ∇ · ((1 −φ)¯
ud¯
ud)] = +∇ · hσdi+ (1 −φ)˜
Fd(1.40)
−D−˜pf∇φ+∇ · hτδdi,
20
1. Equations of multiphase flow
where hτfiis the averaged viscous stress tensor of the fluid, and hσdiis
the averaged total stress tensor of the dispersed phase. An example of more
complete constitutive relations for a dilute liquid-particle suspension is given
in section 1.6.1.
Bubbly flow with mass transfer
If the dispersed phase consists of small gas bubbles instead of solid particles,
the overall structure of the system still remains similar to the liquid-particle
suspension discussed in the previous section. A few things will change,
however. Firstly, the dispersed phase is not incompressible. Instead, the
intrinsic density ˜ρddepends on pressure ˜pdand temperature Tas given by
the equation of state of the gas,
˜ρd= ˜ρd( ˜pd, T ).(1.41)
Secondly, mass transfer between the phases generally occur. The gas phase
usually consists of several gaseous components including the vapor of the
liquid. The mass transfer may take place as evaporation of the liquid or
dissolution of the gas at the surface of the bubbles. Instead of Eqn. (1.32)
we now have
Γf=−Γd= Γ,(1.42)
where the mass transfer rate Γ is a measurable quantity which may depend
on the pressure, on the temperature and on the prevailing vapor content
of the gas etc. In this case, also the second term on the right side of Eqn.
(1.23), which includes the momentum carried by the mass exchanged, is
non-zero. This term can be related to the mass transfer rate Γ by defining
the average velocity ¯
umat the phase interface by the equation
1
VZAd
ρdud(ud−uA)·ˆnddA =¯
umΓ.(1.43)
At this stage, velocity ¯
umis of course unknown and must be modeled sepa-
rately. A natural first choice would be that ¯
umis the mass averaged velocity
of the mixture, i.e.,
¯
um=φ˜ρf¯
uf+ (1 −φ)˜ρd¯
ud
φ˜ρf+ (1 −φ) ˜ρd
.(1.44)
Thirdly, surface tension between the phases may be important. For small
bubbles one may ignore the surface gradient term −∇Aσdf in Eqn. (1.31).
Assuming that 2σdf /|RA|= ˜pd−˜pf(capillary pressure) it is straightforward
to see that, instead of Eqn. (1.33), we now have
Mf+Md=−(˜pd−˜pf)∇φ. (1.45)
21
1. Equations of multiphase flow
Following the analysis given by Eqns. (1.34) through (1.35) we can now
verify that
Mf= ˜pf∇φ+D+¯
umΓd(1.46)
Md= ˜pd∇(1 −φ)−D−¯
umΓd,(1.47)
where the quantity Dis still given by Eqn. (1.36). The flow equations for
bubbly flow with small bubbles may thus be given in the form
˜ρf
∂
∂t φ+ ˜ρf∇ · (φ¯
uf) = Γ (1.48)
∂
∂t [(1 −φ) ˜ρd] + ∇ · [(1 −φ)˜ρd¯
ud] = −Γ (1.49)
˜ρf[∂
∂t (φ¯
uf) + ∇ · (φ¯
uf¯
uf)] = −φ∇˜pf+∇ · hτfi(1.50)
+φ˜
Ff+D+¯
umΓd+∇ · hτδfi
∂
∂t [(1 −φ) ˜ρd¯
ud] + ∇ · [(1 −φ)˜ρd¯
ud¯
ud] = −(1 −φ)∇˜pd+∇ · hτdi(1.51)
+(1 −φ)˜
Fd−D−¯
umΓd+∇ · hτδdi.
Multifluid system
As a final example, we consider a system which consists of several continuous
or discontinuous fluid phases under the simplifying assumptions that there
is no mass transfer between the phases and that the surface tension can be
neglected for each pair of phases. It thus follows that
Γα= 0 for all phases α(1.52)
X
α
Mα= 0.(1.53)
Analogously to Eqn. (1.35) we can decompose the momentum transfer in-
tegrals as
Mα= ˜pα∇φα+Dα,(1.54)
where
Dα=1
VZAα
(−δpα11 + τα)·ˆnαdA. (1.55)
In the special case where the system is at complete rest, the phases must
share the same pressure (this follows from Eqn. (1.5) in the case that σαγ =
0). We shall assume here that this is approximately true also in the general
case where flow is present. We thus have that ˜pα≈˜p, where ˜pis the common
pressure of all phases. Since ∇(Pαφα) = 0, it follows from Eqns. (1.53)
and (1.54) that X
α
Dα= 0.(1.56)
22
1. Equations of multiphase flow
For the present multifluid system, Eqns. (1.27) and (1.28) can now be
written in the form
∂
∂t (φα˜ρα) + ∇ · (φα˜ρα¯
uα) = 0 (1.57)
∂
∂t (φα˜ρα¯
uα) + ∇ · (φα˜ρα¯
uα¯
uα) =
−φα∇˜p+∇ · hταi+φα˜
Fα+Dα+∇ · hτδαi.(1.58)
These equations will be further considered in chapter 2 where various meth-
ods for numerical solution of multiphase flows are discussed.
The apparent restriction to a single common pressure ˜pof phases in
Eqns. (1.58) is not actually a severe limitation of generality. If for any
reason the pressure varies between phases, it is always possible to define
˜pas an appropriate average of the phasial pressures ˜pαand to include the
effect of the deviatoric part ˜pα−˜pin the other terms on the right side of
Eqn. (1.58) such as in ∇ · hταior in Dα.
1.3 Ensemble averaging
In the previous section we derived the equations of multiphase flow (ignor-
ing the energy equation) using volume averaging. In this section we shall
repeat the derivation using a different approach, namely ensemble averaging
[Ish75, Dre83, Buy71, Hwa89, JL90]. As we shall see, the resulting general
equations are formally identical to those derived in the previous section,
Eqns. (1.27) and (1.28). While ensemble averaging, of all the averaging
methods that are commonly used within the Eulerian approach, appears
as the most elegant one, mathematical charm alone would not be a suffi-
cient cause for duplicating our efforts at this point. As discussed in section
1.2.2 however, the major problem within multiphase fluid dynamics is not
derivation of the conservation equations, but the closure of the equations.
Ensemble averaging, as the standard averaging method of also the modern
statistical physics, provides a different view in the physical interpretation of
the interaction terms and of the Reynolds stresses and may thereby provide
a different approach for solving the closure problem.
Conceptually, ensemble averaging is achieved by repeating the measure-
ment at a fixed time and position for a large number of systems with identi-
cal macroscopic properties and boundary conditions, and finding the mean
value of the results. Although the properties and the boundary conditions
are unchanged at the macroscopic level for each system, they differ at the
mesoscopic level. This leads to a scatter of the observed values. We denote
the collection of these macroscopically identical systems by C(the ensem-
ble) and its individual member by µ. If f(r, t;µ) is any quantity observed
23
1. Equations of multiphase flow
for a system µat point rand time t, its ensemble average hfiis defined by
hfi(r, t) = ZC
f(r, t;µ)dm(µ),(1.59)
where the measure dm(µ) is the probability of observing system µwithin
C. If all the necessary derivatives of fexist, it follows from the linearity of
the ensemble averaging that
h∂
∂t fi=∂
∂t hfi(1.60)
h∇fi=∇hfi.(1.61)
Notice that Eqns. (1.60) and (1.61) do not include any additional interfacial
terms, in contrast to Eqns. (1.14) through (1.16) for volume averaging. The
difference lies in the different definitions of variables and averages. The
observable quantity fitself is not associated to any particular phase of the
system. For example if fis density, then f(r, t;µ) is the local value of the
density of the phase that happens to occupy point rat time t. Also the
ensemble average is calculated without paying any attention to the phase
that occupies the point at which the average is calculated. The volume
average of a quantity defined for a certain phase is, on the other hand,
calculated only over the part of the total averaging volume occupied by that
particular phase, which gives rise to the surface integrals.
The partial average of fin phase αis defined by
hΘαfi,(1.62)
where Θαis the phase indicator function defined by Eqn. (1.17). Consider-
ing the phase indicator function as a generalized function (distribution), it
can be shown that it satisfies the equation
DsΘα
Dt ≡∂
∂t Θα+uA· ∇Θα= 0,(1.63)
To see this, consider
ZR3×R∂
∂t Θα+uA· ∇Θαψ dV dt (1.64)
=−ZR3×R
Θα∂
∂t ψ+∇ · (ψuA)dV dt
=−Z∞
−∞ ZVα∂
∂t ψ+∇ · (ψuA)dV dt
=−Z∞
−∞ d
dt ZVα(t)
ψ dV !dt
= 0,
24
1. Equations of multiphase flow
where uAis the velocity of the phase interface, Vα(t) is the volume occupied
by phase αat time t, and ψis a test function, which is sufficiently smooth
and has a compact support both in Vand t. In order that the second line
makes sense we must extend uAsmoothly through phase α. It is now easy
to show that
hΘα∇fi=∇hΘαfi − hf∇Θαi(1.65)
hΘα∇ · fi=∇ · hΘαfi − hf· ∇Θαi(1.66)
hΘα
∂
∂t fi=∂
∂t hΘαfi − hf∂
∂t Θαi(1.67)
=∂
∂t hΘαfi+hfuA· ∇Θαi,
where on the last line we have utilized Eqn. (1.63). The gradient of the phase
indicator function is non-zero only at the phase interfaces, which leads to
the conclusion that Eqns. (1.65) through (1.67) are counterparts of Eqns.
(1.14) through (1.16) and, in particular, that the second terms on the right
side of Eqns. (1.65) through (1.67) are counterparts of the surface integrals
in Eqns. (1.14) through (1.16).
We assume that inside each phase, the normal fluid mechanical equations
for mass and momentum are valid, namely
∂
∂t ρ+∇ · (ρu) = 0 (1.68)
∂
∂t (ρu) + ∇ · (ρuu) = ∇ · σ+F.(1.69)
where ρ, u, τ and Fare local density, flow velocity, stress tensor and external
force density, respectively. In what follows, we do not consider the energy
equation. The hydrodynamic quantities that appear in Eqns. (1.68) and
(1.69) are assumed to be well behaving within each phase but can have
discontinuities at phase interfaces. Multiplying Eqns. (1.68) and (1.69) by
Θα, performing ensemble averaging and using Eqns. (1.65) and (1.67), we
get
∂
∂t hΘαρi+∇ · hΘαρui=hρ(u−uA)· ∇Θαi(1.70)
∂
∂t hΘαρui+∇ · hΘαρuui=∇ · hΘασi+hΘαFi(1.71)
+h(ρu(u−uA)−σ)· ∇Θαi.
In analogy with Eqns. (1.7) and (1.8) we define the intrinsic and Favr´e
averages of any quantity fas
˜
fα=hΘαfi/hΘαi=hΘαfi/φα(1.72)
¯
fα=hΘαρfi/hΘαρi=hΘαρfi/(φα˜ρα),(1.73)
25
1. Equations of multiphase flow
respectively. Here,
φα=hΘαi,(1.74)
which we call the ’volume fraction’ of phase αfollowing the common con-
vention even though ’statistical fraction’ might be a more proper term.
In order to approach a closed set of equations we again use Favr´e aver-
aged velocity ¯
uαand the velocity fluctuation δuαdefined for phase αas
¯
uα=hΘαρui/(φα˜ρα) (1.75)
δuα=u−¯
uα.(1.76)
With these conventions the momentum flux term in Eqn. (1.71) can be
rewritten as
hΘαρuui=φα˜ρα¯
uα¯
uα−τδα,(1.77)
where the Reynolds stress tensor τδα is defined by
τδα =−hΘαρδuαδuαi.(1.78)
Using Eqns. (1.74) through (1.77), the averaged equations (1.70) and (1.71)
finally acquire the form
∂
∂t (φα˜ρα) + ∇ · (φα˜ρα¯
uα) = Γα(1.79)
∂
∂t (φα˜ραuα) + ∇ · (φα˜ρα¯
uα¯
uα)
=∇ · (φα˜σα+τδα ) + (φα˜
Fα) + Mα,(1.80)
where the quantities Γαand Mαare defined by
Γα=hρ(u−uA)· ∇Θαi(1.81)
Mα=h(ρu(u−uA)−σ)· ∇Θαi.(1.82)
These terms are the analogues of the transfer integrals that appear in the
corresponding volume averaged equations (1.22) and (1.23). They include
the rate of mass and momentum transfer between the phases.
As stated before, the general averaged equations obtained using ensem-
ble averaging are formally identical with those derived within the volume
averaging scheme in section 1.2. All the terms that appear in the ensemble
averaged equations have analogous physical content with the correspond-
ing term in the volume averaged equations. The only difference is that the
different formal definitions of terms such as τδα, Γαand Mαwithin these
two approaches may offer different ways of relating these quantities with the
basic averaged variables φα, ˜ρα, ˜pα,¯
uα,etc., and thereby solving the closure
problem for a given system.
26
1. Equations of multiphase flow
1.4 Mixture models
The mixture model (or algebraic slip model) is a simplified formulation of
the multiphase flow equations. We consider a suspension of a dispersed
phase (particles, drops, or bubbles) in a continuous fluid (liquid or gas).
If the dispersed phase follows closely the fluid motion (small particles), it
seems natural to write the balance equations for the mixture of the dispersed
and continuous phases and take the relative motion of the phases into ac-
count as a correction. The mixture model consists then of the continuity
and momentum equations for the mixture and the continuity equations for
the individual dispersed phases. The slip velocity between the dispersed
and continuous phases is taken into account by introducing corresponding
convection terms in the continuity equations.
The essential character of the mixture model is that only one set of
velocity components is solved from the differential equations for momen-
tum conservation. The velocities of the dispersed phases are inferred from
approximate algebraic balance equations. This reduces the computational
effort considerably, especially when several dispersed phases need to be con-
sidered.
The mixture model equations are derived in the literature applying vari-
ous approaches [Ish75, Ung93, Gid94]. The form of the equations also varies
depending on the application. Ishii [Ish75] derives the mixture equations
from a general balance equation. In this section, we derive the mixture
model equations from the original multiphase equations. This approach is
transparent and the required simplifications are clearly shown. Furthermore,
the applicability of the model can be explicitly analysed.
Continuity equation for the mixture
From the continuity equation for phase α(1.27), we obtain by summing over
all phases
∂
∂t
n
X
α=1
(φα˜ρα) + ∇ ·
n
X
α=1
(φα˜ρα¯
uα) =
n
X
α=1
Γα(1.83)
The right hand side of Eqn. (1.83) vanishes due to the conservation of
the total mass, Eqn. (1.30), and we obtain the continuity equation of the
mixture ∂
∂t (ρm) + ∇ · (ρm¯
um) = 0 (1.84)
Here the mixture density and the mixture velocity are defined as
ρm=
n
X
α=1
φα˜ρα(1.85)
¯
um=1
ρm
n
X
α=1
φα˜ρα¯
uα=
n
X
α=1
cα¯
uα(1.86)
27
1. Equations of multiphase flow
The mixture velocity ¯
umrepresents the velocity of the mass center. Notice
that ρmvaries although the component densities are constant. The mass
fraction of phase αis defined as
cα=φα˜ρα
ρm
(1.87)
Eqn. (1.84) has the same form as the continuity equation for single phase
flow.
Momentum equation for the mixture
The momentum equation for the mixture follows from the phase momentum
equations (1.28) by summing over all phases
∂
∂t
n
X
α=1
φα˜ρα¯
uα+∇ ·
n
X
α=1
φα˜ρα¯
uα¯
uα
=−
n
X
α=1 ∇(φα˜pα) + ∇ ·
n
X
α=1
φα(˜τα+ ˜τδα )
+
n
X
α=1
φα˜
Fα+
n
X
α=1
Mα.(1.88)
Here, the stress terms have been written in terms of intrinsic averages of
the stress tensors using Eqn. (1.7). Using the definitions (1.85) and (1.86)
of the mixture density ρmand the mixture velocity ¯
um, the second term of
(1.88) can be rewritten as
∇ ·
n
X
α=1
φα˜ρα¯
uα¯
uα=∇ · (ρm¯
um¯
um) + ∇ ·
n
X
α=1
φα˜ρα¯
umα¯
umα(1.89)
where ¯
umαis the diffusion velocity, i.e., the velocity of phase αrelative to
the center of the mixture mass
¯
umα=¯
uα−¯
um(1.90)
In terms of the mixture variables, the momentum equation takes the form
∂
∂t (ρm¯
um) + ∇ · (ρm¯
um¯
um) = −∇pm+∇ · (hτmi+hτδmi) + ∇ · hτDmi
+Fm+Mm(1.91)
The three stress tensors are defined as
hτmi=−
n
X
α=1
φα˜τα(1.92)
28
1. Equations of multiphase flow
hτδmi=−
n
X
α=1
φαh˜ραδuαδuαi(1.93)
hτDmi=−
n
X
α=1
φα˜ρα¯
umα¯
umα(1.94)
and represent the average viscous stress, turbulent stress, and diffusion stress
due to the phase slip, respectively. In Eqn. (1.91), the pressure of the
mixture is defined by the relation
pm=
n
X
α=1
φα˜pα(1.95)
In practice, the phase pressures are often taken to be equal, i.e., ˜pα=pm.
Accordingly, the last term on the right hand side of (1.91), Mm, comprises
only the influence of the surface tension force on the mixture and depends on
the geometry of the interface. The other additional term in (1.91) compared
to the one phase momentum equation is the diffusion stress term ∇ · τDm
representing the momentum transfer due to the relative motions.
Continuity equation for a phase
We return to consider an individual phase. Using the definition of the diffu-
sion velocity (1.90) to eliminate the phase velocity in the continuity equation
(1.27) gives
∂
∂t (φα˜ρα) + ∇ · (φα˜ρα¯
um) = Γα−∇·(φα˜ρα¯
umα) (1.96)
If the phase densities are constants and phase changes do not occur, the
continuity equation reduces to
∂
∂t φα+∇ · (φα¯
um) = −∇ · (φα¯
umα) (1.97)
The term on the right hand side represents the diffusion of the particles due
to the phase slip.
Diffusion velocity
In the mixture model, the momentum equations for the dispersed phases are
not solved and therefore a closure model has to be derived for the diffusion
velocities. The balance equation for calculating the relative velocity can be
rigorously derived by combining the momentum equations for the dispersed
phase and the mixture. In the following, we consider one dispersed partic-
ulate phase, p, for simplicity. Using Eqn. (1.54) for Mpand the continuity
equation, the momentum equation of the dispersed phase p (1.28) can be
29
1. Equations of multiphase flow
rewritten as follows (here gravity is used for external force , i.e.,˜
Fα= ˜ραg,
Fm=ρmg)
φp˜ρp
∂
∂t ¯
up+φp˜ρp(¯
up· ∇)¯
up=−φp∇(˜pp) + ∇ · [φp(˜τp+ ˜τδp)]
+φp˜ρpg+ Dp(1.98)
The corresponding equation for the mixture is
ρm
∂
∂t ¯
um+ρm(¯
um· ∇)¯
um=−∇pm+∇ · (τm+τδm+τDm) + ρmg(1.99)
Here we have neglected the surface tension forces and therefore Mm= 0.
Assuming that the phase pressures are equal, i.e.,pm= ˜pp, we can eliminate
the pressure gradient from (1.98) and (1.99). As a result we obtain an
equation for Dp
Dp=φp˜ρp
∂
∂t ¯
ump + (˜ρp−ρm)∂
∂t ¯
um
+φp[˜ρp(¯
up· ∇)¯
up−ρm(¯
um· ∇)¯
um]
−∇ · [φp(˜τp+ ˜τδp)] + φp∇ · (τm+τδm+τDm)
−φp(˜ρp−ρm)g(1.100)
In (1.100) we have utilized the definition (1.90) for the diffusion velocity
¯
ump. Next, we will make several approximations to simplify Eqn. (1.100).
Using the local equilibrium approximation, we drop from the first term the
time derivative of ¯
ump. In the second term, we approximate
(¯
up· ∇)¯
up≈(¯
um· ∇)¯
um(1.101)
The viscous and diffusion stresses are omitted as small compared to the lead-
ing terms. The turbulent stress cannot be neglected if we wish to keep the
turbulent diffusion of the dispersed phase in the model. However, all tur-
bulent effects are omitted for the moment. In that case, the final simplified
equilibrium equation for Dpis
Dp=φp(˜ρp−ρm)"g−(¯
um· ∇)¯
um−∂
∂t ¯
um#.(1.102)
Since Dpis a function of the slip velocity ¯
ucp =¯
up−¯
uc, Eqn. (1.102) is an
algebraic formula for the diffusion velocity
¯
ump = (1 −cp)¯
ucp (1.103)
The mixture model consists of Eqns. (1.84), (1.91), (1.96), and (1.102)
together with constitutive equations for the viscous and turbulent stresses.
30
1. Equations of multiphase flow
Validity of the mixture model
The terms omitted in arriving to Eqn. (1.102) from Eqn. (1.100) (except
turbulence terms) can be rewritten in the following form
φp˜ρp"∂
∂t ¯
ump + (¯
ump · ∇)¯
ump#+φp˜ρp[(¯
um· ∇)¯
ump + (¯
ump · ∇)¯
um]
+φp∇ · (τm+τDm)−∇·(φp˜τp) (1.104)
The local equilibrium approximation requires that the particles are rapidly
accelerated to the terminal velocity. This corresponds to setting the first
term in (1.104) equal to zero. Consider first a constant body force, like
gravitation. A criterion for neglecting the acceleration is related to the
relaxation time of a particle, tp. In the Stokes regime tpis given by
tp=˜ρpd2
p
18µm
, Rep<1 (1.105)
and in the Newton regime (constant CD) by
tp=4˜ρpdp
3˜ρcCDut
, Rep>1000 (1.106)
where utis the terminal velocity. Within the time tp, the particle travels the
distance lp=tput/e, which characterizes the length scale of the acceleration.
If the density ratio ˜ρp/˜ρcis small, the virtual mass and Basset terms in the
equation of motion cannot be neglected. The Basset term in particular
effectively increases the relaxation time. The true length scale l0
pof the
particle acceleration can be an order of magnitude larger than lp[MTK96].
An appropriate requirement for the local equilibrium is thus l0
p<< L, where
Lis a typical dimension of the system.
The second term in (1.104) corresponds, in rotational motion, the Cori-
olis force. The radial particle velocity caused by the centrifugal acceleration
causes in turn a tangential acceleration. To the first order, the second term
in (1.104) is proportional to ¯umϕ¯ucp/r, which has to be compared with the
leading term ¯u2
mϕ/r (ris the radius of curvature). Neglecting the second
term thus requires simply that
¯ucp
¯umϕ
<< 1 (1.107)
In the Stokes regime, this condition can be expressed as
dp<< s18µc
ω(˜ρp−˜ρc),(1.108)
where ωis the angular velocity of the rotation.
31
1. Equations of multiphase flow
In the last two terms of Eqn. (1.104), the viscous stresses can obvi-
ously be regarded as small compared to the leading terms, except possibly
inside a boundary layer. The diffusion stress can be neglected within the
approximation of local equilibrium.
In the above analysis, we assumed that the suspension is homogeneous in
small spatial scales. If this is not the case and dense clusters of particles are
formed, the mixture model is usually not applicable. Clustering in a scale
comparable to the length scale of turbulent fluctuations is typical for small
particles (dp<200µm) in gases. The clustering can lead to a substantial
decrease in the effective drag coefficient. Consequently, the particle relax-
ation time becomes large and the local equilibrium approximation is not
valid. Although the mixture model is in principle valid for small particles
(dp<50µm) in gases, it can be used only for dilute suspensions with solids
to gas mass ratio below 1.
1.5 Particle tracking models
Historically Lagrangian tracking models were first introduced in very dilute
flows of particulate suspensions[BH79], characteristic e.g. for electrostatic
precipitators. In these flows, the dispersed phase number density is low
enough such that the flow is dominated by the continuous carrier phase.
Mathematically the delineation between dilute and dense particulate flows
is established by the hydrodynamic response (or relaxation) time of the par-
ticle tpand the mean time between successive collisions between the particles
tc. In a dilute flow tp/tc<1. The particle then has enough time to respond
to the surrounding fluid field before the next collision, and the motion of
the particle is primarily controlled by the fluid flow. This assumption al-
lows separation of the solution for the two phases. Within the most simple
approach to particle tracking one first solves the flow of the carrier fluid
without taking particles into consideration. Next, particles are released at
desired positions in the solution domain. The trajectories of the particles
are then found by integrating an appropriate force balance equation (see
section 1.5.1) for each particle. Particle patches instead of single particles
may also be used at this point in order to reduce computational effort. This
type of approach is also known as one-way coupling as the information is
mainly transfered from the carrier phase to the dispersed phase and not in
the opposite direction.
In flows frequently found, e.g., in pneumatic conveying, the number den-
sity of the particulate phase is relatively high such that the presence of
particles affects the flow of the carrier phase while the collisions between
particles can still be ignored. Then, two-way coupling is said to prevail since
information is transfered from the carrier phase to dispersed phase and vice
versa. Lagrangian particle tracking method can be used to solve also this
32
1. Equations of multiphase flow
type of flow through an appropriate iterative procedure. The principle of
numerical solution for one-way coupled systems and for two-way coupled
systems is illustrated in Fig. 2.5.
In dense particulate flows, such as those found in fluidized beds, we have
that tp/tc>1, whereby the motion of a particle is significantly affected also
by interactions with other particles. These flows represent the case of four-
way coupling where information is also transfered between particles, and are
often solved using Eulerian multiphase models.
