Analytical Description of Two-Phase
Turbulent Flows∗
F. Mashayek† and R.V.R. Pandya
Department of Mechanical Engineering
University of Illinois at Chicago
842 West Taylor Street
Chicago, IL 60607
Abstract
Various existing analytical descriptions for predicting turbulent flows laden with
solid particles or liquid droplets are reviewed here. The main focus is on a collisionless
dispersed phase, however, the two-way coupling effects are considered and discussed.
The review of various methods is conducted by dividing them into two main categories.
The first category includes direct numerical simulation (DNS), large-eddy simulation
(LES), and stochastic modeling, which are collectively called the ‘Lagrangian description’. The second category, under the ‘Eulerian description’, includes Reynolds averaged Navier-Stokes (RANS) and probability density function (pdf) modeling. The emphasis is placed on application of these approaches for both understanding and prediction of turbulent dispersed phase. The discussion is focused on merits and limitations
of these approaches and the nature of predictions offered by them. The mathematical
aspects of RANS and pdf modeling are presented in greater detail. The important
role of DNS generated data in the development and assessment of other approaches is
discussed with the aid of some representative examples in particle-laden homogeneous
turbulent flows.
Keywords: two-phase turbulent flow, particle, droplet, direct numerical simulation, large-eddy simulation, stochastic, second-order closure, probability density function
∗
†
Review article submitted for publication in Prgress in Energy and Combustion Science.
Corresponding author. Email: mashayek@uic.edu, Tel: (312)996-1154.
1
Contents
1 Introduction
1.1
Scope
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Governing Equations
5
7
2.1
Carrier Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Dispersed Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Lagrangian Description
3.1
13
Direct numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.1.1
Isotropic homogeneous flows . . . . . . . . . . . . . . . . . . . . . . .
16
3.1.2
Anisotropic homogeneous flows . . . . . . . . . . . . . . . . . . . . .
21
3.1.3
Inhomogeneous flows . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Large-eddy simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.3
Stochastic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.3.1
Models with implicit velocity autocorrelations . . . . . . . . . . . . .
41
3.3.2
Models with explicit velocity autocorrelations . . . . . . . . . . . . .
46
3.3.3
Low-cost stochastic simulations . . . . . . . . . . . . . . . . . . . . .
50
4 Eulerian Description
4.1
4.2
52
RANS modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.1.1
Second-order moment modeling . . . . . . . . . . . . . . . . . . . . .
56
4.1.2
Algebraic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Pdf modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.2.1
Application of DIA and LHDIA . . . . . . . . . . . . . . . . . . . . .
70
4.2.2
Application of Furutsu-Novikov-Donsker formula . . . . . . . . . . . .
78
4.2.3
Application of Van Kampen’s method . . . . . . . . . . . . . . . . . .
84
4.2.4
Macroscopic equations . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5 Model Assessment via DNS
91
6 Concluding Remarks
95
2
1
Introduction
The term two-phase refers to a physical situation, formed by a fluid phase and a solid phase,
as a mixture of fluid with solid or two immiscible fluids. In two-phase flows, different phases
are separated by interfaces, interact dynamically through interfaces and are in motion in
space due to external force(s) acting on them. In many such flows, one of the fluid phases is
continuously connected (carrier phase) and the other phase is in the form of solid particles
or liquid droplets (hereafter may be simply referred to as ‘particle’ or dispersed phase).
The carrier phase flow can be in laminar or turbulent regime and the discussion in this
review is focused on turbulent flows. These type of flows occur in many important natural
and technological situations, e.g., aerosol transport and deposition,1 spray combustion,2, 3
fluidized bed combustion,4 plasma spray coating and synthesis of nanoparticles.5 Particleladen flows can be broadly classified in three categories determined on the basis of interparticle collisions, namely, 1) collision-free flow (dilute flow) 2) collision-dominated flow
(medium concentration flow) and 3) contact-dominated flow (dense flow).6 These turbulent
flows continue to provide a challenge to engineers and physicists in developing the analytical
description and predictive theory/model from the first principles. An attempt is made here
to review various methodologies, theories/models for the complete description of two-phase
turbulent flows with the main focus on collision-free flow.
The carrier phase flows under the influence of external forces, e.g. pressure force, and is
governed by the Navier-Stokes equation and conservation of mass equation in the Eulerian
framework. The motion of the dispersed phase occurs due to forces generated by the moving
fluid and acting through the interfaces, such as fluid drag force. The particles can be acted
upon by other forces, e.g. gravity, thermophoresis, electrostatic (for charged particles), and
all these forces with Newton’s second law govern the Lagrangian trajectory of motion of
each particle in the fluid flow domain. These Eulerian equations for the carrier phase along
with the Lagrangian equations for the particle phase are referred to as the ‘first-principle’
equations in this review. In general, particles are driven by carrier phase and as a consequence
3
this phase experiences reaction forces from these particles. Inclusion of such forces in the
carrier-phase equations and flow-field modifications due to the presence of these particles
make the first-principle equations coupled in nature.
In principle, the numerical solutions of these equations for turbulent flows, involving a
large number of particles, with appropriate boundary and initial conditions would describe
completely the two-phase flows. This type of direct numerical simulation (DNS) from the first
principles is feasible for turbulent flows in simple geometries and becomes computationally
expensive in cases of inhomogeneous turbulent flow of practical interests. In many such cases
of high Reynolds number flow, resolving all the details of smaller eddies of turbulent flow
becomes impossible by DNS due to the limitation of computational resources. At present,
the computational effort for DNS can be reduced substantially by modeling the average
effect of smaller eddies, which cannot be resolved, on the resolved scales and simulating the
instantaneous flow for large eddies, formally known as large-eddy simulation (LES). The
trajectories for particles are then calculated by using LES generated flow field. Yet another
method, stochastic modeling (SM), for reducing the computational effort is to generate a
‘synthetic’ turbulent flow field having some of its statistical properties of interest identical to
that of the real turbulent flow. The statistical properties of the real flow can be predicted by
single point closure schemes for turbulence, such as, k-² and Reynolds stress closure schemes.7
In this review, these DNS, LES and SM approaches are collectively called “Lagrangian
description,” in which the carrier phase is simulated in an Eulerian frame and the particles
are tracked in a Lagrangian coordinate system.
The analysis in the Lagrangian framework for the particles is accurate, and the more
complex situations of collisions, coagulations, and evaporation if they exist, can be easily
handled in this framework. Despite these advantages, one has to abandon the Lagrangian approach in the prediction of situations where the required number of particles is large enough
to hinder their Lagrangian trajectories simulation. Then, as an alternative, the dispersed
phase can be treated as ‘fluid’ in Eulerian framework and modeling of these fluid equa-
4
tions from the Lagrangian equations continues to create a challenge for researchers. The
existing models can be viewed under the framework of one of the two modeling approaches,
namely, 1) Reynolds averaged Navier-Stokes (RANS) type modeling and 2) kinetic equation or probability density function (pdf) modeling. These approaches result in fluid-like
or continuum governing equations for the statistical properties of the dispersed phase. In
the first approach, modeling involves forming the instantaneous Eulerian equations from
the Lagrangian equations for particles and then modeling of the ensemble average of these
equations due to the involved closure problem. In the second approach, the equation for
pdf transport is obtained from Lagrangian equations via using the Liouville’s theorem and
solving the involved closure problem. The various moments of the pdf equation result in
Eulerian equations for the description of statistical properties of the dispersed phase. These
two approaches are referred to as ‘Eulerian description’ in this review.
1.1
Scope
While this review article was under preparation, several other reviews on two-phase flows
appeared in the literature. Whereas, by itself, this is a clear indication of intensity of the
ongoing research and interest in the general area of two-phase flows, it has been a determining factor in defining the scope of the present article. This review has been prepared to
complement the previous reviews while avoiding overlapping discussions as much as possible.
We will take advantage of these previous works in describing some of the fundamental issues,
such as the governing equations, where necessary and possible.
All the flows discussed in this review involve turbulence which itself is an unsolved problem of classical mechanics and various theoretical and computational approaches for its
understanding and prediction have been evolved after the first formulation of the closure
problem in the late 1800s by Osborne Reynolds. The addition of particles to turbulent flows
further enhances the complexity, and various attempts for complete description of the dispersed phase have led to similar approaches. Writing a thorough and intricate review on the
5
approaches for turbulent flows itself is an enormous task and inclusion of various approaches
to tackle dispersed phase would make the task even more gigantic. Here, we prefer not to
undertake such task, rather limit our scope and focus on recent advances, developed in a
past few decades, in describing mainly the turbulent dispersed phase. We shall assume that
the reader has a previous knowledge of turbulence, at least at a fundamental level.
In view of the above, this review is designed to familiarize the reader with analytical tools
for investigating two-phase turbulent flows that may be considered dilute enough to neglect
the direct interactions and collisions between the particles. We will, however, consider the
two-way coupling by which the presence of the dispersed phase may affect the turbulence in
the carrier phase. Attention is restricted to two-phase flows where the density of the particle
is considerably larger than the density of the carrier phase. These mainly include gas flows
laden by either solid particles or liquid droplets – we will use both ‘particles’ and ‘droplets’
alternatively when referring to the dispersed phase. Various approaches are primarily discussed in the context of isothermal flows and their generalization to non-isothermal flows
and cases of evaporating droplets is reviewed briefly.
The governing equations forming the first-principle equations for the carrier and dispersed phase are given in the Section 2. These equations form the starting point for various analytical approaches which are categorized as direct numerical simulation, large-eddy
simulation, stochastic modeling, Reynolds average Navier-Stokes modeling, and probability density function modeling. DNS, LES and stochastic approaches are collectively put
together and discussed in Section 3, Lagrangian Description, as the dispersed phase is tackled in the Lagrangian frame in all these approaches. When discussing DNS and LES, we
will emphasize on the capabilities of these techniques for both physical understanding and
generating reliable data banks for model assessment. We will also devote some efforts in
describing the numerical aspects as implemented in previous DNS and LES works to provide
the reader with a comprehensive view when comparing various studies. For example, we
note that similar DNS or LES works may differ rather significantly in their treatment of the
6
particle equations or interpolation schemes. We then consider the stochastic modeling in
which different methods for generating a synthetic turbulence, along the particle path, are
discussed.
The RANS and pdf modeling approaches result in continuum equations for the dispersed
phase in the Eulerian frame and are covered in Section 4. In this section, recent works
on RANS-type modeling of particle phase, while properly taking into account the two-way
coupling, are discussed. Further, a detailed account of various closure schemes used in the
one-point probability density function approach is provided and macroscopic or continuum
equations for the dispersed phase are presented.
Due to our own engagement in both numerical simulation and modeling of two-phase
flows, a particular emphasis is placed on the inter-connection between the analytical approaches considered in this review. In this regard, a section ‘Assessment via DNS’ is included
to discuss the implementation of DNS results in the process of development and assessment
of statistical (RANS and pdf) and stochastic models for two-phase turbulent flows. Finally,
some concluding remarks are provided in the last section.
2
Governing Equations
It has been well established through DNS studies that the Navier-Stokes and continuity
equations are capable of generating all the details of real turbulence fluctuations and thus
can be viewed as a first principle for the isothermal turbulent fluid flow. Ideally, at any instant
of time t, the flow field around all the dispersed particles can be obtained by numerically
solving the Navier-Stokes and continuity equations. The knowledge of the flow field enables
us to calculate forces acting on various particles and subsequently the new location of all the
particles at time t + ∆t, can be obtained from Newton’s second law. Then the calculation of
the flow field at time t + ∆t can be repeated with new location of the particles. For a very
small ∆t, this type of calculation can successfully predict the two-phase flow, but it is not
7
feasible for real situations where the number of particles are very large. Therefore, particles
are considered as ‘point’ particles. Instead of computing the forces by solving flow around
them, various existing methodologies use expressions for these forces which are obtained from
either experimental, analytical or DNS studies on a single or ‘lonely’ particle. Certainly, the
obtained expressions do not imitate exactly the real forces acting on the particle surrounded
by other particles, and their improvement is of more importance for the prediction of medium
concentration and dense two phase flows. Despite the shortcoming in accurate description of
forces, the Lagrangian equations for point particle with expressions for the forces on a lonely
particle have been considered as the main starting point in almost all practical applications
thus far and will be discussed below as first-principle equations for the dispersed phase.
2.1
Carrier Phase
b of a Newtonian carrier fluid with density
The flow velocity Ubi , pressure Pb , and enthalpy h
ρ, viscosity µ, coefficient of bulk viscosity λ, and thermal conductivity κ are governed by
continuity, momentum and energy equations. These Eulerian equations in space-time (xi , t)
are written as:8, 9
Continuity Equation:
∂ Ubj
Dρ
+ρ
= Sm ,
Dt
∂xj
(2.1)
b
³ ∂U
∂τij
∂ Ubj ´
∂ Ubk
D(ρUbi )
i
=
+ Sui ; τij = −Pb δij + µ
+
+ δij λ
,
Dt
∂xj
∂xj
∂xi
∂xk
(2.2)
Momentum Equation:
Energy Equation:
b
b
´ ∂U
DPb
∂ ³ ∂T ´ ³
D(ρh)
i
=
+
κ
+ τij + Pb δij
+ Se ,
Dt
Dt
∂xj ∂xj
∂xj
8
(2.3)
where the material derivative D/Dt is defined as
D
∂
∂
=
+ Ubj
.
Dt
∂t
∂xj
(2.4)
Also, T is the fluid temperature at (xi , t), and source/sink terms Sm , Sui , and Se describe
mass, momentum and energy exchange between carrier and dispersed phases. The exact
forms of these source/sink terms depend on the application and various forms have been
suggested in the literature. Equation (2.2) with Sui = 0 and λ = −2µ/3 is the well known
Navier-Stokes equation. When the fluid is a perfect gas, equation of state with R as gas
constant is written as
Pb = ρRT.
(2.5)
Equations (2.1)-(2.5) form the first-principle equations for the flow of compressible, Newtonian, perfect gas as a carrier phase.
2.2
Dispersed Phase
The Lagrangian equations governing the position Xi , velocity Vbi , and temperature Tp along
the trajectory of each spherical particle, of radius a, mass mp , and specific heat coefficient
Cp , in the carrier flow field can be written as
dXi
= Vbi ,
dt
mp
dVbi
= Fi ,
dt
mp Cp
dTp
= Q,
dt
(2.6)
(2.7)
(2.8)
where Fi denotes the summation of all the forces acting on the particle and Q is the net rate
of heat transfer to the particle. While writing equation (2.8), temperature variation inside
the particle is neglected and thus particle temperature Tp is considered uniform.
9
The problem of predicting the forces on a particle moving in a viscous fluid with velocity
field Ubi has been studied in the past 150 years since Stokes first obtained, in the year 1851,
the drag force on a sphere in creeping flow condition. These predictions still remain open for
further improvement.10 Historical background and a detail account on the relevant works
are available in recent lucid review articles by Michaelides,11 Michaelides and Feng12 and
Gouesbet et al.13
Boussinesq and Basset independently obtained an equation of motion11, 12 of a solid spherical particle, starting from rest inside an infinite, viscous, and quiescent fluid. The equation
can be written as
Z t
dVbi
mf dVbi
dVbi /dτ
2
0.5
b
mp
= −6πaµVi −
− 6a (πµρ)
dτ
dt
2 dt
0 (t − τ )0.5
(2.9)
where the first term on the right-hand side is equal to the Stokes drag, second term represents
the added mass force and the last term is the history integral force. These three forces are
accurate for a slowly moving but rapidly accelerating sphere and a new equation was proposed
by Odar and Hamilton for a rectilinear and unrestricted arbitrary motion of a sphere.14 The
proposed equation can be written as
mp
Z t
CD 2 c b
mf dVbi
dVbi /dτ
dVbi
=−
πa ρ|V|Vi − CA
− CH 6a2 (πµρ)0.5
dτ
dt
2
2 dt
0 (t − τ )0.5
(2.10)
where mf is the mass of the fluid displaced by the particle and CD is the drag coefficient.
Also CA and CH are coefficients, which were determined empirically from the experiments,
dependent on the acceleration number A and written as15, 16
CA = 2.1 −
c2
0.52
|V|
0.132
;
C
=
0.48
+
;
A
=
,
H
b
A2 + 0.12
(A + 1)3
|
2a| ddtV
(2.11)
c is the particle velocity vector with magnitude |V|.
c
where V
Equation (2.9) is limited to the situation in which fluid is at rest and this restriction
10
was relaxed by Tchen17 and further improved by Corrsin and Lumley.18 Later, Buevich19
pointed out the strange un-physical behavior of the equation proposed by Corrsin and Lumley
and suggested the correct form of the equation. A more complete derivation taking proper
account of the effect of spatial variation of the flow field was later proposed by Maxey and
Riley.20 The final equation of motion in nonuniform flow is
mp
h
i
a2 ∂ 2 Ubi ¯¯
dVbi
DUbi ¯¯
¯
− 6πaµ Vbi − Ubi (Xi (t), t) −
= (mp − mf )gi + mf
¯
dt
Dt Xi (t)
6 ∂xj ∂xj Xi (t)
i
mf d h b
a2 ∂ 2 Ubi ¯¯
b
−
Vi − Ui (Xi (t), t) −
¯
2 dt
10 ∂xj ∂xj Xi (t)
¯
i
h
b (X (τ ), τ ) − a2 ∂ 2 Ubi ¯¯
Z t d Vbi − U
i
i
dτ
6 ∂xj ∂xj Xi (τ )
0.5
− 6a2 (πµρ)
(t − τ )0.5
0
dτ,
(2.12)
where gi is acceleration due to gravity, and the equation is valid for small particle Reynolds
c − U|/µ
c
numbers Rep = 2aρ|V
< 1. The five terms on the right-hand side of equation (2.12)
represent buoyancy force, fluid acceleration force, drag force, added mass force, and history
force, respectively. The expressions for drag, added mass and history forces are modified due
to the flow curvature effect represented by the term containing
¯
bi ¯
∂2U
¯
.
∂xj ∂xj Xi (t)
In recent years,
equation (2.12) with the drag term replaced by a more general expression valid for higher
values of Rep is used by researchers. The general expression for the drag has been
CD 2 c c b
πa ρ|U − V|(Ui − Vbi ),
2
(2.13)
where CD can be taken as (see Clift et al.21 for a list of CD )
CD =
24
0.42
(1 + 0.15Re0.687
)+
p
Rep
1 + 4.25 × 104 Re−1.16
p
∀ Rep < 3 × 105
(2.14)
and the curvature effect is neglected.
Saffman22, 23 showed that the spherical particle in the shear flow experiences a lift force.
11
This Saffman lift force Fs is given by5
c −0.5 [(U
c − V)
c × (∇ × U)],
c
Fs = 6.44a2 (µρ)0.5 |∇ × U|
(2.15)
and its inclusion along with the modification of the drag given by (2.13) in equation (2.12)
results in an equation which is referred to hereafter as modified Maxey-Riley (MMR) equation.
Other forces which can be included in the MRR equation are 1) Magnus force arising due
to rotation of the particle 2) thermophoretic force 3) Coulomb force for charged particles,
and are discussed in a recent book by Crowe et al.5
Recently, Michaelides and Feng12, 24 derived an expression for Q in an unsteady flow and
temperature field for a rigid sphere with high thermal conductivity and at low Peclet number.
The derived Lagrangian equation governing the temperature of the particle is
"
#
dTp
a2 ∂ 2 T ¯¯
DT ¯¯
mp Cp
= −4πaκ Tp − T (Xi (t), t) −
¯
¯
+ mf Cf
dt
6 ∂xj ∂xj Xi (t)
Dt Xi (t)
¯
h
i
2
2
Z t d Tp − T (Xi (τ ), τ ) − a ∂ T ¯¯
dτ
6 ∂xj ∂xj Xi (τ )
0.5
− 4a2 (πρCf κ)
(t − τ )0.5
0
dτ
(2.16)
in which Cf is the specific heat capacity of the fluid and the three terms on the right-hand side
are analogous to the drag, added mass and history terms in the equation of motion for the
particle.12, 24 The term with
¯
∂2T ¯
¯
∂xj ∂xj Xi (t)
accounts for the curvature effect of the temperature
field. The first term, without the curvature effect, when multiplied by heat transfer factor
N u/2 can make the equation applicable for large Rep and is hereafter referred to as modified
Michaelides-Feng (MMF) equation. The Nusselt number N u for the spherical particle is
given by Ranz-Marshall correlation25 and is written as
N u = 2 + 0.6Rep0.5 P r0.33
12
∀ Rep < 5 × 104
(2.17)
where P r is the Prandtl number for the carrier fluid phase.
Equation (2.6) along with MMR and MMF equations are considered in this review as the
set of first-principle equations for the particle phase. These equations are strictly applicable
for solid spherical particles and the cases of viscous particles (droplets) and bubbles are
discussed in detail elsewhere.11, 12, 26–28
3
Lagrangian Description
The methods discussed in this section all treat the dispersed phase in a similar fashion,
i.e. by tracking a large number of particles in a Lagrangian frame. These methods primarily
differ in their treatment of the continuous (carrier) phase. In DNS, the exact solution
of the Navier-Stokes and continuity equations is attempted by using high-order numerical
schemes on a sufficiently fine grid. LES offers a solution for larger scales of turbulence