As stated above, particle tracking method includes first solving the usual
single-phase flow equations with proper boundary conditions for the carrier
phase, and then an initial value problem for an ordinary differential equa-
tion separately for each particle. (For flows with two-way coupling, this
procedure must be iterated.) In many cases particle tracking method leads
to a marked simplification as compared to the continuum Eulerian method
which requires the solution of a boundary value problem for coupled partial
differential equations, (Eqns. (1.27) and (1.28)), for both phases. Funda-
mental difficulties associated with the closure of Eulerian multiphase models
may also be avoided within the Lagrangian approach. Although the particle
tracking method is conceptually simple, it is not always quite straightfor-
ward in practice. Firstly, the equation of motion of an individual particle in
the surrounding fluid, discussed in the next section, may be quite compli-
cated. Secondly, turbulent flow poses a problem also within particle tracking
method. In particular, the modification of fluid turbulence due to presence
of the dispersed phase still lacks models with adequate theoretical justifica-
tion [Cro93].
Provided that an adequate equation of motion of the particle is given,
the particle tracking method is in general readily applicable in laminar flows
and in cases where only the mean trajectories of particles in a turbulent flow
are of interest. If, however, dispersion of particles in a turbulent flow is con-
sidered, additional modeling is needed in order to relate the dispersion rate
of particles with the statistical characteristics of turbulence of the carrier
fluid. That topic is discussed in section 1.5.2.
1.5.1 Equation of motion for a single particle
The basic equation that describes the motion of a sphere settling in a qui-
escent fluid due to gravity is the well-known equation by Basset [Bas88],
Boussinesq [Bou03] and Oseen [Ose27] (BBO). The original BBO equation
was based on the assumption that the Reynolds number of the particle is
low enough for the disturbance field produced by the motion of the sphere
to be governed by the unsteady Stokes equation. Later, Tchen extended the
33
1. Equations of multiphase flow
BBO equation to an unsteady and non-uniform flow as follows [Tch47]
mp
dv
dt=
I
z }| {
6πrpµf(u−v)−
II
z }| {
mf
ρf∇p+
III
z }| {
mf
2
d
dt(u−v)
+
IV
z }| {
6r2
p√πµfρfZt
t0
d
dτ{u(y(τ), τ)−v(τ)}
√t−τdτ
+
V
z }| {
(mp−mf)g.(1.109)
Here, rpis the radius of the particle, mpand mfare the mass of the particle
and the mass of a fluid sphere of radius rp,ρfand µfare the density and
the dynamic viscosity of the fluid, v(t) is the velocity vector of the particle
instantaneously centred at y(t) and u(y(t), t) is the Eulerian velocity vector
of the fluid at position y(t). It should be noted that urepresents the fluid
velocity at the mass centre of the particle as if the particle would not create
any disturbance field. The convective derivative following the motion of the
particle is given by
d
dt=∂
∂t +vj
∂
∂xjy(t)
.(1.110)
The numbered terms in Eqn. (1.109) are the Stokes drag force (I), the force
by fluid pressure gradient (II), the force by added mass (III), the Basset his-
tory term (IV) and the buoyancy term (V). Notice, that hydrostatic pressure
component is not included in the pressure p, but is taken into account in
the buoyancy term V in Eqn. (1.109).
Several authors have pointed out inconsistencies in Eqn. (1.109) [Lum57,
Buy66, Ril71]. Maxey and Riley [MR83] were the first to derive the equation
of motion of a small particle rationally from basic principles. They formu-
lated the problem for the motion of a rigid Stokes sphere in a nonuniform
flow field following the approach of Riley [Ril71] for the undisturbed flow.
The disturbance field caused by the sphere was calculated generalizing the
results of Basset [Bas88] and extending the work of Burgers [Bur38]. The
equation of motion given by Maxey and Riley is
mp
dv
dt= (mp−mf)g+mf
du
dt−mf
2
d
dt(v−u−r2
p
10∇2u)
−6πrpµf(v−u−r2
p
6∇2u)
−6r2
p√πµfρfZt
t0
d
dτ(v−u−r2
p
6∇2u)
√t−τdτ. (1.111)
The initial conditions are that the sphere is introduced at t=t0and that
there is no disturbance flow prior to this.
34
1. Equations of multiphase flow
By comparing Eqn. (1.111) with the BBO equation, Eqn. (1.109), it is
seen that the Stokes drag, the added mass and the Basset history terms have
been modified by the inclusion of terms containing the Laplacian of the fluid
velocity. These modifications thus include the effects of velocity curvature on
the drag force of a particle at low Reynolds numbers, i.e., Faxen relations
[Fax22]. Other modifications of the BBO equation than those discussed
above can be found in the literature (see, e.g., [Aut83, Buy66]).
In many practical cases, the most important terms on the right side of the
BBO equation or of Eqn. (1.111) are the gravity term and the Stokes drag
term. Especially for rapidly accelerating or oscillating flows also the other
terms may become important, however. In order to select an appropriate
form of the equation, it is thus important to estimate the magnitude of all
the force terms in each particular case. In most applications, the Basset
history term is ignored either as insignificant, or due to excessive numerical
complications brought about by the integral included in this term.
It can be shown that for a laminar time dependent flow (and for the
mean field of a turbulent flow), Eqn. (1.111) is valid provided that [MR83]
rp
L1,rpW
νf1 and r2
p
νf
U
L1,(1.112)
where Lis the characteristic length scale of the system, Wis the character-
istic relative velocity (v−u) and U/L is the scale of fluid velocity gradient
for undisturbed flow.
For a turbulent flow there is no single set of scales but a continuous
spectrum of velocity and length scales. The large scale energetic motions
may be characterized by the integral turbulent length scale Ltand by the
rms velocity urms =pδu2. The dissipative small scale motions are described
by the Kolmogorov microscales of length and velocity
ηk=ν3
f
1/4
and vk= (νf)1/4,(1.113)
respectively. Here, denotes the dissipation rate of turbulent energy. These
two limiting scales are related by
ηk
Lt
=OnRe−1/2
λoand vk
urms
=OnRe−3/2
λo,(1.114)
where Reλis the Reynolds number defined by the Taylor microscale λ, i.e.
Reλ=urmsλ/νf[TL72]. The steepest velocity gradients are found at the
dissipative scales. The appropriate scale of the velocity gradient is thus given
by vk/ηk, or equivalently by urms/λ. For a turbulent field the conditions of
validity of Eqn. (1.111) are thus given by
rp
ηk1,rpW
νf
=Orp
ηk
W
vk1 and r2
p
νf
vk
ηk
=O(r2
p
η2
k)1.(1.115)
35
1. Equations of multiphase flow
These conditions are usually much more restrictive than the conditions given
by Eqn. (1.112) for the mean velocity field of a turbulent flow, and for a
laminar flow.
Provided that the conditions given by Eqn. (1.115) are fulfilled, the
path of an individual particle in a turbulent flow can in principle be solved
using the preferred form of Eqn. (1.111) (or of Eqn. (1.109)). However, this
straightforward approach is not feasible, in general, since the full turbulent
flow field uof the carrier fluid is not known. Instead, we may assume that
the mean (time averaged) velocity ¯
uand the necessary statistical properties
of the turbulence are known (as given by measurements or by an appropriate
turbulence model). The particle tracking method then involves first solving
the mean velocity of the particle ¯
vat a given instant of time tusing Eqn.
(1.111) where the turbulent fluid velocity field uis replaced by the mean
velocity ¯
u=u−δu(and δuis the turbulent velocity fluctuation). The dis-
placement of the particle due to the mean flow during a short time interval
∆tis given by ∆¯
y=¯
v∆t. An additional stochastic displacement δydue
to turbulent dispersion must then be added to yield the total displacement
∆¯
y+δyof the particle at time interval ∆t. The path of the particle during
a finite time is found by iterating this process through subseqent short time
intervals. The pathline of an individual particle thus consists of a smooth
contribution from the mean flow, a possible drift due to gravity or other
external forces, and of a twisted random walk contribution due to turbu-
lence. When applied to a large number of particles, this approach leads to
a convection-diffusion type of behaviour of the dispersed phase.
If conditions of one-way coupling prevail the location of the particle can
be found by the described integration process with the knowledge of the
existing fluid field, available by preceding solution of fluid phase. For con-
ditions of two-way coupling this field is known just after solving the par-
ticle trajectory changing the process inherently to coupled and nonlinear.
Therefore, alternating iterative solution of fluid and dispersed phases until
convergence is necessary.
Calculation of the dispersive displacement yet requires additional mod-
eling to find the propability distribution for the random variable δy. This
can be done making use of Eqn. (1.111) and assuming only the necessary
spectral characteristics of the fluid turbulence to be known. Specific particle
dispersion models are discussed further in the next section.
1.5.2 Particle dispersion
The most important property in characterizing the response of particles to
the fluid flow is their hydrodynamic response time tp. For a rigid sphere in
Stokes flow (Rep<1), tpis given as
tp=ρpd2
p
18µf
,(1.116)
36
1. Equations of multiphase flow
where ρpis the density of particles, dpthe diameter and µfthe fluid dynamic
viscosity. This characteristic time is related to the particle inertia and form
drag. Above the Stokes regime, tpdepends on the particle Reynolds number
Repdefined in terms of the relative velocity between the particle and the
fluid ufp
Rep=ρfdp(u−v)
µf
=ρfdpufp
µf
.(1.117)
Thus, for Rep>1tpis defined as
tp=4ρpdp
3ρfCDufp
,(1.118)
where CD=CD(Rep) is the drag coefficient of the particle. As the particle
is transported by the mean flow and dispersed by the turbulence, scales
for both of them are required. The mean flow can be scaled by a mean
characteristic velocity Uand a characteristic length scale L. The Stokes
number, defined by
St =tpU
L,(1.119)
gives the ratio of the hydrodynamic response time to the time scale of the
mean flow. It thus indicates how well the particle can respond to the mean
flow.
To describe the dispersion by the turbulence both Lagrangian and Eu-
lerian scales of turbulence are required as the dispersion of small neutrally
buoyant particles is governed by the Lagrangian scales and the dispersion
of large heavy particles is dominated by the Eulerian scales. For Eulerian
integral time scales, a scale related to the frame moving with the mean flow
TmEand a scale determined with a fixed frame TfEexist. The former can
be measured and the latter can then be calculated but no device exist for
the direct measurement of Lagrangian integral scales TL. The response of
the particles to turbulence is commonly expressed by a Stokes number in
which TmEis used as the fluid time scale
St =tp
TmE
.(1.120)
The importance of the Stokes number and the above parameters can be
demonstrated by expressing the equation of motion of a single particle Eqn.
(1.109) in non-dimensional form
dv+
dt+=(u+−v+)f
St −γδi3
St .(1.121)
In Eqn. (1.121), for the purpose of simplicity, only the first and the fifth
terms corresponding to form drag and buoyancy have been retained. In
addition, the parameter fexpresses the ratio of the form drag to the Stokes
37
1. Equations of multiphase flow
drag and the gravitational field has been assumed to point in the negative
x3direction. As characteristic values, phδu2ifor velocity, TmEfor time
and phδu2iTmEfor length have been used.
Particles of different material than the surrounding fluid do not follow
equivalent paths with the fluid. This means that the knowledge of the
Lagrangian temporal correlation of velocities for the fluid is not adequate
but the corresponding fluid-particle correlation (tensor) is required. Here,
the possibilities for theoretical treatment are limited on only a few special
cases.
If the properties of the particles are close to that of the fluid, it can be
assumed that a given particle stays inside the same turbulent eddy for the
entire life time of the eddy. This condition can also be stated in the form
that the particles essentially stay in the highly correlated part of the flow
and, consequently, their dispersion follows that of the fluid. For this case
the motion of the particle can be approximated by the linearized form of
the equation of motion of the particle, i.e., the Tchen’s solution [Tch47],
discussed in section 1.6.5. A new theoretical problem, called ’the preferen-
tial concentration of particles’ [EF94], is raised with these almost neutrally
buoyant particles. That indicates a condition where the particles are not
randomly distributed in the fluid but become concentrated on certain areas
of the turbulent structures.
If, instead, particles have a notable relative speed with respect to the sur-
rounding fluid, the particle will leave the eddy before the eddy decays. Such
condition is typically found in gas-particle systems and in the presence of
strong external force field. As a consequence of this ’effect of crossing trajec-
tories’, the particles rapidly loose their velocity correlation to the fluid. This
effect can also lead to strongly unisotropic dispersion. On the other hand,
in many engineering problems of particulate flows the dispersion is mainly
limited by this effect and, consequently, the difficulties related to more de-
tailed description of turbulence and preferential concentration of particles
can be avoided. The effect of crossing trajectories is further discussed in
section 1.6.5 based on the work of Csanady [Csa63b, Csa63a].
The theoretical treatment of particle dispersion rests, in general, on the
assumption of stationary and homogeneous turbulence field. The funda-
mental work concerning the statistical diffusion of fluid points was made
by Taylor [Tay21] and later generalized by Batchelor [Bat53]. This formal-
ism can be directly applied to the dispersion of particles as well. When
Brownian motion is neglected as compared to turbulent contribution, the
random continuous motion of a single particle is defined by the mean square
displacement tensor over an ensemble of realizations of the system
hyL,i yL,j(t)i=qhv2
L,(i)i hv2
L,(j)iZt
0Zt0
0{RLp,ij(τ)+ RLp,ji(τ)}dτdt0
38
1. Equations of multiphase flow
=qhv2
L,(i)i hv2
L,(j)iZt
0
(t−τ){RLp,ij (τ) + RLp,ji(τ)}dτ, (1.122)
where yLstands for the location of the particle in reference to the coordi-
nate system moving with the mean particle velocity (Lagrangian coordinate
system) and vLthe instantaneous Lagrangian velocity of the particle. Here,
parentheses around the subscript are used to note that the Einstein sum-
mation convention must not be used. The latter equality sign holds for
the stationary and homogenous field under study. Further, the Lagrangian
temporal correlation tensor for particle velocities is defined as
RLp,ij(τ) = hvL,i(0) vL,j (τ)i
qhv2
L,(i)i hv2
L,(j)i
.(1.123)
By denoting the symmetric part of RLp,ij(τ) as
RS
Lp,ij(τ) = 1
2{RLp,ij(τ) + RLp,ji(τ)},(1.124)
the mean square displacement tensor can be expressed by
hyL,i yL,j(t)i= 2qhv2
L,(i)i hv2
L,(j)iZt
0
(t−τ)RS
Lp,ij(τ) dτ . (1.125)
Since the Lagrangian temporal correlation tensor RLp,ij (τ) and the power
spectral density ELp,ij (ω) are Fourier transform pairs of each other [TL72]
RLp,ij(τ) = Z−∞∞ELp,ij(ω) exp(−iωτ)dω , (1.126)
where ωcorresponds to the frequency of a temporal harmonic oscillation into
which the motion of the particle is decomposed, it is possible to transform
Eqn. (1.125) into the form
hyL,i yL,j(t)i= 2qhv2
L,(i)i hv2
L,(j)iZt
0
(t−τ)Z∞
0
ES
Lp,ij(ω) cos(ωτ ) dωdτ
=qhv2
L,(i)i hv2
L,(j)iZ∞
0
ES
Lp,ij(ω)2(1 −cos ωt)
ω2dω . (1.127)
To calculate hyL,i yL,j(t)iby Eqn. (1.125) or by Eqn. (1.127) the relationship
either between RLf,ij(τ) and RLp,ij (τ) or between ELf,ij (ω) and ELp,ij (ω)
is needed. The direct relation between the Lagrangian temporal correlation
tensors of the fluid and of the particle is very complicated, in general. Some
approximate solutions are, however, available for the relation between the
two power spectral densities [TL72].
39
1. Equations of multiphase flow
The classical dispersion tensor for a cloud of particles can now be defined
as
Dp,ij =1
2
d
dt{hyL,i yL,j(t)i}
=qhv2
L,(i)i hv2
L,(j)iZ∞
0
ES
Lp,ij (ω)sin ωt
ωdω . (1.128)
The above analysis has been carried out with ensemble averages as is appro-
priate in the general theory of random processes. However, as the analysis
concerns only stationary and homogeneous fields the random fluctuations
are ergodic [TL72] and, consequently, time averages become identical to
ensemble averages for long integration times.
Since it is always possible to use a Cartesian frame of reference where
ES
Lf,ij(ω) and ES
Lp,ij(ω) (and therefore RS
Lf,ij(ω) and RS
Lp,ij(ω)) are simul-
taneously diagonal, the real 3-D problem can be reduced to a combination
of three one-dimensional problems without any further loss of generality.
Therefore, the following treatment utilizes this finding by concentrating only
on a single index for which, e.g., the above correlation tensors and power
spectral densities are denoted as RLf(ω), RLp(ω), ELf(ω) and ELp(ω).
Thus, in order to calculate the mean square displacement, the rms fluctu-
ating velocities of particles and the Lagrangian temporal correlation tensor
must be known. Of these two quantities, the latter poses the main difficulty.
Even for fluid particles this correlation tensor is not generally known as there
is no device available that would be capable of following the tracked particle
and directly measuring it’s velocity. Measurement systems commonly uti-
lize fixed Eulerian coordinate system, and consequently, produce data from
which Eulerian correlations of velocities can be determined. Unfortunately,
no simple relation between these two types of correlation tensor exist, except
in the case of stationary homogeneous fields.
Eddy interaction model
The ’eddy interaction model’ is a customary approach to calculate (or more
accurately, numerically simulate) turbulent dispersion in the Lagrangian
approach. It’s original version is based on the discrete eddy velocity specifi-
cation and constant characteristic scales of eddies throughout the flow field
[HHD71, BH79]. Later on this approach was extended to complex flows
by defining the eddy scales from the turbulent statistics available from the
turbulence model used within the carrier fluid. A large number of articles
on this approach exist, e.g., [GI81a, Fae83, WBAS84, GHN89, SAW92]. It
is also utilized in several commercial flow simulation codes. In the eddy
interaction model, the particle motion is determined by the interaction of a
particle with a succession of turbulent eddies in which the velocity is kept
constant within a finite volume, characterized by the length and time scales
40
1. Equations of multiphase flow
of the eddy. The attraction of this model lies in its conceptual simplicity
and in that the only statistics required is given by the characteristic scales
of velocity, time and length. In the alternative models the forms of either
the temporal Lagrangian or the spatial Eulerian velocity autocorrelation
functions are required [BDG90, BB93, LFA93].
t0t1
x0, y0x1
y1
f l u i d e d d y p a t h
p a r t i c l e p a t h
u
v
2 L e
Figure 1.3: Illustration of the eddy interaction model and related parame-
ters.
An illustration of the eddy interaction model and related properties is
shown in Fig. 1.3. At initial time t0, the particle resides at the center of the
fluid eddy located at x0,i.e.,y0=x0. At some later time t1, the fluid eddy
has traveled to a new location x1=x0+t1u=x0+t1(u+δu). Because of
the velocity difference between the fluid eddy and the particle, their paths
are not identical but the particle will be found at y1=x0+t1(u+v).
The distance between the center of the eddy and the particle is therefore
(y1−x1) = t1v. The particle remains under the influence of the current eddy
of size Leuntil either the life time of the turbulent eddy Teis exceeded, or
it travels out of the eddy due to the velocity difference within an interaction
time Ti(the effect of crossing trajectories). When either of these conditions
is satisfied, a new eddy is generated and the process is repeated.
For Stokesian drag and negligible contributions from fluid pressure gradi-
ent, added mass and Basset history terms, it is possible to derive analytical
expressions for the particle velocity and position at the end of interaction
with an eddy. For that purpose it is assumed that the particle enters the
eddy at x0with velocity of v0. The fluid velocity Uewithin the eddy is con-
stant over the interaction time Ti. The equation of motion of the particle is
now given by
d
dt(v(t)) = Ue−v(t)
tp
,(1.129)
41
1. Equations of multiphase flow
where, as before, tpis the particle hydrodynamic response (relaxation) time,
Eqn. (1.116). Solving this equation gives the updated particle velocity after
the interaction with the eddy as
v1= Ue−(Ue−v0) exp(−Ti/tp).(1.130)
The updated position of the particle is then given by
x(t) = x0+Zt
0
v(t0)dt0(1.131)
= (x0+ TiUe)−(Ue−v0)tp(1 −exp(−Ti/tp)) .(1.132)
The particle crossing time Tcis defined by the equation |x(Tc)−x0−TcUe|=
Le. The solution is given by
Tc=−tplog(1 −Le
|Vfp |tp
),(1.133)
where Vfp = Ue−v0is the fluid-particle relative velocity. This expression is
valid only if Le/(Vfptp)<1, and then Tiis set equal to Tc. If this inequality
does not hold, i.e., the particle stays inside the eddy, the interaction time
Tiis set equal to the eddy life time Te. Because the interaction time can
be determined prior to the interaction, only one integration per eddy is
required.
In the case of small relaxation time tp, the particles are almost always
captured by the eddies. The interaction time will thus be equal to the eddy
life time and the particle velocity will quickly approach the fluid velocity.
Then, in the context of eddy interaction model, the dispersion of particles
will be similar to the dispersion of fluid points. For large relaxation times
the interaction time is more often set by the particle crossing time and,
consequently, fluid-particle interaction will be almost independent of the
eddy life time.
As emphasized above the key problem of the eddy interaction model
is to find the appropriate scales for the velocity, time and length of the
eddies. In general, these scales are random variables and should be given
by the statistical properties of fluid turbulence available from the applied
turbulence model or from experimental data. Thus, the components of, e.g,
eddy velocity can be randomly sampled from a Gaussian velocity distribution
with a zero mean and standard deviation phδu2i=q2
3k, where kis the
kinetic energy of fluid turbulence.
The eddy life time (life time distribution) can be related to the La-
grangian integral time scale of fluid turbulence, given by
TLf=Z∞
0
RLf(τ) dτ, (1.134)
42
1. Equations of multiphase flow
where
RLf(t, τ ) = huL(t)uL(t+τ)i
hu2
L(t)i(1.135)
is the Lagrangian temporal correlation tensor of fluid.
In a stationary turbulence RLfis independent of time, i.e., RLf(t, τ ) =
RLf(τ). Assuming that eddy life times teare distributed according to a
probability distribution function f(te), we can write the Lagrangian tempo-
ral correlation in the form [WS92]
RLf(τ) = R∞
τ(te−τ)f(te) dte
R∞
0tef(te) dte
.(1.136)
The most simple choice for the eddy life time distribution is the delta func-
tion f(te) = δ(te−Te), which corresponds to a constant eddy life time Te
[HHD71, BH79, JHW80, WAW82, GI81a, GHN89]. The Lagrangian tem-
poral correlation then assumes the form
RLf(τ) = 1−τ/Te, τ ≤Te
0, τ > Te.(1.137)
Using Eqn. 1.134, we then obtain the required correlation within the con-
stant life time scheme as.
Te= 2TLf.(1.138)
Another assumption that has been used in the context of eddy inter-
action simulations [KR89] is the exponential distribution of life times, i.e.,
f(te) = exp(−te/Tm)/Tm, where Tmis the mean life time. In this case, the
Lagrangian temporal correlation is given by
RLf(τ) = exp (−τ /Tm).(1.139)
Using Eqn. 1.134 we now obtain
Tm= TLf,(1.140)
which again correlates the eddy life time distribution with the appropriate
turbulent time scale within the exponential distribution scheme.
Notice, that for the eddy interaction model it has been shown [GJ96],
that the mean eddy life time Tmis subject to the general constraint that
TLf≤Tm≤2TLf. The two simple schemes discussed above thus corre-
spond to the limiting cases allowed by this inequality. As the number of
samples that are required for adequate prediction of particle dispersion in
eddy interaction models depends on the mean eddy life time and, therefore,
on the selected life time distribution, the constant eddy life time scheme
requires the lowest, and the exponential scheme the highest number of sam-
ples of all the admissible distributions. Furthermore, to enable a realistic
43
1. Equations of multiphase flow
statistical configuration of particles at the initial time t=t0,i.e., at the mo-
ment of particle release, the life time of the first eddy in constant life time
scheme has to be randomly distributed according to a uniform distribution
f0(te) = 1/Te,0< τ < Te[GJ96]. Within the exponential scheme instead,
a realistic configuration of particles at the initial time is automatically gen-
erated.
For large slip velocity Vfp, the eddy size distribution can be related to
the Eulerian integral size scale of fluid turbulence (similarly with the eddy
life time discussed above). This size scale is defined by
LEf=Z∞
0
REf(λ, 0) dλ, (1.141)
where
REf(λ, τ ) = hu(x, t)u(x+λ, t +τ)i
hu2(x, t)i(1.142)
is the Eulerian fluid velocity autocorrelation function. (In a general case, a
much more complex Lagrangian correlation of fluid along the particle path
and a corresponding Lagrangian scale should be used [Csa63b, Csa63a].)
Analogously with the eddy life time distribution, this autocorrelation func-
tion can be expressed in terms of eddy size distribution g(le). Assuming
constant eddy length Le[HHD71] corresponds to g(le) = δ(le−Le) and, in
the case of stationary turbulence, gives the autocorrelation function of the
form
REf(λ) = 1−λ/Le, λ ≤Le
0, λ > Le.(1.143)
Using Eqn. (1.141) then gives the result Le= 2LEf. In general, the length
scale can be taken to be a random variable with a given distribution func-
tion. An exponential probability distribution, e.g., gives results completely
analogous to Eqns. (1.139) and (1.140) [BM90].
Another possibility for finding the relevant time and length scales is given
by the widely used model of Gosman and Ioannides [GI81a]. Within that
model it is assumed that the eddy time and length scales are related to the
’dissipation scales’ Land T, given by
L=C3/4
µ
k3/2
and T=p3/2C3/4
µ
k
,(1.144)
where kis the kinetic energy of turbulence, its rate of dissipation and Cµ
is the constant that appears in definition of eddy viscosity µT=ρCµk2/ in
the standard k−turbulence model.
Notice, however, that even though we may assume the eddy life time Te
and size Leto be proportional to the corresponding statistical scales given
above, the actual relation depends on the details of the underlying statistical
model.
44
1. Equations of multiphase flow
1.6 Practical closure approaches
In this section we shall give more detailed examples of constitutive relations
for selected systems. The constitutive relations given here finally render the
general flow equations for a given multiphase system derived in section 1.2.2
in a closed form amenable for numerical solution (see Chapter 2 below). The
final equations derived here may not however be generally valid for all such
systems and should merely be taken as examples of plausible constitutive
models that may be used for that particular type of systems. One should
also bear in mind that the existing models in particular for turbulence in
multiphase flows still lack generality and can be considered inadequate to
some extent. For the sake of reliability of the model it is thus essential
that the values of various material parameters, that are left unspecified in
the constitutive relations, are determined by independent measurements in
conditions that closely resemble those of the actual application. It is also
essential that the numerical solution is verified experimentally.
1.6.1 Dilute liquid-particle suspension
As a starting point, we use the Eqns. (1.37) through (1.41) derived in section
1.2.2. We thus have eight equations for the eight unknowns which can be
taken to be the volume fraction of the fluid φ, fluid pressure ˜pfand the
three components of both the velocities ¯
ufand ¯
ud. It remains to specify the
constitutive relations for the viscous stress tensor of the fluid hτfi, the total
stress tensor of the particulate phase hσdi, the momentum transfer integral
D, and the turbulent stresses τδdand τδf.
The constitutive relation for the viscous stress tensor of the fluid hτfican
be derived simply by performing the volume averaging of the mesoscopic
tensor τf=µf((∇uf) + (∇uf)t) and using Eqn. (1.14). We define the
averaged surface velocity of the fluid ¯
USby
1
VZAf
ufˆnfdA =¯
US
1
VZAf
ˆnfdA =−¯
US∇φ, (1.145)
and postulate that ¯
US=b¯
ud−(1 −b)¯
uf, where b=b(φ) is a free parameter
(the ’mobility’ of the dispersed phase) which aquires values between 0 and
1. It is then easy to see that the viscous stress tensor of the fluid can be
given by
hτfi=φ˜τf
=φµf[(∇¯
uf) + (∇¯
uf)t] (1.146)
−bµf[(∇φ)(¯
ud−¯
uf) + (¯
ud−¯
uf)(∇φ)].
For a very dilute liquid-particle suspension where the collisions between par-
ticles can be ignored, the stress state of the particles is determined by the
45
1. Equations of multiphase flow
hydrodynamic forces exerted on the surface of the particles by the surround-
ing fluid. If the velocity difference between the particles and the fluid is not
very high, we may approximate
hσdi=−(1 −φ)˜σd≈ −(1 −φ)˜pf11.(1.147)
The term ∇·hσdi − ˜pf∇φon the right side of Eqn. (1.41) is now reduced to
−(1 −φ)∇˜pf. For a dense suspension instead, the tensor hσdiis contributed
by particle particle collisions and may have a very complicated form which
we do not consider here (see [Hwa89] and references therein).
According to Eqn. (1.36), the transfer integral Dis the force per unit
volume acting on particles due to fluctuations of fluid pressure and due to
viscous stresses. From the standard fluid mechanics we know that specific
hydrodynamic forces act on particles that move through the fluid with con-
stant velocity, with acceleration or with superimposed linear motion and
rotation. In principle, the forces acting on such a particle (with low particle
Reynolds number) are specified by the BBO equation, Eqn. (1.109) or Eqn.
(1.111). Thus, the transfer integral should contain a contribution from all
the force terms that appear on the right side of that equation - except of the
terms that arise from gravitational force and from the fluid pressure gradi-
ent, since these effects are already included in Eqns. (1.39) and (1.41)! For
a dilute suspension where particles can be considered independent of each
other, the terms in Dcorresponding to various terms in the selected version
of the BBO equation can be derived in a rather straightforward manner. For
example, neglegting the effects of fluid velocity gradient, and taking into ac-
count only the Stokes drag force and the added mass term (which arises
since accelerating a particle immersed in a fluid accelerates an amount of
fluid around the particle as well), we may write
D=B(¯
ud−¯
uf) + Ch(∂
∂t (¯
ud−¯
uf) + (¯
ud−¯
uf)· ∇(¯
ud−¯
uf)i,(1.148)
where Band Care unknown but perhaps measurable parameters. For a
very dilute suspension of spherical particles of radius rpand for low particle
Reynolds numbers, Eqn. (1.109) suggest that
B=9µf
2r2
p
(1 −φ) (1.149)
C=1
2˜ρf(1 −φ).(1.150)
Many other interaction mechanisms than those included in Eqn. (1.148)
may be crucial in practical flows of fluid-particle suspensions [Dre83]. For
instance if a particle is rotating, moving in the presence of a strong velocity
gradient of the fluid or, especially, if the particle is moving near a solid wall,
it may experience a ’lift’ or ’side’ force which is perpendicular to its direction
46
1. Equations of multiphase flow
of relative motion with respect to the fluid. A particle moving in a fluid may
also experience hydrodynamic forces which depend on the previous history
of its motion. This effect is taken into account in the BBO equation through
the Basset history term which may become important for instance in the
case of fast oscillatory motions (see [Soo90] and references therein).
The tensors hτδfiand hτδdicontain the momentum transfer due to turbu-
lent and ”pseudo-turbulent” fluctuations of the two phases. Various models
have been proposed for these quantities which are analogous to the ordinary
Reynold’s stresses in single phase flows. The constitutive relations suggested
by Drew and Lahey are [DLJ92]
hτδfi=−φpT
f11 + 2µT
ffΠf+ 2µT
fdΠd
+(1 −φ)ρf[af|¯
uf−¯
ud|211 + bf(¯
uf−¯
ud)(¯
uf−¯
ud)] (1.151)
hτδdi=−(1 −φ)pT
d11 + 2µT
ddΠd+ 2µT
df Πf
+(1 −φ)ρd[ad|¯
uf−¯
ud|211 + bd(¯
uf−¯
ud)(¯
uf−¯
ud)],(1.152)
where Πα=1
2[(∇¯
uα) + (∇¯
uα)t] for α= f,d. The turbulent pressures
pT
α, the eddy viscosities µT
αβ and the coefficients aαand bαare still to be
correlated with appropriate variables that characterise the turbulent and
pseudoturbulent motion of the phases. This might be accomplished through
experiments or through additional turbulence modelling [CTC96]. Here,
we shall consider a simple generalization of the ordinary k−model for
onephase flows.
In this illustrative multiphase version or the k−εmodel we ignore all
the other terms on the right side of Eqns. (1.151) and (1.152) than those
proportional to the eddy viscosities µT
ffand µT
dd. We also assume that the
eddy viscosity hypothesis holds for each individual phase. Thus we can
define the eddy viscosity for phase αin analogy with single phase flows as
µT
αα =Cµ˜ρα
k2
α
εα
,(1.153)
where kαand εαare the turbulent kinetic energy and the dissipation for
phase α, respectively, and Cµis an empirical constant. The transport equa-
tions for kαand εαare postulated to have the form
∂
∂t (φα˜ραkα) + ∇ · (φα˜ρα¯
uαkα)− ∇ · [φα(µα+µT
αα
σk
)∇kα]
=φαSkα + Γkα (1.154)
∂
∂t (φα˜ραεα) + ∇ · (φα˜ρα¯
uαεα)−∇·[φα(µα+µT
αα
σε
)∇εα]
=φαSεα + Γεα.(1.155)
47
1. Equations of multiphase flow
Analogously with single phase flows the source terms are divided into pro-
duction and dissipation terms as follows
Skα =Pα−˜ραεα(1.156)
Sεα =εα
kα
(C1εPα−C2ε˜ραεα) (1.157)
Pα=µT
αα∇¯
uα·[∇¯
uα+ (∇¯
uα)t].(1.158)
Interphasial turbulence exchange terms Γkα,Γεα must be defined separately
for each case. Assuming that the turbulence quantities are equal for both
phases, i.e., that kα≡kand εα=εfor all α= f,d, and summing the above
transport equations, we get
∂
∂t (ρk) + ∇ · [ρ¯
uk−(µ+µT
σk∇k)] = Sk(1.159)
∂
∂t (ρε) + ∇ · [ρ¯
uε−(µ+µT
σε∇ε)] = Sε,(1.160)
where
ρ=X
α
φα˜ρα(1.161)
¯
u=1
ρ(X
α
φα˜ρα¯
uα) (1.162)
µ=X
α
φαµα(1.163)
µT=X
α
φαµT
αα (1.164)
Sk=X
α
φαSkα (1.165)
Sε=X
α
φαSεα.(1.166)
1.6.2 Flow in a porous medium
Most porous materials of practical interest consist either of particles packed
in a more or less distordered manner or of a consolidated irregular porous
structure of some kind. Examples of such materials are numerous: sand,
soil, fractured rock, ceramics, sponge, paper etc. Many important processes
found in geophysics or in various industrial applications involve flow of fluid
through a porous medium. In some cases, such as in slow transport of
ground water through an aquifer, the porous material can be considered
rigid so that the structure of the solid matrix is not significantly deformed
during the process. The basic equation for such a flow is given by the famous
Darcy’s law, which was originally inferred from purely empirical results for a
48
1. Equations of multiphase flow
stationary creeping flow of Newtonian liquid through a homogeneous column
of sand [Bea72]. With processes such as removal of water from a sponge by
squeezing it, the porous structure appears soft and may thus be extensively
deformed by external forces and by hydrodynamical forces exerted on the
solid matrix by the fluid flow.
In this section, we shall utilize Darcy’s experimental formula in the con-
text of the multiphase flow theory and derive the governing equations for
time dependent creeping flow of Newtonian liquid through a soft porous
medium. Formally, we treate the system of the highly deformable solid
matrix and the liquid flowing through the interstities of the matrix as a
binary mixture of two fluids. We assume again that both phases are incom-
pressible, that there is no mass transfer between the two phases and that
surface tension between the solid material and the liquid is negligible. The
situation is thus analogous with the liquid-particle suspension discussed in
section (1.2.2). However, instead of a contiunous liquid phase and a dis-
persed particle phase we now have two continuous phases. Replacing the
subscript ’d’ for ’dispersed’ phase by ’s’ for ’solid’ phase, the Eqns. (1.37)
through (1.41) are thus formally valid also for the present system. Several
simplifications as compared to the liquid-particle suspension can however be
made in this case. Assuming creeping flow indicates that the inertial terms
that appear on the left side of Eqns. (1.39) and (1.41) can be neglected.
Furthermore, the pseudoturbulent stress term ∇·hτδdivanishes for the solid
phase and is expected to be very small also for the fluid phase in this flow
regime. According to Darcy’s early experiments and to innumerable later
experiments, the dominant interaction mechanism in a flow through porous
medium is viscous drag. The results of these experiments, as summarized
by the Darcy’s law, indicate that the momentum transfer integral Dshould
be written in a form
D=−µ
k(¯
uf−¯
us).(1.167)
Here, k=k(φ) is the permeability of the porous material which remains
to be measured. Several experimental correlations for khas been reported
in literature for different types of porous media [Bea72]. Perhaps the most
common formula which can be derived analytically for simplified capillary
models of porous materials and which at least qualitatively grasps the correct
behaviour for many materials, is the Kozeny-Carman relation
k=1
cS2
0
φ
(1 −φ)2.(1.168)
Here, S0is the specific pore surface area and c is the dimensionless Kozeny
constant which aquires values between 2 and 10, in practice. (Notice that
due to the conventions used here, Eqn. (1.168) differs from its more usual
form where φ3instead of φappears in the nominator, see Eqn. (1.177)
below.) Furthermore, if the porosity φis not too close to unity, the viscous
49
1. Equations of multiphase flow
shear stress term ∇ · hτfiis small as compared to the viscous drag term
and can be neglected. Taking gravitation to be the only body force, the
equations for a flow of liquid in a deformable porous medium can thus be
written in a form
∂
∂t φ+∇ · (φ¯
uf) = 0 (1.169)
∂
∂t (1 −φ) + ∇ · [(1 −φ)¯
us] = 0 (1.170)
φ∇˜pf=−µ
k(¯
uf−¯
us) + φ˜ρfg(1.171)
−∇ · hσsi= +µ
k(¯
uf−¯
us)−˜pf∇φ+ (1 −φ) ˜ρsg.(1.172)
Adding Eqns. (1.169) and (1.170) and Eqns. (1.171) and (1.172) we arrive
at the mixture equations
∇ · hqi= 0 (1.173)
∇ · hTi=−hρig,(1.174)
where hqi=φ¯
uf+ (1 −φ)¯
usis the volume flux, hTi=−φ˜pf11 + hσsiis the
total stress, and hρi=φ˜ρf+ (1 −φ) ˜ρsis the density of the mixture.
For linearly elastic materials, the stress tensor hσsiis readily given as a
function of local strain by Hooke’s law. For viscoelastic materials instead,
hσsimay depend both on the strain and on the rate of strain (i.e. on ¯
us).
Since the solid phase is actually not a fluid in an ordinary sense, a finite
stress implies finite strain on the solid matrix. It follows that the velocity
of the solid phase can be non-zero only in a transient state. In a stationary
state (and in the case of rigid porous material) we have ¯
us= 0. The porosity
φis then independent of time, and the flow equations are reduced to
∇ · qf= 0 (1.175)
qf=−¯
k
µ(∇˜pf+ ˜ρfg),(1.176)
and one of Eqns. (1.172) or (1.174). Here qf=φ¯
ufis the volume flux of
the fluid (the ’seepage’ velocity), and
¯
k=1
cS2
0
φ3
(1 −φ)2.(1.177)
Eqn. (1.176) is the Darcy’s formula in its conventional form.
50
1. Equations of multiphase flow
1.6.3 Dense gas-solid suspensions
The behaviour of solid particles in a dense gas-solid suspension is strongly
affected by the binary interparticle collisions. The kinetic theory of granular
flow is derived for this special case of twophase flow in analogy with the
kinetic theory of dense gases.
A conservation equation (the Bolzmann equation) for the particulate
phase is formulated in terms of the single particle velocity distribution func-
tion f1(x,u, t)
∂
∂t f1+∂
∂xi
(uif1) + ∂
∂ui
(Fif1) = ∂
∂t f1!coll
(1.178)
where Fis the external force per unit of mass acting on a sphere and the right
side describes the rate of change of the distribution function due to particle
collisions. In the kinetic theory of granular flow the ensemble-average (1.59)
of a function ψ(u) is defined by
hψ(u)i=1
npZψ(u)f1(x,u, t)du (1.179)
where npis the number of particles per unit volume. A transport equation
for hψ(u)ican be derived from Eqn. (1.178) by multiplying it by ψ(u) and
integrating it over the velocity domain [CC70]:
∂
∂t (nphψi) + ∂
∂xi
(nphψuii) (1.180)
−np *∂ψ
∂t ++*ui
∂ψ
∂xi++*Fi
∂ψ
∂ui+!=C(ψ),
where the collisional rate of change for ψis defined by ([JR85], cited in
[Pei98])
C(ψ) = χ(ψ)−∂
∂xi
θi(ψ)−∂us,j
∂xi
θi*∂ψ
∂δuj+(1.181)
where us,j is the mean velocity of the discrete phase. Alternative formu-
lations of C(ψ), where the last term is missing, can be found in litera-
ture (e.g. [JS83, DG90]). The source term χ(ψ) describes loss of ψdue
to inelastic collisions and θi(ψ) transport of property ψduring collisions.
These terms are defined by integrals involving the pair distribution function
f2(x1, x2, u1, u2, t) and can be calculated analytically. In the derivation,
the pair distribution function is given as a product of the single velocity
distribution functions and a correction function g0.
Kinetic theory yields the continuity equation and the momentum equa-
tion of a phase in a form similar with the traditional multifluid equations.
51
1. Equations of multiphase flow
The continuity equation is obtained using ψ= 1 and the momentum equa-
tion by using ψ=u. In the resulting equations, the kinetic and colli-
sional contributions of the particulate phase stress are treated together.
The isotropic parts are described as a solid pressure and the rest as a shear
stress term. In addition to the continuity and the momentum equations,
a field equation for the particle fluctuating kinetic energy must be solved.
The following formulation can be found, e.g., in [Boe97].
The continuity equation without mass transfer can be written in the
form ∂
∂t (φα˜ρα) + ∇ · (φα˜ρα¯
uα) = 0.(1.182)
Using the notation of total averaged stresses the momentum equations can
be written in the form
∂
∂t (φα˜ρα¯
uα) + ∇ · (φα˜ρα¯
uα¯
uα) = ∇ · hσαi+Mα+φα˜
Fα.(1.183)
Averaged total stresses are given by
hσpi=φp˜τp−φp˜pp11−φp˜pg11 (1.184)
hσgi=φg˜τg−φg˜pg11,(1.185)
where ˜ppis an ’extra stress’ due to collision of particles and
˜τα= 2µαΠα+ (µα,b −2
3µα)·tr(Πα)11,(1.186)
where µαis the shear viscosity and µα,b is the bulk viscosity of phase αand
Πα=1
2h∇¯
uα+ (∇¯
uα)ti.(1.187)
Furthermore assuming that the interfacial momentum exchange term consist
of bouyancy and viscous drag term, i.e.,
Mg= ˜pg∇φg+B(¯
up−¯
ug)
Mp=−Mg
we get
∂
∂t (φg˜ρg¯
ug) + ∇ · (φg˜ρg¯
ug¯
ug) = −φg∇˜pg+∇ · (φg˜τg)
+Dg+φg˜
Fg(1.188)
∂
∂t (φp˜ρp¯
up) + ∇ · (φp˜ρp¯
up¯
up) = −φp∇˜pg+∇ · [φp(˜τp−˜pp11)]
−Dg+φp˜
Fp,(1.189)
where
Dg=B(¯
up−¯
ug).(1.190)
52
1. Equations of multiphase flow
Both the shear viscosity and the solids ’extra stress’ consist of a kinetic part
and a collisional part. The extra stress can be written as follows
˜pp= ˜ρpΘp[1 + 2(1 + e)φpgo],(1.191)
where Θpis the granular temperature describing the fluctuating kinetic en-
ergy of the solid material
Θp=1
3trhδupδupi(1.192)
and eis the coefficient of restitution for particle collisions and gois the radial
distribution function given by [DG90]
go=3
5"1− φp
φp,max !1/3#−1
,(1.193)
where φp,max is the maximum packing of the solids. Several alternative forms
of the radial distribution function have been proposed in the literature.
The shear viscosity can be written as a sum of the kinetic part and the
collisional part as follows
µp=µp,kin +µp,col,(1.194)
where [LSJC84]
µp,col =4
5φp˜ρpdpgo(1 + e)rΘp
π(1.195)
and [GBD92]
µp,kin =10√π˜ρpdppΘp
96φp(1 + e)go"1 + 4
5goφp(1 + e)#2
(1.196)
Also for the shear viscosity several alternatives exist, with biggest differences
in the dilute regions.
The bulk viscosity expresses the resistance against compression and is
given by
µp,b=4
3φp˜ρpdpgo(1 + e)rΘp
π.(1.197)
The solids pressure, the shear viscosity and the bulk viscosity above are all
given as functions of the granular temperature Θ. The following transport
equation has been derived for the fluctuating kinetic energy thus yielding
the granular temperature [DG90]
3
2"∂
∂t (φp˜ρpΘp) + ∇ · (φp˜ρpΘpup)#(1.198)
= (−pp11 + τp) : ∇up+∇ · (kΘ∇Θp)−γΘ+φΘ,
53
1. Equations of multiphase flow
where the first term on the right hand side presents the generation by local
acceleration of particles, the second term the diffusion of Θ, the third term
the dissipation of Θ and the fourth term the exchange between gas and solid
phases. Several different closure relations for these different terms have been
suggested (see the review in [Boe97]).
The diffusion coefficient can be divided as follows [GBD92]
kΘ=kΘ,dilute +kΘ,dense,(1.199)
where
kΘ,dense = 2φ2
p˜ρpdpg0(1 + e)rΘp
π(1.200)
and
kΘ,dilute =75
192
˜ρpdppπΘp
(1 + e)g0"1 + 6
5(1 + e)g0φp#2
.(1.201)
The dissipation of fluctuating energy can be described as [JS83]
γΘ= 3(1 −e2)φ2
p˜ρpg0Θp 4
dprΘp
π− ∇ · ¯
up!(1.202)
and the interphase exchange term as [DG90]
φΘ=−3BΘp.(1.203)
Due to the time consumption of the solution of an extra field equation, an
algebraic equation is often used for calculation of the granular temperature.
It is based on the assumption that there is a local equilibrium and all terms
but the generation and dissipation of granular temperature can be neglected.
The resulting algebraic equation is ([SRO93], cited in [Boe97])
pΘp=−K1φptr(Πp)
2K4φp
(1.204)
+r(K1φp)2tr2(Πp) + 4K4φphK2tr2(Πp) + 2K3tr(Π2
p)i
2K4φp
,
where
K1= 2(1 + e) ˜ρpgo+˜ρp
φp
(1.205)
K2=4dp˜ρp(1 + e)φpgo
3√π−2
3K3(1.206)
K3=dp˜ρp
2(√π
3(3 −e)[1 + 0.4(1 + e)(3e−1)φpgo] + 8φpgo(1 + e)
5√π)(1.207)
54
1. Equations of multiphase flow
K4=12(1 −e2)˜ρpgo
dp√π(1.208)
The typical applications of the kinetic theory of granular flow are in bubbling
and circulating fluidized beds. In dense flows of this type the inter-phase
drag forces require special closure relations that are based on measurements
in dense suspensions. For bubbling beds the models are based on the mod-
els for packed beds (Ergun equation [Erg52]) and on the measurements of
Richardson and Zaki who studied liquid-solid fluidization [RZ54]. For cir-
culating fluidized beds it is often necessary to apply clustering corrections
to the drag forces [Boe97]. In bubbling beds the inter-particle distances
are short and particle-particle collisions dominate the flow. In very dense
regions the inter-particle friction can dominate over the fluctuating motion
and special description of the shear stress term is required [Boe97]. In circu-
lating fluidized beds the gas phase turbulence and particle-gas interactions
are important and corrections to the drag force and several others of the
closure relations above may be necessary [BBS95].
1.6.4 Constitutive equations for the mixture model
The equations of the mixture model were derived from the general mul-
tiphase equations in Section 1.4. For practical applications, constitutive
relations are needed for the diffusion velocity and the viscous and turbulent
stresses. Some formulations of the constitutive relations are given below.
Diffusion velocity
In a liquid-particle suspension, the momentum source term Dpcan be writ-
ten in the form
Dp=φpFp
Vp
,(1.209)
where Vpis the particle volume. Neglecting all other effects except the
viscous drag and assuming spherical particles, we can write in the Stokes
regime
Fp= 3πdpµcucp.(1.210)
Using Eqns. (1.209) and (1.210) together with (1.102), the expression for
the slip velocity is
ucp =d2
p(ρp−ρm)
18µc"g−(¯
um· ∇)¯
um−∂
∂t ¯
um#(1.211)
and the diffusion velocity follows from Eqn. (1.103). For larger particle
Reynolds numbers, Eqn. (1.210) must be replaced by a corresponding model
for the drag force.
55
1. Equations of multiphase flow
Viscous stress
The viscous stress tensor is approximated in the mixture model by an ex-
pression analogous to incompressible single phase flow
τm=µm[∇¯
um+ (∇¯
um)t] (1.212)
It should be noted that the apparent viscosity of a suspension µmis not a
well defined property of the mixture, but depends on many factors, including
the method of measurement. However, it turns out that, at reasonably low
concentrations, it can be correlated in a simple way to the concentration.
In mixture model applications, the most often used correlation for the
mixture viscosity is that according to Ishii and Zuber [IZ79]; see also [IM84]
for a summary of the results. The general expression for the mixture vis-
cosity, valid for solid particles as well as bubbles or drops, is given by
µm=µc 1−φp
φpm !2.5φpmµ∗
(1.213)
where φpm is a concentration for maximum packing. For solid particles
φpm ≈0.62. In Eqn. (1.213), µ∗= 1 for solid particles and
µ∗=µp+ 0.4µc
µp+µc
(1.214)
for bubbles or drops. Numerous other correlations for the viscosity of solid
suspensions are presented in the literature. Rutgers [Rut62] presented a
review of various empirical formulas for the relative viscosity. One of the
correlations with a theoretical foundation is due to Mooney (cited in [Rut62])
ln µm
µc!=2.5φp
1−1.4φp
(1.215)
Turbulence
In the mixture model the effects of turbulence appear in the mixture mo-
mentum equation as part of the general stress term. Additionally, turbulent
effects appear in the fluid-particle interaction term and in the fluctuating
components of particle velocity, i.e., as a turbulent stress in the particle
momentum equation.
In turbulent multiphase flow at low loadings, it can be assumed that
both the viscous stresses of the carrier phase and the turbulent stresses of
the particulate phase are negligible. The effective viscosity of the continuous
phase can then be calculated directly from a turbulence model for the con-
tinuous phase, e.g., from the k−model. There a correction can be applied
56
1. Equations of multiphase flow
to reduce the turbulence intensity. The models for correcting turbulent vis-
cosity are unfortunately very uncertain. In some cases the results with the
corrections can be less accurate than those obtained using the standard k−
model. Therefore, simulations should be performed without the corrections
and with various correction methods.