while the smaller scales are modeled. Both of these methods attempt to produce a ‘real’
turbulence, whereas in the third approach considered in this section, the stochastic simulation
of turbulence is mainly designed for the dispersed phase and provides a ‘synthetic’ turbulence
along the particle path. While all three methods heavily rely on statistical averaging for final
interpretation of the results, it is important to note that averaging (over a large number of
particles) is the only way to produce ‘meaningful’ results in case of stochastic simulations.
We will elaborate more on this point in Section 3.3.
3.1
Direct numerical simulation
The term DNS is extensively used to describe simulations of turbulent flows without implementation of turbulence models. The basic concept in this methodology is to resolve
all the scales of the flow by solving the Navier-Stokes equations on sufficiently fine grids.
Nevertheless, the fact is that one may still argue the true ‘directness’ of these simulations
13
due to the assumptions involved in the Navier-Stokes equations and/or in the treatment of
the fluid properties. This argument gains a much stronger impetus when DNS term is used
in conjunction with two-phase flows as the treatment of the dispersed phase always involves
physical models. This is mainly due to the fact that it is not possible to resolve the flow
around (and inside, in case of droplets) the dispersed phase and one must resort to physical
models to describe the interactions between the two phases. The situation becomes even
more restrictive when the two-way coupling between the phases is considered. Despite our
acknowledgement of the above fact, we continue to use the term DNS here as it has become
a common terminology when referring to simulations in which the carrier-phase equations
are solved on a fine grid and no turbulence model is implemented for the description of this
phase.
Direct numerical simulations of two-phase turbulent flows have been conducted in three
configurations: (i) isotropic homogeneous, (ii) anisotropic homogeneous, and (iii) inhomogeneous. An ‘isotropic’ turbulence is defined as one in which the average quantities are
independent of position in the fluid and do not change under rotation or reflection of the
coordinate system. In ‘homogeneous’ turbulence, the average properties of random motion
are independent of position in the fluid. Therefore, for example, the Reynolds stress tensor
reduces to only one component in isotropic turbulence, whereas in homogeneous turbulence
all components can be present and have different values. If these values are also allowed to
differ from one location in the flow to another, then the flow is called ‘inhomogeneous’.29–31
In almost all of the reported DNS studies, the carrier phase is simulated in the Eulerian frame while particles are treated in a Lagrangian context. Recently, Druzhinin and
Elghobashi32 implemented a two-fluid formulation for DNS of a bubble-laden turbulent flow.
This methodology is only applicable to cases where the density of the dispersed phase is
much smaller than the density of the carrier phase. These cases are beyond the scope of this
review, therefore the two-fluid formulation for DNS will not be further discussed here.
An important issue in Lagrangian treatment of the dispersed phase is the number of
14
particles tracked. In simulations with one-way coupling the restriction on the number of
particles is imposed primarily from a statistical viewpoint. Since each particle is considered
as one realization, it is required that the number of particles Nd be large enough to provide
accurate statistics (generated via ensemble averaging over the number of particles). Appealing to the central limit theorem, it is shown33 that the statistical sampling error decreases
as Nd0.5 . Therefore, using 104 particles, for instance, results in an error of the order of 1%.
In simulations with two-way coupling, the number of particles is not chosen arbitrarily, and
is rather determined from the mass loading ratio. The number of particles found in this
manner usually suffices to provide a reasonable accuracy for statistical analysis. However,
due to limitations in computational resources, it is not always possible to consider such a
large number of particles in one simulation. One remedy is to consider a smaller number of
‘computational’ particles, each representing several ‘physical’ particles. This, however, may
result in significant errors if the ratio of the computational to real particles is too small. This
issue was addressed by Elghobashi34 by considering the sensitivity of the magnitude of the
particle slip velocity to this ratio.
In simulations with two-way coupling the effects of the particles on the fluid must be
properly accounted for in the formulation. In most of the applications these effects are
included as point sources concentrated at the center of the particle. In earlier studies, an
Eulerian representation for these Lagrangian point sources were obtained by adding the
effects of all of the particles residing inside a cell volume centered around each node. When
implemented in conjunction with spectral methods, this procedure is successful provided
that the number of particles is large and/or the spatial fluctuations in the magnitude of the
source term is small. Otherwise, large variations in the magnitude of the source term, from
one node to the other, may cause resolution problems. Recently, Maxey et al.35, 36 adopted a
somewhat different approach which provides a more smooth distribution of the source term
within the simulation domain. This approach is also based on the concept of point source,
however, in transforming from the Lagrangian to the Eulerian description, the source term is
distributed within an envelope centered at the particle position. The envelope gives a local
15
spatial average or filtering which can be controlled by the magnitude of a parameter. If the
magnitude of this parameter is chosen to be significantly larger than the particle diameter,
then the envelop provides a local volume average similar to the continuum two-fluid approach.
3.1.1
Isotropic homogeneous flows
The simplest configuration that can be considered for DNS is an isotropic flow for which
periodic boundary conditions can be applied in all directions. These flows have been studied
in either decaying or stationary frameworks. The decaying isotropic turbulence is a reasonable approximation of the flow in a wind tunnel, and for its numerical simulation an initial
velocity field is usually generated as a solenoidal random field with a specified form for the
energy spectrum. This initial random field is then allowed to evolve in time according to the
Navier-Stokes equations. Since no energy is added to the system, the dissipation at small
scales of the flow results in the decay of the turbulence kinetic energy while increasing the
length scales of the flow. The simulations are usually continued till the size of the largest
scales becomes comparable to the size of the simulation box, or the Reynolds number becomes so low that the flow is nearly laminar. The statistical analysis of these flows is usually
performed after the effects of the initial conditions are vanished and the energy spectra are
self-similar.
The first DNS of particle-laden turbulent flows was carried out by Riley and Patterson.37
The carrier phase was simulated via a Fourier spectral method on 323 grid points and a
relatively small number of solid particles (432) was used for the dispersed phase. In these
simulations the effects of the particles on the carrier phase were neglected (i.e. one-way
coupling) and the equation for motion of the particles included the Stokes drag, the gravity,
and the apparent mass terms. The fluid velocity at the particle location was calculated using
a three-dimensional linear interpolation scheme. The particles were injected into the flow
with the same velocity as that of the surrounding fluid element, and the statistics of the
dispersed phase was calculated after the effects of the initial conditions were vanished. Riley
16
and Patterson37 used the temporal evolution of the mean square relative velocity to determine
the appropriate time for starting the statistical analysis. During the early simulation times,
the mean square relative velocity increased indicating that the particles were still under
the influence of the initial conditions. After reaching peak values, the mean square relative
velocity began to decrease. All of the dispersed phase statistics were measured after the
peak time. The DNS results of Riley and Patterson37 were mainly used to calculate various
autocorrelation coefficients and indicated that an increase of the particle inertia increases its
velocity autocorrelation.∗
A more detailed investigation on the statistics of solid particles in decaying turbulence
was carried out by Squires and Eaton38 using a pseudo-spectral method on 1283 grid points
for the fluid. The results showed that, at long times the eddy diffusivity of the particles
reached stationary values which, in the absence of gravity, were larger than that of the
fluid particle. In the presence of gravity, the DNS results were widely used to study the
effects of the crossing trajectories39 and the (associated) continuity effects.40 Elghobashi
and Truesdell41 also investigated the dispersion of solid particles in decaying turbulence by
performing simulations on 963 grid points using a second-order finite difference method. They
considered the full equation for the particle motion and showed that the Stokes drag and the
inertia are of primary importance when the density ratio is large. After considering proper
scalings for time and velocity, Elghobashi and Truesdell41 found good agreements between
the DNS results and the measurements, from the experiment of Snyder and Lumley,42 for
temporal evolution of the mean-square displacement of the particles. The effects of the
particles on the carrier phase were investigated by performing simulations with two-way
coupling in decaying turbulence.43, 44 It was shown that in the presence of both gravity
and two-way coupling, energy was transferred from the gravity direction to other directions
through the pressure-strain correlation. More recently, Boivin et al.45 conducted similar
∗
The Lagrangian aurocorrelation of the particle velocity fluctuation, vα , is defined as hhvα (t0 )vα (t0 + t)ii
hhvα (t0 )vα (t0 +t)ii
and the Lagrangian aurocorrelation coefficient is Rαα (t) = hhv2 (t
0.5 hhv 2 (t +t)ii0.5 for α = 1, 2, 3, with no
α 0 )ii
α 0
summation over the repeated Greek indices. Here hh ii denotes the ensemble average associated with the
dispersed phase.
17
simulations and showed that the increase of the high-wavenumber portion of the fluid energy
spectrum can be attributed to transfer of the fluid-particle covariance by the fluid turbulence.
This also explained the relative increase of small-scale energy caused by small particles as
observed in previous simulations.43, 46
The simulations of stationary isotropic flows are also started from a random field with
a specified form for the energy spectrum. However, to compensate for dissipation at small
scales of the flow and to maintain stationarity, the larger scales of the flow are artificially
forced. As the initial flow field evolves in time, the flow reaches a stationary condition in
which the energy added to the low wavenumbers cascades down in the energy spectrum and
dissipates at higher wavenumbers. Provided that there is enough separation between the
small and large scales of the flow, it is expected that the smaller scales are not affected by
the external forcing. A variety of forcing techniques have been implemented to generate
stationary flows. The early techniques were based on keeping the energy contents in the
first few modes constant.47 This type of forcing does not properly account for temporal and
spatial correlations of turbulence. More sophisticated techniques are based on an UhlenbeckOrnstein stochastic process and provide a more flexible forcing at low wavenumbers.48
Squires and Eaton,38, 46, 49 simulated stationary, two-phase turbulent flows with one- and
two-way couplings. In the case with two-way coupling they found that the fraction of energy
at high wave numbers of the spatial energy spectrum of turbulence increased relative to
that at low wave numbers as the mass loading ratio was increased. They also found that
particles tend to collect preferentially in regions of low vorticity and high strain rate. This
was also true for the case with one-way coupling. Recently, Squires and Eaton50 and Eaton51
used the results of these simulations to study the effects of preferential distributions of the
particles on turbulence modifications in dilute two-phase flows. The settling velocity of heavy
particles in isotropic turbulence was studied by Wang and Maxey52 for different particle time
constants and drift velocities. The results showed an increase of the settling velocity for all
cases. The maximum increase in settling velocity was obtained when both the particle
18
time constant and the drift velocity were comparable to the Kolmogrov scales of the flow.
More recently, Maxey et al.35, 36 implemented their force-coupled method to investigate the
interaction between the particles and the carrier phase in three homogeneous flows, i.e. a
suspension of uniformly distributed particles with no initial turbulence in the carrier phase,
a forced turbulence, and a decaying turbulence. It was concluded that the particles were
more influential on the vorticity of the fluid than its kinetic energy. The analysis of the
spectra of the kinetic energy, the dissipation rate of the kinetic energy, the particle force,
and the force-fluid coupling indicated that the interactions between the phases were mostly
concentrated at intermediate and high wave numbers.
DNS has also been implemented to study dispersion and polydispersity (i.e. size variation)
of evaporating droplets in turbulent flows by Mashayek et al.53, 54 The carrier phase was
considered to be incompressible, and the rate of evaporation of the droplets was modeled
by the conventional d2 -law.2 These assumptions were plausible as long as the mass loading
of the droplets was small and the carrier phase was not influenced by the presence of the
dispersed phase (i.e. one-way coupling). The DNS results were mainly used to investigate the
distribution of droplet sizes under various conditions, and it was shown that, for practical
purposes, the distribution of the droplet diameter may be reasonably described by a Gaussian
pdf. However, when the initial size of the spray was small compared to the large scales of
the flow, then the pdf of the droplet diameter deviated from Gaussian. In a related study,55
DNS of incompressible, isotropic turbulence was used to investigate temperature statistics
in both dispersed and carrier phases. Solid (non-evaporating) particles were considered and
an energy equation was solved for the carrier phase with the zero-Mach-number assumption.
In this manner, the continuity and momentum equations were decoupled from the energy
equation and the change of kinetic energy to internal energy was not taken into account.
Various cases with both one- and two-way couplings were considered to study the effects of
the particle time constant, the mass loading ratio, the Prandtl number, and the ratio of the
specific heats of the two phases.
19
In a series of papers,56–58 Mashayek and coworkers presented a somewhat more realistic
framework to study two-way coupling effects for solid particles and evaporating droplets.
A compressible turbulent flow was considered which allowed for density variations in the
presence of evaporating droplets. Further, reacting droplets were investigated by assuming
that the chemical reaction occurred in the gas phase between the fuel (droplets) vapor and
the oxidizer. A one-step reaction was considered to simplify the investigation, however, the
effects of two-way coupling on the carrier phase mass, momentum, and energy were included
in the formulation.
The concept of stationarity in compressible flows assumes a slightly different definition
than its counterpart for incompressible flows. Here, the energy added to the system by
forcing is continuously converted to internal energy, a process that results in an increase
of the ‘mean’ temperature in time. Nevertheless, the fluctuations in temperature reach a
statistically stationary state after a sufficient time.59 Mashayek and Jaberi56 showed that
a similar ‘quasi-stationary’ state is attainable for two-phase flows when droplets are not
evaporating or reacting. In the presence of evaporating droplets, such a quasi-stationary
state is not achieved as the energy exchange due to evaporation (or reaction) would be
considerably larger than that added to the system by artificially forcing the large scales of
the flow.
The results of simulations56–58 were extensively utilized to investigate the effects of a
wide variety of parameters, including the droplet time constant, the mass loading ratio, the
specific heats ratio, the evaporation and reaction rates, etc. For solid particles, it was found
that the ratio of the root mean square Mach number to the mean Mach number was nearly
constant (∼ 0.41) for all of the cases considered. The results for evaporating droplets showed
that the pdfs of the droplet diameter were skewed towards smaller sizes, however, they could
be approximated as Gaussian for small mass loading ratios. This observation was in accord
with the results of Mashayek et al.53 for incompressible flows. As there was no major source
of energy for the two-phase system, the energy utilized by evaporation lowered the mean
20
temperature of both phases, and a nearly saturated state was observed for the mixture at
long times. In reacting cases, the final fate of the system was primarily dictated by the rate
of heat release, and a strong correlation was observed between the droplet concentration and
the reaction rate.
3.1.2
Anisotropic homogeneous flows
These flows are characterized by the presence of one (or more) mean velocity gradient(s)
that could result in an anisotropic Reynolds stress tensor. While still homogeneous, the
normal (or diagonal) components of the Reynolds stress tensor are not necessarily equal in
these flows and as such they provide a convenient framework for a preliminary assessment of
various turbulence models. Although these simple flows do not exhibit all the complexities
involved in practical situations, they can be considered as basic flows locally prevailing in
more general inhomogeneous configurations. Consequently, model validation in these flows
can be considered as a logical first step. It is realized that establishing a good agreement
between the model and the data in these flows does not, necessarily, guarantee acceptable
performance in more complex flows, however, the reverse may be true. That is, if a model
fails when compared against basic flows, it stands little chance of success in a flow of practical
interest. As an example, the schematic in Fig. 1 shows the correspondence between three
basic flows and various regions in a backward-facing step flow which could be used as a
good model for many applications such as a dump combustor. The basic flows shown in
the figure include isotropic homogeneous, anisotropic homogeneous shear, and anisotropic
homogeneous plane strain flows. The former was discussed in the previous section, below we
discuss the last two (anisotropic) flows along with their numerical simulations.
From a numerical point of view, DNS of anisotropic inhomogeneous flows is not as
straightforward as isotropic flows due mainly to the fact that periodic boundary conditions
cannot be applied in all directions. While simulations of single-phase anisotropic homogeneous flows are now routinely conducted in a variety of configurations, their application
21
to two-phase flows has just recently been initiated.60–62 To configure an anisotropic homogeneous turbulent flow, a mean velocity with linear profile in one or more direction(s) is
superimposed on an initially isotropic turbulent field. The components of the mean velocity gradient tensor are, therefore, constant and, depending on the magnitudes of these
component(s), various flow configurations are feasible.
The condition for establishing a homogeneous turbulence can be more rigorously discussed
by considering the coordinate transformation
ξi = Bij (t)xj ,
i = 1, 2, 3,
j = 1, 2, 3
(3.18)
as proposed by Rogallo.63 Starting from the momentum equation for instantaneous velocity
of an incompressible carrier phase:
∂ Ubi ∂(Ubi Ubj )
∂ Pb
1 ∂ 2 Ubi
+
=−
+
,
∂t
∂xj
∂xi Re0 ∂xj ∂xj
(3.19)
and by decomposing the velocity as
Ubi = Ui + ui ,
(3.20)
a transport equation for the fluctuating (turbulent) velocity can be described as:
∂ui
∂ui
∂ Pb
1 ∂ 2 ui
+ Ui,j uj + Uj,m xm
=−
+
,
∂t
∂xj
∂xi Re0 ∂xj ∂xj
(3.21)
where we have substituted Ui = Ui,j xj , Ui,j = ∂Ui /∂xj . In these equations, b denotes the
instantaneous quantity and ui is the fluctuating carrier-phase velocity. The mean velocity
Ui =< Ubi > is calculated by (Eulerian) ensemble averaging (denoted by < >) over the
number of grid points. Here, for convenience, all variables are normalized by reference scales
for length (L0 ), density (ρ0 ), and velocity (U0 ), resulting in the reference Reynolds number
Re0 = ρ0 U0 L0 /µ.
22
The dependency on the coordinates xi is explicit in (3.21), but can be removed by applying
the coordinate transformation (3.18) to obtain
"
#
∂ui
dBmn
∂ui
∂ Pb
1
∂ 2 ui
+
+ Bmj Uj,n
xn = −Bmi
+
Bmj Bnj
− Ui,j uj .
∂t
dt
∂ξm
∂ξm Re0
∂ξm ∂ξn
(3.22)
Therefore, to establish a homogeneous turbulence whose statistics are independent of the
coordinate system, the second term on the left-hand side must vanish. For constant mean
velocity gradients, this requires the transformation tensor to satisfy:
dBmn
+ Bmj Uj,n = 0.
dt
(3.23)
It is noted that (3.23) also applies to the limiting case of isotropic flows where Uj,n = 0 and
Bmn can be taken as a constant (time-independent) value.
Among the possible configurations to study anisotropic flows, the homogeneous shear flow
has received the greatest attention and serves as a good model for most of inhomogeneous
flows such as turbulent jets and wakes which develop regions of thin shear. In this flow, the
instantaneous velocity of the carrier phase is described as, with no summation over repeated
Greek indices,
Ubα = U1,2 x2 δα1 + uα ,
α = 1, 2, 3
(3.24)
where δij is the Kronecker delta function, and U1,2 = dU1 /dx2 = constant, with x1 and x2
indicating the streamwise and cross-stream flow directions, respectively. It is noted that the
homogeneous shear flow is identified by only one nonzero (shear) component in the mean
velocity gradient tensor. In contrast, other homogeneous flows may be formed with normal
components of the mean velocity gradient tensor only, without any shear component. These
include plane strain, axisymmetric contraction, and axisymmetric expansion flows, with an
instantaneous carrier-phase velocity described as
Ubα = Uα,α xα + uα ,
23
α = 1, 2, 3
(3.25)
where Uα,α = constant. These simple irrotational flows are good models for a variety of
inhomogeneous flows such as those along a stagnation streamline, through a contraction or
expansion, etc.
The intensity (or rapidity) of the mean strain rate, referred to as ‘equivalent mean strain
rate’, can be measured in terms of S = (Sij Sij /2)1/2 where Sij = (Ui,j + Uj,i ) /2 is the
strain-rate tensor. With this definition,