In turbulent flows the fluid particle interaction force should be written
in the form
Dp=−B¯
ucp +D0
p(1.216)
where D0
pis the fluctuating part of the force that causes particle dispersion.
In turbulent flow the term D0
pshould be added to the right hand side of
Eqn. (1.100).
In the original multiphase equations, the turbulent dispersion terms are
included in the momentum equations of dispersed phases, Eqn. (1.28). In
the mixture model, the influence of turbulence must be contained in the
equation for the diffusion velocity. This implies additional terms in (1.211)
due to the turbulent stresses and the fluctuation part of Dp. Instead of
developing a model for those terms directly, we adopt a simpler and more
intuitive method. The main effect of the slip velocity is the diffusion term
introduced in the continuity equation of the dispersed phase. If the turbulent
terms are taken into account, other terms will appear in the continuity
equation representing the turbulent diffusion. The simplest way is to model
these terms as Fickian diffusion. The continuity equation of the particulate
phase is then
∂
∂t (φp˜ρp) + ∇ · (φp˜ρp¯
up) = ∇ · (Dmp∇φp)−∇·(φp¯ρp¯
ump),(1.217)
where Dmp is a dispersion (or diffusion) coefficient.
One simple way to estimate the dispersion coefficient is to estimate it
from the turbulent particle viscosity as follows ([SCQM96])
Dmp =µT
p
σT
p˜ρp
,(1.218)
where σT
pis the turbulent particle Schmidt number for which values of, e.g.,
0.34 and 0.7 have been suggested.
Csanady’s model [Csa63b] for the turbulent diffusion takes into account
the crossing trajectory effect. It was developed for a special boundary layer
application under special assumptions. Csanady’s model is still today the
best model available for predicting particle dispersion and has shown to be
fairly accurate even for other applications. The following equation is based
on Csanady’s work and the k−model [PBG86]
Dmp =νT
c 1 + 0.85 ¯u2
cp
2k/3!−1/2
,(1.219)
57
1. Equations of multiphase flow
where 0.85 is an empirical value determined from particle dispersion data.
The equation above is based on the assumption that the carrier phase tur-
bulent Schmidt number is equal to one. In addition, the turbulent Schmidt
number of the dispersed phase was defined using the carrier phase diffusivity
as a reference.
1.6.5 Dispersion models
The theoretical treatment of particle dispersion was discussed earlier in Sec-
tion 1.5.2. There, this treatment was further supplemented with the most
common approach utilized in numerical simulations involving turbulent par-
ticle dispersion, i.e., the eddy interaction model. In that context the two
extreme conditions of particles almost following the turbulent fluid motions,
and heavy particles crossing the eddies were mentioned. In the following,
an analytic approach of particle dispersion is given for these two limiting
cases. They constitute the only theoretical conditions for which practical
numerical models can be compared.
Linearized equation of motion
To examine the relation between fluid and particle correlations or spectral
densities we need to solve the equation of motion of a particle. To this end
we study the BBO-equation (1.109), which is a nonlinear integro-differential
equation. For the Stokes regime Rep<1 the nonlinearity is caused mainly
by the pressure gradient term (II). By using the Lumley’s approximation
for this term [Lum57] the BBO-equation for the velocity component ican
be written as
dvL,i
dt−bduL,i
dt−2
3νf
∂2uL,i
∂xj∂xj
+ab (vL,i −uL,i) + 2
3a(vL,j −uL,j)∂uL,i
∂xj
+br3a
πZt
t0
d
dτ{vL,i(τ)−uL,i (τ)}
√t−τdτ+fi= 0 .(1.220)
In this equation fidenotes an external field force per unit effective mass
(real mass + virtual mass) of the particle. The parameters aand bare given
by
a=3νf
r2
p
and b=3ρf
2ρp+ρf
.(1.221)
Hinze [Hin75] pointed out that the nonlinearity becomes negligible if the
term including derivative in the third term of Eqn. (1.220) is very small,
i.e.,2
3a
∂uL,i
∂xj1.(1.222)
58
1. Equations of multiphase flow
This requirement is fulfilled if the particle size is small as compared to the
Taylor microscale. However, the restrictions of validity of the particle equa-
tion of motion in turbulent field, set by Eqns. (1.115), already includes this
requirement. It was further realized by Hinze that additional simplification
is possible if the latter part of also the second term of Eqn. (1.220) is neg-
ligible. It is reasonable to assume that this requirement is also met if the
particle size is sufficiently small as compared to the turbulent structures.
Implicit in both these simplifying conditions is the assumption that the par-
ticle resides in a locally uniform fluid field. In other words, it is restricted
to move in the strongly correlated region of the turbulent field (inside an
eddy) for a time comparable to the eddy decay time.
In what follows we consider an arbitrary coordinate direction iin a coor-
dinate system where the correlation tensors and spectral densities are diag-
onal (see discussion at the end of Sect. 1.5.2), and omit subscript i. Under
the assumptions made above, the linearized BBO-equation for a velocity
component can be written as
dvL
dt−bduL
dt+ab(vL−uL)
+br3a
πZt
t0
d
dτ{vL(τ)−uL(τ)}
√t−τdτ+f= 0 .(1.223)
By defining the Fourier transforms of the fluid and particle velocity compo-
nents as
ˆuL(ω) = Z∞
−∞
uL(t) exp(−iωt) dt
ˆvL(ω) = Z∞
−∞
vL(t) exp(−iωt) dt , (1.224)
and by applying them in Eqn. (1.223), leads to simple relationship between
the velocities [Cha64]
ˆvL=na+q3aω
2o+inω+q3aω
2o
na+q3aω
2o+inω
b+q3aω
2oˆuL.(1.225)
Multiplying each side of this equation by its complex conjugate and applying
ensemble averaging on the resulting equation gives
hˆvL(ω) ˆv?
L(ω)i
hˆuL(ω) ˆu?
L(ω)i=Ω(1)
Ω(2)
,(1.226)
where
Ω(1) ω
a=ω
a2+√6ω
a3/2+ 3 ω
a+√6ω
a1/2+ 1 (1.227)
Ω(2) ω
a, b=1
b2ω
a2+√6
bω
a3/2+ 3 ω
a+√6ω
a1/2+ 1 .
59
1. Equations of multiphase flow
The Lagrangian spectral densities (see Eqn. (1.126)) can be given in terms
of the Fourier transformed velocities as
ELf(ω) = hˆuL(ω) ˆu?
L(ω)i
hu2
Li(1.228)
ELp(ω) = hˆvL(ω) ˆv?
L(ω)i
hv2
Li.
It thus follows that ELp
ELf
=hu2
Li
hv2
Li
Ω(1)
Ω(2)
.(1.229)
Consequently, the (diagonal elements of) the mean square displacement and
dispersion tensors of the particle, Eqns. (1.127) and (1.128), can be given
in terms of variables defined by the turbulent field of the fluid as
hy2
L(t)i=hu2
LiZ∞
0
Ω(1)
Ω(2)
ELf(ω)2(1 −cos ωt)
ω2dωand (1.230)
Dp=hu2
LiZ∞
0
Ω(1)
Ω(2)
ELf(ω)sin ωt
ωdω . (1.231)
A number of conclusions concerning Eqns. (1.230) and (1.231) can be made
without actually solving the spectral densities or calculating the integrals.
For the low and high frequency limits, the amplitude ratio of the phases is
seen to behave as
ω
a→0,Ω(1)
Ω(2) →1 and (1.232)
ω
a→ ∞,Ω(1)
Ω(2) →b2.(1.233)
For the ratio of the effective masses of the phases b, it is observed that in
the case of very heavy particles, where ρf/ρp→0, and in the case of equal
densities, where ρf=ρp, the amplitude ratio is given by
b→0,Ω(1)
Ω(2) →0 and (1.234)
b= 1 ,Ω(1)
Ω(2)
= 1 ,(1.235)
respectively. These results are all physically very plausible. Furthermore,
the asymptotic values of the dispersion coefficient for short and long disper-
sion times are given by
lim
t→0Dp=hu2
Litand lim
t→∞ Dp=π
2hv2
LiELp(0) .(1.236)
60
1. Equations of multiphase flow
The relative degree of the dispersion of particles and fluid is given by
Dp
Df
=hv2
Li
hu2
LiRt
0RLp(τ) dτ
Rt
0RLf(τ) dτ=hv2
Li
hu2
LiR∞
0
sin ωt
ωELp(ω) dω
R∞
0
sin ωt
ωELf(ω) dω,(1.237)
For short and long dispersion times, the limiting behavior of that ratio is
given by
lim
t→0
Dp
Df
=hv2
Li
hu2
Liand lim
t→∞
Dp
Df
= 1 .(1.238)
The latter limit results since for ω= 0, ELp(0) = ELf(0) and Ω(1) = Ω(2)
(see Eqn. (1.232)). These asymptotic expressions for both short and long
dispersion times should hold regardless of the detailed form of either the
Lagrangian temporal correlation or power spectral density.
The equality of the particle dispersion and the fluid diffusion at long
dispersion times is a consequence of the assumptions made while lineariz-
ing the particle equation of motion. As stated above, these assumptions
indicate that the particle moves in the strongly correlated area of the fluid
eddy, since it’s motion has been related to the surrounding fluid through the
Lagrangian correlation coefficient. This is plausible if the particle is very
small as compared to the fluid eddy or if the densities of the phases are close
to each other. Considerably heavier or lighter particles can not follow the
fluid motion. However, if the particles are very much smaller than the fluid
eddies, e.g. dust particles in the athmosphere, the assumptions may still be
valid.
Crossing trajectories
In general, a heavy particle can have a significant mean velocity relative to
the fluid, while a particle with a density comparable with that of the fluid
tends to follow the motion of the fluid more closely. Therefore, a heavy
particle continuously changes it’s fluid neighborhood and drifts away from
the fluid eddy in a time scale that is small as compared with the eddy decay
time. The velocity correlation of heavy particles with the surrounding fluid
thus decreases rapidly and their dispersion rate is low. This phenomenon is
known as the effect of crossing trajectories.
The approach of Csanady [Csa63b, Csa63a] is essentially based on an ex-
tended form of Taylor’s hypothesis according to which the Eulerian temporal
velocity correlation in the streamwise direction approaches the correspond-
ing Eulerian spatial velocity correlation as the ratio of turbulent velocity
fluctuations to mean velocity approaches zero. Within Csanady’s extension
it is assumed that in a frame of reference moving with the mean flow, the La-
grangian spatial velocity correlation can be approximated by the Eulerian
temporal velocity correlation. Both theoretical and experimental support
to this approximation exist. However, by analogy with Taylor’s hypothesis,
61
1. Equations of multiphase flow
mean relative velocity of particles exceeding the standard deviation of veloc-
ity fluctuations at least by a factor of four is required for the approximation
to be valid.
In hes study of atmospheric dispersion Csanady used a coordinate system
where x1is the horizontal component (direction of atmospheric wind), x2is
the span-wise component and x3is the vertical component (upwards). Based
mainly on experimental observations [Csa63a] he expressed the Lagrangian
spatial correlation functions in the three directions as
Rλ
Lp,11 = exp −W3τ
L(1.239)
Rλ
Lp,22 =1−W3τ
2Lexp −W3τ
L(1.240)
Rλ
Lp,33 = exp
−τ
LsW2
3+hw2
3i
β2
.(1.241)
Here Lis the characteristic size of an eddy (assumed to be approximately
the same in all directions), W3is the vertical component of mean relative
velocity and w3it’s fluctuating part. The coefficient βis given by
β=phw2
3iTL
L,(1.242)
where TLthe Lagrangian time scale. Using the exponential approximation
of the correlation function Eqn. (1.241) and Taylor’s theorem Eqn. (1.122)
the mean square displacement in the vertical direction can be expressed in
the non-dimensional form as
η= 2ξ1−1
ξ(1 −exp(−ξ)),(1.243)
where the dimensionless displacement ηand the dimensionless time ξare
defined by
η=hy2
L,3(t)i
hw2
3iL2W2
3+hw2
3i
β2
ξ=t
LsW2
3+hw2
3i
β2.
The asymptotic long time vertical dispersion coefficient now becomes (see
Eqn. (1.128))
lim
t→∞ Dp,3=D∞
p,3=hw2
3iL
qW2
3+hw2
3i
β2
.(1.244)
62
1. Equations of multiphase flow
Therefore, at large mean relative velocity the mean square displacement and
the long time dispersion coefficient are seen to be proportional to 1/W3. By
dividing Eqn. (1.244) with the long time diffusivity of the fluid we finally
obtain D∞
p,3
D∞
f,3
=1
r1 + β2W2
3
hw2
3i
.(1.245)
Similarly, for dispersion in the horizontal and span-wise directions we find
D∞
p,1=hw2
1iL
W3
(1.246)
D∞
p,2=1
2hw2
2iL
W3
,(1.247)
and
D∞
p,1
D∞
f,1
=1
r1 + β2W2
3
hw2
3i
(1.248)
D∞
p,2
D∞
f,2
=1
r1 + 4β2W2
3
hw2
3i
.(1.249)
Notice, however, that the equation for horizontal direction is derived here
exactly as for vertical direction, Eqn. (1.245), whereas the equation of lateral
direction is written purely by analogy so as to give the correct behavior in
the limits W3→0 and W3→ ∞.
63
2. Numerical methods
2.1 Introduction
Since the various multiphase flow models introduced in the previous chapter
are, in general, complex set of heavily coupled nonlinear equations it is
impossible, without dramatic simplifications, to get an analytic solution to
any of those models. Without an exception numerical methods are needed.
Most of the numerical work done on the multiphase flow equations has
been focused to particle tracking algorithms (see section 1.5 for the theory
of particle tracking) and solution of multifluid equations (1.57) and (1.58).
We thus restrict ourselves to these traditional topics in this monoghraph.
We also give a brief introduction to novel numerical methods that are espe-
cially suitable for direct simulation of certain types of multiphase flows in a
’mesoscopic’ scale. These include the lattice-gas, the lattice-Boltzmann and
the dissipative particle dynamics methods.
Until now, the numerical development of multifluid equations is done
mainly within the finite difference method (FDM) or within the finite volume
method (FVM). Applications using finite element method (FEM) seem to be
less extensively studied. Independently of the applied discretization method
the nature of the multifluid equations will end up to the same difficulties
in the numerical solution procedures. While having the same mathematical
form as the one phase Navier-Stokes equations also the same problems are
encountered, including the problems of pressure-velocity coupling and domi-
nating convection. Furthermore in multifluid flow equations, the inter-phase
coupling terms pose specific demands to the numerical algorithms.
Contrary to the multifluid flow models the particle tracking reveals no
specific theoretical difficulties in numerical methods. The particle tracks are
computed by numerical integration of the equation for particle motion and
then the set of ordinary differential equations is solved along these tracks.
It is a relatively easy method with some technical tricks and can be adopted
to any existing one phase flow solver.
As compared to conventional methods discussed above, mesoscopic sim-
ulation methods provide a completely different approach to multiphase flows
as they do not resort to solving the averaged continuum equations. Instead,
they are based on describing the fluid in terms of a large number of ’particles’
64
2. Numerical methods
that move and collide in a discrete lattice.
In sections 2.2 and 2.3 numerical solution algorithms based on the finite
volume method (FVM) and finite element method (FEM) of the continuum
multifluid flow equations are represented. In section 2.2 FVM algorithms
and schemes are given in detail using Body-Fitted Coordinates. Also the
methods, like pressure correction algorithm SIMPLE, mimicked from the
one phase context are extended to cover multifluid equations. In section
2.3 the stabilized FEM algorithms are introduced. The presentation covers
twofluid flow equations but can easily be extended to multifluid flow equa-
tions. In section 2.4 an overview of the basic numerical methods needed in
particle tracking and finally in section 2.5 a brief introduction to mesoscopic
numerical methods is given.
Balance equations
In what follows we will consider the following continuum multifluid flow
equations
∂
∂t (φα˜ρα) + ∇ · (φα˜ρα¯
uα) = 0 (2.1)
∂
∂t (φα˜ρα¯
uα) + ∇ · (φα˜ρα¯
uα¯
uα)−∇·φαµα[∇¯
uα+ (∇¯
uα)t]
=−φα∇˜p+
Np
X
β=1
Bαβ(¯
uβ−¯
uα) + φα˜
Fα.
These equations are derived from the general multifluid equations (1.57)
and (1.58) neglecting the pseudo-turbulent stress and assuming that all the
phases behave like Newtonian fluids, i.e.,
∇ · hταi=φαµα[∇¯
uα+ (∇¯
uα)t].(2.2)
Furthermore the momentum exchange term Dαis assumed to be due to
inter-phase drag only, i.e.,
Dα=
Np
X
β=1
Bαβ(¯
uβ−¯
uα).(2.3)
Using a general dependent variable Φαthe flow equations for phase αcan
be given in a generic form
∂
∂t (φα˜ραΦα) + ∇ · [φα( ˜ρα¯
uαΦα−Γα∇Φα)] (2.4)
=
Np
X
β=1
BI
αβ(Φβ−Φα) + Sα.
The continuity equation and the momentum equation for phase αfollow
from Eqn. (2.4) by setting Φα= 1 and Φα=¯
uα, respectively.
65
2. Numerical methods
2.2 Multifluid Finite Volume Method
The development of numerical methods for multifluid flow equations within
FVM methods is based heavily on the extensions of single phase flow algo-
rithms, such as the pressure-correction algorithm SIMPLE and its improve-
ments to the multifluid context [Spa77, Spa80, Spa83, Kar02].
As discussed in the previous chapter, multifluid flow equations typically
include for each phase equations that are formally similar to the conven-
tional single fluid equations (unless simplified models such as the mixture
model is considered), except of additional terms that arise due to the pres-
ence of other phases. These terms provide coupling between the equations
and pose one of the major problems for modelling and for numerical solu-
tion of the multifluid flow equations. The coupling terms are often expressed
in the form of a coupling constant times the difference between the values
of the dependent variable of the two phases. When the coupling is strong
the coupling constant is large and the value of the dependent variables in
different phases are nearly equal. This condition with a large number multi-
plying a very small number often leads to convergence problems. The Partial
Elimination Algorithm (PEA) developed by Spalding [Spa80] for twophase
equations and the SINCE algorithm [Lo89, Lo90] for multifluid equations
handles this problem by separating the solution of the phases of the depen-
dent variable. This can be done effectively in the cell level. Further details
of SINCE algorithm can be found in Ref. [KL99].
Until now the multifluid algorithms in FVM are based on structured
grid approach. Complex geometries are handled by changing to Body Fit-
ted Coordinates (BFC) and by using multi-block grids. A better way to
approximate the physical geometry would be to use an unstructured grid
where local refinements and adaptation are possible. Also, the ease of grid
generation in complex geometries using automatic generation algorithms is
a tempting feature of unstructured grids. Within FVM, the development of
unstructured grid solvers has been distinct from the development of struc-
tured solvers where the benefits of the grid structure are exploited. More
recently, development of unstructured grid FVM solvers has followed same
guidelines as the development of structured grid solvers. In this method
[MM97] the values are stored, like in structured solvers, in the cell centers
and the use of the so called reconstruction gradient enables the construction
of higher order schemes. Earlier, the problem of pressure oscillation was
dealt with a staggered grid for velocity components and pressure. Recently
this unwanted staggering (especially in complex geometries) is overcome
using the Rhie-Chow algorithm [RC83, BW87], where the cell-boundary ve-
locity components are obtained by interpolating the discrete momentum
equations instead of the velocity values. In Control Volume Finite Element
Method (CVFEM) by Baliga and Patankar [BP80] finite volume equations
and element by element assembling, typical to FEM, are combined. In this
66
2. Numerical methods
method the variables are attached to discretization points similarly to FEM.
For this method some development is done also in twophase context [MB94].
In this Section the FVM method in general Body-Fitted Coordinates
(BFC) for multifluid equations is represented. Pressure stabilization is dealt
with the Rhie-Chow algorithm, where the cell-boundary velocity compo-
nents are obtained by interpolating the discrete momentum equations in-
stead of the velocity values. Three different treatments of the inter-phase
coupling are given including explicit algorithm, PEA algorithm for two phase
situation and an extension of the PEA algorithm to multifluid equations.
The solution of the volume fraction equations is obtained from the scheme
which ensures the volume fraction sum to unity. The pressure-velocity cou-
pling is handled with the SIMPLE method which extension to multifluid
equations is given in detail. In the end the overall algorithm called IPSA
(Inter Phase Slip Algorithm) for the solution of the system of multifluid flow
equations is given.
2.2.1 General coordinates
In order to facilitate the numerical solution of the macroscopic balance equa-
tions (2.1), (2.3) and (2.4) in a general 3D geometry using Body-Fitted Co-
ordinates (BFC), these equations are expressed in the covariant tensor form.
The coordinate system used in this context is the local non-orthogonal coor-
dinate system (ξ1, ξ2, ξ3), referred to as the computational space, obtained
from the Cartesian coordinate system (x1, x2, x3), i.e. the physical space,
by the curvilinear coordinate transformation xi(ξj) (Fig. 2.1). Because
there is no danger of confusion, the phase and mass-weighted phase aver-
age notations above, e.g., ˜ραand ¯
uα, have been omitted in the subsequent
treatment in order to enhance the readability of the equations. Thus the
balance equations in the computational space are
∂
∂t (|J|φαρα) + ∂
∂ξi(ˆ
Ii
mα) = 0 (2.5)
∂
∂t (|J|φαραuk
α) + ∂
∂ξi(ˆ
Ii
uk
α) = −φαAi
k
∂p
∂ξi+∂
∂ξi(φαµα
Ai
mAj
k
|J|
∂um
α
∂ξj)
+
Np
X
β=1 |J|Bαβ (uk
β−uk
α) + |J|φαFk
α(2.6)
∂
∂t (|J|φαραΦα) + ∂
∂ξi(ˆ
Ii
Φα) =
Np
X
β=1 |J|Bαβ (uk
β−uk
α) + |J|Sα,(2.7)
where |J|is the Jacobian determinant of the mapping xi→ξj(xi) and the
total normal phasic fluxes ˆ
Ii
mα,ˆ
Ii
uk
αand ˆ
Ii
Φαare defined as
ˆ
Ii
mα=φαραˆui
α,(2.8)
67
2. Numerical methods
ξ1
ξ1
ξ2
ξ2
x→ξ(x)
x1
x2
Figure 2.1: A single block of a collocated BFC-grid in the physical space
(x1, x2, x3) and in the computational space (ξ1, ξ2, ξ3) with the dummy cells
on the boundaries.
ˆ
Ii
uk
α=φα(ραˆui
αuk
α−µα
Ai
mAj
m
|J|
∂uk
α
∂ξj),(2.9)
ˆ
Ii
Φα=φα(ραˆui
αΦα−µα
Ai
mAj
m
|J|
∂Φα
∂ξj).(2.10)
The notations ˆui
α=Ai
mum
αand Ai
min the above equations stand for the
normal flux velocity component and the adjugate Jacobian matrix of the
mapping xi→ξj(xi). The area vectors of the surfaces of an elementary
grid cell A(i)(Fig. 2.2), i.e., the vectors pointing to the outward normal
direction of the cell faces and having the magnitude of the length equal to
the magnitude of the area of these faces, are obtained as
A(1) =e(2) ×e(3),A(2) =e(3) ×e(1) and A(3) =e(1) ×e(2),(2.11)
where e(i)are the contravariant basis vectors associated with the curvilinear
frame ξi(xj) (tangential to the coordinate curves). Covariant basis vectors
(normal to the coordinate surfaces) are denoted by e(i). The cartesian com-
ponents of contravariant vectors are given by
e(i)k=Jk
i.(2.12)
With the help of these area vectors the volume of elementary grid cell VPis
then related to the Jacobian determinant as
A(i)·e(j)=e(1) ·e(2) ×e(3)δi
j=VPδi
j=|J|δi
j.(2.13)
68
2. Numerical methods
Because the two frames of basis vectors are dual to each other e(i)·e(j)=δi
j
the Eqn. (2.13) can be written in the form
A(i)·e(j)=|J|e(i)·e(j).(2.14)
It can be shown that the cartesian components of the area vectors A(i)
kare
determined by the adjugate Jacobian matrix as
A(i)
k=Ai
k.(2.15)
The necessary information to perform the coordinate transformation in-
cludes only the volumes and the cartesian components of the area vectors
of the grid cells in the physical space. This information is completed by cal-
culation of the Jacobian determinant |J|and the adjugate Jacobian matrix
Ai
kof the transformation.
P
A(1)
A(2)
A(3)
x→ξ(x)
ξ1
ξ2
ξ3
e(1)
e(2)
e(3)
u
d
n
s
e
w
U
D
N
S
E
W
1
1
1
Figure 2.2: Area vectors on the faces of the finite-volume cell in the physical
space and the notation of the neighboring cell centers and cell faces in the
computational space.
2.2.2 Discretization of the balance equations
Within the coordinate transformation all the derivatives of the macroscopic
balance equations in the physical space have been expressed with the cor-
responding terms in the computational space, in which the transformed
macroscopic balance equations are discretized. Now we are free to use any
discretization method we prefer. In what follows the conservative finite vol-
ume approach with collocated dependent variables is applied.
69
2. Numerical methods
Integration of the generic balance equation over a single finite volume
cell (Fig. 2.2) in the computational space and the use of Gauss’ law results
in
ZVc
∂
∂t (|J|φαραΦα) dVc+ZAc
ˆ
Ii
Φα·ndAc=ZVc
Np
X
β=1 |J|BI
αβ (Φβ−Φα) dVc
+ZV c |J|SαdVc,(2.16)
where Vcis the volume of cell and Acis the surface of that cell in the
computational space.