 0 1 0 



Sij = S 
 1 0 0 




(3.26)
0 0 0
for shear flow,


 1

Sij = S 
 0


0
for plane strain flow,
0
0 


−1 0 


0
(3.27)
0


0
0
 1
2S 

Sij = √ 
0 −1/2
0
3

0
0
−1/2






(3.28)
for axisymmetric contraction flow, and


0
0
 −1
2S 
Sij = √ 
 0 1/2
0
3

0
0 1/2






(3.29)
for axisymmetric expansion flow. It can also be shown that:


 1 −U1,2 t 0 




Bij (t) = 
,
1
0 
 0



0
0
24
1
(3.30)
for homogeneous shear flow, and




Bij (t) = 



0
B11
exp(−U1,1 t)
0
0
0
0
B22
exp(−U2,2 t)
0
0
0
0
B33
exp(−U3,3 t)



.


(3.31)
for plane strain and axisymmetric flows with their respective Uα,α values. Here the superscript 0 indicates the initial value, i.e. Bij0 = Bij (0).
The coordinate transformation (3.18) has been extensively implemented, in conjunction
with Fourier spectral methods, to simulate anisotripic homogeneous flows in the moving
coordinate ξi . The extension of this numerical procedure to two-phase flows requires some
further considerations. To elaborate on this issue, we begin by ensemble averaging the
particle momentum equation (which, for dilute two-phase flows with large density ratio,
may also be viewed as an Eulerian equation for the dispersed phase64 ):
V
1
∂vi
D Vi
=¿ (Ubi∗ − Vbi ) À − ¿ vj
À,
Dt
τp
∂xj
(3.32)
where τp = Re0 ρp d2p /18 is the particle time constant and the superscript ∗ denotes the fluid
velocity at the particle location. Here, ρp and dp are the particle density and diameter,
respectively,
V
D
Dt
=
∂
∂t
+ Vj ∂x∂ j , the notation ¿ À denotes the ensemble average associated
with the dispersed phase, and Vi (=¿ Vbi À) and vi are the particle mean and fluctuating
velocities, respectively. It must be emphasized that only the Stokes drag is considered in
writing (3.32).
For a homogeneous dispersed phase, it can be shown65 that the last correlation in (3.32)
vanishes. Then, substituting from (3.24) for the carrier-phase mean velocity (assuming
Ui∗ ' Ui ) shows that Vi = Ui is a solution to (3.32) for the homogeneous shear flow. In
contrast, substituting (3.25) in (3.32) indicates that Vi = Ui is not a solution for the particle
mean velocity in plane strain and axisymmetric flows. This indeed complicates the analysis
for these flows and limits the discussion to small particle Reynolds numbers (Rep ) for which
25
a Stokes drag can be considered without the correction term due to large Rep (see equation
(2.14)). Under this condition, by considering one-way coupling only and assuming that the
initial fluctuating velocity of the dispersed phase is isotropic, it can be shown65, 66 that the
particle mean velocity in plane strain and axisymmetric flows can be described as Vα =
σα (t)xα , where
σα (t) =
0
0
(βα − Vα,α
)ηα exp(ηα t) − (ηα − Vα,α
)βα exp(βα t)
,
0
0
(βα − Vα,α ) exp(ηα t) − (ηα − Vα,α ) exp(βα t)
with
βα =
−1 +
q
1 + 4Uα,α τp
2τp
,
for 1 + 4Uα,α τp > 0,
ηα =
−1 −
³
σα (t) =
1
2τp
0 +
1 + Vα,α
1
2τp
³
q
1 + 4Uα,α τp
2τp
0
0
Vα,α
− Vα,α
+
´
´
t
2τp
t
(3.33)
,
,
(3.34)
(3.35)
for 1 + 4Uα,α τp = 0, and
σα (t) =
0
0
2ωτp Vα,α
− (Vα,α
− 2Uα,α ) tan(ωt)
,
0 ) tan(ωt)
2ωτp + (1 + 2τp Vα,α
(3.36)
q
ω=
|1 + 4Uα,α τp |
2τp
.
(3.37)
0
for 1 + 4Uα,α τp < 0. Here Vα,α
is the initial value of σα (t).
It is clear that for irrotational flows the mean velocity of the dispersed phase is a function
of time and depends on a ‘critical’ value for the particle time constant:
τpcr =
1
.
4|Uα,α |
(3.38)
For τp > τpcr , the particle motion is oscillatory with the frequency ω. The physical explanation for this phenomenon is that the particles with large inertia are able to cross the
coordinate axis where then they are faced with an opposing flow direction. This enhances
the drag force significantly and directs the particles back towards the axis.
26
Applying the coordinate transformation to the particle momentum equation, in normalized form, one obtains
dvi
1
= (u∗i − vi ) − Vi,j vj ,
dt
τp
(3.39)
for the particle fluctuating velocity, vi . For a (spatially) constant particle mean velocity
gradient, the right-hand side of this equation does not show any dependency on position.
Therefore, if the initial particle fluctuating velocity is homogeneous, the evolution of vi is
independent of ξi and the turbulence remains homogeneous. In the shear flow, the particle
mean velocity is also constant in time and the same as that of the fluid, thus the moving
coordinate (and the homogenous domain associated with that) evolves with the same rate
for both phases. For the plane strain and axisymmetric flows, however, the particle phase
evolves with a different rate than that of the carrier phase. This has been demonstrated
for the plane strain flow in Barré et al.65 by plotting the instantaneous locations of all the
particles at various times. Figure 2 clearly shows a ‘compressibility’ effect associated with
the dispersed phase, despite the incompressible carrier phase. This effect and several other
subtle differences among various homogeneous flows can be used for assessing various aspects
of two-phase turbulence models. These differences are mainly due to the presence of a mean
relative velocity between the two phases in irrotational flows. The study of these flows creates
an opportunity to address several fundamental phenomena in two-phase turbulent flows,
such as the so-called ‘crossing-trajectories effect’. Capturing the fundamental phenomena
such as this, could be considered as a stringent test for statistical models which are usually
derived following a long and tedious mathematical procedure while making many simplifying
assumptions.
The above formulation has been extended to compressible turbulent flows laden with
solid particles or evaporating/reacting droplets.60, 61, 67 This extension was only possible for
homogeneous shear flow where there was no relative mean velocity between the two phases.
This allowed the inclusion of all two-way coupling terms to account for modifications in
mass, momentum, and energy of the carrier phase. The gravity effects, however, were not
27
included as it would complicate the computational methodology significantly. Under these
circumstances, and as explained above, for homogeneous shear flow the computational domain was the same for both phases and periodic boundary conditions were applied. This
implied that there was no net transfer of heat, mass, or momentum from the computational
domain and thus the computations provided a good model of a constant-volume process.
In the absence of chemical reaction, the results of the simulations60 were used to investigate the evolution of the turbulence kinetic energy and the internal energy. Various terms in
the transport equations of these energies were discussed and useful data were provided for
the evolution of the Reynolds stresses in both phases. Non-evaporating droplets decreased
the turbulence kinetic energy by causing an extra dissipation due to drag. Evaporation,
however, increased both the (mass-weighted) turbulence kinetic energy and the mean internal energy of the carrier phase by mass transfer. Evaporation was controlled by vapor mass
fraction gradient around the droplet, when the initial temperature difference between the
phases was negligible. In cases with small initial droplet temperature, the convective heat
transfer was more important in the evaporation process. At long times the evaporation rate
approached asymptotic values depending on the values of various parameters. It was also
found that evaporation rate was larger for droplets residing in high-strain-rate regions of the
flow, mainly due to larger droplet Reynolds numbers in these regions.
In a recent work, Mashayek and Jacobs68 consolidated the results from several studies58, 61, 68 on reacting droplets in both isotropic and homogeneous shear turbulence to provide
a general view. A simple one-step reaction scheme was considered to make the computations
feasible and the reaction was assumed to occur in the gas phase between the fuel vapor and
the oxidizer gas. The main feature observed in the simulations, was that the structure of the
reaction zone was significantly affected by interaction of the droplets with the carrier phase.
As a result, there was a tendency for the reaction zone to align with the mean flow direction
in shear flow. In the absence of the mean flow, the preferential distribution of the droplets
by small scales of the flow played a determining role in the construction of the reaction zone.
28
The correlations between the reaction rate and different flow variables were used to reveal
several interesting features of two-phase turbulent reactive flows. Although some of these
features were in agreement with physical intuition, others were somewhat unexpected and
indicated the delicate relative importance of various time scales involved in these flows. For
example, despite the strong two-way coupling expected between the carrier-phase temperature and the reaction rate, the correlation coefficient of these two variables assumed negative
values for cases with low initial droplet temperature.
3.1.3
Inhomogeneous flows
It appears that the most popular inhomogeneous configuration chosen for investigation has
been the fully developed channel flow,69 and in almost all of the applications the fluid phase
has been simulated via psuedospectral method.70 One of the first DNS studies of two-phase
channel flows is due to McLaughlin71 who studied aerosol particle deposition. The channel
was vertical so that the gravity effects on the deposition of the particles on the wall could
be neglected. Therefore, only the inertia, the Stokes drag, and the Saffman lift forces were
included in the equation for the particle motion. The full spectral representation was used
to calculate the fluid velocity at the particle location, and 16 × 64 × 65 grid points were used
for all simulations. It was also assumed that a particle remains attached to the wall after
contact. The results showed that the particles tend to accumulate in the viscous sublayer,
and that the rate of deposition was increased as the density ratio was increased. The Saffman
lift force did not affect the particle trajectory, except within the viscous sublayer.
Using a similar technique to simulate the carrier phase, Brooke et al.72 attempted to
identify the eddies that were primarily responsible for the deposition of aerosols. The dispersed phase, however, was simulated with more simplified assumptions. The equation of
motion included only the Stokes drag force, and the wall effects were completely ignored in
the calculations and a geometrical criterion was used for deposition. It was mentioned that
these simplifications did not significantly affect the results, which showed that the particles
29
collided the wall at approximately 45o angle for all particle sizes. An analysis of the particle
trajectories indicated that a particle wandered sidewise until it was trapped in a strongly
correlated motion that brought it to the wall. Brooke et al.73 used the results of similar
simulations to study the free-flight mixing and deposition of aerosols. Two groups of particles were identified; one group was entrained in the carrier phase turbulence while the other
group moved in free flight from one location to the other. A portion of the particles of the
second group exhibited a free flight to the wall.
Ounis et al.74 used a simulated turbulent channel flow to study dispersion and deposition
of Brownian submicrometer-size particles. The equation of motion of the particles included
the Stokes-Cunningham drag, inertia, and Brownian forces. The Brownian force was simulated as a Gaussian white noise random process, and the numerical technique for simulating
the carrier phase was similar to that used by McLaughlin.71 The interpolation scheme to
evaluate the fluid velocity at the particle location involved a partial Hermite method in the
downstream and spanwise directions and a direct spectral method in the cross-stream direction. The results indicated that the dispersion of the particles near the wall was significantly
affected by the Brownian motion. Also, the effect of the variation in the density ratio on
the dispersion and deposition rate was small, in contrast to the results of McLaughlin71 for
larger particles. A similar study was conducted by Ounis et al.75 to provide a fundamental
understanding of the mechanisms that control the deposition process, with a particular attention to the deposition rate. It was shown that the number of deposited particles on the
wall and the deposition velocity increases with the decrease of the particle diameter.
The DNS results of particle-laden channel flow was used by Chen and McLaughlin76 to
generate empirical correlations for aerosol deposition rate in vertical ducts. The effects of
wall on the drag and lift forces were accounted for, and molecular slip and Brownian effects
were included in the equation of motion. The simulations were performed for monodisperse
particles with a wide range of particle relaxation times. By combining the results from
various simulations, the authors also produced correlations for the deposition rate of log-
30
normally distributed polydisperse particles. The results indicated that the deposition rate
strongly depended on polydispersity and monodispersed particles always had the lowest
deposition rate when compared with polydispersed particles. The deposition rate strongly
depended on polydispersity, thus the correlations for the deposition rate were expressed in
terms of both the particle relaxation time and the particles’ geometric standard deviation. In
general, the simulated deposition rates were smaller than most of the available experimental
measurements. In an attempt to explain this deviation, Chen et al.77 accounted for collisions
among droplets in their DNS of vertical channel flows. It was assumed that the droplets
coalesce upon collision resulting in a loss in the kinetic energy. The results showed that the
deposition rate was increased by as much as four times when the collision of the particles
led to coalescence.
Rouson et al.78, 79 performed simulations of particle-laden channel flows with somewhat
different objectives than those followed in the above studies. They considered larger particles with the intention of investigating the preferential concentration of particles. Also,
turbulence modifications were included in the simulations and the particles were assumed to
collide with the channel walls elastically. The results generated by these simulations were
used by Eaton80 to discuss turbulence modification by particles in shear flows. Pan and
Banerjee81 also investigated the modification of turbulence by particles in a channel flow. A
low volume fraction was considered so that inter-particle collisions could be neglected. The
fluid phase was simulated by a pseudo-spectral method and the particles were transported
by considering the Stokes drag force, the force due to the acceleration of the undisturbed
fluid flow, and the body force. A different approach was adopted when accounting for the
effects of the particles on the carrier phase. The approach directly incorporated the local
disturbance velocity field, due to each particle, into the undisturbed fluid velocity field by
assuming that the flow around the particle was locally Stokesian. The authors argued that
this approach was necessary to ensure the correct implementation of the velocity boundary
conditions at the wall. It was found that particles smaller than the dissipative length scale
reduced turbulence intensities and Reynolds stresses, whereas particles that were somewhat
31
larger increased intensities and stresses.
All of the above contributions have considered vertical channels. In contrast, Pedinotti
et al.82 simulated the particle-laden flow in a horizontal channel in which the gravity effects
were accounted for. The particle density was very close to the fluid density which did not
justify neglect of other terms in the particle equation of motion. The particle-wall collisions
were inelastic and frictionless, allowing for the tangential movement of particles on the wall.
The results showed that, due to gravity, the particles ejected from the wall fell back to the
near wall region faster than the fluid which was simultaneously injected. The authors pointed
out that two (inner and outer) scales were associated with the particle dynamics. Therefore,
it was not possible to match the results of flows at different Reynolds numbers. Since DNS
was limited to low Reynolds numbers, the experimental data could not be directly compared
to DNS results.
In recent years, the temporally-developing mixing layer has also emerged as a popular
configuration for DNS studies of two-phase flows. This flow is a simplification of its spatial
counterpart which has been extensively reproduced in laboratory. Both of these flows have
been the subject of intensive investigations, theoretically, experimentally, and numerically,
within the context of reacting turbulent flows in the past three decades. The temporal model
of the spatial flow is generated by considering a coordinate frame moving with the mean
flow velocity in the streamwise direction. From a numerical point of view, this simplifies
the problem significantly as one would only need to simulate a computational domain large
enough to accommodate for a small number (as little as a pair) of vortices. This is possible
by implementing periodic boundary conditions, thus neglecting the widening of the mixing
layer, in the streamwise direction. The periodic boundary conditions are also implemented for
particles in the streamwise direction. However, these convenient conditions are not applicable
to boundaries in the cross-stream direction, which could result in a loss of particles as they
cross these boundaries. These particles are usually not replaced, a treatment that translates
into a decrease of the mass loading of the dispersed phase. This is particularly important
32
when studying two-way coupling effects as they are directly proportional to particle mass
loading.
In a series of papers, Ling et al.83–85 reported results obtained from DNS of a threedimensional, temporally-developing mixing layer laden with particles or droplets. The carrier phase was simulated using a pseudospectral method and the particle motion was governed by inertia and a modified Stokes drag. The fluid velocity at the particle location
was found using a third-order Lagrange interpolation scheme. The results showed that the
three-dimensional mixing layer was characterized by both two-dimensional and streamwise
large scale structures. The particles with Stokes number of order unity were found to have
the largest concentration on the circumference of the two-dimensional large-scale structures.
However, the presence of the streamwise large-scale structures caused the variation of the
particle concentration along the spanwise and transverse directions, resulting in a ‘mushroom’ shape of the particle distribution. In the presence of two-way coupling,84 it was found
that for particles at an intermediate Stokes number, higher mass loading ratio resulted in
lower energy of the large-scale vortex structures and less particle dispersion. While particles
at intermediate Stokes numbers stabilized the flow, particles at small Stokes numbers at a
particular mass loading destabilized the flow and enhanced the mixing. To study thermal
two-way coupling, a mixture of hot air and cool water droplets was considered.85 Higher
air density was observed near the interface of the two streams and eventually resulted in
an increase of the magnitude of the vorticity field, a more unstable flow, a higher droplet
concentration and a lower droplet dispersion across the mixing layer.
Miller and Bellan86 conducted DNS of a temporally-developing mixing layer with one
stream laden with evaporating hydrocarbon droplets. An eighth-order finite difference spatial
discretization was used in conjunction with an explicit fourth-order Runge-Kutta temporal
integration scheme. Properties of air (carrier phase) and decane (dispersed phase) at 350K
were used, however, a much lower viscosity value had to be considered for air in order to
make the DNS feasible. A complete two-way coupled formulation was implemented and the
33
effects of the initial mass loading ratio, initial Stokes number, and initial droplet temperature
on the mixing layer growth and development were discussed. The results showed that the
mixing layer growth rate and kinetic energy were increasingly attenuated for increasing
mass loading ratios up to 0.35. The laden stream became saturated before evaporation was
completed for large mass loading ratios. Later, Miller and Bellan87 implemented a parallel
version of the same code, which allowed them to consider larger numbers of grid points and
droplets. The complete transition to turbulence was captured for several of the high Reynolds
number simulations, and it was observed that the increase in the droplet mass loading ratio
resulted in a more ‘natural’ turbulence characterized by increased rotational energy. An a
priori subgrid analysis, based on the DNS results, showed that neglecting subgrid velocity
fluctuations in the context of LES may result in significant errors in predicting the droplet
drag force for Stokes numbers of order one. (A flow time scale based on the mean velocity
difference and initial velocity thickness was used to define the Sotkes number as the ratio of
the particle time constant to the flow time scale.)
The above conclusion is despite the fact that the majority of LES studies tend to (conveniently) neglect the subgrid velocity fluctuations, and indicates the need for more comprehensive subgrid models for two-phase flows. These two-phase models may also be able
to properly account for the effects of two-way coupling and turbulence modification due to
the presence of the second phase. This observation trigerred more a priori subgrid analysis
by Okong’o and Bellan88 using the DNS data generated in the study of Miller and Bellan.87
It was proposed to model the ‘unfiltered’ gas-phase variables at the droplet locations by assuming the gas-phase variables to be the sum of the filtered variables and a correction based
on the subgrid scale standard deviation. Three methods were investigated for modeling the
standard deviation: the Smagorinsky approach, the gradient model and the scale-similarity
formulation, and it was found that the latter two yield results in excellent agreement with
the DNS.
Recently, Miller89 extended the above studies to the case of a reacting mixing layer
34
laden with non-reacting solid particles or liquid droplets. To perform DNS and fully resolve
all scales of the flow, several simplifying assumptions were made, among them one-step
reaction scheme and two-dimensional configuration. Given the limitations in computational
resources, this was necessary to achieve the transition to turbulence. It was shown that
cold solid particles, entrained into the mixing zone, cooled the flame in the braid regions,
and caused flame suppression and, under certain conditions, local flame extinction. For
cases with evaporating droplets, the flame extinction was substantially enhanced due to the
use of the gas energy for latent heat of evaporation as well as a reduction in the reaction
rate as a result of the dilution of the reactants by non-reacting vapor. The extent of flame
suppression and local extinction was increased with increasing reaction activation energy
and/or dispersed phase mass loading, and also by deceasing particle Stokes number.
Tackling two-phase flows in more complex inhomogeneous configurations requires numerical codes with geometrical flexibility of traditional schemes, such as finite volume, while
preserving a high degree of accuracy and a low level of numerical diffusion as customary for
DNS. Due to tremendous amount of computations expected for such simulations, it is very
clear that such a numerical scheme must efficiently bend itself for parallel implementation.
While spectral methods have been very successful for DNS in simple flows, they usually lack
the capability of dealing with complex geometries. This perspective has recently began to
change with the emergence of spectral element methods, in particular with their ‘multidomain’ treatment that shows a significant potential for parallel coding.90 As noted in recent
studies of Jacobs et al.,91, 92 the implementation of these methods for two-phase flows requires
a fresh look at various steps involved in Lagrangian tracking of particles. In multidomain
spectral element method, the carrier-phase equations are solved on a computational domain
that is divided into non-overlapping elements (referred to as ‘subdomains’) with high orders
of approximation (typically, 5th through 20th). These elements are treated separately, and,
unlike its finite element method counterpart, no global assembly is required for the multidomain spectral method (see Kopriva93 for details of the method). While this feature makes
the method desirable for parallel implementation, it imposes some restrictions on the treat35
ment of particles. For example, it would be more efficient and convenient to evaluate fluid
properties at the particle location using the information from only the subdomain that contains the particle. Jacobs et al.,91 reported a detail study on various interpolation schemes
and concluded that a sixth-order Lagrangian polynomial would be sufficiently accurate for
this purpose. The practical implementation of multidomain methods for two-phase flows
still requires many modifications and will likely remain a subject of research for some time
to come.
3.2
Large-eddy simulation
In the majority of applications of turbulent flows, the Reynolds number is much higher than
those feasible by DNS. With the current status of supercomputers and with their rate of
progress, it is not expected that, at least in a foreseeable future DNS will become applicable
to flows of practical interest. On the other hand, when dealing with complex flows, the
details of the small structures of the flow may not be of main interest. Therefore, one may
seek for a solution for larger scales of the flow by accounting for the effects of small scales only
in a ‘filtered’ sense. This has been the fundamental principle in implementing the method
of large-eddy simulation for turbulent flows. The theoretical and numerical aspects of LES
are not discussed here, they are available in recent reviews by Lesieur and Métais94 and
Meneveau and Katz.95 Here, we focus our attention on the applications of LES to two-phase
turbulent flows.
In early studies, Lagrangian equations of particles were computed with a flow-field generated via LES96, 97 and the effect of subgrid scales on the particles were not accounted for.
These subgrid scale effects were taken into account in later studies and a detail assessment
on the effects was recently reported by Armenio et al.98 The two-way coupling effects were
incorporated by Boivin et al.99 where the coupling was restricted only to the resolved scale,
i.e. the particle effects were included in the calculation of the resolved scale through a source
term. Only very recently, the two-way coupling was treated rigorously by also taking into
36
account the effects of particles on the subgrid scale viscosity.100 Below, we provide a review
of various LES works on two-phase flows.
One of the early contributions is due to Yeh and Lei96 who investigated the motion of
small particles in an isotropic decaying turbulence generated by LES. The subgrid scale
Reynolds stress was related to large scale flow field via the eddy viscosity model in which
the length scale was a constant Cs times the filter size. The filtered (large scale) carrierphase equations were solved using a Fourier pseudospectral method on 323 grid points. The
effect of the particles on turbulence was neglected and their motion was described by inertia,
modified Stokes drag, and gravity forces. The large scale velocity resolved in LES was used
in the particle equations, thus the effects of the velocity fluctuations in the subgrid scale were
neglected. To assess the capability of LES in predicting the flow statistics, the simulations
results were compared with the experimental data of Snyder and Lumley42 and Wells and
Stock.101 By performing simulations based on various values for Cs and the domain size,
Yeh and Lei96 concluded that the particle motion was mainly controlled by large scales of
turbulence.
Deutsch and Simonin102 utilized LES to study the motion of particles in a stationary
isotropic turbulence. The subgrid scale was modeled with the Smagorinsky model and a
finite difference with fractional step method was used to simulate the carrier phase on 643
grid points. By considering small volume fractions, the effects of the particles on turbulence
was neglected, and the equation of motion for the particles accounted for drag, fluid pressure
and added mass forces. Simulations were performed for three different values of the density
ratio (ρp /ρf = 0.001, 2, and 2000). The results of the simulations were used by Simonin et
al.103 and Pozorski et al.104 to assess their models for particle-laden turbulent flows.
The motion of small particles in homogeneous turbulent shear flows was also studied via
LES. Yeh and Lei97 used a Fourier pseudospectral method to solve the filtered governing
equations of the carrier phase. To apply periodic boundary conditions, the equations were
transformed to a coordinate deforming based on the mean velocity gradient of the carrier
37
phase.63 The particle equations, however, were integrated in a (non-deforming) coordinate
translating with the mean streamwise velocity. The equation of motion for the particles
included the inertia, modified Stokes drag, and Saffman lift forces. However, the simulation
results indicated that, the Saffman lift force was not important for the particle dispersion.
The simulations were performed on 323 grid points and the statistics of the clean flow was
compared to experimental data of Champagne et al.105 and Harris et al.106 The results were
used to investigate the effects of the particle’s inertia and the drift velocity on the particle
dispersion and settling velocity. Two different mean shear rates (S = 12.9 and 44 sec−1 )
were considered. It was found that for the smaller shear rate < X12 >∝ t3 , < X22 >∝ t,
and < X32 >∝ t when 3 < St < 5, and for large shear rate < X12 >∝ t4 , < X1 X2 >∝ t3 ,
< X22 >∝ t2 , and < X32 >∝ t2 for large values of St. The heavy particles led the fluid in the
streamwise direction but fell less rapidly in a homogeneous shear flow than in a still fluid.
Simonin et al.107 also considered the motion of particles in a homogeneous shear turbulent
flow generated via LES. It was assumed that the turbulence was not affected by the presence
of the particles and the subgrid scale stresses were modeled according to Smagorinsky’s
eddy viscosity model with Cs = 0.12. The filtered equations were solved by finite difference
method on a grid with 643 nodes. The particle’s equation of motion included the inertia,
modified Stokes drag and gravity forces, and was solved in a fixed computational domain.
It was argued that, when the particles start from the same instantaneous velocity as that of
their surrounding fluid, the particles mean velocity remains the same as that of the carrier
phase. This information was needed in applying the boundary conditions for the dispersed
phase. The initial conditions for both phases were taken from numerical simulations of
stationary isotropic particle-laden flows of Deutsch and Simonin.102 The results showed that
the dispersed phase is more anisotropic than the carrier phase, and were mainly used to
assess the performance of a two-fluid turbulence model for particle-laden flows with one-way
coupling.
Wang and Squires108, 109 studied particle-laden, fully-developed turbulent channel flows.
38
A one-way coupling between the phases was assumed and the filtered equations governing
the carrier phase were discretized using second-order central differences on a fully staggered
grid with 64 × 65 × 64 nodes. The subgrid scale stresses were related to resolved velocity field
by a gradient type model and the dynamic approach of Germano et al.110 was implemented
to determine the model coefficient Cs . A fourth-order Lagrangian interpolation scheme was
used to calculate the fluid velocity at the particle location, and deposition of the particle
on the wall was assumed to occur when the particle was within one radius of the wall.
The main focus in the study of Wang and Squires109 was to assess the performance of LES
in simulating particle-laden flows via comparisons with the DNS results of McLaughlin.71
Therefore, the particle equation of motion and the flow conditions were chosen to be similar
to those used by McLaughlin.71 The LES and DNS results were found to be in reasonable
agreement. Only the LES-resolved velocities were used in the particle equation, and it was
shown that the subgrid scale velocity fluctuations did not have a large effect on deposition.
Wang and Squires108 also evaluated the accuracy of LES results via comparisons with DNS
results of Rouson and Eaton111 and experimental data of Kulick et al.112 In this work, the
Saffman’s lift force was not included in the particle equation of motion, however, the gravity
force was accounted for. A large number (2.5 × 105 ) particles were tracked in order to study
the statistics of the dispersed phase as well as the preferential distribution of the particles.
Armenio et al.98 studied the effect of subgrid scales on the dispersed phase by comparing
their DNS and LES results for the case of channel flow. The tracer particles and inertial particles whose motions were governed by the Stokes drag force were considered. It was found
that subgrid scales contributions to the particle dispersion, particle velocity autocorrelation,
Lagrangian integral time scale and other properties of interest were significant when a substantial portion of the energy was contained in the subgrid scale. Also, the inertial particles
were found to be less sensitive as compared to the tracer particles which had smaller particle
time constants.
Boivin et al.99 studied the two-way coupling in the LES framework for the case of station-
39
ary isotropic turbulence in which the particles were acted upon by fluid drag force. A priori
tests were conducted for LES models using the DNS data and the obtained results were
then confirmed through the actual LES calculations. Various subgrid scale models, namely,
Smagorinsky, dynamic subgrid, scale similarity and mixed models were assessed against the
DNS data. The model coefficient Cs was shown as a function of mass loading of particles
and the mixed model results were found to be more accurate in predicting the overall subgrid scale dissipation. More recently, Yuu et al.100 studied the situation of particle-laden jet
flow by taking into account the two-way coupling and suggested modifications in the subgrid
scale model due to the presence of particles moving under the influence of fluid drag force.
They compared the LES results against their experimental data and a detail assessment of
their modified model using the DNS data remains to be done. In view of the above, LES
application to two-phase flow is still in the developing stage when the proper and accurate
account of two-way coupling of large and subgrid scales with particles is to be included.
3.3
Stochastic modeling
The main idea in the application of stochastic methods to two-phase turbulent flows is to
introduce a large number of particles into the flow of interest and to generate a ‘synthetic’
turbulence with some known statistical properties, such as the mean and the variance of the
fluid fluctuating velocity. These known properties can, in general, be obtained by solving
any single-point closure scheme such as those developed by Reynolds averaging the NavierStokes equations for the carrier fluid. A large number of particles are then introduced into
the flow and their trajectories are simulated while updating various particle properties, such
as velocity and temperature, in time. It is important to realize that the data generated using
the synthetic turbulence may represent the physics of the real turbulence only after averaging
over a large number of particles (i.e. samples) in a statistical sense. The statistical analysis
on the particles is performed by collecting samples from a small control volume within the
flow field. This procedure is similar to that implemented in laboratory experiments where
40
the particles crossing a small ‘probe’ volume are sampled in time.
While in generating a synthetic turbulence stochastic models heavily rely on randomlysampled values for the fluid velocity fluctuations, it is noted that turbulent flows are not
purely random fields and exhibit strong temporal and spatial correlations among velocity
fluctuations. As a result, the level of accuracy in representation of these correlations by
a stochastic model can be expected to have a major impact on its performance. Various
stochastic models greatly differ in their method of implementation of these correlations,
in particular the temporal ones which are widely referred to as autocorrelations. In this
section we review these models by dividing them into two groups. The first group of models
include the fluid velocity autocorrelation in an ‘implicit’ manner only, whereas the second
group ‘explicitly’ implements the autocorrelation in the construction of the model. Despite
the possible lower accuracy of the first group, their simplicity of implementation and lower
computational cost have greatly promoted their use for simulation of practical two-phase
turbulent flows. Another important issue is that the form of the autocorrelation function
is not always well defined for a flow of interest. Therefore, although the possibility of
explicit incorporation of various autocorrelations adds to the robustness of the models in the
second group, the actual improvement in the performance may yet depend on the ability to
accurately describe the autocorrelations for various flows.
3.3.1
Models with implicit velocity autocorrelations
These models primarily consider the interaction of particles with random-velocity fluid eddies
and are mostly recognized as eddy-interaction models (EIMs). Various forms of these models
have appeared in the literature, and they have been modified from their original form to
account for different physical phenomena, such as crossing-trajectories and continuity effects.
One of the early eddy-interaction models is due to Gosman and Ioannides.113 In this
model, the turbulence is assumed to be isotropic and to have a Gaussian pdf with the
variance of 2k/3, where k is the turbulence kinetic energy. The fluctuating fluid velocity
41
along the particle trajectory is randomly sampled from the Gaussian pdf and the particle
is allowed to interact with an eddy for a certain period of time. This time is determined
by considering two critical scales associated with the turbulent eddy: the eddy life time, te ,
and the eddy length, Le . The latter may also be used to determine another time scale called
the eddy crossing time, tc . The turbulence properties are assumed to be the same within an
eddy and change randomly when the particles move to another eddy.
More specifically, a particle is assumed to interact with an eddy for a time which is the
minimum of either the eddy lifetime or the time required for a particle to cross an eddy. In the
framework of statistical turbulence models for the carrier phase, these times are estimated
by assuming that the characteristic size of an eddy is the dissipation length scale
3
3
Le = Cµ4 k 2 /²,
(3.40)
where Cµ is the empirical constant from the k-² model, with k and ² denoting the turbulence
kinetic energy and its rate of dissipation, respectively. The eddy lifetime is then computed
by
1
te = Le /(2k/3) 2 .
(3.41)
The particle is assumed to interact with an eddy as long as both the time (∆tp ) and the
relative distance (∆xp ) of interaction satisfy the following criteria:
∆tp ≤ te ,
|∆xp | ≤ Le .
(3.42)
From the eddy lenght scale Le , Gossman and Ioannides113 have derived the eddy crossing
time