In a same way the macroscopic balance equations (2.5), (2.6) and (2.7)
can be formally represented as
{|J|φαρα}
∆tt+∆t
t
+hˆ
I1
mαiu
d+hˆ
I2
mαin
s+hˆ
I3
mαie
w= 0 (2.17)
{|J|φαραuk
α}
∆tt+∆t
t
+hˆ
I1
uk
αiu
d+hˆ
I2
uk
αin
s+hˆ
I3
uk
αie
w=−φα|J|∂p
∂ξi
+"φαµα
A1
mAj
k
|J|
∂um
α
∂ξj#u
d
+"φαµα
A2
mAj
k
|J|
∂um
α
∂ξj#n
s
+"φαµα
A3
mAj
k
|J|
∂um
α
∂ξj#e
w
+{|J|φαFk
α}
+
Np
X
β=1{|J|Bαβ (uk
β−uk
α)}(2.18)
{|J|φαραΦα}
∆tt+∆t
t
+hˆ
I1
Φαiu
d+hˆ
I2
Φαin
s+hˆ
I3
Φαie
w
=
Np
X
β=1{|J|BI
αβ (Φβ−Φα)}+{|J|Sα}.(2.19)
In the above equations the notation { } indicates that the operand is in-
tegrated over the volume of the finite volume cell in question. Because the
computational space cells all are of unit volume, the resulting mean value
is directly expressed per unit volume basis. In addition, the notation [ ]u
d
expresses a difference between the values of the operand at the specified
faces or time, e.g.,
[Ii
Φα]u
d=Ii
Φαu−Ii
Φαd.(2.20)
When spatial integration over cell face is considered the right hand side
notation Ii
Φαuimplies that the operand is integrated over the respective
face (face corresponding to point uin this case) of the finite volume cell.
70
2. Numerical methods
Integration over finite volume cell
It is seen from the Eqns. (2.17), (2.18) and (2.19) that the integration
of the operand over the finite- volume cell noted as {Φ}is applied to the
terms including time derivatives and to the source terms. To perform the
integration the real field of the operand inside the cell is approximated with
a constant value found at the center of the cell. So we apply a one point
integration rule. Because the discretization is done in the computational
space the integration results simply in
{Φ} ≡ Φ|PVc= Φ|P= ΦP.(2.21)
Integration over cell surface
The rest of the terms expressing the flux of the operand through the faces
of the finite volume cell are integrated over the cell face. For the averaging
the real profile of the operand over the face is approximated with a constant
value located at the center of the face. Thus we apply again one point
integration rule which results in
ZAi
c
Φ dAc≡Φ|cAi
c= Φ|c= Φc.(2.22)
Now the integral forms of the phasic fluxes can be represented as
[ˆ
Ii
mα]h
l=Ci
αh−Ci
αl(2.23)
[ˆ
Ii
uk
α]h
l=Ci
αhuk
αh−Ci
αluk
αl−Dij
αh
∂uk
α
∂ξjh
+Dij
αl
∂uk
α
∂ξjl
[ˆ
Ii
Φα]h
l=Ci
αhΦα|h−Ci
αlΦα|l−Dij
Φαh
∂Φα
∂ξjh
+Dij
Φαl
∂Φα
∂ξjl
,
where
i=
1 ; h=u, l =d
2 ; h=n, l =s
3 ; h=e, l =w
(2.24)
indicate the pair of faces in question. The phasic convection and diffusion
coefficients Ci
α,Dij
αand Dij
Φαare specified by the relations
Ci
αc=φα|cρα|cˆui
αc(2.25)
Dij
αc=φα|cµα|cGij |c(2.26)
Dij
Φαc=φα|cΓα|cGij |c,(2.27)
where
i=
1 ; c=u, d
2 ; c=n, s
3 ; c=e, w
(2.28)
71
2. Numerical methods
and
Gij =Ai
mAj
m
|J|.(2.29)
Values on cell faces
In order to calculate the flux terms [Φ]u
d, the material properties and the
values of the dependent variables at the centers of the cell faces are needed.
These values are found by a weighted linear interpolation scheme from the
cell center values. The weight factors used in this context are based on the
distances between cell centers and corresponding cell faces on the physical
space (Fig. 2.3). The value on the cell face is then given by
Φ|c= Φc= (1 −Wc)ΦP+WcΦC,(2.30)
where
Wc=∆P c
∆P c + ∆Cc (2.31)
and where c= (u, d, n, s, e, w) and C= (U, D, N, S, E , W). This scheme is
second order accurate in rectangular non-uniform meshes.
x→ξ(x)
u
u
u
u
u
d
dn
n
s
s
U
U
U
U
U
U
U
U
U
D
D
D
D
D
D
N
N
N
NN
N
S
S
S
S
S
S
P
P
P
P
P
∆
∆
∩
Figure 2.3: Illustration of the notations used in weighted interpolation.
Calculation of gradient at cell center
The source terms include the pressure gradients calculated at the cell centers.
These gradients are approximated by the same type of weighted interpola-
tion scheme like the one for values on the cell faces, but in order to enhance
72
2. Numerical methods
the accuracy the weight factors are in this case based on the arc lengths be-
tween the cell centers (Fig. 2.3). Although the calculation of the arc lengths
is computationally expensive, the Rhie-Chow interpolation method which is
used to provide the normal flux velocity components ˆui
αon the cell faces
requires second order accuracy also on the non-uniform curvilinear meshes.
Thus the gradient at the cell center is obtained as
∂Φ
∂xiP
=WLΦH+ (WH−WL)ΦP−WHΦL,(2.32)
where the weight factors are defined to be
WH=∩HP
∩P L(∩HP +∩P L)(2.33)
WL=∩P L
∩H P (∩HP +∩P L).(2.34)
Above the length of the arc for example from the point Pto point Lis
denoted by ∩P L. According to the Eqn. (2.32), the pressure gradient
source term in the discretized momentum equation is not calculated on the
computational space but directly from the physical space gradient.
Calculation of gradient on cell faces
The total normal phasic fluxes include gradients of the dependent variable
on the cell faces in their terms involved with diffusion fluxes. These gradients
are all discretized using central differences on computational space. For the
gradients normal to the cell faces the central difference method is applied as
∂Φ
∂ξic
= ΦC−ΦPwith i=
1; C=U, c =u
2; C=N, c =n
3; C=E, c =e.
(2.35)
In the case of cross-derivatives, i.e., derivatives on the plane of the cell face,
the gradients are approximated as the mean of the two central differences
calculated at the cell centers on the both sides of the face in question. As
an example, the cross-derivatives on the cell face uare (Fig. 2.3)
∂Φ
∂ξ2u
=1
4(ΦN−ΦS+ ΦUN −ΦUS ) (2.36)
∂Φ
∂ξ3u
=1
4(ΦE−ΦW+ ΦUE −ΦUW ).(2.37)
Approximation of the cross-derivatives connects eighteen neighboring cells
to the treatment of every finite volume cell. To reduce the bandwidth of
the coefficient matrix the treatment is reduced back to the usual one involv-
ing only the eight neighboring cells sharing a cell face with the cell under
73
2. Numerical methods
consideration by the deferred correction approach, i.e., treating the cross-
derivatives as source terms by using values from the previous iteration for
the dependent variables in question. Thus in addition to the terms related
to the gradients normal to the cell faces (the diagonal terms of the diffusion
tensor), the deferred correction approach results to the following additional
source terms expressing the effect of non-orthogonality
SD
Φα=D12
Φα
∂?Φα
∂ξ2+D13
Φα
∂?Φα
∂ξ3u
d
+D21
Φα
∂?Φα
∂ξ1+D23
Φα
∂?Φα
∂ξ3n
s
+D31
Φα
∂?Φα
∂ξ1+D32
Φα
∂?Φα
∂ξ2e
w
,(2.38)
where ?Φαdenotes the value of the dependent variable from the previous
iteration.
Following the details given above the formally discretized macroscopic
balance equations (2.17), (2.18) and (2.19) can now be written in the final
form. To enhance the readability and to keep more physical nature in the
discretized equations they are next given in a partly discretized form retain-
ing most of the terms in their previous form. Thus the macroscopic balance
equations of mass, momentum and generic dependent scalar variables are
C1
αu−C1
αd+C2
αn−C2
αs+C3
αe−C3
αw=−φα|Pρα|PVP
∆tt+∆t
t
(2.39)
X
c
auk
αcuk
αP=X
c,C
auk
αcuk
αC−φα|P
∂p
∂xkP
VP
+"φαµα
A1
mAj
k
|J|
∂um
α
∂ξj#u
d
+"φαµα
A2
mAj
k
|J|
∂um
α
∂ξj#n
s
+"φαµα
A3
mAj
k
|J|
∂um
α
∂ξj#e
w
+φα|P0Fk
αPVP+
Np
X
β=1
Bαβ |P(uk
βP−uk
αP)VP
+SD
uk
α−"φα|Pρα|Puk
αPVP
∆t#t
(2.40)
X
c
aΦα|cΦα|P=X
c,C
aΦα|cΦα|C+
Np
X
β=1
BI
αβ P(Φβ|P−Φα|P)VP
+0SαPVP
+SD
Φα−φα|Pρα|PΦα|PVP
∆tt
.(2.41)
In the Eqn. (2.41) 0Fk
αPand 0SαPare the constant part of the linearized
source terms. The summation symbols are defined as
X
c
Φ|c= Φ|u+ Φ|d+ Φ|n+ Φ|s+ Φ|e+ Φ|w(2.42)
74
2. Numerical methods
X
c,C
a|cΦ|C=a|uΦ|U+a|dΦ|D+a|nΦ|N
+a|sΦ|S+a|eΦ|E+a|wΦ|W.(2.43)
With hybrid differencing scheme the matrix coefficient from convective and
diffusive transport above can be written as
aΦα|h= max 1
2Ci
αh, Dii
Φαh−1
2Ci
αhi=
1; h=u
2; h=n
3; h=e
aΦα|l= max 1
2Ci
αl, Dii
Φαl+1
2Ci
αli=
1; l=d
2; l=s
3; l=w
(2.44)
In the hybrid differencing scheme the central differencing is utilized when the
cell Peclet number (Peα=Cα/Dii
Φα) is below two and the upwind differenc-
ing, ignoring diffusion, is performed when the Peclet number is greater than
two. This scheme together with the deferred correction approach associated
with the gradients on the cell faces guarantees the diagonal dominance of
the resulting coefficient matrix. For that reason hybrid differencing scheme
serves as the base method for which other more accurate schemes can be
built upon by the deferred correction approach [Com, LL94].
2.2.3 Rhie-Chow algorithm
In calculating the convection fluxes across cell faces, velocities have to be
inferred from those calculated at cell centers. A straightforward linear in-
terpolation, i.e.,
ui
αe=ui
α|e= (1 −We)ui
αP+Weui
αE(2.45)
would lead to the well known chequer-board oscillations in the pressure
field, since the use of 2δξ-centered differences for the computation of pres-
sure gradients at control cell centers effectively decouples the even and odd
grids. This problem is overcome by the Rhie-Chow algorithm where the cell
boundary velocities are obtained by interpolating the discrete momentum
equations.
The Cartesian velocity components uk
αP,uk
αEat control volume cen-
tered at Pand at Ecorrespondingly obey the discretized momentum equa-
tion (2.40) given in short hand form as
ui
αP+bαk
iP
∂p
∂xkP
=Si
αP(2.46)
ui
αE+bαk
iE
∂p
∂xkE
=Si
αE,(2.47)
75
2. Numerical methods
where
bαk
iC=φα|CAk
i
Pcaui
αc
(2.48)
Si
αC= "φαµα
A1
mAj
i
|J|
∂um
α
∂ξj#u
d
+"φαµα
A2
mAj
i
|J|
∂um
α
∂ξj#n
s
+"φαµα
A3
mAj
i
|J|
∂um
α
∂ξj#e
w
+φα|P0Fi
αPVP+
Np
X
β=1
Bαβ |P(ui
βP−ui
αP)VP+SD
ui
α
−"φα|Pρα|Pui
αPVP
∆t#t+∆t
t
. X
c
aui
αc!.(2.49)
In a staggered grid approach the pressure oscillation is avoided by discretiz-
ing the momentum equation on the grid whose centers are the faces of the
original grid. In this case the face velocity component obeys the discretized
momentum equation
ui
αe+bαk
ie
∂p
∂xke
=Si
αe,(2.50)
where the pressure gradients are calculated using 1δξ-centered differences.
In the Rhie-Chow method the solution of Eqn. (2.50) is approximated
interpolating linearly momentum equations (2.46) and (2.47). Approximat-
ing the right side term in Eqn. (2.50) by weighted linear interpolation of
the corresponding terms in Eqns. (2.46) and (2.47) we get
ui
αe+bαk
ie
∂p
∂xke
=Si
α|e=ui
α|e+bαk
ie
∂p
∂xke
.(2.51)
Assuming that bαk
ie≈bαk
ie, we obtain
ui
αe=ui
α|e+bαk
ie ∂p
∂xke−∂p
∂xke!.(2.52)
For the normal velocity components ˆui
α=Ai
kuk
αwe get
ˆui
αe= ˆui
α|e+ˆ
bik
αe ∂p
∂xke−∂p
∂xke!,(2.53)
where
ˆ
bik
αe=Ai
jbαk
je.(2.54)
Since all pressure gradients are computed using central differences, the cross-
derivative terms in Eqn. (2.53) cancel leaving the formula
ˆui
αe= ˆui
α|e+ˆ
bii
αe ∂p
∂xie−∂p
∂xie!.(2.55)
76
2. Numerical methods
2.2.4 Inter-phase coupling algorithms
The macroscopic phasic balance equations (2.39), (2.40) and (2.41) are cou-
pled to the corresponding balance equations of the other phases with the
transport related to the change of phase and with the interfacial force den-
sity. In many cases these couplings are strong which results into a very slow
convergence when iterative sequential solution methods are used. To solve
this problem some degree of implicitness is required in the treatment of the
interfacial coupling terms. The approaches are introduced here using the
equation of the generic dependent variable (2.41).
Explicit treatment
The only necessary operation in this approach is to transfer all the terms
depending on the variable to be solved, Φα, to the left side of Eqn. (2.41)
(implicit handling of Φα) and leave the other terms on the right side (explicit
handling of Φβ). The resulting linear equation can be expressed in the
following compact form
AΦαΦα|P=CΦα(2.56)
AΦα=X
c
aΦα|c+
Np
X
β=1
BI
αβPVP−1SαPVP
+φα|Pρα|PVP
∆t(2.57)
CΦα=X
c,C
aΦα|cΦα|C+
Np
X
β=1
BI
αβPΦβ|PVP
+0SαPVP+SD
Φα+φα|Pρα|PΦα|PVP
∆tt
.(2.58)
Partial Elimination Algorithm (PEA)
When there are only two phases to be considered simultaneously, the phasic
balance equations of the dependent variables Φαand Φβcan be written in
the form where they can be mathematically eliminated from the balance
equations of each other. The equations for the phases αand βare first
written as
AΦαΦα|P=BI
αβ(Φβ|P−Φα|P) + CΦα(2.59)
AΦβΦβ|P=BI
αβ(Φα|P−Φβ|P) + CΦβ,(2.60)
where the coefficient AΦα,AΦβ,Bαβ ,CΦαand CΦβare defined by the equa-
tions
AΦα=X
c
aΦα|c−1SαPVP+φα|Pρα|PVP
∆t(2.61)
77
2. Numerical methods
AΦβ=X
c
aΦβc−1SβPVP+φβ|Pρβ|PVP
∆t(2.62)
BI
αβ =BI
αβ PVP(2.63)
CΦα=X
c,C
aΦα|cΦα|C+0SαPVP
+SD
Φα+φα|Pρα|PΦα|PVP
∆tt
(2.64)
CΦβ=X
c,C
aΦβcΦβ|C+0SβPVP
+SD
Φβ+φβ|Pρβ|PΦβ|PVP
∆tt
.(2.65)
Eqns. (2.59) and (2.60) can be rearranged to the form
(AΦα+BI
αβ) Φα|P=BI
αβ Φβ|P+CΦα(2.66)
(AΦβ+BI
αβ ) Φβ|P=BI
αβ Φα|P+CΦβ.(2.67)
Now it is clearly seen that when the inter-phase transfer coefficient BI
αβ is
very large, the values of the the two phases are very close to each other.
Furthermore, in the sequential iterative solution the variables would have
changed very little from their starting values and thereby the rate of conver-
gence is slow. PEA algorithm manipulates the above equations to eliminate
Φαand Φβfrom the Eqns. (2.59) and (2.60) giving
AΦα+BI
αβ
AΦβ
(AΦα+AΦβ)!Φα|P
=BI
αβ
AΦβ
(CΦα+CΦβ) + CΦα(2.68)
AΦβ+BI
αβ
AΦα
(AΦα+AΦβ)!Φβ|P
=BI
αβ
AΦα
(CΦα+CΦβ) + CΦβ.(2.69)
Clearly the Eqns. (2.68) and (2.69) are now fully decoupled from each other
and the problem of slow convergence related to the strong coupling of phases
is eliminated. Though it should be noticed that in this approach only the
interfacial force density can be incorporated implicitly in the algorithm but
the transfer related to the change of phase would be treated as in the explicit
method above.
78
2. Numerical methods
Simultaneous solution of Non-linearly Coupled Equations (SINCE)
The PEA algorithm, developed for twophase flows above, can not be gener-
alized for multifluid conditions. A straightforward method to include some
implicitness in the treatment of the interfacial coupling terms also in mul-
tifluid flows has been developed by Lo [Lo89]. The generic macroscopic
balance law for Npphases can be arranged like Eqns. (2.66) and (2.67) in
the twophase case to the following system of linear equations
DΦ1Φ1|P=BI
12 Φ2|P+BI
13 Φ3|P+...+BI
1NPΦNP|P+CΦ1
DΦ2Φ2|P=BI
21 Φ1|P+BI
23 Φ3|P+...+BI
2NPΦNP|P+CΦ2
.
.
. (2.70)
DΦNPΦNP|P=BI
NP1Φ1|P+BI
NP2Φ2|P+...
+BI
NP(NP−1) ΦNP−1|P+CΦNP,
where
DΦα=AΦα+
NP
X
β=1 BI
αβ .(2.71)
This can be written in a matrix form as
EΦΦ|?
P=CΦ,(2.72)
where Φ|?
Pis the solution vector, EΦis the coefficient matrix and CΦis the
right hand side vector. Solving this system cell by cell the new estimates
are obtained. These are next substituted for the interfacial coupling terms
on the right side of Eqns. (2.70) resulting to the following explicit balance
equation
DΦαΦα|P=CΦα+
NP
X
β=1 BI
αβ Φβ|?
P.(2.73)
2.2.5 Solution of volume fraction equations
In the simplified form the continuity equations (2.39) can be written for the
solution of volume fractions as
φα|P=Nα
Dα
(2.74)
where
Dα=X
c
aφα|c+ρα|PVP
∆t(2.75)
Nα=X
c,C
aφα|cφα|C+φα|Pρα|PVP
∆tt
.(2.76)
79
2. Numerical methods
The coefficient aφα|c,bφα|cabove are defined as
aφα|h=1
2ρα|h|ˆui
αh| − ˆui
αh, i =
1 ; h=u
2 ; h=n
3 ; h=e
(2.77)
aφα|l=1
2ρα|l|ˆui
αl|+ ˆui
αl, i =
1 ; l=d
2 ; l=s
3 ; l=w
(2.78)
To ensure that the volume fractions always sum to unity, i.e.,
NP
X
α=1
φα|P= 1 (2.79)
we can solve instead of Eqn.(2.74) the equation
φα|P=Nα
Dα(1 + PNP
β=1
Nβ−φβDβ
Dβ).(2.80)
This equation is equivalent to the original equation when the constraint
equation (2.79) is used. Furthermore, we can see that Eqn. (2.80) reduces
to the form of Eqn. (2.74) when the residuals Nα−φαDαof all the phases
αvanish.
2.2.6 Pressure-velocity coupling
Because of the iterative nature of the solution procedure, the velocities uk
α
obtained from the solution of the phasic momentum equations do not usu-
ally fulfill the phasic (2.39) and total mass balances when substituted into
the corresponding continuity equations. Therefore a correction to the pres-
sure δp is desired which produces new estimates of the phasic velocities uk?
α
obeying the total mass balance. These new estimates are denoted as a sum
of the current value and a correction
uk?
α=uk
α+δuk
α(2.81)
ˆuk?
α= ˆuk
α+δˆuk
α(2.82)
p?=p+δp. (2.83)
To limit the interdependency of the cells the velocity corrections to the
neighbors of the cell under consideration are discarded. Accordingly, these
velocities are approximated by
uk?
αC=uk
αC+Wpc uk?
αP−uk
αP,(2.84)
80
2. Numerical methods
where the weight factor Wpc = 0 for the SIMPLE algorithm and Wpc = 1 for
the SIMPLEC algorithm [VDR84]. Substituting the Eqns. (2.81), (2.82),
(2.83) and (2.84) to the momentum equation (2.40) results in
Auk
αP−Wpc X
c
auk
αc!uk?
αP=X
c,C
auk
αcuk
αC
−φα|PAi
kP
∂p
∂ξiP
+sR
uk
α−Wpc X
c
auk
αcuk
αP
−φα|PAi
kP
∂δp
∂ξiP
,(2.85)
where
Auk
αP=X
c
auk
αc+
Np
X
β=1
Bαβ|PVP+φα|Pρα|PVP
∆t(2.86)
sR
uk
α="φαµα
A1
mAj
k
|J|
∂um
α
∂ξj#u
d
+"φαµα
A2
mAj
k
|J|
∂um
α
∂ξj#n
s
+"φαµα
A3
mAj
k
|J|
∂um
α
∂ξj#e
w
+φα|PFk
αPVP
+
NP
X
β=1
Bαβ|Puk
βPVP+SD
uk
α+φα|Pρα|Puk
αPVP
∆tt
.(2.87)
Using the original equation (2.40) some terms are canceled from the Eqn.
(2.86) and the desired corrections to the phasic Cartesian velocity compo-
nents can now be written in a compact form
δuk
αP=− Hαi
kP
∂δp
∂ξiP
,(2.88)
where
Hαi
kP=φα|PAi
kP
VP
hk
αP,(2.89)
and
hk
αP=VP
Auk
αP−Wpc Pcauk
αc
.(2.90)
The corresponding corrections to the phasic normal flux velocity components
are according to the definition of phasic normal flux
δˆui
αP=ˆ
Hij
αP
∂δp
∂ξiP
,(2.91)
81
2. Numerical methods
where
ˆ
Hij
αP=Ai
kPHαj
kP=
3
X
k=1
φα|PAi
kPAj
kP
VP
hk
αP.(2.92)
The corrections to the phasic normal flux velocity components produce also
new estimates for the phasic convection coefficients (2.25), which can be
expressed as
Ci
α
?c=Ci
αc+δCi
αc=φα|cρα|cˆui
αc+φα|cρα|cδˆui
αc.(2.93)
Finally to get the desired equation for the pressure correction the total mass
balance is formed by adding all the phasic mass balances (2.39) together.
Utilizing Eqn. (2.93) the joint continuity is given by
NP
X
α=1 hδC1
αu−δC1
αd+δC1
αn−δC1
αs
+δC1
αe−δC1
αw+Rmα/ ρα|Pi= 0,(2.94)
where Rmαdenotes the phasic mass residual with the current convection
coefficients Ci
α
Rmα=φα|Pρα|PVP
∆tt+∆t
t
+C1
αu−C1
αd+C2
αn−C2
αs+C3
αe
−C3
αw.(2.95)
As shown by Eqn. (2.94), the mass balance of each phase is normalized
by its phasic density before the summation in order to avoid bias of the
correction procedure towards the heavier fluid. Evidently the total mass
balance (2.94) can now be written for the corrections of the phasic normal
flux velocity components as
NP
X
α=1 hφα|uρα|uδˆu1
αu−φα|dρα|dδˆu1
αd+φα|nρα|nδˆu2
αn
−φα|sρα|sδˆu2
αs+φα|eρα|eδˆu3
αe−φα|wρα|wδˆu3
αw
+Rmα)/ ρα|Pi= 0.(2.96)
In Eqn. (2.96) the corrections of the phasic normal flux velocity components
are introduced on the cell faces. This necessitates interpolation of these
values from the cell centers where they are originally determined (see Eqn.
(2.91). The adjugate Jacobian matrix Ai
kis naturally defined on the cell face
but the coefficient hk
αhas to be interpolated following the weighted linear
interpolation procedure introduced by Eqn. (2.30). If the pressure correction
82
2. Numerical methods
gradients on the cell faces are approximated using central differences on
the computational space (2.35)–(2.37), Eqn. (2.91) can be substituted in
the joint continuity equation (2.96) to give the desired pressure correction,
namely
Ap|Pδp|P=X
c,C
ap|cδp|C+SD
p−
NP
X
α=1
Rmα
ρα|P
,(2.97)
where
Ap|P=X
c
ap|c(2.98)
ap|c=
NP
X
α=1
φα|cρα|cˆ
Hij
αc
ρα|P
.(2.99)
The term SD
pin Eqn.(2.97) denotes the cross-derivatives of the shared pres-
sure corrections treated according to the deferred correction approach. Using
the formal discretization notation it can be written as
SD
p=E12
p
∂?δp
∂ξ2+E13
p
∂?δp
∂ξ3u
d
+E21
p
∂?δp
∂ξ1+E23
p
∂?δp
∂ξ3n
s
(2.100)
+E31
p
∂?δp
∂ξ1+E32
p
∂?δp
∂ξ2e
w
,
where the pressure correction coefficients are defined as
Eij
p=
NP
X
α=1
φα|cρα|cˆ
Hij
αc
ρα|P
.(2.101)
2.2.7 Solution algorithm
The discretized macroscopic balance equations of mass, momentum and
generic scalar dependent variable (2.39), (2.40) and (2.41) are solved itera-
tively in a sequential manner. The solution methods studied are extensions
of the well known single phase solution algorithm SIMPLE (Semi-Implicit
Method for Pressure-Linked Equations) of Patankar & Spalding [PS72].