tc = −τp ln 1.0 −
Le
c
c − V|
τp |U
.
(3.43)
The particle is allowed to interact with the same eddy over an eddy interaction time ti which
42
is the minimum of two time scales:
ti = min(te , tc ).
(3.44)
c − V|
c < L /τ , then the particle is said to be ‘captured’ and t = t is used.
If |U
e p
i
e
This model was also implemented by Faeth and coworkers114–116 to predict particle-laden
jets, and more applications were reported in Refs. 117, 118. Wang and Stock119 showed
that the velocity autocorrelation for ‘discontinuous’ stochastic processes, such as the one
described above, can be obtained by transforming the ensemble average into time average
for a stationary random process and employing physical reasoning to find the contribution
of individual velocity pairs to the velocity autocorrelation function. The form of the autocorrelation depends on whether a constant time step T :
u(0 → T ) = u1 ,
u(T → 2T ) = u2 ,
u(2T → 3T ) = u3 ,
···,
(3.45)
or random time steps ti :
u(0 → t1 ) = u1 ,
u(t1 → t1 + t2 ) = u2 ,
u(t1 + t2 → t1 + t2 + t3 ) = u3 ,
···,
(3.46)
are used for sampling the random fluid velocity u at the particle location. Here, T is a
constant whereas ti ’s are random with a given pdf. The autocorrelation was found to be
independent of the fluid velocity pdf, but to be related to the time-step pdf. For constant
time step, the autocorrelation linearly decreased from 1 to 0 in a time interval equal to T
which, for self consistency, must be considered to be twice the fluid Lagrangian integral time
scale. When the time step was chosen randomly with an exponential pdf distribution, the
resulting velocity correlation was an exponential function.
Etasse et al.120 wrote (3.44) as ti = 2 min(te , tc ), noting that the factor 2 is necessary to
ensure the equality between the integral time scale of the model and the postulated interac-
43
tion time, since the stochastic process generates a linear autocorrelation function.119 It was
also argued that this choice of ti for sampling a new fluid velocity fluctuation at the particle
location, misrepresents several physical effects: (i) As τp increases and approaches or exceeds
the eddy turn-over or eddy-crossing time, the particle motion is known to become less responsive to the turbulence and leads to increased dispersion. This effect is not accounted for
in the stochastic model. (ii) As the mean relative velocity increases, particles have less time
to interact with the eddies, and the effective agitation seen by the particles should weaken
thus leading to a decrease of the dispersion coefficient. This effect is not properly accounted
for in the model. (iii) The model also fails to accommodate for the fact that for large trajectory crossing effect, the dispersion in the directions transverse to the relative velocity is
less than that in the longitudinal direction.40 To alleviate these deficiencies, Etasse et al.120
determined ti in terms of the particle dispersion coefficient described as a function of two
parameters related to Stokes number and the mean relative velocity. Previous theoretical121
and LES results were used to describe the functionality of the dispersion coefficient, and
application to dispersion in decaying turbulence showed acceptable agreement with experimental data. The limitations of the model were also pointed out, among which the fact that
the parameterization was based on theoretical and LES results for isotropic turbulence, thus
errors may occur when the model is implemented in inhomogeneous environments.
Graham and James122 examined the performance of EIMs in homogeneous isotropic stationary turbulence and investigated the relationships between the eddy time and length
scales. A relationship was derived between the eddy length distribution and the Eulerian spatial velocity correlation, and a method was introduced to ensure consistency of length scales
by a suitable choice of the eddy length distribution so that the Eulerian integral length scale
of the model turbulence was equal to that of the actual turbulence. Graham123–125 proposed
further improvements to the standard EIM to account for the effects of inertia, crossingtrajectories, and continuity.123–125 Recently, Graham126 studied spectral characteristics of
EIMs and concluded that each of the resulting spectra exhibits unphysical characteristics. In
particular, it was shown that the dissipation from all EIMs is infinite and no EIM is capable
44
of demonstrating completely satisfactory spectral characteristics. Nevertheless, as long as
EIMs are used to predict the dispersion characteristics, which is controlled by the large-scale
fluid motions, satisfactory results may be expected.
As pointed out above, the original model of Gosman and Ioannides,113 although widely
promoted due to its simplicity, does not account for directional anisotropy associated with
anisotropic turbulent flows. Various remedies have been suggested in the literature to improve the performance of this basic model. Wang and James127 tried to account for some of
the effects of anisotropy in near wall regions, within the framework of k-² model calculation,
by introducing damping functions into the standard eddy-interaction model. These functions
were defined as
(ux ux )1/2
,
fu =
(2k/3)1/2
(uy uy )1/2
fv =
,
(2k/3)1/2
q
fw =
3 − fu2 − fv2 ,
(3.47)
and used to modify the isotropic components of fluid velocity fluctuations as extracted from
the turbulence kinetic energy calculated by k-² model. Specific forms for fu and fv were
suggested by curve-fitting the DNS data of Kim et al.70 These forms were strictly valid for
the particular low-Reynolds-number flow against which they were calibrated.
A more straightforward approach for including the anisotropy effect, is through the use of
Reynolds stress models where different values for normal stresses are provided. These values
can then be directly used in conjunction with EIMs to generate non-equal values for the
fluid velocity fluctuations at the particle location. Several studies have reported successful
implementation of this procedure. One of the recent studies is due to Chen128 that also
accounts for the effect of turbulence inhomogeniety. By keeping the sampled fluctuating
velocity unchanged for a period of time, according to equation (3.44), the standard eddyinteraction model tends to produce an artificial transfer of fluid turbulence from the high- to
the low-intensity region.129 To remedy this problem, Chen128 assumed that the particle was
always influenced by the local turbulence. The normalized fluctuating velocities obtained at
the beginning of a particle-eddy encounter were multiplied by the local turbulence intensity
45
calculated with the Reynolds stress model.
The main characteristic of EIMs is that the sampled velocity remains constant in space
and time throughout the interaction of the particle with the eddy. However, the fluid and
the heavy partilces do not remain at the same location and follow separate paths as a
result of the particle finite inertia. Noting this, a somewhat more rigorous improvement,
at a more fundamental level, was proposed by Ormancey and Martinon.130 In this model,
the trajectories of massless fluid particles are constructed by integrating their Lagrangian
equations. Associated with each fluid particle is a ‘fluid domain’ centered at the fluid particle
location. A heavy particle can follow a fluid domain or can move from one fluid domain to
another one, accounting for the effect of crossing trajectories. Within the fluid domain,
the fluid velocity fluctuation at the particle location is found from a random sampling with
specified one- and two-point correlations. A particle remains within one fluid domain as long
as its distance from the fluid particle is smaller than some pre-defined length, or until the
turbulent structure around the fluid particle is vanished by exceeding the random lifetime
of the fluid domain. The sizes and lifetimes of fluid domains are determined by length and
time scales of turbulence.
3.3.2
Models with explicit velocity autocorrelations
The models in this group are capable of producing a more ‘continuous’ turbulence in time
as opposed to the models in the first group that only implicitly include a fluid velocity autocorrelation. As pointed out by Parthasarathy and Faeth,131 these models can be developed
rigorously within the framework of the time series analysis as described by Box and Jenkins.132 In this framework, one can implement any number of statistical properties of the
carrier phase, including mean velocities, velocity fluctuations, Lagrangian time correlations,
and higher-order moments. For practical purposes, however, the implementation of only
mean and fluctuating velocities, and time correlations suffice for treatment of the particle
turbulent dispersion. Further inclusion of higher-order correlations, although possible from
46
a theoretical point of view, could substantially complicate the models and increase the cost
of the computations. Parthasarathy and Faeth131 used this approach to develop a time-series
model to simulate liquid velocities along the particle path. The predictions of the model were
assessed against the authors’ laboratory measurements for turbulent dispersion of particles
in a self-generated homogeneous turbulence.
Along the lines of Ormancey and Martinon’s130 model, Berlemont et al.133, 134 proposed
a stochastic model based on building a correlation matrix which simulates the Lagrangian
time correlations along the entire particle trajectory. This technique allows for explicitly
introducing any form for the Lagrangian autocorrelation function. To track individual particles, first a fluid particle trajectory is simulated for which the components of the fluctuating
velocity correlation (i.e. Reynolds stress) tensor are determined from a set of second-order
algebraic relations. Various components of the fluctuating fluid velocity are calculated from
the Reynolds stresses by assuming a Gaussian pdf and by satisfying the specified form of the
Lagrangian velocity autocorrelation. Once the fluid particle trajectory is constructed, the
heavy particle trajectory is simulated in a similar manner to that in Ormancey and Martinon’s130 model. Chauvin et al.135 and Berlemont et al.136 discuss the extension of this model
to account for particle-particle interactions. Another variation of this model was proposed
by Blumcke et al.137
There are several other models131, 138–143 that are not as complicated and require less
bookkeeping than the models proposed in Refs. 130, 133. Various applications reported in
the literature144–146 indicate satisfactory performance of these models, which also account
for temporal and spatial correlations. However, instead of constructing the trajectories for
fluid particles through the correlations at several time steps, only the correlation between
two successive time steps is considered. One of these models is described below to highlight
the procedure for implementation of turbulence correlations in time and space.
The model proposed by Lu143 implements the Eulerian (as opposed to the Lagrangian)
fluid velocity autocorrelation, and is capable of producing the same trend for the variation of
47
the particle diffusivity coefficient as those predicted by theory (e.g. Pismen and Nir147 ). The
rudiments of the model are taken from the methodology of time series analysis.132 Consider
a coordinate system moving with the mean fluid velocity, such that only the fluctuating
velocities are of interest in the computations. The initial particle position (Xi (0), i = 1, 2, 3)
and velocity (vi (Xi (0), 0)) are known. The initial fluid velocity ui (Xi (0), 0) at the particle
location is obtained by sampling from a random seed with Gaussian pdf with the standard
deviation u0 (assumed to be known a priori). Then, the particles are moved within the
time interval of ∆t to new positions Xi (∆t). In order to proceed to the next time level,
the fluid velocity ui (Xi (∆t), ∆t) at the new particle location must be determined. By the
time the particles arrive at their new location, the fluid velocity at the old particle location
has changed to ui (Xi (0), ∆t). To relate the old and the new fluid velocities at Xi (0), the
Eulerian velocity autocorrelation
Fαα (∆t) =
hhwα (Xα (0), 0)wα (Xα (0), ∆t)ii
,
hhwα (Xα (0), 0)wα (Xα (0), 0)ii
α = 1, 2, 3
(3.48)
is used, where wα = uα /u0 is the normalized fluid velocity and hh ii indicates the Lagrangian
ensemble average. It is also necessary to account for the spatial separation between the fluid
particle and the heavy particle through the Eulerian spatial correlation
Gαα (∆s) =
hhwα (Xα (0), ∆t)wα (Xα (∆t), ∆t)ii
,
hhwα (Xα (0), ∆t)wα (Xα (0), ∆t)ii
α = 1, 2, 3
(3.49)
where ∆s = |X(∆t) − X(0)| is the distance between the old and the new particle locations.
Before (3.49) can be used, it is necessary to reorient the coordinate system such that one of
the axes aligns with the vector X(∆t) − X(0). By defining autoregressive processes132 (in
time) for wi (Xi (0), 0) and wi (Xi (0), ∆t), and (in space) for wi (Xi (0), ∆t) and wi (Xi (∆t), ∆t),
the fluid velocity at the new particle location is given by143
wα (Xα (∆t), ∆t) = aα bα wα (Xα (0), 0) + γα ,
48
α = 1, 2, 3
(3.50)
where aα = Fαα (∆t), bα = Gαα (∆s), and γα is the Wiener process parameterized by its variance σγα =
q
1 − a2α b2α . Once the fluid velocity at the new particle location is determined,
the procedure above is repeated until the desired time level is reached. In the implementation of the model, the following correlations were recommended143 Fαα (∆t) = exp(−∆t/τE ),
G11 (∆s) = exp(−∆s/Λ1 ), and G22 (∆s) = G33 (∆s) = exp(−∆s/Λ2 ), where τE is the Eulerian integral time scale and Λ1 and Λ2 are the Eulerian integral length scales in the longitudinal and transverse directions, respectively. In isotropic incompressible flows, these are
q
estimated by τL = C1 (u0 )2 /², τE = τL /C2 , Λ1 = C3 τL (u0 )2 , and Λ3 = Λ2 = 0.5Λ1 , where
τL is the Lagrangian integral time scale, with C1 = 0.212, C2 = 0.73, and C3 = 2.778.
Mashayek146 performed a detailed comparison between the predictions of Lu’s143 model
with DNS data of Mashayek et al.53 and theoretical results of Mei et al.121 in an isotropic
incompressible particle-laden turbulent flow. The stochastic model predicted most of the
trends as portrayed by DNS and theory. However, the continuity effect was not captured
properly. Further, the peaking in the variation of the particle asymptotic diffusivity coefficient with the particle time constant was not observed. For evaporating droplets, the
stochastic model predicted thinner pdfs for the particle diameter as compared with DNS
generated pdfs. The model was also implemented to study the gravity effects on evaporation, and was shown that the depletion rate increases with increase of the drift velocity at
short and intermediate times, but an opposite trend was observed at long times. Dispersion
of evaporating droplets decreased with respect to that of non-evaporating particles at small
drift velocities; an opposite trend was observed at large drift velocities.
Pascal and Oesterlé148 also proposed a stochastic model based on autoregressive processes
to predict the fluid instantaneous velocity at the particle location. This model was based on
the model of Burry and Bergeles,149 and its basic principles were similar to those discussed
above for the model of Lu.143 However, instead of the isotropic autocorrelation function
(3.48) an autocorrelation tensor was used which allowed the model application be extended
to anisotropic turbulent flows. Pascal and Oesterlé148 also showed that the use of Frenkiel
49
correlations by Burry and Bergeles149 is not consistent with the principles of generating the
fluid instantaneous velocity by means of first-order time series model – In fact, it was shown
that only exponential correlation matrices can be used in such autoregressive processes.
Recently, Moissette et al.150 proposed a model that also calculates temperature fluctuations for discrete particles. The model was constructed using first-order autoregressive
processes for both velocity and temperature fluctuations of the particles and accommodated
for anisotropy effects. An important aspect of the model was the inclusion of the correlation
between temperature and velocity fluctuations in a systematic manner within the framework
of autoregressive processes. The model was implemented to simulate solid particles dispersed
in a homogenous shear turbulence and good agreements with analytical results of Zaichik151
were found.
3.3.3
Low-cost stochastic simulations
A major difficulty associated with stochastic (or, in general, any Lagrangian) approach is
the need for tracking a large number of particles in order to generate accurate statistics. In
complex flows and in the presence of more sophisticated phenomena, including two-way coupling, this could become computationally inhibiting. Various remedies have been suggested
to alleviate this problem and reduce the total number of particles that must be tracked.
Zhou and Yao152 proposed a ‘group’ modeling approach where only the center of a group
of particles is tracked using the Lagrangian equations of motion. Each group represents a
number of particles which disperse spatially with respect to the center of the group under the
influence of turbulence. In Zhou and Yao’s model, the turbulent flow was assumed locally
isotropic and the dispersion of the particles originated from a point source located at the
center of the group as it left the nozzle. The probability of a droplet transport by turbulent
diffusion was assumed to be similar to Brownian diffuison which could be closely represented
by a normal Gaussian distribution. Numerical simulations indicated that reasonable results
can be generated using only 15 groups of particles as opposed to 500 particles needed for
50
generating similar results via the traditional stochastic approach. The authors pointed out
that the growth of the group in time eventually results in the failure of the assumption of
locally isotropic turbulence and hence inaccurate results. It was proposed that larger groups
could be split into smaller ones to overcome this problem.
Following a similar concept, Chen and Pereira153, 154 also proposed a method to decrease
the number of particles required for accurate calculation of statistics. Their stochasticprobabilistic efficiency-enhanced dispersion (SPEED) model was built upon the stochastic
model of Gossman and Ioannides, enhanced with an additional probabilistic model to account for the probabilistic distribution of a physical particle in space. This model is capable
of accounting for anisotropy of the turbulence, when implemented in conjunction with a
Reynolds stress model that provides various normal components of the fluid velocity fluctuations. The SPEED model also takes into account the finite inertia of the particles when
applying the probabilistic approach. Consequently, the dispersion of the particles, due to
turbulence, is not described by Brownian diffusion, as was the case in Zhou and Yao’s group
model. The probabilistic computation is based on a computed trajectory variance (σpi in
the direction xi ) and an assumed probability density function. The equation governing the
particle trajectory variance is derived from its definition and written tensorially as:153
Z t
2
dσpα
= 2 hvα (t)vα (t1 )idt1 .
dt
0
(3.51)
The particle velocity autocorrelation between times t and t1 is not known but can be estimated from
hvα (t)vα (t1 )i = Ωpα huα (t)uα (t1 )i,
(3.52)
where Ωpα accounts for the slipping effects between the two phases and is estimated based
on the fluid normal stress, hu2α (t)i, following the work of Rizk and Elghobashi.155 The fluid
velocity autocorrelation huα (t)uα (t1 )i, in (3.52), is also related to the predicted fluid normal
51
stress through a Lagrangian autocorrelation function:
RLα (τ ) =
huα (t)uα (t1 )i
hu2α (t)i
(3.53)
where τ = t1 − t. The autocorrelation function is then estimated using a Frenkiel correlation
function, and, under the assumption that a very small time step ∆t is used, the final equation
for σpi is derived as:
·
µ
2
2
(t) + 2Ωpα hu2α iTLα ∆t − 2TLα exp −
σpα
(t + ∆t) = σpα
¶
µ
∆t
∆t
sin
2TLα
2TLα
¶¸
,
(3.54)
where the integral time scale of each component is determined by
TLα = 0.235hu2α i/².
(3.55)
In the computations, while the center of the particle group is tracked using the Lagrangian
equations of motion, the particle trajectory variances are also determined simultaneously to
account for turbulence dispersive effects.
4
Eulerian Description
In this section we consider two modeling approaches, namely, RANS and pdf. In RANS
modeling, the analysis starts from the instantaneous Eulerian equations for particle phase,
which are similar to equations for the fluid phase. Whereas in pdf, the analysis starts from
the Lagrangian equations and a kinetic equation is obtained which governs the probability
density of particle’s position, velocity, and other variables of interest at time t. The pdf
equation if solved using the Mote Carlo method would have characteristics of Lagrangian
framework. Instead of solving the pdf equation, one can derive ‘macroscopic’ or ‘fluid’
equations by taking various moments of the pdf equation. These equations govern the
statistical properties of the particle phase and are in Eulerian framework. Therefore, we note
52
that the pdf approach may be discussed under either Lagrangian or Eulerian descriptions.
Here, we have chosen to discuss it under the Eulerian description due to the final framework
of the macroscopic equations. The Eulerian description is also known as the ‘two-fluid’
approach as the particle phase is described by fluid-like equations.
4.1
RANS modeling
The starting Eulerian equations for particle instantaneous variables in RANS are associated
with the cloud of particles present in a unit volume for a single realization and are obtained
by various methods of averaging.64, 156, 157 Jackson’s method of spatial (volume) averaging
and Zhang and Prosperetti’s156 method of ensemble averaging have led to very similar results.
The process of averaging creates some unknown terms which can be neglected in dilute flows
of interest in this review. The final Eulerian equations for particle volume fraction and
particle cloud momentum in the limit of dilute flow, such as those proposed by Jackson, are
taken here as the instantaneous equations. The volume fraction, a new variable introduced
during the averaging, is defined as the ratio of the volume of the particles to the total volume
of the mixture, and may be interpreted similar to the density of a compressible gas. A typical
set of particle Eulerian equations is given by equations (4.59) and (4.60) in Section 4.1.1.
The ensemble average of instantaneous equations in case of turbulent flows further poses
closure problems due to the appearance of unknown correlations similar to the Reynolds
stress for fluid turbulence. Utilizing various closure schemes of fluid turbulence, different
models have been proposed by researchers to close the unknown terms. The final closed
set of equations governs the statistical properties of turbulent flow of particle phase. This
section provides a review of various RANS methods as applied for modeling of the closure
terms within the two-fluid framework.
The early two-fluid treatments of particle-laden turbulent flows were primarily based on
the isotropic ‘eddy-viscosity’ concept. Pourahmadi and Humphrey158 constructed a twofluid model on the basis of the (widely-used) k-² model in single-phase flows. The Reynolds
53
stresses in both phases and the turbulent fluxes of the volume fraction were modelled via
the eddy viscosity approach. The turbulent diffusion coefficient of the dispersed phase was
related to that proposed for the fluid phase (in the context of the single-phase k-² model).
The turbulence kinetic energy of the dispersed phase and the trace of the fluid-particle
covariance tensor were calculated from the turbulence kinetic energy of the fluid, i.e.
kp = k
TL
,
τp + TL
ui vi = 2k
TL
,
τp + TL
(4.56)
which were valid for high Reynolds numbers and small particle time constants. In (4.56),
TL = CT k/² is the fluid Lagrangian integral time scale with the suggested value CT = 0.41.
It is noted that the model proposed for kp only considers the fluid turbulence as the source of
energy in the particulate phase, and neglects the production of the dispersed phase turbulence
kinetic energy due to the mean velocity gradient of the particles.
Elghobashi et al.155, 159–161 proposed two-equation models based on transport equations
for turbulence kinetic energy of the carrier phase and its rate of dissipation. These equations
accounted for turbulence modifications (i.e. two-way coupling) and involved models for up
to third-order correlations. In all of these works, a dilute two-phase flow was considered such
that interparticle collisions could be neglected, and the continuum equations used for the description of the dispersed phase were based on volume averaging. The Eulerian equations for
both phases were time averaged, following a Reynolds decomposition, resulting in unclosed
correlations for Reynolds stresses and volume fraction fluxes of both phases. For the volume
fraction flux, Elghobashi and Abou-Arab159 used a model which was originally developed for
the flux of a passive scalar. This model accounted for inhomogeniety of the flow, however, its
use for the volume fraction flux may be questionable as the transport equation for a passive
scalar flux differs from that for the void fraction flux. An eddy viscosity approach was used
to model the Reynolds stresses with νt = Cµ k 2 /² for the carrier phase. The turbulent viscosity of the dispersed phase was related to that of the carrier phase through a droplet Schmidt
number. Elghobashi et al 160 assessed the model predictions for a turbulent axisymmetric
54
jet laden with monosize particles, and reported good agreements with experimental data.
The model was later extended161 to evaporating droplets by assuming isothermal flow and
negligible density fluctuations in the carrier phase. To calculate the mass transfer due to
evaporation, a transport equation was also considered for the vapor mass fraction. Rizk and
Elghobashi155 modified the model for low Reynolds number turbulence and compared the
model predictions for a turbulent two-phase pipe flow with experimental data.
Chen and Wood162 presented a two-fluid model based on the formulation of Marble.163
The Reynolds stresses and the volume fraction fluxes were modeled using a gradient hypothesis, and the eddy viscosity of the carrier phase was calculated using the k-² model.
The dispersed phase eddy viscosity was determined using the Hinze-Tchen relation νtp =
νt /(1 + τp /τe ) where τe = 0.165k/² was the time scale of the energetic turbulent eddies. The
eddy viscosity for the flux of volume fraction was obtained from νtφ = νt /Sct with Sct = 0.7.
The two-equation k-² model was also implemented by Issa and Oliveira164 in conjunction
with the dispersed phase Eulerian equations derived by a double-averaging procedure. The
turbulent stresses of each phase was modelled by eddy viscosity. However, the main assumption in the modeling of the dispersed phase was that the velocity fluctuations of the two
phases were proportional. Picart et al.165 used a k-² model along with Rodi’s166 implicit
algebraic relations for Reynolds stresses to calculate the carrier phase. The dispersed phase
was simulated using a dispersion tensor which was computed in the framework of Tchen’s17
theory. Analogous to its single-phase counterpart, modeling of two-phase flows was not limited to two-equation models. Abou-Arab and Roco167 presented a model using one transport
equation for the turbulence kinetic enery. A four-equation model was proposed by Kashiwa
and Gore168 which involved transport equations for the turbulence kinetic energy and its
dissipation rate for both phases.
55
4.1.1
Second-order moment modeling
The eddy viscosity approach is very convenient in application and has produced reasonably
accurate predictions for a number of turbulent single-phase flows. However, as pointed
out by Viollet et al.,169 some laboratory measurements170 and DNS results60, 65, 67 indicate
that the dispersed phase is highly anisotropic, thus the eddy viscosity approach is not well
justified for particles. This has motivated the use of more sophisticated models based on
the transport equations for various statistical moments up to the second order, coupled
with different closures for higher-order moments. Referred to as second-order closures, these
schemes offer extensive advantages over the Boussinesq type approximations based on the
isotropic eddy diffusivities (such as k-² model in single-phase flows). However, the modeling
is understandably more difficult as the number of unclosed terms is more than that in
isotropic closures. The literature of turbulence modeling in single-phase flows is very rich
with predictive schemes based on second-order closures7, 171–176 and will not be reviewed here.
Simonin et al.169, 177–179 proposed transport equations for Reynolds stresses in the dispersed phase with and without consideration of interparticle collision. The effects of the
particles on the carrier phase were accounted for by including source terms in the transport
equations for the fluid turbulence kinetic energy and its dissipation rate. The fluid-particle
covariance appearing in the formulation, was treated via either the solution of its transport
equation or an eddy viscosity approach. Nevertheless, the eddy viscosity model used in the
formulation, did allow for anisotropy and antisymmetry of the covariance tensor. When
interparticle collision was present, the transport equations for Reynolds stresses contained a
complementary term which was obtained following Grad’s180 approach for monosized elastic
hard sphere collision. This term resulted in destruction of the off-diagonal stresses and redistribution of energy among the normal components, thus had no effect on the turbulence
kinetic energy. Other terms in the transport equation accounted for turbulent transport, production by the particle mean velocity gradient, and the effects of the drag. The third-order
correlation in the turbulent transport term was modeled in the frame of Grad’s approach
56
and by neglecting convective transport and mean gradient effects. When the interparticle
collision was inelastic, the extension of Grad’s theory resulted in an extra dissipation term
in the transport equation for the Reynolds stresses of the dispersed phase.
Whereas in the above-mentioned contributions, the use of second-order closures for the
dispersed phase has resulted in the improvement of the model predictions, there is an inconsistency in the overall treatment of the two-phase flow as the continuous phase is still
modeled with the eddy viscosity. To alleviate this inconsistency, Simonin et al.107 proposed
second-order closures for both phases. The dispersed phase Eulerian equations were derived
via ensemble averaging181 of the Lagrangian equations for the individual particles. The
transport equations for the fluid Reynolds stress and the fluid-particle velocity covariance
were derived in a Lagrangian context starting from the generalized Langevin equation as
proposed by Haworth and Pope.182 The equation for the fluid-particle velocity covariance
tensor contained production terms due to mean velocity gradients in both phases, and further
accounted for the antisymmetry of this tensor. Simonin et al.107 reported good agreement
between the model predictions and LES results.
Shih and Lumley183 proposed a set of second-order equations for particle-laden flows
by assuming that the particle time constant and size were smaller than the Kolmogorov
time and length scales, respectively. They neglected the convective term in the particle
Eulerian momentum equation, and accounted for the effect of inertia by expressing the
dispersed phase velocity in terms of an expansion of the fluid velocity. It was noted that
the resulting equations were suitable for the study of correlations between particle velocity
and other quantities, but not for particle Reynolds stress. A dissipation term was added to
the transport equation for the variance of the particle-density fluctuations. This dissipation
term was necessary to bound the magnitude of the variance.
A main advantage of Eulerian (as opposed to Lagrangian) methodology for the dispersed
phase is that it allows the effects of particles on turbulence (i.e. two-way coupling) to be
directly included in various stages of model development. In this situation, one includes
57
the relevant source term(s) in the carrier phase equations from the onset. For example, in
case of an incompressible carrier phase laden with monosize particles, these (carrier phase)
equations may be described as64, 156, 157
∂ Ubj
= 0,
∂xj
(4.57)
b
1 ∂ Pb
∂ 2 Ubi
λΦ
∂ Ubi ∂(Ubj Ubi )
=−
+ν
−
(Ubi − Vbi ) + gei ,
+
∂t
∂xj
ρf ∂xi
∂xj ∂xj
τp
(4.58)
along with the Eulerian equations for the dispersed phase
b
b
∂Φ
∂(Vbj Φ)
+
= 0,
∂t
∂xj
(4.59)
∂ Vbi
∂ Vbi
1
+ Vbj
= (Ubi − Vbi ) + gei ,
∂t
∂xj
τp
(4.60)
where g denotes the gravity constant, and ei is the unit vector in the gravity direction. Here,
Ui , Vi , P , and Φ represent the carrier-phase velocity, the dispersed-phase velocity, the carrierphase pressure, and the volume (or void) fraction, respectively. Thebshows the instantaneous
quantity, and ρf and ν are the fluid viscosity and density, respectively. Equations (4.57)b ∼ O(10−4 ) or
(4.60) have been simplified for gas-solid flows with low volume fraction (Φ
less) and large density ratio (λ = ρp /ρf ∼ O(103 )), and are valid for particle time constants
sufficiently larger than zero. The Stokes drag relation is used for phase interactions and there
is no mass transfer between the two phases. Under these conditions, the effects of particles
on the carrier phase is exhibited through the drag only and is included as the last term on
the right-hand side of (4.58).
We use this set of equations as a starting point to elaborate on various stages involved
in modeling of two-phase flows within the second-order moment framework. Following the
standard Reynolds decomposition procedure, all the instantaneous variables are decomposed
58
into the ensemble-mean and fluctuations about the mean:
Ubi = Ui + ui ,
Vbi = Vi + vi ,
b = Φ + φ,
Φ
Pb = P + p,
where uppercase and lowercase show the mean and the fluctuating parts, respectively. The
equations governing the transport of the mean variables are obtained by ensemble averaging
(4.57)-(4.60):
∂Uj
= 0,
∂xj
(4.61)
1 ∂P
∂ 2 Ui
∂ui uj
λ
λ
∂Ui ∂(Uj Ui )
=−
+ν
−
− Φ(Ui − Vi ) − (ui φ − vi φ) + gei , (4.62)
+
∂t
∂xj
ρf ∂xi
∂xj ∂xj
∂xj
τp
τp
∂Φ ∂(Vj Φ)
∂vj φ
+
=−
,
∂t
∂xj
∂xj
(4.63)
∂Vi
∂Vi
∂vi vj
1
∂vj
+ Vj
=−
+ (Ui − Vi ) + vi
+ gei ,
∂t
∂xj
∂xj
τp
∂xj
(4.64)
where the overbar indicates the ensemble-averaged value.
It is noted that the averaging procedure leading to equations for mean variables ((4.61)(4.64)) involves two steps; a volume averaging followed by an ensemble averaging. As pointed
out in Ref. 184, the volume averaging results in Reynolds stresses due to small deviations
of the particle velocity from its volume averaged value. However, these volume averaged
stresses are usually small in comparison to those resulting from the time averaging. The
reason is that particles, due to their large inertia, interact with larger scales of the flow185
while the volume averaging is performed over volumes which are normally much smaller than
the large scales of the flow in order to satisfy the conditions required in volume averaging. For
the equations used here, Jackson64 argues that the contribution of the Reynolds stress type
terms resulted from the first averaging is small except possibly in a short interval following
the initial conditions. This situation of double-averaging is avoided in pdf modeling as
discussed in Section 4.2.
The closure problem in the averaged equations (4.62)-(4.64), is exhibited by the Reynolds
59
stress tensors associated with both phases (ui uj and vi vj ), and the void fraction flux vectors
(ui φ and vi φ).† To proceed within the context of second-order moment modeling, differential
transport equations are considered for these correlations. These equations can be derived
following standard procedures similar to those widely used for Reynolds stress transport
equation.7, 175 Mashayek and Taulbee186 have presented these transport equations for the
simplified case of homogeneous turbulence. In this case, turbulent fluxes of the volume
fraction are zero and the spatial derivatives of all second- and third-order correlations also
vanish. Consequently,
∂Uj
∂Ui
λΦ
dui uj
= −ui ul
− uj ul
+ ψij − ²ij −
[ui (uj − vj ) + uj (ui − vi )],
dt
∂xl
∂xl
τp
(4.69)
for the carrier-phase Reynolds stress,
dvi vj
1
∂Vj
∂Vi
= −vi vl
− vj vl
− (2vi vj − ui vj − uj vi ),
dt
∂xl
∂xl τp
(4.70)
for the dispersed-phase Reynolds stress. It is now noted that a transport equation for the
fluid-particle velocity covariance (ui vj ) is needed. This equation is also derived similarly,
†
The modeling of these fluxes can be avoided if a weighted average is used instead of the Reynolds average.
The procedure is similar to that adopted in Favre averaging of compressible flows, with the volume fraction
here playing the role of the density. If we define
00
Vbi = Vei + vi ,
with
00
denoting the fluctuation about the Favre average velocity
Z
1 bb
1 1 τ bb
Vei = Φ
Vi =
ΦVi dt,
Φ
Φτ 0
(4.65)
(4.66)
then the dispersed phase continuity equation reads:
∂Φ ∂(ΦVei )
+
= 0.
∂t
∂xi
(4.67)
A comparison of (4.67) with (4.63) shows that the need for modeling of vi φ has been eliminated by Favre
averaging. This information, however, is needed to recover the Reynolds average velocity, Vi , which can be
b Vbi = ΦVi + vi φ to write:
calculated using (4.66) and Φ
Vi =
1 e
(ΦVi − vi φ).
Φ
60
(4.68)
and reads:
dui vj
∂Vj
∂Ui
1
λΦ
= −ui vl
− ul vj
+ χij + (ui uj − ui vj ) −
(ui vj − vi vj ).
dt
∂xl
∂xl
τp
τp
(4.71)
In (4.69), ψij = (p/ρf )(∂ui /∂xj + ∂uj /∂xi ) is the pressure-strain correlation, and ²ij =
2ν(∂ui /∂xl )(∂uj /∂xl ) indicates the dissipation rate. In (4.71), χij = (p/ρf )(∂vj /∂xi ) is the
pressure-dispersed phase velocity gradient correlation. In these equations, the third-order
correlations involving both the velocity and the void fraction fluctuations are neglected.
The pressure terms, ψij and χij can be modeled similarly to the widely utilized closure
of Launder-Reece-Rodi (LRR).187 First, the dissipation rate tensor (²ij ) is expressed as the
sum of an isotropic part ( 23 ²δij ) and a deviatoric part (²D
ij ). The latter is combined with the
pressure-strain correlation, and the resulting tensor (Πij = ψij − ²D
ij ) is modeled. Utilizing
Fourier transform methods the Poisson’s equation for the pressure can be solved resulting in
expressions for Πij and χij in terms of integrals of two-point velocity correlations. The result
for Πij contains extra terms, involving fluid-particle velocity correlations, arising from the