Inter Phase Slip Algorithm (IPSA)
The IPSA method [Spa77, Spa80, Spa83] is based on the following sequential
solution steps repeated until the convergence is achieved:
Algorithm 1
1. Solve momentum equations for all the phases using values for the de-
pendent variables obtained from the initial guess or from the previous
iteration cycle.
83
2. Numerical methods
2. Solve the pressure correction equation (2.97) based on the joint conti-
nuity equation (2.94).
3. Update the pressure (2.83).
4. Correct the phasic Cartesian velocity components (2.88).
5. Correct the normal flux velocities at cell centers (2.91).
6. Calculate the normal flux velocity components at cell faces with the
Rhie-Chow interpolation method (2.55).
7. Calculate the phasic convection (2.25) coefficients.
8. Solve the phasic continuity equations for the phasic volume fractions
from Eqn.(2.80).
9. Solve the conservation equations for the other scalar dependent vari-
ables.
As shown by Eqn. (2.85), in the IPSA method the interphase coupling terms
are treated as in the explicit algorithm (2.56)– (2.58). In the IPSA-C method
[KL99] the performance of the pressure correction step has been improved
by incorporating the interphase coupling terms into this step following the
idea of the SINCE method. In cases where the coupling of the phases is
strong this improves convergence.
2.3 Multifluid Finite Element Method
In finite element methods the development of multifluid flow algorithms
must also be based on the one phase algorithms. For one phase flow it is
well known that the standard Galerkin methods may suffer from spurious
oscillations when applied directly to Navier-Stokes equations. This is due
to two main reasons. The first reason is the advection-diffusion character
of the equations, when oscillations contaminate the dominating advection.
The second reason is the mixed formulation character which, with unsuit-
able choice of pressure-velocity interpolations, may lead to pressure oscil-
lations, similarly to FVM. These shortcomings can be dealt with proper
choice of interpolation pair fulfilling the ’inf-sup’ condition and modifying
the advective operator to include some ’upwinding’ effect. Recently, great
interest have been taken in the so-called stabilized finite element methods
[FFH92, FF92, FHS92], where these two problems can be handled by adding
extra stabilizing terms into the system of variational equations. The con-
sistency is preserved within these methods since the stabilizing terms will
vanish with the residual of the governing equations. This stabilizing al-
gorithm can be extended to cover twofluid flows [Hil97]. When applied a
84
2. Numerical methods
strongly coupled system arises. Besides the inter-phase coupling we have
an extra coupling, although not so tight, appearing through the stabilizing
terms. The use of PEA like algorithms reducing the inter-phase coupling is
not possible because of the nonlocal character of elementwise equations. To
be able to solve this system properly all equations must be solved simulta-
neously. This amounts to need of more memory and computation time for a
single iteration step but on the other hand when Newton based linearisation
is used gives an optimal convergence rate and therefore less iteration steps
are needed.
In this Section we will represent a stabilized finite element algorithm for
the solution of isothermal steady-state twofluid continuum equations. Let
Ω∈RN,N = 2,3 be the flow domain and let the carrier fluid be denoted
by subscript f and the dispersed phase be denoted by subscript A. Let the
phases be intrinsically incompressible, i.e., ˜ρA= constant,˜ρf= constant.
Assuming stationary flow we get from Eqns. (2.1) and (2.3)
∇ · (φA¯
uA) = 0 (2.102)
∇ · (φf¯
uf) = 0
−2µA∇ · (φAΠ(¯
uA)) + ˜ρAφA(¯
uA· ∇)¯
uA=−φA∇˜p+B(¯
uA−¯
uf)
+φA˜
FA
−2µf∇ · (φfΠ(¯
uf)) + ˜ρfφf(¯
uf· ∇)¯
uf=−φf∇˜p+B(¯
uf−¯
uA)
+φf˜
Ff,
where Π(¯
u) = 1
2(∇¯
u+(∇¯
u)t) is the rate of deformation tensor. For simplic-
ity, viscosities µαare assumed to be constant. For the inter-phase momen-
tum transfer coefficient we have B=BAf =BfA. Next we will write Eqns.
(2.102) in dimensionless form. To this end we introduce the dimensionless
coordinates and independent variables. Let Ube some characteristic veloc-
ity and L some characteristic length of the system. We define x?=x/L to be
the dimensionless coordinate, u?
α=¯
uα/U to be the dimensionless velocity
and p?= ˜p/(˜ρfU2) to be the dimensionless pressure. With these notations
Eqns. (2.102) can be written in a dimensionless form as
∇?·(φAu?
A) = 0 (2.103)
∇?·(φfu?
f) = 0
−(2/ReA)∇?·(φAΠ(u?
A)) + φA(u?
A· ∇?)u?
A=β[−φA∇?p?+B?(u?
f−u?
A)
+φAF?
A]
−(2/Ref)∇?·(φfΠ(u?
f)) + φf(u?
f· ∇?)u?
f=−φf∇?p?−B?(u?
f−u?
A)
+φfF?
γ,
where Reα= ˜ραUL/µαis the Reynolds number for phase α,β= ˜ρf/˜ρAis
the density ratio, ∇?is the dimensionless gradient operator, B?= L/(¯ρfU)B
is the dimensionless momentum transfer coefficient and F?
α= L/(U2˜ρf)˜
Fα
is the dimensionless body force.
85
2. Numerical methods
2.3.1 Stabilized Finite Element Method
The traditional sequential methods, which are frequently used with the fi-
nite volume methods suffer from poor convergence when the inter-phase
coupling is significant enough. In order to attain better convergence in such
cases fully coupled equations must be solved. Although this will lead to a
large algebraic system of equations and therefore will demand more mem-
ory and computation time, the achieved gain in convergence, especially when
Newton-type linearization is used, will yield lower total cost in computing
time. This is even more plausible in cases where efficient iterative solvers
and especially matrix free algorithms are used. About the algorithmic pro-
cedures for effective solution strategies see the paper by Hughes and Jansen
[HJ95]. Another advantage of the full coupling is the increased stability of
the method. For Eqns. (2.103) stabilization must be applied at two levels,
namely the stabilization of velocity pressure pair and the stabilization of
phase volume fractions.
In what follows a stabilized finite element method for solving the system
of Eqns. (2.103) is introduced [Hil97]. Within the method the above men-
tioned problems of coupling and stabilization of the equations are resolved.
To understand the following presentation of the method the reader needs
some basic knowledge of the functional spaces. One can study these things
from the standard FEM text book, for example [KN90]. In the sequel L2(Ω)
is the space of square integrable functions in Ω and L2
0(Ω) the space of func-
tions in L2(Ω) with zero mean value in Ω. Furthermore (·,·) stands for the
L2-inner product in Ω, e.g., for functions u,v∈Vhthe L2-inner product is
defined as
(u,v) =
N
X
i=1 ZΩ
uividx. (2.104)
Also, C0(Ω) is the space of continuous functions in Ω and H1
0(Ω) is the
Sobolev space of functions with square integrable value and derivatives in Ω
with zero value on the boundary of Ω.
In order to formulate the stabilized finite element method we first specify
the required functional spaces as follows
Vh={v∈H1
0(Ω)N:v|K∈Pk(K)N,K∈Πh},
Ph={p∈ C0(Ω) ∩L2
0(Ω) : p|K∈Pl(K),K∈Πh},
where N is the dimension of the problem and k,l≥1 are integers. The
symbol Πhdenotes the partition of ¯
Ω into elements and h is the size of that
partition. The symbol Pk(K) denotes the space of polynomial functions of
degree k or less defined on element K. Furthermore for volume fractions we
introduce the space
Φh={φ∈H1(Ω) : φ|K∈Pm(K),K∈Πh},
86
2. Numerical methods
where the integer m ≥1. In what follows, we drop the superscript ?in
Eqns. (2.103). The stabilized method considered for Eqns. (2.103) can now
be stated formally as:
Method 2.3.1 Find (uAh,ufh, φAh, ph)∈Vh×Vh×Φh×Phsuch that
A(uAh,ufh, φAh, ph;vA,vf, qf, qA) = F(vA,vf, qA, qf),(2.105)
(vA,vf, qA, qf)∈Vh×Vh×Φh×Ph,
with
A(uA,uf, φA, p;vA,vf, qA, qf) = X
α=A,f {Cα(uα;vα)
| {z }
convection
+Dα(uα;vα)
| {z }
viscous stress
−Pα(vα;p)
| {z }
pressure
−Pα(uα;qα)
| {z }
continuity
−Iα(uβ−uα;vα)
| {z }
interaction
+Sα(uα, p, uβ−uα;vα, qα)
| {z }
momentum stabilization
+SM
α(uα, φα;uα, qα)
| {z }
continuity stabilization
}+¯
DA(φA;qA)
| {z }
discont. capturing
}
F(vA,vf, qA, qf) = X
α=A,f
Bα(vα, qα)
where
Cα(u;v) = (φαu· ∇u,v)
Dα(u;v) = ((2/Reα)φαΠ(u),Π(v))
¯
DA(φ;q) = (κ∇φ, ∇q)
Pα(v;q) = (∇ · (βαφαv), q)
Iα(w;v) = (Bβαw,v)
Sα(u, p, w;v, q) = (Resα(u, p, w), τα(φαuα· ∇v−φαβα∇q))
Resα(u, p, w) = −(2/Reα)∇ · (φαΠ(u)) + φαu· ∇u+φαβα∇p−βαBw
SM
α(u, φ;v, q) = (ResM
α(φ, u), δα∇ · (βαqv))
ResM
α(φ, u) = ∇ · (βαφu)
Gα(v, q) = (φαβα˜
Fα,v+τα(φαuα· ∇v−φαβα∇q))
βα= ˜ρf/˜ρα.
The coefficient κfor discontinuity capturing operator and the stability pa-
rameters τα, δαare defined as
κ=cdif hKResTOT/(|∇U|p+ hK)
ResTOT =|ResA|1+|Resf|1+|ResM
A|+|ResM
f|
87
2. Numerical methods
|∇U|p=
N
X
i=1
(|∇uAi|p+|∇ufi|p) + |∇φA|p+|∇p|p
δf=λf|uf|phKξ(ReK
f(x))
δA=λA|φA|phKξ(PeK
A(x))
τα=hK
2|uα|pξ(ReK
α(x))
ReK
α(x) = 1
4mK|uα|phKReα
PeK
A(x) = 1
2mK|uA|phK/κ
ξ(γ) = γ, 0≤γ < 1
1, γ ≥1
|uα|p=(PN
i=1 |uαi|p1/p,1≤p<∞
maxi=1,N|uαi|,p = ∞,
where λα, cdif and mKare positive constants.
Notice that the variational system (2.105) is nonlinear due to the con-
vection and interaction terms and due to the discontinuity capturing terms
and the stabilization terms. The stabilization terms Sαfor momentum equa-
tions are similar to those used for incompressible Navier-Stokes equations
(see, e.g., [FF92]). The only difference is that here the volume fractions are
involved. The contribution arising from the lower order terms in stabiliza-
tion, i.e., the interaction terms in the momentum equations, is neglected.
The inclusion of interaction terms would lead to different stability parame-
ters. Such a case is studied for diffusion-convection type scalar equation for
example in ref. [LFF95]. Since we shall restrict ourselves to linear approx-
imations here, the terms including second order derivatives are neglected.
The stabilization terms SM
αarising from the continuity equations are differ-
ent for each phase. For the carrier phase it affects directly the momentum
equation while for dispersed phase it affects the continuity equation. That
is also reason for the different normalization of the stabilization parame-
ters δα. Since steep gradients can be expected in many applications of the
method, we have introduced the discontinuity capturing operator ¯
D(φA, qA)
to the equations. In the above method this term is added only for continuity
equation of the dispersed phase. The corresponding diffusivity coefficient κ
depends on the discrete residual of the system. The form of the coefficient
equals to the one introduced by Hansbo and Johnson [HJ91]. The value of
the coefficient is largest near steep gradients and vanishes when the solution
is smooth.
Stability analysis can be carried out for the linearized form of Eqns.
(2.105) following the same guidelines as for incompressible Navier-Stokes
equations (see, e.g., [FF92]), except of additional complications which arise
from the interaction terms and from the compressible nature of the flow.
88
2. Numerical methods
The analysis is nevertheless nontrivial and is skipped here.
In Galerkin Finite Element Method the solution functions
(uAh,ufh, φAh, ph) are approximated as a linear combination of basis func-
tions (˜
vA,˜
vf,˜qA,˜qf) in a finite-dimensional product space Vh×Vh×Φh×Ph.
In other words, the phase velocities uk
αh∈Vh,Vh= (Vh)N(componentwise)
are approximated as
uk
αh(x) =
dim Vh
X
j=1
uk
αj˜vj(x), k = 1,...,N (2.106)
where uk
αj∈Rare the unknowns in discretization points xj,j= 1,...,dim Vh.
In a same way the dispersed phase volume fraction φAhand pressure phare
approximated as
φAh(x) =
dim Φh
X
j=1
φAj˜qAj(x) (2.107)
ph(x) =
dim Ph
X
j=1
pj˜qfj(x),(2.108)
where φAjand pjare the nodal values of the dispersed phase volume fraction
and pressure respectively. In the following we will assume that the unknowns
are approximated using equal order elements and therefore the nodal values
are collocated and if boundary effects (elimination of degrees of freedoms) are
neglected the dimensions of the spaces equals, i.e., we can denote dim Vh=
dim Φh= dim P≡ndof s.
In Galerkin approximation, the test functions (vA,vf, qA, qf) and basis
functions (˜
vA,˜
vf,˜qA,˜qf) are taken from the same space. Let us then denote
ϕj≡˜vj= ˜qAj= ˜qfj. Using definitions (2.106)–(2.108) the variational
system of equations (2.105) can be written in semi-discretized matrix form
AU =F,(2.109)
where Ais an (2 ×(N + 1) ×ndof s)×(2 ×(N + 1) ×ndof s) matrix and F=
(F1(k), F2(k),...,F2×(N+1)(k))tis the right hand side vector, where Fi(k)
are ndofs dimensional subvectors. The vector U= (u1
f,...,uN
f,p,u1
A,...
,uN
A, φA)tis the nodal unknown, where the subvectors are defined by
ui
α= (ui
α1,...,ui
αndof s ) (2.110)
p= (p1, p2,...,pndof s ) (2.111)
φA= (φA1, φA2,...,φAndofs ).(2.112)
Note that the above system of equations is nonlinear in the sense that the
coefficient matrix Aand the right hand side vector Fdepend on the solution
U.
89
2. Numerical methods
2.3.2 Integration and isoparametric mapping
The construction of the coefficient matrix Aand right hand side vector F
requires the computation of integrals over interpolation functions ϕj. In-
stead of the global integration and global equations and global interpolation
functions we can integrate in a single element with local equations and lo-
cal interpolation functions. These local equations can then be assembled
to a global system using special assembling algorithm. Details concern-
ing these standard finite element procedures one can find in any textbook
concerning finite element methods. One further standard procedure in fi-
nite element methods is the use of isoparametric mapping converting the
physical coordinates into computational coordinates (compare with BFC in
FVM). In finite element method the relationship between the physical co-
ordinates (x1, x2, x3) and the computational coordinates (ξ1, ξ2, ξ3) for any
element K in the partitioning Πhis obtained using parametric concept, i.e.,
a coordinate transformation defined by
xi(ξj) =
nK
nodes
X
k=1
λk(ξj)xi
k,(2.113)
where λkare the interpolation functions over the element Kand xi
kthe nodal
coordinate values of that element. The symbol nK
nodes denotes the number of
the nodes in element K. The above defined transformation is quite general.
When the interpolation functions defining the dependent variables are of the
same order as the functions defining the element geometry, i.e.,ϕi=λi, the
element is called isoparametric. If the order of λiis greater or less than that
of ϕithe element is superparametric or subparametric correspondingly. For
example the four node quadrilateral element in 2 dimension is isoparametric
and we get
x1(ξ1, ξ2) =
4
X
k=1
˜ϕk(ξ1, ξ2)x1
k
x2(ξ1, ξ2) =
4
X
k=1
˜ϕk(ξ1, ξ2)x2
k
Note that the interpolation functions ˜ϕk(ξ1, ξ2) above are local functions in
reference element, i.e., they are the same for every physical element (see
Fig. 2.4).
The derivatives are transferred according to the chain rule. In a matrix
form we get
∂˜ϕi
∂ξ1
∂˜ϕi
∂ξ2
∂˜ϕi
∂ξ3
=J
∂ϕi
∂x1
∂ϕi
∂x2
∂ϕi
∂x3
,(2.114)
90
2. Numerical methods
where Jis the Jacobian matrix of the mapping ξj→xi(ξj). From this we
get
∂ϕi
∂x1
∂ϕi
∂x2
∂ϕi
∂x3
=J−1
∂˜ϕi
∂ξ1
∂˜ϕi
∂ξ2
∂˜ϕi
∂ξ3
,(2.115)
where J−1is the inverse of the Jacobian.
When isoparametric elements are used, integration can be done over the
reference element. Let us denote the reference element by E. Now we have
for example for the local convection coefficient
C(K)
α(k, l) = (φαuα· ∇ϕl, ϕk)K
=ZK
φα(x)uα(x)· ∇ϕl(x)ϕk(x)dx
=ZE
φα(ξ)uα(ξ)· ∇ϕl(ξ)ϕk(ξ)|J|dξ
=ZE
φα(ξ)ui
α(ξ)(J−1)im ∂˜ϕl(ξ)
∂ξm˜ϕk(ξ)|J|dξ.
In practise this integration must be done numerically. Gauss numerical
integration formulae are used. The order of the integration is naturally kept
the same as the interpolation order of the unknowns.
ξ→x(ξ)
x1
x2
ξ2
ξ1
+1
+1
−1
−1
Figure 2.4: Mapping of the local reference element in ξ-coordinates to global
element in physical x-coordinates for four point quadrilateral element
2.3.3 Solution of the discretized system
The system of nonlinear equations (2.109) is transferred to algebraic system
using the integration formula given above. To solve this resulting nonlin-
ear system of algebraic equations a variety of methods exists. Nonlinear-
ity results to iterative methods. The most naive approach would be to
use fixed point iteration (Picard iteration), where the nonlinearity is elimi-
nated taking the coefficient values from the previous iteration step. Unfor-
tunately this method results to slow convergence. More effective way is to
91
2. Numerical methods
use Newton-type methods, where the system is expanded using Taylor series
in the neigborhood of the solution. This results to method with asymptot-
ically quadratic convergence. Unfortunately the radius of convergence of
this method can sometimes be quite small and care must be exercised in the
choice of the initial solution vector. Convergence radius can be expanded
using for example the so called incremental method where the right side load
is incremented step by step. Another way is to take a few Picard iteration
steps before proceeding to the Newton steps. We might also have to control
the magnitude of the Newton step. This is needed when the Jacobian is not
calculated exactly like in the Modified Newton-Raphson method, where the
Jacobian is calulated only once in the beginning of the iterations or like in
the Quasi-Newton method where the Jacobian is updated in a more simple
manner than calculating it exactly or like in the inexact Newton methods
where the Jacobian is evaluated from the residual vector using difference
approximations. An effective way is to use line-search backtracking.
For every nonlinear step we have to solve a linear system. When large
systems are considered, like the system (2.109) mostly is, we have to use
iterative solvers. Iterative solvers start with an initial guess and compute a
sequence of approximate solutions that converge to the exact solution. The
accuracy of the solutions depend on the number of iterations performed.
When the linear equation solver is part of the nonlinear iteration loop, like
we have, exact convergence is not required. The amount of work used with
iterative solvers depends on the convergence rate and the desired accuracy.
Depending on the system to be solved the convergence of the iterative meth-
ods can be slow and irregular. The rate of the convergence depends on the
spectrum of the coefficient matrix and hence the condition number of the
matrix. The rate of the convergence can often be increased by the technique
of preconditioning. When the system of linear equations is nonsymmetric,
like system (2.109), special methods must be used. Methods like Conjugate
Gradient Squared (CGS) [SWd85] and the Generalized Minimum Resid-
ual Method (GMRES) [SS83] are succesively used. For the solution of the
(2.109) the GMRES method is succesfully used [Hil97].
In the following the Newton-GMRES algorithm with the linesearch back-
tracking is introduced. Let the nonlinear system of equations be R(U) =
A(U)U−F(U) = 0. Then the method as expressed in algorithmic form is
as follows,
Algorithm 2
U0given, n = −1
REPEAT
n = n + 1
Solve J(Un)δ(n) = −R(Un) by GMRES
Un+1 =Un+ωnδ(n)
92
2. Numerical methods
UNTIL convergence
Above ωnis computed by line-search backtracking to decrease residual norm
r(U) = 1
2R(U)tR(U).
Above the superscript ndenotes the nthiteration step, i.e., previous
iteration step. Matrix J(Un) is the Jacobian associated with R(Un), i.e.,
Jij (n) = Aij +∂Aik (Un)
∂Uj
Un
k+∂F i
∂Uj
,(2.116)
where Ujis the jthcomponent of the vector Uin (2.109). Furthermore δ(n)
is the desired search direction, ωnis the length of the step and R(Un) is
the residual from the previous step. As a preconditioner for the GMRES
the ILU preconditioning is preferred, which seems to work fairly well in the
present problem of twofluid flow. For detailed analysis of Newton-GMRES
algorithms (including also inexact Newton algorithms), see Ref. [CE93].
2.4 Particle tracking
From the algorithmic point of view the particle tracking is a considerably
simple method as compared to multifluid algorithms. For the solution of
the carrier fluid with low particle volume fractions standard one-phase algo-
rithms can be used with extra source terms emanating from the appearance
of the particulate phase. Particle trajectories are given by the kinematic
equation. The treatment of this equation involves only technical difficulties.
Equations of motion are reduced to ordinary differential equations and are
therefore easy to solve. The coupling between the carrier phase and the par-
ticulate phase is taken into account through source terms. These terms can
be obtained elementwise (or cellwise) keeping track of trajectories passing
through the considered element.
The solution strategy used in the particle tracking is an iterative one.
First the solution of the carrier phase is obtained. Next the ordinary differ-
ential equations for the dispersed phase are solved for a number of particles
using the continuum phase solution. The computed trajectories (and also
other scalar quantities such as temperature and masses of particles) are
combined into source terms of momentum equations (and of energy and
continuity equations). These source terms are then used in the next solu-
tion of the continuum equations. The process is iterated until convergence
is attained. In this procedure, two-way coupling is said to prevail since in-
formation is transfered from the carrier phase to dispersed phase and vice
versa. When the particles are not affecting to the carrier phase and the
tracking is such a postprocessing after the flow field of the carrier phase is
solved one speaks of one-way coupling. The principle of numerical solution
93
2. Numerical methods
Start
Solution of fluid
phase flow field
Lagrangian tracking
of dispersed phase
entities
Equation set
converged
Statistics adequate
End
No
Yes
Yes
No
End
Solution of fluid
phase flow field with
phase coupling terms
Equation set
converged
1
Start
Solution of fluid
phase flow field
without coupling to
dispersed phase
Lagrangian tracking
of dispersed phase
entities
Equation set
converged
Statistics adequate
1
No No
Yes Yes
Yes
No
Figure 2.5: Solution schemes for problems corresponding to one-way and
two-way coupling.
for one-way coupled systems and for two-way coupled systems is illustrated
in Fig. 2.5.
In this section the numerical realization of the particle tracking is shortly
reviewed. All numerous technical details are excluded as well as special
considerations for turbulent flows. In turbulent tracking we refer here to the
widely used work by Gosman and Ioannides [GI81b], where k−εturbulence
model for the carrier fluid and a one-step-per-eddy, Monte-Carlo simulation
for the particles were used.
2.4.1 Solution of the system of equation of motion
To solve the momentum equation (1.109) several numerical methods for
ODE can be used. Let us write equation (1.109) in a form
d
dtV=F(V, t).(2.117)
94
2. Numerical methods
In order to solve this system, an explicit second-order predictor-corrector
method could be used, i.e.,
Vn+1 =Vn+ ∆tF(Vn+1
2, tn+1
2) (2.118)
Vn+1
2=Vn+1
2∆tF(Vn, tn).
It is well known however that explicit methods perform poorly when system
of equations becomes stiff, i.e., when relaxation times becomes small. In such
cases unconditionally stable implicit methods are preferred. An example is
the following Crank-Nicholson second order scheme
Vn+1 =Vn+ ∆tF(Vn, tn) + F(Vn+1, tn+1 )
2.(2.119)
2.4.2 Solution of particle tra jectories
Particle trajectories are given by the kinematic equation
d
dtY=V,(2.120)
where Y(t) is the position of the particle at time tand V(t) the correspond-
ing velocity.