effects of the particles on the fluid. Following Launder et al.,187 the pressure-strain term Πij
is formulated as a linear polynomial function of the velocity correlations. The coefficients
of this polynomial are obtained by applying the constraints of symmetry, incompressibility,
and normalization. The final result for homogeneous flows is
Πij = −Cf 1 ²afij + ²
−
³
´
2
f
f
f
Sijf − 6Cf 2 afik Skj
+ afjk Ski
− afmn Snm
δij
5
3
h4
i
λΦ
4 + 14Cf 2 f f
f
(aik ωkj + afjk ωki
) + Cf 3
[2kafij − um vm (bfijp + bfjip )],
3
τp
(4.72)
where Cf 1 , Cf 2 , and Cf 3 are empirical constants, afij = ui uj /k−2δij /3 and bfijp = ui vj /um vm −
δij /3 are the normalized form of the fluid-fluid and the fluid-particle Reynolds stresses,
respectively, and Sijf = (k/2²)(∂Ui /∂xj + ∂Uj /∂xi ) and ωijf = (k/2²)(∂Ui /∂xj − ∂Uj /∂xi )
are the normalized strain rate and rotation tensors of the carrier phase, respectively. On
the right-hand side of (4.72), the first two terms are the LRR closure for the slow and
61
fast pressure-strain effects, respectively, and the last term represents the particle effects.
Following a similar procedure, a model is derived for the pressure-dispersed phase velocity
gradient correlation:
χij = −Cf p1
ui v j
∂Ui
λ
+ Cf p2 ul vj
+ Cf p3 Φ(ui vj − vi vj ),
τ
∂xl
τp
(4.73)
where τ = k/² is the carrier-phase turbulence time scale, and Cf p1 , Cf p2 , and Cf p3 are empirical constants. Equations (4.72) and (4.73) indicate that the proposed closures involve six
empirical constants which must be determined. Mashayek and Taulbee186 used a combination of DNS and laboratory data to find empirical values for these constants. A discussion
on the use of DNS for determination of model constants is given in Section 5, and table 1
shows the recommended values.
Equations (4.69)-(4.71) constitute a set of Reynolds stress models for homogeneous,
particle-laden turbulent flows. To close, an equation must be provided for the dissipation
rate of the fluid turbulence kinetic energy. The modeled dissipation rate equation can be
rationalized from the exact equation for the dissipation rate as obtained from (4.58). For
the present case this equation includes an additional term,
"
#
∂ui ∂vi
2λΦ ∂ui ∂ui
ν
−ν
,
τp
∂xj ∂xj
∂xj ∂xj
(4.74)
due to coupling with the dispersed phase. In this expression, the first term is the dissipation
rate, ², and the second term is modeled as the relaxation of the trace of the fluid-particle
velocity correlation tensor, i.e. C²3 um vm /τ . The final form of the modeled transport equation
for the dissipation rate for homogeneous flow is expressed as:186
²
∂U1
²2
² λΦ
∂²
= −C²1 u1 u2
− C²2 −
(2k − C²3 um vm ),
∂t
k
∂x2
k
k τp
(4.75)
where C²1 , C²2 , and C²3 are constants. The values for C²1 and C²2 are taken from their
single-phase equivalents; C²1 = 1.45 and C²2 = 1.85. The value of constant C²3 = 0.8 is
62
found by comparison with DNS data.186
Before we discuss the algebraic models for second-order correlations in the next subsection, we mention here the work of Elperin and coworkers on the closure problem posed by the
ensemble average of equation (4.59) with molecular diffusion term added on the right-hand
side of it, written as
∂vj φ
∂ 2Φ
∂Φ ∂(Vj Φ)
=−
+D
,
+
∂t
∂xj
∂xj
∂xj ∂xj
(4.76)
where D is the molecular diffusion coefficient. Elperin et al.188, 189 solved the closure problem,
posed by the term vj φ, by using the method of stochastic calculus developed for the Wiener
process.190 The final equation, when vj changes very rapidly and is correlated over a very
short period of time, is
"
Ã
∂Φ
∂
∂vj
+
Φ Vi − ϑvi
∂t
∂xi
∂xj
!#
"
∂ 2Φ
∂
∂Φ
=D
+
ϑvj vm
∂xj ∂xj ∂xj
∂xm
#
(4.77)
where ϑ is the momentum relaxation time of random velocity field vi . For the particle with
∂vj
time constant τp ¿ 1, Elperin et al.188 have shown that the term ϑvi ∂x
yields a new phej
nomenon of turbulent thermal diffusion of particles in incompressible gas and an additional
new phenomenon of turbulent barodiffusion in compressible gas flow.189, 191 The condition of
rapidly changing particle velocity field and thus ϑ being small has been relaxed in their recent work.192 These two phenomena have also been recently obtained by solving the closure
problem posed by equation (4.76) using Kraichnan’s direct interaction approximation.193
4.1.2
Algebraic modeling
Despite a more accurate representation, the need for solving additional moment transport
equations makes second-order modelling potentially less attractive for practical applications.
The increase is naturally expected to be higher in two-phase flows. A remedy to overcome the
high computational cost associated with the second-order modeling is to utilize ‘algebraic’
closures.194–200 Such closures are either derived directly from the transport equations for
63
second-order moments, or other types of representations201–204 which lead to anisotropic
eddy diffusivities. One of the original contributions in the development of algebraic Reynolds
stress models (ARSM) is due to Rodi.195 In this work, all the stresses were determined from
a set of ‘implicit’ equations which must be solved in an iterative manner. Pope194 offered an
improvement of the procedure by providing an ‘explicit’ solution for the Reynolds stresses.
This solution was generated with the use of the Cayley-Hamilton theorem (CHT), but was
only applicable for predictions of two-dimensional (mean) flows. The extension of Pope’s
formulation to three-dimensional flows was done by Taulbee,198 Taulbee et al.200 and Gatski
and Speziale.199 In these efforts, CHT was used to generate explicit algebraic Reynolds stress
models which are valid in both two- and three-dimensional flows.
The contributions in particle-laden flows are, understandably, more limited and recent.
Zhou205 and Zhou et al.206 derived a set of ‘implicit’ algebraic models by neglecting the total
time derivative of second-order quantities. The resulting algebraic models were augmented
by transport equations for the turbulence kinetic energies of both phases and the dissipation
rate of the carrier phase turbulence kinetic energy.
Following a similar procedure to that used for single-phase flows, Mashayek and Taulbee207
derived ‘explicit’ algebraic models for second-order moments in nonhomogeneous, turbulent
gas-solid flows. The procedure involves the followings: (1) construction of differential transport equations for the second order moments, (2) modeling of various terms in the transport
equations, specially the pressure-strain and the pressure-void fraction gradient correlations,
(3) simplification of the transport differential equations to implicit ‘algebraic equations’, and
(4) the solution of the algebraic equations to generate ‘explicit’ algebraic models. Steps (1)
and (2) were described in the previous subsection for homogeneous flows. Step (3) is carried
out following two fundamental assumptions: (i) the flow is in equilibrium state, and (ii) the
difference in the transport terms is negligible. The final form of the implicit algebraic models
64
are presented in tensor forms as:207
³
´
2
af − C1f q f − C2f af S f + S f af − {S f af }I 3 − C3f (af ω f − ω f af ) = 0,
3
(4.78)
³
´
2
ap − C1p q p − C2p ap S p + S p ap − {S p ap }I 3 − C3p (ap ω p − ω p ap ) = 0,
3
(4.79)
ϕf + Df Af ϕf + B f ϕp + Υf = 0,
(4.80)
ϕp + Dp Ap ϕp + B p ϕf + Υp = 0,
(4.81)
where a single underline indicates a vector, a double underline denotes a second-order tensor,
I 3 is the three-dimensional identity tensor, af = afij , ap = apij , ϕf = ui φ, ϕp = vi φ, S f = Sijf ,
S p = Sijp , ω f = ωijf , and ω p = ωijp . In (4.78) and (4.79), q f and q p are second order tensors
representing the effects of two-way coupling between the phases. The expressions for these
tensors and coefficients Af , Ap , B f , B p , Df , Dp , Υf , and Υp are given in Ref. 207.
These equations are solved analytically to produce explicit algebraic models. The solution
of the coupled equations for ϕf and ϕp is obtained via direct utilization of CHT, and is given
in Ref. 207. The solution procedures for af and ap are identical, and may be explained by
considering the general form of the tensor a governed by:
³
´
2
a = C1 q + C2 a S + S a − {S a}I 3 + C3 (a ω − ω a).
3
(4.82)
The solution for a is also obtained via CHT, and is analogous to (but much more involved
than) that in single-phase turbulent flows.194, 198, 199 The complexity is due to the fact that
the tensor a here is a function of three tensors (S, ω, and q) in contrast to the case of a singlephase flow where a is a function of only two tensors (S and ω). As a result, new integrity
basis and irreducible matrix polynomials must be specified. Using various theorems208, 209
originated from CHT, Mashayek and Taulbee207 expressed a via a finite set of polynomials
65
for a two-dimensional mean flow with three-dimensional turbulence:
1
1
a = G1 ( I 3 − I 2 ) + G2 S + G3 (S ω − ω S) + G4 q + G5 (q ω − ω q).
3
2
(4.83)
The coefficients Gλ of the matrix polynomials are functions of the invariants η1 = {S 2 },
η2 = {ω 2 }, η3 = {S q}, and η4 = {S ω q} + {ω q S}, which are obtained directly via
CHT. Substituting (4.83) into (4.82), after extensive algebraic manipulations, the following
expressions for G’s were obtained:207
G1 =
G3 =
−6C1 C2 C3 η4 + 6C1 C2 η3
,
2C22 η1 + 6C32 η2 − 3
2C1 C22 C32 η4 − 2C1 C22 C3 η3
,
(1 − 2C32 η2 )(2C22 η1 + 6C32 η2 − 3)
G2 =
2C1 C22 C3 η4 − 2C1 C22 η3
,
(1 − 2C32 η2 )(2C22 η1 + 6C32 η2 − 3)
G4 =
C1
,
1 − 2C32 η2
G5 =
C1 C3
.
1 − 2C32 η2
(4.84)
To close the algebraic models, Mashayek and Taulbee210 provided transport equations
for k, ², kp , and kf p ; thus proposing a four-equation model:
∂k
∂k
∂ ³ 0´
∂Uj
+ Ui
= −
ui k − ui uj
−²
∂t
∂xi
∂xi
∂xi
i
λh
−
2Φ (k − kf p ) + (Ui − Vi )ui φ ,
(4.85)
τp
∂²
∂²
∂ ³ 0´
²
∂Uj
²2
+ Ui
= −
ui ² − C²1 ui uj
− C²2
∂t
∂xi
∂xi
k
∂xi
k
h
io
²λn
−
2Φk − C²3 2Φkf p − (Ui − Vi )ui φ ,
(4.86)
k τp
∂kp
∂kp
∂ ³ 0´
∂Vj
2
+ Vi
= −
vi k p − vi vj
− (kp − kf p ) ,
(4.87)
∂t
∂xi
∂xi
∂xi
τp
´
∂kf p
∂ ³ 0
1
∂Uj
1
∂Vj
∂kf p
+ (Ui + Vi )
= −
ui kf p + vi kf0 p − (1 − Cf p2 ) ui vj
− uj v i
∂t
∂xi
∂xi
2
∂xi
2
∂xi
kf p
1
λΦ
− Cf p1
+ (k − kf p ) + (1 − Cf p3 )
(kp − kf p )
τ
τp
τp
λ
− (1 − Cf p3 ) (Uj − Vj )vj φ,
(4.88)
τp
∂ui ∂ui
where k 0 = 12 ui ui , ²0 = ν ∂x
, kp0 = 12 vi vi , and kf0 p = 12 ui vi . The third-order correlations apj ∂xj
66
pearing in these equations, were modeled following a similar procedure to that implemented
in single-phase flows.211 For thin-shear flows, where only the derivatives in the cross-stream
direction are significant, the following gradient-type models were proposed:
10
∂k
Cs τ k
,
9
∂x2
∂²
= −0.5C² τ k
,
∂x2
10
∂kp
,
= − τp kp
27
∂x2
Ã
!
8
∂kf p 2
∂kp
∂k
= − Cs τ (k + kp )
− C s τ kp
+k
.
9
∂x2
9
∂x2
∂x2
u2 k 0 = −
(4.89)
u2 ² 0
(4.90)
v2 kp0
u2 kf0 p + v2 kf0 p
(4.91)
(4.92)
These models were valid for λΦ < 0.4, τp > τ , and kp > k.
4.2
Pdf modeling
The kinetic or pdf approach has its base in the study of kinetic theory of gases212, 213 and
Brownian motion.214, 215 The main task in this approach is to obtain a transport equation
governing the probability distribution of particle velocity v and position x at time t where v
c and position
and x represent the phase space variables corresponding to particle velocity V
X respectively. This single point distribution is the simplest case for the isothermal flow and
for the case of non-isothermal flow with evaporating spherical droplets, temperature and
particle radius also need to be included as phase space variables, corresponding to particle
temperature TP and radius a. In a more general single point pdf description, fluid variables,
e.g. fluid velocity vf and temperature θf along the particle trajectory or ‘seen’ by the
particle,216, 217 can be included in the phase space variables list. Further, various moments of
the pdf equation results in ‘macroscopic’ Eulerian equations governing the temporal evolution
of statistical properties of the particle phase in physical space x. The more general description
and other two-point fluid-particle pdf description in which the pdf of fluid and dispersed
particle is obtained simultaneously are recently discussed in greater detail by Minier and
67
Peirano.217 In this review, we restrict our discussion to the single point pdf for particle
phase.
The main task of obtaining single point pdf equation requires 1) forming an equation
for the phase space density of the variables from the Lagrangian equations describing the
motion of the variables and 2) obtaining the closed form for the ensemble average of the phase
space density equation after tackling the closure problems arose due to ensemble averaging.
We denote all the N phase space variables by θi (i = 1, 2, 3, ..., N ) corresponding to the
properties Θi , along the particle trajectory, which are governed by the Lagrangian equations
of motion
dΘi
= Ki (Θ1 , Θ2 , ..., ΘN , Sn , t).
dt
(4.93)
Here Ki is a function of Θi and some stochastic variables which are denoted here by Sn . The
‘fine grained’ phase space density W (θi , t) ≡ W (θ1 , θ2 , ..., θN , t) is defined using Dirac delta
function as218, 219
W (θi , t) = ΠN
i=1 δ[Θi (t) − θi ],
(4.94)
and its ensemble average over all the realizations of the stochastic variables appearing in
(4.93) represents the probability density function hW i. The governing equation for W for
collisionless particles can be obtained by differentiating both sides of (4.94) (see for example
Ref. 219) or using the Liouville’s theorem, written as
∂W
∂
+
[Ki (θ1 , θ2 , ...θN , Sn , t)W ] = 0
∂t
∂θi
(4.95)
which indicates that the phase space density remains constant along the particle path in
the absence of particle collisions. It should be emphasized that W is stochastic in nature
due to the presence of stochastic variables present in Ki functions. The ensemble average of
equation (4.95)
∂hW i ∂hKi iW
∂hKi0 W i
+
+
=0
∂t
∂θi
∂θi
(4.96)
poses a closure problem due to the unknown correlation hKi0 W i between the fluctuation Ki0 ,
68
over the mean hKi i, and phase space density W . When the stochastic part Ki0 has the form
of a white noise, closure problem can be solved exactly215, 220 to result in a Fokker-Planck
equation for hW i. In the situations under consideration, that is not the case and the closure
problem, which is similar in nature to the well known closure problem of fluid turbulence,
could be tackled by the theories developed for turbulence.
An attempt to solve the turbulence closure from the first principle led Kraichnan to
propose direct interaction approximation (DIA)221, 222 as a pioneering renormalized perturbation theory (RPT), followed by other RPTs223–225 in the field of theory of turbulence. An
energetically consistent DIA, which obtained an exact solution for closure problem of a Random Coupling Model (RCM),226 failed to reproduce the result of similarity theory proposed
by Kolmogorov.227 This similarity theory has also received strong experimental support.228
Kraichnan attributed the failure to the Eulerian framework of DIA and interpreted it in
terms of the failure to maintain the symmetry of the phenomenon under random Galilean
transformation (RGT). This led Kraichnan to propose a Lagrangian counterpart, Lagrangian
history direct interaction approximation (LHDIA)229 partly compatible with RGT. DIA and
LHDIA have also been applied by Orszag and Kraichnan230, 231 in the phase space of probability density method to analyze the turbulence in a Vlasov plasma. The work on DIA and
LHDIA has a pronounced influence on the solution to the closure problem of equation (4.96)
that is central to the pdf modeling of dispersed phase as is evident from Reeks works.232–236
Another approach, namely functional approach, to obtain expression for unknown correlation hKi0 W i has its root in an important result from functional calculus derived independently by Novikov237 in turbulence theory, Furutsu,238 and Donsker.239 The FurutsuNovikov-Donsker formula was first used by Derevich and Zaichik240, 241 and is a basis in their
work on two-phase turbulent flows.151, 242–245 Recently, Hyland et al.219 used the formula in
their functional approach and arrived at a pdf equation that is identical to the equation
obtained by Reeks235 in LHDIA framework. This pdf equation has also been derived by
Pozorski and Minier216 by applying yet another closure scheme, due to Van Kampen,220 pro-
69
posed to study nonlinear stochastic differential equations of quantum and statistical physics.
These important contributions by Reeks, Zaichik, Derevich and Hyland et al. along
with some recent works on pdf modeling of evaporating droplets in isotropic turbulence246
and of non-isothermal two-phase flow247 are now discussed in more details in the following
subsections.
4.2.1
Application of DIA and LHDIA
Reeks232 considered the problem of forming the equation for W (xi , vi , t), (also written as
W (x, v, t)), describing the pdf for particle moving under the influence of fluid drag force
governed by the Stokes formula. In this case, the Lagrangian Maxey-Riley equation reduces
to
dVbi
= β(Ubi − Vbi ),
dt
β=
6πaµ
,
mp
(4.97)
where β is the inverse of the particle time constant τp . Using Lagrangian equations (2.6)
and (4.97), the Liouville equation (4.95) for W (xi , vi , t), with xi , vi as phase space variables
corresponding to Xi , Vbi , reduces to
∂W
∂
∂
+
[vi W ] +
[β(Ubi − vi )W ] = 0,
∂t
∂xi
∂vi
(4.98)
and its ensemble average is written as
∂
∂
∂hW i
∂
[vi hW i] +
[β(hUbi i − vi )hW i] = −
[βhu0i W i],
+
∂t
∂xi
∂vi
∂vi
(4.99)
where Ubi = hUbi i + u0i . The unknown correlation βhu0i W i, known as phase space diffusion
current, and present on the right-hand side of equation (4.99), poses the closure problem.
Reeks obtained the expression for this term using DIA232 and LHDIA233, 235 and now we
present Reeks’ derivations.
70
Equation (4.98) is linear in W and with a known initial W (x0i , vi0 , t0 ) at t = t0
Z
Ĝ(x, v, t; x0 , v0 , t0 )W (x0 , v0 , t0 )dx0 dv0 .
W (xi , vi , t) =
(4.100)
and its ensemble average, when W at initial time t0 is not correlated with Ĝ,
Z
hW (xi , vi , t)i =
G(x, v, t; x0 , v0 , t0 )hW (x0 , v0 , t0 )idx0 dv0 ,
(4.101)
suggest that hW i can be completely determined by the average Green’s function G. Here
G = hĜi and Ĝ is the Green’s function which satisfies
(
)
∂
∂
∂
+
vi +
[β(hUbi i − vi )] Ĝ(x, v, t; x0 , v0 , t0 ) =
∂t ∂xi
∂vi
−λ
∂
[βu0i Ĝ] + δ(v − v0 )δ(x − x0 )δ(t − t0 ).
∂vi
(4.102)
Here δ is Dirac delta function, λ is a perturbation expansion parameter which is introduced
for further convenience and is set equal to one at the end of the analysis. The ensemble
average of equation (4.102) ∀ t > t0 is written as
(
)
∂
∂
∂
∂
+
vi +
[β(hUbi i − vi )] G(x, v, t; x0 , v0 , t0 ) = −λ
[βhu0i Ĝi],
∂t ∂xi
∂vi
∂vi
(4.103)
which poses the closure problem due to the presence of unknown term hu0i Ĝi. Expanding Ĝ
in powers of λ
Ĝ = G0 + λĜ1 + λ2 Ĝ2 + . . . ,
(4.104)
substituting this expansion in equation (4.102), and equating the terms with similar powers
in λ result in the equations for Ĝn , n = 0, 1, 2, . . .. As G0 is not a stochastic function, we
71
have used G0 = Ĝ0 . The solution for n ≥ 1 can be written as
0
0
0
Ĝn (x, v, t; x , v , t ) = −
Z tZ
t0
G0 (x, v, t; y, w, s)
∂
[βu0i (y, s)Ĝn−1 (y, w, s; x0 , v0 , t0 )]dydwds,
∂wi
(4.105)
which suggests that Ĝn ∀ n ≥ 1 can be written in terms of G0 . Reeks232 referred to
Ĝ(x, v, t; x0 , v0 , t0 ) and
∂
βu0i (x, t)
∂vi
by Ĝ(1, 2) and l(1) respectively. In Ĝ(1, 2), argument
1 represents (x, v, t) and argument 2 represents (x0 , v0 , t0 ). Using these same notations,
equations (4.104)-(4.105) with λ = 1 give
Ĝ(1, 2) = G0 (1, 2) − G0 (1, 3) ∗ l(3)G0 (3, 2) + G0 (1, 3) ∗ l(3)G0 (3, 4) ∗ l(4)G0 (4, 2)
−G0 (1, 3) ∗ l(3)G0 (3, 4) ∗ l(4)G0 (4, 5) ∗ l(5)G0 (5, 2) + . . .
(4.106)
where ∗ means a convolution over repeated arguments. The ensemble average of expansion
equation (4.106) gives G in terms of G0 and further reversion, using the procedure given by
Kraichnan,248 of the equation results in an expansion for G0 in terms of G.232 The first two
terms of the resulted expansion are written as
G0 (1, 2) = G(1, 2) − hl(3)l(4)iG(1, 3) ∗ G(3, 4) ∗ G(4, 2).
(4.107)
Multiplying equation (4.106) by l(1), taking the ensemble average and then substituting for
G0 from equation (4.107) give
h
hl(1)Ĝ(1, 2)i = −hl(1)l(3)iG(1, 3) ∗ G(3, 2) − hl(1)l(3)l(4)l(5)i − hl(1)l(5)ihl(3)l(4)i
i
−hl(1)l(3)ihl(4)l(5)i G(1, 3) ∗ G(3, 4) ∗ G(4, 5) ∗ G(5, 2) + . . . . (4.108)
The first term in this series corresponds to the DIA expression and when written out in full
72
gives
hu0i Ĝi
=−
Z tZ
t0
G(x, v, t; y, w, s)
∂
[βhu0i (x, t)u0j (y, s)iG(y, w, s; x0 , v0 , t0 )]dydwds.
∂wj
(4.109)
Reeks232 showed that this DIA expression for hu0i Ĝi along with equation (4.103) did not obtain correct solution in the situation where hUbi i = 0 and u0i is constant in each realization but
randomly distributed from one realization to the next. This situation when the distribution
of u0i is isotropic and Gaussian was later associated with random Galilean transformation
(RGT)234 and thus the DIA expression was shown not to be compatible with RGT.
The application of RGT concept further led Reeks234 to derive the general form for the
phase space diffusion current written as
βhu0i W i = −µji
∂hW i
∂hW i
− λji
∂vj
∂xj
(4.110)
where µji and λji are phase space diffusion tensors. Later, Reeks235 formally derived, by
using LHDIA, the closed expression for βhu0i W i compatible with RGT in case of homogeneous turbulent flow. Reeks further generalized these results and the LHDIA method to
nonhomogeneous flows.
Application of LHDIA : Consider the case when hUbi i = 0 and equation (4.103) with λ = 1
is now written as
"
#
∂ Ĝ
∂
∂
∂
+ vi
−
βvi G(x, v, t; x0 , v0 , t0 ) = −hβu0i
i.
∂t
∂xi ∂vi
∂vi
(4.111)
In LHDIA, the closure problem is solved for the equation for generalized Green’s function
Ĝ(x, v, t|s ; x1 , v1 , t1 |s1 ) which has characteristics of both Eulerian and Lagrangian viewpoints. The arguments define that the perturbation is applied at time s1 to a trajectory in
phase space passing through x1 , v1 at time t1 and perturbation’s effect is measured at time
s ≥ s1 along the trajectory which passes through the point (x, v, t) in phase space. Yet
73
another generalized function is introduced for the velocity Ubi as Ubi (x, v, t|s) which defines
the fluid velocity at time s along the trajectory of the particle which passes through x at
time t with velocity v. Kraichnan named t and t1 as labeling and s and s1 as measuring
times.
Consider the case when hUbi i = 0. Introducing the transformation
wi = vi eβt ,
yi = xi + β −1 vi (1 − eβt ),
(4.112)
and making use of u0i (y, w, t|t) = u0i (x, t) give the original equation (4.102) in the form
∂
Ĝ(y, w, t|s; y1 , w1 , t1 |s1 ) = λβu0i (y, w, t|t)li Ĝ(y, w, t|s; y1 , w1 , t1 |s1 ),
∂t
(4.113)
with s = t, s1 = t1 ,
li (y, w, t) = −eβt
∂
(1 − eβt ) ∂
−
,
∂wi
β
∂yi
(4.114)
and the relationship
Ĝ(y, w, t|t; y1 , w1 , t1 |t1 ) = e−βδii (t−t1 ) Ĝ(x, v, t|t; x1 , v1 , t1 |t1 )
(4.115)
holds. The term δ(w − w1 )δ(y − y1 )δ(t − t1 ) is to be added to the right-hand side of
equation (4.113) when s = s1 . Equation (4.113) also remains valid when s 6= t, s1 6= t1 and
u0i (y, w, t|s) is governed by
∂ 0
u (y, w, t|s) = λu0j (y, w, t|t)lj u0i (y, w, t|s),
∂t i
∀ s 6= t.
(4.116)
The ensemble average of equation (4.113) with G = hĜi,
∂
G(y, w, t|s; y1 , w1 , t1 |s1 ) = λβhu0i (y, w, t|t)li Ĝ(y, w, t|s; y1 , w1 , t1 |s1 )i,
∂t
(4.117)
poses the closure problem due to unknown term present on the right-hand side. Also,
74
equations (4.113) and (4.117) suggest that Ĝ and G are independent of measuring time
so that
G(y, w, t|s; y1 , w1 , t|s1 ) = G(y, w, t|t; y1 , w1 , t|t) = δ(w − w1 )δ(y − y1 ).
(4.118)
Here we prefer to present the closure solution in the framework proposed by Kraichnan229
and do not repeat the derivation given by Reeks235 which is based on a more formal framework
of LHDIA proposed by Kraichnan.248 The unknown term is approximated by the lowestorder term in its series expansion and thus
βhu0i (y, w, t|t)li Ĝ(y, w, t|s; y1 , w1 , t1 |s1 )i = βhu0i (y, w, t|t)li Ĝ1 (y, w, t|s; y1 , w1 , t1 |s1 )i,
(4.119)
where Ĝ1 is the solution of equation
∂
0(0)
Ĝ1 (y, w, t|s; y1 , w1 , t1 |s1 ) = βui (y, w, t|t)li G0 (y, w, t|s; y1 , w1 , t1 |s1 ).
∂t
0(0)
Here G0 and ui
(4.120)
represent the zeroth-order solutions for Ĝ and u0i , respectively, and can be
obtained from equations (4.113) and (4.116) with λ = 0. Integrating equation (4.120) in the
standard LHDIA manner along the path t1 |s1 → s1 |s1 → s|s → t|s, we have
Ĝ1 (1, t|s; 2, t1 |s1 ) =
+
+
Z s
s1
Z t
s
Z s1
t1
0(0)
dt2 d3G0 (1, t|s; 3, t2 |s1 )li (3, t2 )βui (3, t2 |t2 )G0 (3, t2 |s1 ; 2, t1 |s1 )
0(0)
dt2 d3G0 (1, t|s; 3, t2 |t2 )li (3, t2 )βui (3, t2 |t2 )G0 (3, t2 |t2 ; 2, t1 |s1 )
0(0)
dt2 d3G0 (1, t|s; 3, t2 |s)li (3, t2 )βui (3, t2 |t2 )G0 (3, t2 |s; 2, t1 |s1 )
(4.121)
0(0)
where arguments 1, 2 and 3 refer to (y, w), (y1 , w1 ) and (y2 , w2 ) respectively, and li βui
−
0(0)
βui li = 0 is used while writing the equation. Substituting equation (4.121) in equation
0(0)
(4.119), changing every labeling time from t2 to t in every G0 and ui
75
and using the property
given by equation (4.118), we obtain for s1 = t1 and s = t
βhu0i (y, w, t|t)li Ĝ(y, w, t|t; y1 , w1 , t1 |t1 )i
2
= β li (y, w, t)
Z t
t1
dt2 hu0i (y, w, t|t)lj (y, w, t2 ) ×
0(0)
uj (y, w, t|t2 )iG0 (y, w, t|t; y1 , w1 , t1 |t1 ).
0(0)
Replacing G0 and ui
(4.122)
by G and u0i in equation (4.122) would produce a renormalized expres-
sion for the unknown term and on substitution in equation (4.117) with (λ = 1, s = t, s1 = t1 )
produces an equation identical with equation (40) of Reeks.235 The final equation for hW i
in the original phase space of (x, v) is written as
"
#
∂
∂ h ∂ Zt
∂
∂
−
βvi hW (x, v, t)i =
dsheβ(s−t) β 2 u0i (x, t)u0j (x, v, t|s)i
+ vi
∂t
∂xi ∂vi
∂vi ∂vj 0
Z t
i
∂ 0
+ dshβ(1 − eβ(s−t) )u0i (x, t)
uj (x, v, t|s)i hW i. (4.123)
∂xj
0
For hUbi i = 0, comparison of equation (4.123) with equation (4.99) suggests
Ã
βhu0i W i
!
∂
∂
=−
λji +
µji + γi hW i,
∂xj
∂vj
(4.124)
where
λji = h∆xj (x, v, t|0)βu0i (x, t)i,
µji = h∆vj (x, v, t|0)βu0i (x, t)i,
*
γi = − ∆xj (x, v, t|0)
and
∆xj (x, v, t|0) =
Z t
0
∆vj (x, v, t|0) =
(4.125)
+
∂
βu0 (x, t) ,
∂xj i
ds (1 − eβ(s−t) )u0j (x, v, t|s),
Z t
0
ds eβ(s−t) βu0j (x, v, t|s).
(4.126)
(4.127)
(4.128)
(4.129)
Here ∆xj (x, v, t|0) and ∆vj (x, v, t|0) represent changes in the position and velocity due to
the fluctuating acceleration βu0i (x, v, t|s) along the particle trajectory starting at time zero
76
and passing through (x, v) at time t. The closed expression given by equation (4.124) is
compatible with RGT.
Reeks235 further generalized the LHDIA method to nonhomogeneous flows and his work
suggests
βhu0i W i
= −
Z t
0
"*
+
β 2 u0i (x, t)gjk (t1 |t)
dt1
*
+
dgjk (t1 |t)
β 2 u0i (x, t)
dt
∂ 0
u (x, v, t|t1 )
∂xk j
+#
∂ 0
u (x, v, t|t1 )
∂vk j
hW i,
(4.130)
where gjk (t1 |t) is governed by
d2
d
∂hUbk i
g
+
β
g
−
βg
= δjk δ(t − t1 ).
jk
jk
ji
dt2
dt
∂xi
(4.131)
For the case when gjk is independent of x and v, equation (4.130) can be written in the
form identical to equation (4.124) with λji , µji , and γi as given by equations (4.125)-(4.127),
respectively. Further, ∆xi and ∆vi are now given by more general expressions
∆xi (x, v, t|0) =
Z t
0
t
Z
∆vi (x, v, t|0) =
0
ds gji (s|t)βu0j (x, v, t|s),
(4.132)
dgji (s|t) 0
βuj (x, v, t|s).
dt
(4.133)
ds
Computations of µji and λji require a prior knowledge of the velocity correlation hu0i (x, t)
u0j (x, v, t|s)i. The prediction of this correlation and other correlations of fluid properties
along the particle path from the first principles remain as yet another challenge.249 In the
case of homogeneous flows, hu0i (x, t)u0j (x, v, t|s)i can be given by
hu0i (x, t)u0j (x, v, t|s)i = hu0i (x, t)u0j (x, t)iΨ(t − s),
(4.134)
and Ψ(t − s) = e(s−t)/T̃L ∀ s ¿ t with Lagrangian fluid integral time scale T̃L = 0.482k/²
represent the correlation within the acceptable range of error.151, 250
77
4.2.2
Application of Furutsu-Novikov-Donsker formula
Furutsu-Novikov-Donsker formula suggests that for any Gaussian random function fi (p), its
correlation with functional R[f ] can be exactly given by
Z
hfi (p)R[f ]i =
*
+
δR[f ]
hfi (p)fj (p )i
dp0 ,
δfj (p0 )dp0
0
(4.135)
where the integral extends over the region of arguments p in which the function fi is defined
and
D
δR[f ]
δfj (p0 )dp0
E
represents the functional derivative of R with respect to fj . This formula
forms the basis of Derevich,244 Zaichik,151 Hyland et al.219 and Pandya and Mashayek247
works on the derivation of closed kinetic or pdf equation for turbulent particle phase.
Now we present the derivation in the framework proposed by Hyland et al .219 In the case
when particle trajectory is governed by equations (2.6) and (4.97), W (x, v, t) is a functional
of u0 and using the formula, an expression for unknown correlation hu0i W i can be written as
*
Z
hu0i W i
Here
δW (x,v,t)
δu0j (x1 ,t1 )dx1 dt1
=
hu0i (x, t)u0j (x1 , t1 )i
+
δW (x, v, t)
dx1 dt1 .
0
δuj (x1 , t1 )dx1 dt1
(4.136)
is functional derivative and is given by
"
#
∂
δW (x, v, t)
δXk (t)
∂
δ Vbk (t)
=
−
+
W.
δu0j (x1 , t1 )dx1 dt1
∂xk δu0j (x1 , t1 )dx1 dt1 ∂vk δu0j (x1 , t1 )dx1 dt1
(4.137)
The functional derivatives of Xk (t) and Vbk (t) appearing in (4.137) are written as
δXk (t)
0
βδuj (x1 , t1 )dx1 dt1
= Gjk (x1 , t1 ; X(t), t)δ(X(t1 ) − x1 ),
d
δ Vbk (t)
= Gjk (x1 , t1 ; X(t), t)δ(X(t1 ) − x1 ),
0
βδuj (x1 , t1 )dx1 dt1
dt
78
(4.138)
(4.139)
where the generalized response function Gjk is defined as
Gjk (X1 (t1 ), t1 ; X(t), t) =
δXk (t)
,
0
βδuj (X1 , t1 )dt1
(4.140)
and its governing equation can be obtained from the Lagrangian equations (2.6) and (4.97),
written as
d2
d
∂hUbk i
= δjk δ(t − t1 ) + Ajk δ(t).
G
+
β
G
−
βG
jk
jk
ji
dt2
dt
∂Xi
(4.141)
This equation suggests that Gjk is statistically sharp function. Here Ajk accounts for the
correlation between particle velocity and fluid velocity at initial time t = 0. Substituting
from (4.138)-(4.139) in (4.137) and then using (4.137) in (4.136), result in
"
Z
hu0i W i
= −β
dx1 dt1 hu0i (x, t)u0j (x1 , t1 )i
∂
hGjk (x1 , t1 ; X(t), t)δ(X(t1 ) − x1 )W i
∂xk
#
∂
hĠjk (x1 , t1 ; X(t), t)δ(X(t1 ) − x1 )W i ,
+
∂vk
(4.142)
where
Ġjk (x1 , t1 ; X(t), t) =
d
Gjk (x1 , t1 ; X(t), t).
dt
(4.143)
The terms in the angular brackets can be further simplified219 and written in terms of the
conditional probabilities as
hGjk (x1 , t1 ; X(t), t)δ(X(t1 ) − x1 )W i = Gjk (x1 , t1 ; x, t)hδ(X(t1 ) − x1 )ic hW i,
(4.144)
hĠjk (x1 , t1 ; X(t), t)δ(X(t1 ) − x1 )W i = Ġjk (x1 , t1 ; x, t)hδ(X(t1 ) − x1 )ic hW i.
(4.145)
and
Here the second term
hδ(X(t1 ) − x1 )ic = hδ(X(t1 ) − x1 )|X(t) = x; V(t) = vi,
79
(4.146)
on the right-hand side of equations (4.144)-(4.145) is conditioned on the particle position
and velocity being equal to x and v respectively, at time t. Substituting (4.144)-(4.145) in
equation (4.142) and performing the integration over x1 , we have
"
βhu0i W i
#
∂
∂
=−
λki +
µki + γi hW i,
∂xk
∂vk
(4.147)
where λki , µki and γi are given by Hyland et al.219
λki = β 2
µki = β
2
γi = −β
Z t
0
Z t
0
2
dt1 hu0i (x, t)u0j (x, v, t|t1 )iGjk (x1 , t1 ; x, t),
d
Gjk (x1 , t1 ; x, t),
dt
(4.149)
∂u0i (x, t) 0
uj (x, v, t|t1 )iGjk (x1 , t1 ; x, t).
∂xk
(4.150)
dt1 hu0i (x, t)u0j (x, v, t|t1 )i
Z t
0
dt1 h
(4.148)
Equation (4.147) is identical to equation (4.124) and thus the functional method yields results
identical to the LHDIA method.
Recently, Pandya and Mashayek247 have extended Hyland et al.219 formulation to the
case of non-isothermal flow in which particles exchange heat with the surrounding fluid and
the temperature of the particle is governed by the Lagrangian equation
dTp
Nu
= −2
πaκ[Tp − T (Xi (t), t)],
dt
mp Cp
(4.151)
where N u is given by equation (2.17). The ensemble average equation governing the pdf
W (x, v, θ, t) is written as
"
#
∂
∂
∂
∂
+
vi +
{β(hUbi i − vi )} + {βθ (hT i − θ)} hW i =
∂t ∂xi
∂vi
∂θ
−
∂
∂
{βhu0i W i} − {βθ ht0 W i},
∂vi
∂θ
(4.152)
where βθ = 2hN uiπaκ/mp Cp , t0 is the temperature fluctuation of the fluid in the vicinity of
80
the particle and θ is the phase space variable corresponding to particle temperature Tp . The
expressions derived for phase space diffusion current βhu0i W i and phase space heat current
βθ ht0 W i are written in the form247
"
βhu0i W i
#
∂
∂
∂
=−
λki +
µki + ωi + γi hW i,
∂xk
∂vk
∂θ
"
(4.153)
#
∂
∂
∂
Λk +
Πk + Ω − Γ hW i,
βθ ht0 W i = −
∂xk
∂vk
∂θ
(4.154)
where the remaining tensors ωi , Λk , Πk , Ω, and Γ are given by
ωi = β
Z t
2
0
+ βv βθ
Πk = ββθ
Z t
0
Z t
Λk = ββθ
0
Z t
0
Ω = ββθ
+
Γ = ββθ
0
dt1 ht0 (x, t)u0j (x, v, t|t1 )iGjk (x1 , t1 ; x, t),
Z t
0
t
0
Z t
dt1 hu0i (x, t)t0 (x, v, t|t1 )iGθ (x1 , t1 ; θ, t),
dt1 ht0 (x, t)u0j (x, v, t|t1 )i
Z
βθ2
dt1 hu0i (x, t)u0j (x, v, t|t1 )iGj (x1 , t1 ; θ, t)
d
Gjk (x1 , t1 ; x, t),
dt
(4.155)
(4.156)
(4.157)
dt1 ht0 (x, t)u0j (x, v, t|t1 )iGj (x1 , t1 ; θ, t)
dt1 ht0 (x, t)t0 (x, v, t|t1 )iGθ (x1 , t1 ; θ, t),
dt1 h
∂t0 (x, t) 0
uj (x, v, t|t1 )iGjk (x1 , t1 ; x, t).
∂xk
(4.158)
(4.159)
Here the generalized response functions Gj and Gθ are governed by the following equations:
∂hT i
d
Gj − βθ Gjk
+ βθ Gj = 0,
dt
∂Xk
(4.160)
d θ
G + βθ Gθ = δ(t − t1 ) + Cθ δ(t),
dt
(4.161)
where Cθ accounts for the correlation between particle temperature and fluid temperature at
initial time. To complete the description, a prior knowledge is needed of various correlations
81
of fluid properties along the particle path appearing in the expressions for ωi , Λk , Πk and Ω.
Following equation (4.134), we assumed
hu0i (x, t)t0 (x, v, t|s)i = hu0i (x, t)t0 (x, t)iΨui t (t − s),
(4.162)
ht0 (x, t)t0 (x, v, t|s)i = ht0 (x, t)t0 (x, t)iΨtt (t − s),
(4.163)
ht0 (x, t)u0i (x, v, t|s)i = hu0i (x, t)t0 (x, t)iΨtui (t − s),
(4.164)
where Ψui t , Ψtt and Ψtui were approximated by exponential function with Lagrangian integral
time scale T̃ui t , T̃tt and T̃tui , respectively, in case of homogeneous shear flow with constant
temperature gradient.247
Zaichik’s151 method is identical up to equation (4.139) and then incorporates certain approximations to obtain analytical expressions for functional derivatives. Although Zaichik151
considered the case of non-isothermal flow, here we present his results for the case of isothermal flow. The final approximate expression for the functional derivative of Xk is
δXk (t)
=
0
βδuj (x1 , t1 )dx1 dt1
with
Ã
Ã
δXk (t)
0
δuj (x1 , t1 )dx1 dt1
δXk (t)
0
δuj (x1 , t1 )dx1 dt1
Ã
δXk (t)
0
βδuj (x1 , t1 )dx1 dt1
!
!
Ã
δXk (t)
+
0
βδuj (x1 , t1 )dx1 dt1
1
h
!
(4.165)
2
i
= δkj 1 − e−β(t−t1 ) δ(X(t1 ) − x1 )H(t − t1 ),
(4.166)
1
!
Z th
= δ(X(t1 ) − x1 )
t1
2
ih
1 − e−β(t−t2 ) 1 − e−β(t2 −t1 )
i ∂hU (X(t ), t )i
k
2
2
∂xj
dt2 .
(4.167)
The functional derivative of Vbk is given by
δ Vbk (t)
= βδkj e−β(t−t1 ) δ(X(t1 ) − x1 )H(t − t1 )
δu0j (x1 , t1 )dx1 dt1
+ β
+
Z t
t1
b
−β(t−t2 ) ∂hUk (X(t2 ), t2 )i
dt2 e
∂ Vbk (X(t), t)
∂xn
Ã
∂xn
δXn (t)
0
δuj (x1 , t1 )dx1 dt1
82
Ã
!
δXn (t2 )
0
δuj (x1 , t1 )dx1 dt1
.
2
!
1
(4.168)
Using these functional derivatives and the relation
hu0i (x, t)u0j (X1 (t1 ), t1 )i = Ψ(t − t1 )hu0i (x, t)u0j (x, t)i,
(4.169)
in equations (4.136)-(4.137), Zaichik obtained the closed expression for hu0i W i as
"
hu0i W i
=
−hu0i u0k i
#
lu ∂hUbn i ∂
hu ∂hUbn i ∂hVbj i ∂
gu ∂
hu ∂hUbn i ∂
∂
+
+ 2
+
+ 2
hW i.
fu
∂vk β ∂xk ∂vn β ∂xk ∂xn ∂vj β ∂xk β ∂xk ∂xn
(4.170)
Here fu , gu , lu and hu are given by
fu = β
Z ∞
0
Ψ(s)e−βs ds,
gu = β
and
hu = β 2
Z ∞
0
Z ∞
0
l u = gu − β 2
Ψ(s)ds − fu ,
Ψ(s)se−βs ds + β 2
Z ∞
0
Z ∞
0
Ψ(s)se−βs ds, (4.171)
Ψ(s)sds − 2gu .
(4.172)
Derevich244 preferred phase space variable for fluctuating part of the particle velocity
instead of the total particle velocity and wrote the pdf equation in the form
i ∂hW i
∂
∂hW i h ∂hVbi i
∂hVbi i
hW (x, v0 , t)i + (hVbk i + vk0 )
−
+ (hVbk i + vk0 )
− β(hUbi i − hVbi i)
=
∂t
∂xk
∂t
∂xk
∂vi0
∂
−β 0 (hu0i W i − vi0 hW i), (4.173)
∂vi
where v0 is phase space variable for fluctuating particle velocity. Hereafter we use v to
represent v0 while discussing Derevich’s work. Derevich derived the expression for hu0i W i in
the functional approach and the resulting expression is given by
hW i
hW i
1
− Qp hu0i u0j i
,
hu0i W i = − Gp hu0i u0j i
β
∂xj
∂vj
(4.174)
where
Gp hu0i u0j i = gp hu0i u0j i +
∂hVbi i i
1 h 0 0 ∂hVbj i
hp hui uk i
+ hu0j u0k i
,
2β
∂xk
∂xk
83
(4.175)
Qp hu0i u0j i = qp hu0i u0j i−
∂hu0i u0j i ∂hu0i u0j u0k i
1 h ∂hu0i u0j i
∂hVbj i
∂hVbi i i
pp
+hVbk i
+
+hu0i u0k i
+hu0j u0k i
.
2β
∂t
∂xk
∂xk
∂xk
∂xk
(4.176)
The functions gp , hp , pp , and qp are written as
gp hu0i u0j i =
Z t
0
(1 − A)hEij iβds,
pp hu0i u0j i =
Z t
0
hp hu0i u0j i =
Z th
Aβ 2 shEij ids,
0
i
βs(1 + A) − 2(1 − A) hEij iβds, (4.177)
qp hu0i u0j i =
Z t
0
AβhEij ids,
(4.178)
where A = e−βs and hEij i represents the correlation along the particle path and is defined
as
hEij i ≡ hEij (x, t; X(s), s)i,
(4.179)
hEij (x − 0.5X(s), t − 0.5s; X(s), s)i = hu0i (x, t)u0j (x − X(s), t − s)i.
4.2.3
(4.180)
Application of Van Kampen’s method
Van Kampen251, 252 originally proposed a cumulant expansion method for the solution of
linear stochastic differential equations written for a vector process Z
dZ(t)
= [A0 + αA1 (t)]Z(t)
dt
(4.181)
where the linear operators A0 and A1 are deterministic and stochastic in nature, respectively,
and α is the level of fluctuations. A0 is constant in time, and when ensemble average of A1
is time dependent, the proposed solution is written as
·
¸
Z t
dhZ(t)i
2
= A0 + αhA1 (t)i + α
hA01 (t)esA0 A01 (t − s)ie−sA0 ds hZ(t)i,
dt
0
(4.182)
for ατc ¿ 1. Here A01 (t) = A1 − hA1 i and τc is the autocorrelation time for A01 . In case of
a single random oscillator problem where hA1 (t)i = 0 and A01 (t) has Gaussian distribution,
this approximate equation (4.182) yields the exact solution.251
84
The closure problem of nonlinear stochastic equation (4.93) can be linearized to the
closure problem of linear stochastic equation (4.95) for W through the use of Liouville
theorem and the result given by equation (4.182) remains applicable.220 Comparison of
equations (4.95) and (4.181) with Ki replaced by Ki0 + Ki1 suggests that
A0 = −
∂
(K 0 ),
∂θi i
αA1 = −
∂
(K 1 ),
∂θi i
(4.183)
where Ki0 is stationary and not stochastic and Ki1 is the stochastic part of Ki . Substituting
A0 and A1 from equation (4.183) into equation (4.182) and replacing Z by W , yields
*
+
∂hW i
∂
∂ Zt
sA0 ∂
0
1
1
ds Ki (θ, t)e
=−
(K hW i) +
K (θ, t − s) e−sA0 hW i,
∂t
∂θi i
∂θi 0
∂θj j
(4.184)
when hKi1 i = 0. The action of esA0 on any function f (θ) is given by the following:216, 220
esA0 f (θ) = f (θ −s )
d(θ −s )
,
d(θ)
(4.185)
where θ represents the value at time t and θ s represents the value at time t + s along the
trajectory which is determined from
dθi
= Ki0 (θ1 , θ2 , ..., θN ),
dt
(4.186)
and d(θ −s )/d(θ) stands for the Jacobian. Repeated application of equation (4.185) allows
us to simplify equation (4.184), and the final general form of the transport equation for the
probability density function can be written as216, 220
+
*
∂ Zt
d(θ −s ) ∂
∂hW i
∂
d(θ)
−s
0
1
1
+
(Ki hW i) =
ds Ki (θ, t)
hW i.
−s Kj (θ , t − s)
∂t
∂θi
∂θi 0
d(θ) ∂θj
d(θ −s )
(4.187)
85
In the case when particle trajectory is governed by equations (2.6) and (4.97),