Using the finite element method for the equation (2.120) we get compo-
nentwise d
dtYi=X
k
Vi
k(t)φk(ξj).(2.121)
The relationship between local coordinates ξjand the global coordinates Yi
is given by the isoparametric mapping (2.113). Now we get for the reference
element d
dtYi=∂Y i
∂ξj
d
dtξj=Vi,(2.122)
or in matrix form d
dtY=Jd
dtξ=V,(2.123)
where Jis the Jacobian of the parametric transformation. So we can trans-
form the original problem to a problem in reference element,
d
dtξ=J−1V.(2.124)
Furthermore using second order Crank-Nicholson scheme for time derivative
we get
ξj(tn+1) = ξj(tn) + ∆t(J−1)jkVk(tn) + (J−1)j kVk(tn+1 )
2.(2.125)
For finite volume methods the calculation of the trajectories is performed
more or less the same way as within finite element method. Like in FEM,
instead of dealing with physical coordinates computational coordinates are
used.
95
2. Numerical methods
2.4.3 Source term calculation
In the PSIC approach [CSS77], the source terms are calulated from the
residence time of the each individual particle on each cell or element.
Let nj
pbe the number of particles per unit time traversing the jth trajec-
tory and δtj
Ebe the residence time of a particle on trajectory jwith respect
to element E. Then the contribution to the ith momentum transfer source
to element Eis
Si(E) = 1
VE
nE
X
j=1
nj
ph(mj
pVi)out −(mj
pVi)ini,(2.126)
where nEis the number of trajectories passing through element Eand VEis
the volume of that element and (mj
pVi)out is the momentum of the particle
leaving the element and (mj
pVi)in the momentum of the particle entering
the element Ewith respect to the trajectory j.
2.4.4 Boundary condition
When a particle reaches the domain boundary, the distribution of source
terms between the carrier phase and the boundary depends on whether
the particle penetrates, rebounds or attaches the boundary. A penetrating
particle exchanges momentum (and, in general, energy and mass) with the
carrier phase up to the boundary and then exits the domain carrying a
residual of momentum (and of energy and mass). If the particle rebounds
from the boundary, it exchanges momentum with the carrier phase and with
the boundary face. If the particle attaches the boundary, it simply loses all
its momemtum to the boundary.
2.5 Mesoscopic simulation methods
Numerical solution of flow has traditionally been based on finding a solution
for partial differential continuum equations such as continuity and Navier-
Stokes equations that govern the fluid flow. In principle, it would be possible
to solve any fluid flow problem on microscopic level with direct molecular-
dynamical simulations. It is quite clear that practical fluid flow problems
cannot be solved with this approach at the present due to the large number of
particles, and thus large amount of computer resources that would be needed
for such a simulation. Another possibility for direct microscopic simulation
of flow would be using the standard kinetic theory. Here the basic object
is the particle distribution function f(r,v) which gives the probability of
finding at position ra particle with velocity v. The time evolution of this
function is given by the Boltzmann equation
(∂t+v1·∇r+F
m·∇v1)f1=ZdΩZd3v2δ(Ω)|v1−v2|(f0
2f0
1−f1f2).(2.127)
96
2. Numerical methods
This approach is also far too complicated in most practical problems even
for gaseous fluids. Many dynamical systems can however be modeled with
radically simplified microscopic or mesosocopic models. This has been uti-
lized, e.g., in the famous Ising models for magnetic materials, and since
1950’s in various cellular-automaton models for biological and physical sys-
tems. Encouraged by these models, similar models (i.e. models that were
not based on continuum mechanics) for fluid flows were also developed, first
with a limited success, though. In 1986 it was realised that fluid flow could
be successfully simulated with very simple discrete models provided that
the simulation lattice was carefully chosen. This discovery was the start-
ing point of the lattice-gas methods [FdH+87, RZ97, Kop98] and later of
the lattice-Boltzmann methods [RZ97, Kop98, QdL92, BSV92]. We refer
these methods as ’mesoscopic’ indicating that the size scale of the basic
constituents (’particles’) or variables of the model is large as compared to
molecular scale of the fluid, but small as compared to the typical size scale
of macroscopic flow, and the local hydrodynamical quantities are defined as
suitable averages of the basic variables.
In the lattice-gas method fluid is modeled with identical particles which
move in a discrete lattice interacting with each other only at the lattice
nodes. A ’particle’ is not considered here as a physical molecule, but rather
as a ’fluid particle’ that contains a large number of molecules. The space,
the time and the velocity of particles are all discrete. Notice that the method
is not based on explicitly solving any governing equation of particle motion.
It is simply an algorithm of moving particles in the lattice with a set of
collision rules that conserve mass (number of particles) and momentum.
The basic hydrodynamic quantities such as local flow velocity can be defined
as the velocity of particles averaged over a specified volume, time or both.
Provided that the simulation lattice fulfills certain symmetry requirements,
the lattice-gas automaton indeed develops macroscopic flow field that is
close to the solution of the Navier-Stokes equations for incompressible flow
in the same macroscopic conditions. The number of particles needed for a
realistic simulation of many practical systems is well in the limits of present
computational capabilities.
The boolean and local nature of the lattice-gas model gives several tech-
nical advantages in computing. Perhaps the most important advantage is
the ease of introducing complex flow geometries, which follows from the reg-
ular lattice and from the extremely simple and strictly local updating rules
of the method. Since particles can be represented by single bits in computer,
the method does not require very large memory as compared to other meth-
ods. Furthermore, rounding errors are not involved in bit manipulations,
and unconditional numerical stability is guaranteed. The computations are
also inherently parallel thus being ideal for massive parallel computers. In
recent years many sophistications have helped to get rid of most of the early
problems [RZ97, BSV92], and the lattice-gas method is now a viable tool
97
2. Numerical methods
for computational fluid dynamics.
The early deficiencies of the lattice-gas models inspired the formula-
tion of the more advanced lattice-Boltzmann models. The idea behind this
model was to track a population of (fluid) particles instead of a single par-
ticle, a reasonable modification justified by the Boltzmann molecular-chaos
assumption from kinetic theory of gases. This mean-value representation
of particles eliminates much of the statistical noise present in lattice-gas
methods. The basic variable to be solved within lattice-Boltzmann methods
is the discrete distribution function that fulfils a discretised version of the
Boltzmann equation (see eqn. (2.128) below). The hydrodynamic variables
are defined in terms of the distribution function as averaged quantities anal-
ogously with the manner in which those variables are defined in the usual
kinetic theory of gases. In the next section we discuss in more detail one of
the simplest versions of the lattice-Boltzmann models that is regularly used
in practical simulations, namely the lattice-BGK (Bhatnagar-Gross-Krook)
model [QdL92].
Recently a third mesoscopic method, dissipative particle dynamics (DPD),
has been developed [MBE97]. This method is a combination of molecular
dynamics, Brownian dynamics and lattice-gas automata. The DPD algo-
rithm models a fluid with fluid particles out of equilibrium and conserves
mass and momentum. The fluid particles interact with each other through
conservative and dissipative forces. In contrast to the lattice-gas and lattice-
Boltzmann methods, the position and velocity of these fluid particles are
continuous, as in molecular dynamics, but time is discrete. This method is
very suitable e.g. for simulating rheological properties of complex fluids on
hydrodynamic time scales.
2.5.1 The lattice-BGK model
In the lattice-Boltzmann method a simplified version of the Boltzmann equa-
tion, eqn. (2.127), is solved on a discrete, regular lattice. Fluid particles,
described by the discrete distribution function, move synchronously along
the bonds of this lattice, and interact locally according to a given set of
rules. A single iteration step (time step) within this method consists of the
following two phases:
1. Propagation; particles move along lattice bonds to the neighboring
lattice nodes.
2. Collision; particles on the same lattice node shuffle their velocities
locally conserving mass and momentum.
In the lattice-BGK (Bhatnagar-Gross-Krook) model the collision operator
is based on a single time relaxation to the local equilibrium distribution
[QdL92]. The lattice structure can be chosen in several ways (see Fig. 2.6
98
2. Numerical methods
for two possible realizations). In the D3Q19 lattice-BGK model, each lattice
point is linked in three dimensional space with its six nearest neighbors at
a unit distance and with twelve diagonal neighbors at a distance of √2.
Including the rest state, the particles can thus have 19 different velocity
states. The dynamics of the D3Q19 model is given by the equation [RZ97,
QdL92],
fi(r+ci, t + 1) = fi(r, t) + 1
τ(f(0)
i(r, t)−fi(r, t)) ,(2.128)
where ciis the i-th link, fi(r, t) is the density of particles moving in the
ci-direction, τis the BGK relaxation parameter, and f(0)
i(r, t) is the equi-
librium distribution function towards which the particle populations are
relaxed. The hydrodynamic fields such as the density ρ, the velocity vand
the momentum tensor Π are obtained from moments of the discrete velocity
distribution fi(r, t) as
ρ(r, t) =
18
X
i=0
fi(r, t),v(r, t) = P18
i=0 fi(r, t)ci
ρ(r, t),
Παβ (r, t) =
18
X
i=0
ciαciβ fi(r, t).
The equilibrium distribution function can be chosen in several ways. A
common choice is
f(0)
i=tiρ(1 + 1
c2
s
(ci·v) + 1
2c4
s
(ci·v)2−1
2c2
s
v2),
where tiis a weight factor depending on the length of the link vector and
csis the speed of sound. The weight factors can be chosen as 1
3,1
18 and
1
36 for the rest particle, nearest neighboring and diagonal neighboring links,
respectively. These values yield to a correct hydrodynamic behavior for an
incompressible fluid in the limit of low Mach and Knudsen numbers. The
speed of sound and the kinematic viscosity of the simulated fluid in lattice
units are given by cs=1
√3and ν=2τ−1
6, respectively [QdL92].
2.5.2 Boundary conditions
In lattice-gas and lattice-Boltzmann simulations the no-slip boundary con-
dition is usually realized using the bounce-back condition [RZ97, Kop98]. In
this approach the momenta of particles that meet wall points are simply re-
versed. The bounce-back boundary may generate errors which in some cases
violate the second-order spatial convergence of these methods and more so-
phisticated boundaries have been proposed [FH97]. For practical simulations
the bounce-back boundary is however very attractive, because it is a simple
99
2. Numerical methods
Figure 2.6: Two possible realizations of lattice structures.
and computationally efficient method for imposing no-slip walls in irregular
geometries. Also, the bounce-back method can easily be generalized to allow
moving boundaries [Lad94a, Lad94b].
Successful numerical simulation of practical fluid flow problems requires
that the velocity and pressure boundary conditions of the system have been
imposed in a consistent way. So far most of the practical simulations have
used a body force [Kop98] instead of pressure or velocity boundaries. When
a body force is used, a pressure gradient acting on the fluid is replaced by a
uniform external force. (Usually periodic boundaries are imposed at least in
the direction of the flow.) The use of a body force is based on the assumption
that the effect of an external pressure gradient is approximately constant all
over the system, and that it can be replaced by a constant force that adds at
every time step a fixed amount of momentum to fluid particles. In a simple
tube flow the body-force approach is exact. In more complicated geometries
this approach is supposed to work best with small Reynolds numbers where
nonlinear effects on the flow are small.
2.5.3 Liquid-particle suspensions
The bounce-back condition can easily be modified [Lad94a] to allow moving
boundaries with no-slip boundary condition. We assume that a boundary
moving with velocity uwis located in the halfway of links between the last
fluid points and the first solid points. The distribution function for particles
moving along such a link towards the solid point is given by
fi(r+ci, t + 1) = fi0(r+ci, t+) + 2ρfBi(uw·ci).(2.129)
Here t+is used to indicate the post-collision distribution, i0denotes the
bounce-back link, and Bi=1
3,1
12 for the diagonal and non-diagonal links,
respectively. The last term in Eqn. 2.129 is added in order to account for the
momentum transfer between the fluid and the moving solid boundary. For
computational convenience the fluid is usually made to fill also the suspended
solid particles. This trick removes the need for creating and destroying fluid
when particles move. Models without interior fluid have also been used.
100
2. Numerical methods
The lattice-Boltzmann method for suspensions is based upon Newtonian
dynamics of solid particles that move in a continuous space. The discretized
images of these particles interact with the lattice-Boltzmann fluid at their
boundary nodes. The technique takes advantage of the fact that the hy-
drodynamic interactions are time dependent and develop from purely local
interactions at the solid-liquid interfaces. Thus it is not necessary to consider
the global system, but one can update one particle at a time. The method
scales linearly with the number of suspended particles and, therefore, allows
far larger simulations than the conventional methods. The hydrodynamic
interactions between solid particles are fully accounted for, both at zero
and finite Reynolds numbers [Lad94a]. Furthermore, there is no need for
solution of the linear systems. Therefore, lattice-Boltzmann method can
be efficiently implemented on parallel processors. The electrostatic interac-
tions, the flow geometry, the Peclet number, the shear and particle Reynolds
numbers, as well as the size and shape of the suspended particles can easily
be varied.
2.5.4 Applicability of mesoscopic methods
The most important property of the mesoscopic methods discussed above is
the simplicity with which models for many complex flows such as multiphase
flows and flows in porous media can be implemented. Numerical simulation
is based on a simple algorithm, or on numerical solution of a single trans-
port equation instead of a system of coupled partial differential equations
of conventional methods. Another important property of the lattice-gas
and lattice-Boltzmann methods is the inherent spatial locality of their up-
dating rules, which makes these methods ideal for parallel processing. Both
methods are also numerically relatively stable. In the lattice-gas and lattice-
Boltzmann methods a uniform lattice is usually used, and they can thus be
easily and quickly applied to new geometries. Also, time-dependent simu-
lations can be carried out relatively easily, as no time is lost for remeshing.
Moreover, these methods spontaneously generate hydrodynamic instabili-
ties which make them useful for simulating fluid flow at moderately large
Reynolds numbers. At very large Reynolds numbers mesoscopic methods
encounter the problem of turbulence like the more conventional methods:
in a fully turbulent flow wave lengths of all scales are present, and the so-
lution would require a very large lattice or additional turbulence modelling.
Furthermore, some features of the methods such as use of irregular lattice
for improved accuracy, implementation of velocity and pressure boundary
conditions and models for finite-temperature systems are still under devel-
opment.
So far, successful implementations of mesoscopic methods have included
e.g. multiphase flows [RZ97, MC96], suspension flows [Lad94a] and flows
in complex geometries [KKHA98, Kop98, MC96]. In Ref. [KVH+99] a
101
2. Numerical methods
Figure 2.7: Couette flow of liquid-particle suspension as solved using the
lattice-BGK-method in two dimensions. The upper wall is moving to right
and the lower wall is fixed. The velocity of the liquid is indicated by gray
scale (light colour indicating high velocity) and the motion and rotation of
particles by arrows.
detailed comparison between conventional methods and lattice-Boltzmann
methods was given for 3D fluid flow in an industrial static mixer. In addition
to basic flow simulation, these methods are convenient in finding numeri-
cal correlations for closure relations and constitutive relations necessary in
conventional methods for solving multiphase flows, as discussed in chapter
1. The essential property of the mesoscopic methods in this respect is the
simplicity of constructing complicated geometries and implementing mov-
ing boundaries. Due to this feature it is straightforward, e.g., to generate a
large ensemble of macroscopically identical systems of particles suspended in
a liquid, solve the flow and the motion of the particles for each system (pos-
sibly with a given interaction between the suspended particles), and finally
compute the properly averaged quantities and transfer integrals as defined
in chapter 1. The results can then be used to correlate the unknown terms
such as Γαand Mαin averaged flow equations (see eqns. (1.27)–(1.28) or
eqns. (1.79)–(1.80)) with basic averaged flow quantities. Similarly, these
methods could be helpful in finding rheological properties and boundary
conditions of multiphase fluids to be used in conventional non-Newtonian
simulation of such flows.
As an example of a typical solution obtained by the lattice-Boltzmann
method, we show in Figure 2.7 the instantaneous flow pattern of a two-
dimensional liquid-particle suspension in a shear flow [RSMK+00]. Here, the
flow of the carrier fluid is calculated using the lattice-BGK -model while the
motion of the suspended spherical particles is given by Newtonian mechanics
including the impulsive forces due to direct collisions between particles, and
the hydrodynamic forces due to fluid flow. The numerical solution can be
102
2. Numerical methods
Figure 2.8: Viscosity of two-dimensional liquid-particle suspension as a func-
tion of consistency (area fraction) calculated by lattice-Boltzmann simula-
tions (see Fig. 2.7). The open squares are the calculated values. The
semiempirical result by Krieger and Dougherty [KD59], and the analytical
results by Batchelor [BG72, Bat77] and Einstein [Ein06, Ein11] valid for low
consistencies are given by lines as indicated.
used to compute, e.g., the time averaged total shear force acting on the walls,
and thereby to find the dependence of the nominal viscosity on the mean
shear velocity and on the properties of the suspension. Figure 2.8 shows the
viscosity of the two dimensional liquid-particle suspension as a function of
consistency of particles as calculated using lattice-Boltzmann simulations.
The result is also compared with previous analytical results by Einstein
[Ein06, Ein11] and Batchelor [BG72, Bat77], and with the semiempirical
results by Krieger and Dougherty [KD59].
103
Bibliography
[Aut83] Auton, T.R. The dynamics of bubbles, drops and particles in
motion in liquids. PhD thesis, University of Cambridge, 1983.
[Bas88] Basset, A.B. Treatise on hydrodynamics. Pp. 285–297.
Deighton Bell, London, 1888.
[Bat53] Batchelor, G.K. The theory of homogeneous turbulence. Cam-
bridge University Press, London, 1953.
[Bat77] Batchelor, G.K. The effect of brownian motion on the bulk
stress in a suspension of spherical particles. Journal of Fluid
Mechanics, 83:97–117, 1977.
[BB93] Burry, D. and Bergeles, G. Dispersion of particles in anisotropic
turbulent flows. Int. J. Multiphase Flow, 19:651–664, 1993.
[BBS95] Bo¨elle, A., Balzer, G., and Simonin, O. Second order prediction
of the particle phase stress tensor of inelastic spheres in simple
shear dense suspensions. Gas-Solid Flows, ASME FED, 228:9–
18, 1995.
[BDG90] Berlemont, A., Desjonqueres, P., and Gousbet, G. Particle
lagrangian simulation in turbulent flows. Int. J. Multiphase
Flow, 16:19–34, 1990.
[Bea72] Bear, J. Dynamics of Fluids in Porous Media. American Else-
vier, New York, 1972.
[BG72] Batchelor, G.K. and Green, J.T. The determination of the bulk
stress in a suspension of spherical particles to order c2.Journal
of Fluid Mechanics, 56:401–427, 1972.
[BH79] Brown, D.J. and Hutchinson, P. The interaction of solid or
liquid particles and turbulent fluid flow fields — a numerical
simulation. Journal of Fluids Engineering, 101:265–269, 1979.
104
Bibliography
[BM90] Burnage, H. and Moon, S. Predetermination de la dispersion
de particules materielles dans un ecoulement turbulent. C. R.
Acad. Sci. Paris, 310:1595–1600, 1990.
[Boe97] Boemer, A. Euler/Euler-Simulation der Fluiddynamik Blasen-
bildender Wirbelschichten. PhD thesis, Aachen, 1997.
[Bou03] Boussinesq, J. Theorie analytique de la chaleur. L’´
Ecole Poly-
technique, Paris, 1903.
[BP80] Baliga, B.R. and Patankar, S.V. A new finite-element for-
mulation for convection-diffusion problems. Numerical Heat
Transfer, 3:393–409, 1980.
[BS78] Buyevich, Y.A. and Shchelchkova, I.N. Flow of dense suspen-
sions. Prog. Aerospace Sci., 18:121–150, 1978.
[BSV92] Benzi, R., Succi, S., and Vergassola, M. Phys. Rap., 222:145,
1992.
[Bur38] Burgers, J.M. 2nd report on viscosity and plasticity. K¨on. Ned.
Akad. Wet., Verhand, 1938.
[Buy66] Buyevich, Y.A. Motion resistance of a particle suspended in
a turbulent medium. Izv. AN SSSR. Mechanika Zhidkosti i
Gaza, 1:182–183, 1966. English translation in Fluid Dynamics.
[Buy71] Buyevich, Y.A. Statistical hydromechanics of disperse systems.
Part 1. Physical background and general equations. J. Fluid
Mech., 49(3):489–507, 1971.
[Buy92a] Buyevich, Y.A. Heat and mass transfer in disperse media
- I. Averaged field equations. Int. J. Heat Mass Transfer,
35(10):2445–2452, 1992.
[Buy92b] Buyevich, Y.A. Heat and mass transfer in disperse media
- II. Constitutive equations. Int. J. Heat Mass Transfer,
35(10):2453–2463, 1992.
[BW87] Burns, A.D. and Wilkes, N.S. A finite difference method for
the computation of fluid flows in complex three dimensional
geometries. Technical Report AERE-R 12342, Harwell Labo-
ratory, 1987.
[CC70] Chapman, S. and Cowling, T.G. The mathematical theory of
non-uniform gases. Cambridge Mathematical Library, Cam-
bridge, 1970.
105
Bibliography
[CE93] Choquet, R. and Erhel, J. Some convergence results for the
Newton-GMRES algorithm. Internal publication 755, IRISA,
September 1993.
[Cha64] Chao, B.T. Turbulent transport behavior of small particles in
dilute suspension. ¨
Osterreichisches Ingenieur Archiev, 18(7):7–
21, 1964.
[Com] Computational fluid dynamics services, Oxfordshire. CFDS-
FLOW3D; User Manual, Release 3.3.
[Cro93] Crowe, C.T. Modeling turbulence in multiphase flows. In W.,
Rodi and F., Martelli (editors), Engineering Turbulence Mod-
elling and Experiments 2. Pp. 899–913. 1993.
[Csa63a] Csanady, G.T. An atmospheric dust fall experiment. J. Atm.
Sci., 21:222–225, 1963.
[Csa63b] Csanady, G.T. Turbulent diffusion of heavy particles in the
atmosphere. J. Atm. Sci., 20:201–208, 1963.
[CSS77] Crowe, C.T., Sharma, M.P., and Stock, D.E. The particle
Source-in-Cell Method for gas-droplet flows. J. Fluid Eng.,
99:325–332, 1977.
[CTC96] Crowe, C.T., Troutt, T.R., and Chung, J.N. Numerical models
for two-phase turbulent flows. Annu. Rev. Fluid. Mech., 28:11–
43, 1996.
[DALJ90] Drew, D.A., Arnold, G.S., and Lahey Jr., R.T. Relation of
microstructure to constitutive equations. In Joseph, D.D. and
Schaeffer, D.G. (editors), Two phase flows and waves. Pp. 45–
56, 1990.
[DG90] Ding, J. and Gidaspow, D. A bubbling fluidization model using
kinetic theory of granular flow. AIChE Journal, 36(4):523–538,
1990.
[DLJ79] Drew, D.A. and Lahey Jr., R.T. Application of general con-
stitutive principles to the derivation of multidimensional two-
phase flow equations. Int. J. Multiphase Flow, 5:243–264, 1979.
[DLJ92] Drew, D.A. and Lahey Jr., R.T. Chapter 16: Analytical model-
ing of multiphase flow. In Roco, M. (editor), Particulate Two-
Phase Flows. Butterworth-Heinemann, 1992.
[Dre71] Drew, D.A. Averaged field equations for two-phase media.
Studies in Applied Mathematics, 50:133–166, 1971.
106
Bibliography
[Dre76] Drew, D.A. Two-phase flows: Constitutive equations for lift
and Brownian motion and some basic flows. Arch. Rat. Mech.
Anal., 62(117), 1976.
[Dre83] Drew, D.A. Mathematical modeling of two-phase flow. Ann.
Rev. Fluid Mech., 15:261–291, 1983.
[DS71] Drew, D.A. and Segel, L.A. Averaged equations for two-phase
flows. Studies in Applied Mathematics, 50(3):205–231, 1971.
[EF94] Eaton, J.K. and Fessler, J.R. Preferential concentration of
particles by turbulence. Int. J. Multiphase Flow, 20 Suppl.:169–
209, 1994.
[Ein06] Einstein, A. Ann. Phys., 19:289, 1906.
[Ein11] Einstein, A. Ann. Phys., 34:591, 1911.
[Erg52] Ergun, S. Fluid flow through packed columns. Chemical Engi-
neering Progress, 48(2):89–94, 1952.
[Fae83] Faeth, G.M. Recent advances in modeling particle transport
properties and dispersion in turbulent flow. In Proc. ASME-
JSME Thermal Engineering Conference, Vol. 2. Pp. 517–534,
1983.
[Fax22] Faxen, H. Die Bewegung einer starren Kugel l¨ngs der Achse
eines mit z¨aher Fl¨ussigkeit gef¨ullten Rohres. Ark. Mat., Astron.
Fys., 17:1–28, 1922.
[FdH+87] Frisch, U., d’Humi´eres, D., Hasslacher, B., Lallemand, P.,
Pomeau, Y., and Rivet, J.-P. Complex Syst., 1:649, 1987.
[FF92] Franca, L.P. and Frey, S.L. Stabilized finite element meth-
ods: II. The incompressible Navier-Stokes equations. Computer
Methods in Applied Mechanics and Engineering, 99:209–233,
1992.
[FFH92] Franca, L.P., Frey, S.L., and Hughes, T.J.R. Stabilized fi-
nite element methods: I. Application to the advective-diffusive
model. Computer Methods in Applied Mechanics and Engineer-
ing, 95:253–276, 1992.
[FH97] Filippova, O. and Haenel, D. Computers in Fluids, 26:697,
1997.
[FHS92] Franca, L.P., Hughes, T.J.R., and Stenberg, R. Stabilized fi-
nite element methods. In Gunzburger, M. D. and Nicolaides,
107
Bibliography
R. A. (editors), Incompressible Computational Fluid Dynam-
ics Trends and Advances. Pp. 87–107. Cambridge University
Press, 1992.
[GBD92] Gidaspow, D., Bezburual, R., and Ding, J. Hydrodynamics
of circulating fluidized beds, kinetic theory approach. In Flu-
idization VII, Proceedings of the 7th Engineering foundation
conference on fluidization. Pp. 75–82, 1992.
[GHN89] Govan, A.H., Hewitt, G.F., and Ngan, C.F. Particle motion
in a turbulent pipe flow. Int. J. Multiphase Flow, 15:471–481,
1989.
[GI81a] Gosman, A.D. and Ioannides, E. Aspects of computer simu-
lation of liquid-fuelled combustors. In AIAA 19th Aerospace
Sciences Meeting, Paper AIAA-81-0323. 1981.
[GI81b] Gosman, A.D. and Ioannides, E. Aspects of computer simu-
lation of liquid-fuelled combustors. Paper AIAA-81-0323, pre-
sented at AIAA 19th Aerospace Science Meeting, St Louis, MO,
1981.
[Gid94] Gidaspow, D. Multiphase flow and fluidization. Continuum and
kinetic theory descriptions. Academic Press, San Diego, 1994.
[GJ96] Graham, D.I. and James, P.W. Turbulent dispersion of par-
ticles using eddy interaction models. Int. J. Multiphase Flow,
22(1):157–175, 1996.
[HHD71] Hutchinson, P., Hewitt, G.F., and Dukler, A.E. Deposition
of liquid or solid dispersions from turbulent gas streams: a
stochastic model. Chem. Engineering Sci., 26:419–439, 1971.
[Hil97] Hiltunen, K. A stabilized finite element method for particulate
two-phase flow equations: Laminar isothermal flow. Computer
Methods in Applied Mechanics and Engineering, 147:387–399,
1997.
[Hin75] Hinze, J.O. Turbulence. McGraw-Hill, New York, 1975.
[HJ91] Hansbo, P. and Johnson, C. Adaptive streamline diffusion
methods for compressible flow using conservation variables.
Computer Methods in Applied Mechanics and Engineering,
87:267–280, 1991.
[HJ95] Hughes, T.J.R. and Jansen, K. A stabilized finite element for-
mulation for the Reynolds-averaged Navier-Stokes equations.
Surveys on Mathematics for Industry, 4(4):279–317, 1995.
108
Bibliography
[HS89] Hwang, G.-J. and Shen, H.H. Modeling the solid phase stress in
a fluid-solid mixture. Int. J. Multiphase Flow, 15(2):257–268,
1989.
[HS91] Hwang, G.-J. and Shen, H.H. Modeling the phase interaction
in the momentum equations of a fluid-solid mixture. Int. J.
Multiphase Flow, 17(1):45–57, 1991.
[Hwa89] Hwang, G.-J. Modeling two-phase flows of a fluid and solid
mixture. PhD thesis, Clarkson University, 1989.
[IM84] Ishii, M. and Mishima, K. Two-fluid model and hydrodynamic
constitutive relations. Nucl. Eng. & Des., 82:107–126, 1984.
[Ish75] Ishii, M. Thermo-fluid dynamic theory of two-phase flow. Ey-
rolles, 1975.
[IZ79] Ishii, M. and Zuber, N. Drag coefficient and relative veloc-
ity in bubbly, droplet or particulate flows. AIChE Journal,
25(5):843–855, September 1979.
[JHW80] James, P.W., Hewitt, G.F, and Whalley, P.B. Droplet motion
in two-phase flow. In Proc. ANS/ASME/NRC International
Topical Meeting on Nuclear Reactor Thermal-Hydraulics. Pp.
1484–1503, 1980.
[JL90] Joseph, D.D. and Lundgren, T.S. Ensemble averaged and mix-
ture theory equations for incompressible fluid-particle suspen-
sions. Int. J. Multiphase Flow, 16:35–42, 1990.
[JR85] Jenkins, J.T. and Richman, M.W. Grad’s 13 moment system
for a dense gas of inelastic spheres. Arch. Ratio. Mech. Anal.,
87:355–377, 1985.
[JS83] Jenkins, J.T. and Savage, S.B. A theory for the rapid flow of
identical, smooth, nearly elastic, spherical particles. J. Fluid
Mech., 130:187–202, 1983.
[Kar02] Karema, H. Application of pressure correction method to multi-
pressure systems of dispersed flows. PhD thesis, Tampere Uni-
versity of Technology, 2002.
[KD59] Krieger, I.M. and Dougherty, T.J. Trans. Soc. Rheol., 3:2817,
1959.
[KKHA98] Koponen, A., Kandhai, D., Hell´en, E., and Alava, M. Phys.
Rev. Lett., 80:716, 1998.
109
Bibliography
[KL99] Karema, H. and Lo, S. Efficiency of interphase coupling al-
gorithms in fluidized bed conditions. Computers & Fluids,
28:323–360, 1999.
[KN90] Krizek, M. and Neittaanm¨aki, P. Finite Element Approxima-
tion of Variational Problems and Applications. Longman Sci-
entific & Technical, 1990.
[Kop98] Koponen, A. Simulations of fluid flow in porous media by lattice
gas and lattice-Boltzmann methods. PhD thesis, University of
Jyv¨askyl¨a, 1998.
[KR89] Kalio, G.A. and Reeks, M.W. A numerical simulation of parti-
cle deposition in turbulent boundary layers. Int. J. Multiphase
Flow, 15:433–446, 1989.
[KVH+99] Kandhai, D., Vidal, D., Hoekstra, A., Hoefsloot, H., Iedema,
P., and Sloot, P. Int. J. Num. Meth. Fluids, 31:1019, 1999.
[Lad94a] Ladd, A.J.C. J. Fluid. Mech., 271:285, 1994.
[Lad94b] Ladd, A.J.C. J. Fluid. Mech., 271:311, 1994.
[LFA93] Lu, Q.Q., Fontaine, J.R., and Aubertin, G. A lagrangian model
for solid particles in turbulent flows. Int. J. Multiphase Flow,
19:347–367, 1993.
[LFF95] Lesoinne, M., Farhat, C., and Franca, L.P. Unusual stabi-
lized finite element methods for second order linear differential
equations. In Cecchi, M. M., Morgan, K., Periaux, J., Schre-
fler, B. A., and Zienkiewicz, O. C. (editors), Finite elements
in fluids, New trends and applications. Pp. 377–386. IACM,
October 1995.
[LL94] Lien, F.S. and Leschziner, M.A. A general non-orthogonal
collocated finite volume algorithm for turbulent flow at all
speeds incorporating second-moment turbulence-transport clo-
sure, part 1: Computational implementation. Computer Meth-
ods in Applied Mechanics and Engineering, 114:149–167, 1994.
[Lo89] Lo, S.M. Mathematical basis of a multi-phase flow model. Tech-
nical Report AERE R 13432, Harwell Laboratory, Oxfordshire,
1989.
[Lo90] Lo, S.M. Multiphase flow model in the Harwell-Flow3D com-
puter code. unclassified AEA-InTec-0062, Harwell Laboratory,
Computational fluid dynamics section, Thermal hydraulics di-
vision, Oxfordshire, June 1990.
110
Bibliography
[LSJC84] Lun, C.K.K., Savage, F.B., Jeffrey, D.J., and Chepurniy, N.
Kinetic theories for granular flow: Inelastic particles in cou-
ette flow and slightly inelastic particles in a general flowfield.
Journal of Fluid Mechanics, 140(4):223–256, 1984.
[Lum57] Lumley, J.L. Some problems connected with the motion of small
particles suspended in turbulent fluid. PhD thesis, the Johns
Hopkins University, 1957.
[MB94] Masson, C. and Baliga, B.R. A control-volume finite-element
method for dilute gas-solid particle flows. Computers & Fluids,
23(8):1073–1096, 1994.
[MBE97] Marsh, C., Backs, G., and Ernst, M. Phys. Rev. E, 56:1676,
1997.
[MC96] Martys, N. and Chen, H. Phys. Rev. E, 53:743, 1996.
[MM97] Mathur, S.R. and Murthy, J.Y. A pressure-based method for
unstructured meshes. Numerical Heat Transfer, part B, 31:195–
215, 1997.
[MR83] Maxey, M.R. and Riley, J.J. Equation of motion for a small
rigid sphere in a nonuniform flow. Phys. Fluids, 26:883–889,
1983.
[MTK96] Manninen, M., Taivassalo, V., and Kallio, S. On the mixture
model for multiphase flow. Research report 288, VTT Technical
Research Centre of Finland, Espoo, 1996.
[Nig79] Nigmatulin, R.I. Spatial averaging in the mechanics of het-
erogeneous and dispersed systems. Int. J. Multiphase Flow,
5:353–385, 1979.
[Ose27] Oseen, C.W. Hydrodynamik. Akademische Verlagsgesellschaft
M.B.H., Leipzig, 1927.
[PBG86] Picart, A., Berlemont, A., and Gouesbet, G. Modelling and
predicting turbulence fields and the dispersion of discrete par-
ticles transported by turbulent flows. Int. J. Multiphase Flow,
12:237–261, 1986.
[Pei98] Peirano, E. Modelling and simulation of turbulent gas-solid
flows applied to fluidization. PhD thesis, Chalmers University
of Technology, 1998.
[PS72] Patankar, S.V. and Spalding, D.B. A calculation procedure for
heat, mass and momentum transfer in parabolic flows. Int. J.
Heat Mass Transfer, 15:1787–1806, 1972.
111
Bibliography
[QdL92] Qian, Y.H., d’Humi´eres, D., and Lallemand, P. Europhys. Lett.,
17:479, 1992.
[RC83] Rhie, C.M. and Chow, W.L. Numerical study of the turbu-
lent flow past an airfoil with trailing edge separation. AIAA
Journal, 21(11):1525–1532, 1983.
[Ril71] Riley, J.J. PhD thesis, The Johns Hopkins University, 1971.
[RSMK+00] Raiskinm¨aki, P., Shakib-Manesh, A., Koponen, A., J¨asberg,
A., Kataja, M., and Timonen, J. Simulations of non-spherical
particles suspended in a shear flow, to be published in. Comput.
Phys. Comm., 2000.
[Rut62] Rutgers, I.R. Relative viscosity and concentration. Rheologica
Acta, Band 2, Heft 4:305–348, 1962.
[RZ54] Richardson, J.F. and Zaki, W.N. Sedimentation and Fluidi-
sation: Part I. Transactions of the Institution of Chemical
Engineers, 32(1):35–53, 1954.
[RZ97] Rothman, D.H. and Zaleski, S. Lattice-gas cellular automata.
Cambridge University Press, Cambridge, 1997.
[SAW92] Sommerfeld, M., Ando, A., and Wennerberg, D. Swirling,
particle-laden flows through a pipe expansion. ASME J. Fluids
Engineering, 114:648–655, 1992.
[SCQM96] Shirolkar, J.S., Coimbra, C.F.M., and Queiroz McQuay, M.
Fundamental aspects of modeling turbulent particle dispersion
in dilute flows. Progr. Energy Combust. Sci., 22:363–399, 1996.
[Soo90] Soo, S.L. Multiphase Fluid Dynamics. Science Press, Beijing,
1990.
[Spa77] Spalding, B.D. The calculation of free-convection phenomena
in gas-liquid mixtures. In Spalding B. D., Afgan N. (editor),
Heat transfer and turbulent buoyant convection studies and ap-
plication. Pp. 569–586. Hemisphere, 1977.
[Spa80] Spalding, D.B. Numerical computation of multi-phase fluid
flow and heat transfer. In Taylor, C. and Morgan, K. (editors),
Recent Advances In Numerical Methods In Fluids. Pp. 139–168.
Pineridge Press, Swansea U.K., 1980.
[Spa83] Spalding, D.B. Developments in the IPSA procedure for nu-
merical computation of multiphase-flow phenomena with inter-
phase slip, unequal temperatures, etc. In Shih, T. M. (editor),
112
Bibliography
Numerical Methodologies in Heat Transfer. Pp. 421–436. Sec-
ond National Symposium, Hemisphere, 1983.
[SRO93] Syamlal, M., Rogers, W., and O’Brien, T.J. MFIX Documen-
tation, Theory Guide, Technical Note DOE/METC-94/1004.
DOE, Morgatown, 1993.
[SS83] Saad, Y. and Schultz, M.H. GMRES: A generalized minimum
residual algorithm for solving nonsymmetric linear systems.
Technical Report Research report 254, Department of com-
puter science, Yale University, 1983.
[SWd85] Sonneveld, P., Wesseling, P., and de Zeeuw, P.M. Multigrid
and conjugate gradient methods as convergence acceleration
techniques. In Paddon, D.J. and Holstein, H. (editors), Multi-
grid methods for integral and differential equations. Clarendon
Press, Oxford, 1985.
[Tay21] Taylor, G.I. Diffusion by continuous movements. Proceedings
of the London Mathematical Society, Series 2, 20:196, 1921.
[Tch47] Tchen, C.M. Mean value and correlation problems connected
with the motion of small particles suspended in a Turbulent
fluid. Martinus Nijhoff, The Hague, 1947.
[TL72] Tennekes, H. and Lumley, J.L. A first course in turbulence.
The MIT Press, Cambridge, 1972.
[Ung93] Ungarish, M. Hydrodynamics of suspensions: Fundamentals
of centrifugal and gravity separation. Springer-Verlag, Berlin,
1993.
[VDR84] Van Doormal, J.P. and Raithby, G.D. Enhancements of the
SIMPLE method for predicting incompressible fluid flows. Nu-
merical Heat Transfer, 7:147–163, 1984.
[WAW82] Wilkes, N.S., Azzopardi, B.J., and Willets, I. Drop motion and
deposition in annular two-phase flow. Research report HTFS
RS425 AERE R10571, AERE Engineering Sciences Division,
Harwell, 1982.
[WBAS84] Weber, R., Boysan, F., Ayers, W.H., and Swithenbank, J. Sim-
ulation of dispersion of heavy particles in confined turbulent
flows. AICHE J., 30:490–493, 1984.
[WS92] Wang, L-P. and Stck, D.E. Stochastic trajectory models for
turbulent diffusion: Monte carlo process versus markov chains.
Atmospher. Envir., 26A:1599–1607, 1992.
113
Appendix 1
Submatrices and vectors for the finite-element matrix system (2.109):
A11
kl = 2K11
f(k, l) + K22
f(k, l) + K33
f(k, l) + Cf(k, l) + If(k, l)
+Sc
f(k, l) + SI
f(k, l) + SM11
f(k, l)
A12
kl =K12
f(k, l) + SM12
f(k, l)
A13
kl =K13
f(k, l) + SM13
f(k, l)
A14
kl =−P1
f(k, l) + Sp1
f(k, l)
A15
kl =−If(k, l)−SI
f(k, l)
A1i
kl = 0, i = 6,7,8
A21
kl =K21
f(k, l) + SM21
f(k, l)
A22
kl =K11
f(k, l) + 2K22
f(k, l) + K33
f(k, l) + Cf(k, l) + If(k, l)
+Sc
f(k, l) + SI
f(k, l) + SM22
f(k, l)
A23
kl =K23
f(k, l) + SM23
f(k, l)
A24
kl =−P2
f(k, l) + Sp2
f(k, l)
A25
kl = 0
A26
kl =−If(k, l)−SI
f(k, l)
A2i
kl = 0, i = 7,8
A31
kl =K31
f(k, l) + SM31
f(k, l)
A32
kl =K32
f(k, l) + SM32
f(k, l)
A33
kl =K11
f(k, l) + K22
f(k, l) + 2K33
f(k, l) + Cf(k, l) + If(k, l)
+Sc
f(k, l) + SI
f(k, l) + SM33
f(k, l)
A34
kl =−P3
f(k, l) + Sp3
f(k, l)
A3i
kl = 0, i = 5,6
A37
kl =−If(k, l)−SI
f(k, l)
A38
kl = 0
A41
kl =−P1
f(l, k)−Sqc1
f(k, l)−SqI1
f
1/1
Appendix 1
A42
kl =−P2
f(l, k)−Sqc2
f(k, l)−SqI2
f(k, l)
A43
kl =−P3
f(l, k)−Sqc3
f(k, l)−SqI3
f(k, l)
A44
kl =−Sqp
f(k, l)
A45
kl =SqI1
f(k, l)
A46
kl =SqI2
f(k, l)
A47
kl =SqI3
f(k, l)
A48
kl = 0
A51
kl =−IA(k, l)−SI
A(k, l)
A5i
kl = 0, i = 2,3
A54
kl =−P1
A(k, l) + Sp1
A(k, l)
A55
kl = 2K11
A(k, l) + K22
A(k, l) + K33
A(k, l) + CA(k, l) + IA(k, l)
+Sc
A(k, l) + SI
A(k, l)
A56
kl =K12
A(k, l)
A57
kl =K13
A(k, l)
A58
kl = 0
A61
kl = 0
A62
kl =−IA(k, l)−SI
A(k, l)
A63
kl = 0
A64
kl =−P2
A(k, l) + Sp2
A(k, l)
A65
kl =K21
A(k, l)
A66
kl =K11
A(k, l) + 2K22
A(k, l) + K33
A(k, l) + CA(k, l) + IA(k, l)
+Sc
A(k, l) + SI
A(k, l)
A67
kl =K23
A(k, l)
A68
kl = 0
A71
kl = 0
A72
kl = 0
A73
kl =−IA(k, l)−SI
A(k, l)
A74
kl =−P3
A(k, l) + Sp3
A(k, l)
A75
kl =K31
A(k, l)
A76
kl =K32
A(k, l)
A77
kl =K11
A(k, l) + K22
A(k, l) + 2K33
A(k, l) + CA(k, l) + IA(k, l)
+Sc
A(k, l) + SI
A(k, l)
A78
kl = 0
A81
kl =SqI1
A(k, l)
1/2
Appendix 1
A82
kl =SqI2
A(k, l)
A83
kl =SqI3
A(k, l)
A84
kl =−Sqp
A(k, l)
A85
kl =−Sqc1
A(k, l)−SqI1
A(k, l)
A86
kl =−Sqc2
A(k, l)−SqI2
A(k, l)
A87
kl =−Sqc3
A(k, l)−SqI3
A(k, l)
A88
kl =−DM(k, l)−¯
P1
A(l, k)−¯
P2
A(l, k)−¯
P3
A(l, k)
+¯
SM11
A(k, l) + ¯
SM12
A(k, l) + ¯
SM13
A(k, l)
+¯
SM21
A(k, l) + ¯
SM22
A(k, l) + ¯
SM23
A(k, l)
+¯
SM31
A(k, l) + ¯
SM32
A(k, l) + ¯
SM33
A(k, l),
where the indices get values k, l = 1,...,ndofs. The integral forms above
are defined as
Dij
α(k, l) = (1/Reα)(φα
∂ϕl
∂xi
,∂ϕk
∂xj
)
Cα(k, l) = (φαuα· ∇ϕl, ϕk)
Pi
α(k, l) = (ϕl, βα(φα
∂ϕk
∂xi
+ϕk
∂φα
∂xi
))
¯
Pi
α(k, l) = (ϕl, βα(ϕk
∂uαi
∂xi
+uαi
∂ϕk
∂xi
))
Iα(k, l) = (gβαϕl, ϕk)
Sc
α(k, l) = (φαuα· ∇ϕl, ταuα· ∇ϕk)
Spi
α(k, l) = (βαφα
∂ϕl
∂xi
, ταuα· ∇ϕk)
SI
α(k, l) = (gβαϕl, ταuα· ∇ϕk)
Sqci
α(k, l) = (φαuα· ∇ϕl, ταβα
∂ϕk
∂xi
)
Sqp
α(k, l) = (βαφα∇ϕl, ταβα∇ϕk)
SqIi
α(k, l) = (gβαϕl, ταβα
∂ϕk
∂xi
)
SMij
α(k, l) = (βα(φα
∂ϕl
∂xj
+ϕl
∂φα
∂xj
), δαβα(φα
∂ϕk
∂xi
+ϕk
∂φα
∂xi
))
¯
SMij
α(k, l) = (βα(ϕl
∂uαj
∂xj
+uαj
∂ϕl
∂xj
), δαβα(ϕk
∂uαi
∂xi
+uαi
∂ϕk
∂xi
))
DM(k, l) = (βAκ∇ϕl,∇ϕk).
1/3
Appendix 1
The subvectors of the right hand side vector Fare defined as
Fi(k) = (φfβf˜
Fi
f, ϕk+τf(φfuf· ∇ϕk)), i = 1,2,3
F4(k) = (φfβf˜
Ff,−τfφfβf∇ϕk)
Fi(k) = (φAβA˜
Fi−4
A, ϕk+τA(φAuA· ∇ϕk)), i = 5,6,7
F8(k) = (φAβA˜
FA,−τAφAβA∇ϕk),
where k= 1,...,ndof s.
1/4
Series title, number and
report code of publication
VTT Publications 722
VTT-PUBS-722
Author(s)
Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja,
Antti Koponen, Mikko Manninen & Veikko Taivassalo
Title
Multiphase Flow Dynamics
Theory and Numerics
Abstract
The purpose of this work is to review the present status of both theoretical and
numerical research of multiphase flow dynamics and to make the results of that
fundamental research more readily available for students and for those working
with practical problems involving multiphase flow.
Flows that appear in many of the common industrial processes are intrinsically
multiphase flows. The advanced technology associated with these flows has
great economical value. Nevertheless, our basic knowledge and understanding of
these processes is often quite limited as, in general, is our capability of solving
these flows. In the first part of this publication we give a comprehensive review of
the theory of multiphase flows accounting for several alternative approaches. We
also give general guidelines for solving the ’closure problem’, which involves,
e.g., characterising the interactions between different phases and thereby deriv-
ing the final closed set of equations for the particular multiphase flow under con-
sideration. The second part is devoted to numerical methods for solving those
equations.
ISBN
978-951-38-7365-3 (soft back ed.)
978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp)
Series title and ISSN Project number
VTT Publications
1235-0621 (soft back ed.)
1455-0849 (URL: http://www.vtt.fi/publications/index.jsp)
33861
Date Language Pages
December 2009 English, Finnish abstr. 113 p. + app. 4 p.
Name of project Commissioned by
Dynamics of Multiphase Flows, Finnish national
Computational Fluid Dynamics Technology
Programme 1995–1999
Keywords Publisher
Multiphase flows, volume averaging, ensemble
averaging, mixture models, multifluid finite
volume method, multifluid finite element method,
particle tracking, the lattice-BGK model
VTT Technical Research Centre of Finland
P.O. Box 1000, FI-02044 VTT, Finland
Phone internat. +358 20 722 4520
Fax +358 20 722 4374
Julkaisun sarja, numero ja
raporttikoodi
VTT Publications 722
VTT-PUBS-722
Tekijä(t)
Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja,
Antti Koponen, Mikko Manninen & Veikko Taivassalo
Nimeke
Monifaasivirtausten dynamiikka
Teoriaa ja numeriikkaa
Tiivistelmä
Työssä tarkastellaan monifaasivirtausten teoreettisen ja numeerisen tutkimuksen
nykytilaa, ja muodostetaan tuon perustutkimuksen tuloksista selkeä kokonaisuus
opiskelijoiden ja käytännön virtausongelmien kanssa työskentelevien käyttöön.
Monissa teollisissa prosesseissa esiintyvät virtaukset ovat olennaisesti moni-
faasivirtauksia – esimerkiksi kaasu-partikkeli-, neste-partikkeli- ja neste-kuitu-
suspensioiden virtaukset, sekä kuplavirtaukset, neste-neste-virtaukset ja virtaus
huokoisen aineen läpi. Julkaisun ensimmäisessä osassa tarkastellaan kattavasti
monifaasivirtausten teoriaa ja esitetään useita vaihtoehtoisia lähestymistapoja.
Toisessa osassa käydään läpi monifaasivirtauksia kuvaavien yhtälöiden numeeri-
sia ratkaisumenetelmiä.
ISBN
978-951-38-7365-3 (nid.)
978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp)
Avainnimeke ja ISSN Projektinumero
VTT Publications
1235-0621 (nid.)
1455-0849 (URL: http://www.vtt.fi/publications/index.jsp)
33861
Julkaisuaika Kieli Sivuja
Joulukuu 2009 Englanti, suom. tiiv. 113 s. + liitt. 4 s.
Projektin nimi Toimeksiantaja(t)
Dynamics of Multiphase Flows, Finnish national
Computational Fluid Dynamics Technology
Programme 1995–1999
Avainsanat Julkaisija
Multiphase flows, volume averaging, ensemble
averaging, mixture models, multifluid finite
volume method, multifluid finite element method,
particle tracking, the lattice-BGK model
VTT
PL 1000, 02044 VTT
Puh. 020 722 4520
Faksi 020 722 4374
VTT CREATES BUSINESS FROM TECHNOLOGY
ISBN 978-951-38-7365-3 (soft back ed.) ISBN 978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp)
ISSN 1235-0621 (soft back ed.) ISSN 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp)
The purpose of this work is to review the present status of both theoretical and
numerical research of multiphase flow dynamics and to make the results of that
fundamental research more readily available for students and for those working with
practical problems involving multiphase flow.
Flows that appear in many of the common industrial processes are intrinsically
multiphase flows. The advanced technology associated with these flows has great
economical value. Nevertheless, our basic knowledge and understanding of these
processes is often quite limited as, in general, is our capability of solving these flows.
In the first part of this publication we give a comprehensive review of the theory of
multiphase flows accounting for several alternative approaches. We also give general
guidelines for solving the ’closure problem’, which involves, e.g., characterising the
interactions between different phases and thereby deriving the final closed set of
equations for the particular multiphase flow under consideration. The second part
is devoted to numerical methods for solving those equations.