 x 
θ=
,

K0 = 
v

v
c − v)
β(hUi

,


 0 
K1 = 
βu0
,
(4.188)
and hW i ≡ hW (x, v, t)i. And for any function h = h[x(x−s , v−s ), v(x−s , v−s )],
∂h
∂h ∂xj
∂h ∂vj
.
−s =
−s +
∂vi
∂xj ∂vi
∂vj ∂vi−s
(4.189)
Using equations (4.188)- (4.189), equation (4.187) can be simplified and written as
#
"
∂
∂hW (x, v, t)i
∂
+
vi +
β(hUbi i − vi ) hW i =
∂t
∂xi
∂vi
*
+
∂ Zt
∂xk 2 0
∂ 0 −s −s
ds
β ui (x, t)
u (x , v , t − s) hW i
∂vi 0
∂vj−s
∂xk j
*
+
∂ Zt
∂vk 2 0
∂ 0 −s −s
+
ds
β ui (x, t)
u (x , v , t − s) hW i.
∂vi 0
∂vj−s
∂vk j
(4.190)
For t − s = t1 , u0j (x−s , v−s , t − s) reduces to u0j (xt1 −t , vt1 −t , t1 ) representing the value of
u0j at time t1 along the particle trajectory which passes through x, v at time t – in LHDIA
notation this is written as uj (x, v, t|t1 ). Introducing the notation gjk (t1 |t) and dgjk (t1 |t)/dt
for
∂xk
t −t
∂vj1
and
∂vk
t −t ,
∂vj1
respectively, and using LHDIA notation, equation (4.190) is written as
"
#
∂hW (x, v, t)i
∂
∂
+
vi +
β(hUbi i − vi ) hW i =
∂t
∂xi
∂vi
*
+
∂ Zt
∂
dt1 β 2 u0i (x, t)gjk (t1 |t)
u0 (x, v, t|t1 ) hW i
∂vi 0
∂xk j
*
+
dgjk (t1 |t) ∂ 0
∂βhu0i W i
∂ Zt
2 0
dt1 β ui (x, t)
uj (x, v, t|t1 ) hW i ≡ −
. (4.191)
+
∂vi 0
dt
∂vk
∂vi
Equation (4.191) suggests closed expression for βhu0i W i which is identical to equation (88)
in Reeks.235
Recently, Van Kampen’s method was used by Pandya and Mashayek246 to obtain an
86
approximate equation for pdf to predict the statistical properties of interest of collisionless
evaporating droplets suspended in isothermal, isotropic turbulent flows. It was assumed that
the time constant τd = 1/β = (2ρd a2 /9µ) of the droplet changes along its trajectory due to
the evaporation and is governed by the Lagrangian equation
dτd
ρd
=−
κ = −A − Bτd0.25 S 0.5 ,
dt
18µ
c − V|,
c
S = |U
where
4ρd
A=
Γ1 ln(1 + BM );
9µ
Ã
B=
0.3ASc.333
d
18ρ
νρd
(4.192)
!0.25
,
(4.193)
are obtained from Ranz and Marshall formula for evaporating droplets.25 Here ρd is droplet
density, Γ1 is mass diffusivity coefficient, BM is the transfer number, Scd is the droplet
Schmidt number. Also S represent the magnitude of the droplet slip velocity and is stochastic
in nature. The ensemble average of Liouville equation for phase space density W (τd , t) can
be written as
i
i
∂
∂
∂ h 1/4
∂ h 1/4 0
hW (τd , t)i −
[AhW (τd , t)i] =
Bτd RhW (τd , t)i +
Bτd hR W (τd , t)i ,
∂t
∂τd
∂τd
∂τd
(4.194)
with the terms, present on the right-hand side of (4.194), that pose the closure problem.
These terms are 1) hR0 W (τd , t)i and 2) R, where R is the ensemble average value of S 0.5
and R0 is the fluctuating part of S 0.5 . Application of Van Kampen’s method gives the
following closed equation for hW (τd , t)i:246
∂
1 ∂2
∂
hW (τd , t)i = −
[K(τd , t)hW (τd , t)i] +
[Q(τd , t)hW (τd , t)i] ,
∂t
∂τd
2 ∂τd2
(4.195)
where
1/4
K(τd , t) = −A − Bτd R +
+
Z t
0
Z t
0
dτ 0.25B 2 τd−0.75 [(τd−τ )0.25 hR0 (τd , t)R0 (τd , t)ie
dτ B 2 τd0.25 (τd−τ )0.25 h
∂R0 (τd , t) 0 −τ
R (τd , t − τ )i,
∂τd
87
− Tτ
R
]
(4.196)
Z t
1
(− τ )
Q(τd , t) =
dτ B 2 τd0.25 (τd−τ )0.25 [hR0 (τd , t)R0 (τd , t)ie TR ],
2
0
τd−τ = τd + Aτ.
(4.197)
In equation (4.196)-(4.197), the approximation
hR0 (τd , t)R0 (τd−τ , t − τ )i ∼ (− Tτ )
= e R
hR0 (τd , t)R0 (τd , t)i
(4.198)
is used, where TR represents the integral time scale for the correlation of R0 . Approximate
expressions for hR0 (τd , t)R0 (τd , t)i and R are given in terms of hu0i u0i i, hu0i vi0 i and hvi0 vi0 i where
vi0 is particle velocity fluctuation.246
4.2.4
Macroscopic equations
The closed equation for probability density function hW i along with the proper boundary and
initial conditions contains all the necessary information for the dispersed phase and could be
solved by using existing finite difference methods or the path-integral method.217, 246 In many
engineering applications, the detailed information as provided by the distribution function
is more than is needed, and the statistical properties obtained by taking various moments of
the pdf sufficiently serve the purpose. In view of this, instead of first solving for the pdf and
then obtaining various moments, equations for statistical properties can be obtained first
by taking moments of the closed pdf equation. These macroscopic equations have Eulerian
forms and are easier to solve numerically as compared to the pdf equation. The macroscopic
or statistical quantities of interest are the mean concentration, mean velocity and velocity
fluctuation correlation of the particles. These properties are given by
Z
N =
hW (x, v, t)idv,
(4.199)
Z
1
vj hW idv,
NZ
1
=
(vi − V i )(vj − V j )hW idv,
N
Vj =
vi0 vj0
88
(4.200)
(4.201)
and their governing equations as obtained from Reeks’ pdf equation are235, 236
∂N
∂
+
[V i N ] = 0,
∂t
∂xi
∂V j
∂ 0 0 vi0 vj0 ∂N
λkj ∂N
∂λkj
∂V j
+Vi
+
vi vj +
= β(hUbj i − V j ) −
−
− γj ,
∂t
∂xi
∂xi
N ∂xi
N ∂xk
∂xk
(4.202)
(4.203)
∂ 0 0
1 ∂
∂V n
∂ 0 0
∂V j
vj vn + V i
vj vn +
[N vi0 vj0 vn0 ] = −vi0 vj0
− vi0 vn0
− 2βvj0 vn0
∂t
∂xi
N ∂xi
∂xi
∂xi
∂V j
∂V n
− λkn
+ µjn + µnj .(4.204)
−λkj
∂xk
∂xk
While writing these equations here, λij , µij , and γi are considered as independent of v.
The term
∂ 0 0
vv
∂xi i j
in equation (4.203) accounts for the ‘turbophoresis’ phenomenon which
was first recognized by Caporaloni et al.253 and formally derived by Reeks.233 Recently,
Pandya and Mashayek193 have shown that the last two terms on the right-hand side of
equation (4.203) account for the turbulent thermal diffusion and barodiffusion phenomena
in compressible gas flows. The process of taking various moments of pdf equation poses
another closure problem due to the unknown correlation vi0 vj0 vn0 in equation (4.204). One
method to obtain a closed expression for this third-order correlation is to write a transport
equation for it, express the fourth-order correlations, appearing in the equation, in terms of
the second-order correlation by invoking the quasi-normality condition and then obtaining
the algebraic relation by neglecting the time evolution and the convection generation due
to the mean velocity terms.151 Another method is to solve analytically the pdf equation by
using Chapman-Enskog method and then obtain an expression for vi0 vj0 vn0 from the known
approximate expression for pdf.244, 254
Zaichik151 and Derevich244 obtained an equation for N identical to equation (4.202).
Zaichik’s equations for V j and vj0 vn0 are
∂V j
∂ 0 0
∂V j
Djk ∂N
+Vi
+
,
vi vj = β(hUbj i − V j ) −
∂t
∂xi
∂xi
N ∂xk
89
(4.205)
∂ 0 0
∂ 0 0
1 ∂
∂V n
∂V j
[N vi0 vj0 vn0 ] = −vi0 vj0
− vi0 vn0
− 2βvj0 vn0 + 2βfu hu0j u0n i
vj vn + V i
vj vn +
∂t
∂xi
N ∂xi
∂xi
∂xi
"
#
"
#
∂hUbn i
∂V n
∂V j
∂hUbj i
0 0
0 0
+huj uk i lu
− gu
+ hun uk i lu
− gu
, (4.206)
∂xk
∂xk
∂xk
∂xk
where
"
#
hu
1 0 0
∂hUbj i
Dij =
vi vj + gu hu0i u0j i + hu0i u0k i
,
β
β
∂xk
and
(4.207)
#
"
1
∂ 0 0
∂ 0 0
∂ 0 0
vi0 vj0 vk0 = − Din
vj vk + Djn
vi vk + Dkn
vv .
3
∂xn
∂xn
∂xn i j
(4.208)
Derevich’s equations244 for V j and vj0 vn0 are
Dp ∂N
∂ 0 0
∂V j
∂V j
+Vi
+
vi vj = β(hUbj i − V j ) − jk
,
∂t
∂xi
∂xi
N ∂xk
(4.209)
∂ 0 0
∂V j
∂ 0 0
1 ∂
∂V n
vj vn + V i
vj vn +
[N vi0 vj0 vn0 ] = −vi0 vj0
− vi0 vn0
− 2βvj0 vn0 + 2βQp hu0j u0n i,
∂t
∂xi
N ∂xi
∂xi
∂xi
(4.210)
where
p
Dij
=
i
1h 0 0
vi vj + Gp hu0i u0j i .
β
(4.211)
The expression for mixed correlation moment between velocity fluctuations of the fluid
and dispersed phase which are required to simulate the ‘back-effect’ of particles on the fluid
turbulence can be obtained by using
ui0 vj0
1 Z 0
=
hui (vj − V j )W idv.
N
(4.212)
This equation along with the Reeks235 and Hyland et al.219 expression for hui W i give an
algebraic relation255
βu0i vj0 = −λki
90
∂V j
+ µji .
∂xk
(4.213)
Zaichik’s expression for u0i vj0 is written as151
"
u0i vj0
5
=
fu hu0i u0j i
#
1
∂hUbj i
∂V j
+ hu0i u0k i lu
− gu
.
β
∂xk
∂xk
(4.214)
Model Assessment via DNS
In the previous sections, we discussed various approaches, namely, DNS, LES, stochastic
modeling, RANS modeling, and pdf modeling, to extract useful information from the firstprinciple equations. The approach of DNS allows us to obtain details of turbulence fluctuations of carrier and dispersed phases from the first principles and the error involved in the
obtained data occurs only due to the numerical schemes, and some of the physical models
utilized for description of the Lagrangian equations for the dispersed phase. The error can
be controlled by using the well tested schemes and physical models. In the other remaining
approaches, modeling has been involved at different levels starting from the first principles, for example, modeling of subgrid scale contribution to obtain governing equations for
large scales in LES, generation of synthetic turbulence fluctuations along the particle path
in stochastic approach, closure approximations for unknown correlations in RANS and pdf
approaches. The modeling induces certain assumptions and approximations in the first principles, which can be validated and assessed by using the wealth of data generated by DNS.
This is despite the fact that currently DNS is limited to relatively low Reynolds number
flows with simple configurations, and the final validations and adjustments must be made
via comparisons with experimental data. Recent works related to development of LES for
two-phase flows88, 98, 99 are good examples of the importance of DNS data in developing the
necessary LES models by performing a priori analysis of subgrid scale models. From these
examples it is clear that the required data on details of fluctuations of fluid velocity in the
flow domain can be generated by DNS and their generation through experiments, if possible,
is certainly an enormous and extremely difficult task. Here, instead of giving the details of
various cases where the DNS is used to develop and assess the models, we present a few cases
91
representative of different approaches considered in this review. These cases are taken from
our own work on DNS and modeling.
First, we consider the case related to RANS modeling where the DNS data is first used to
obtain values for different constants appearing in the model and then the model predictions
are compared with the data for the purpose of assessment. In the RANS modeling, DNS
generated data are particularly important in the modeling of the pressure terms as laboratory
experiments are unable of providing any direct information for such terms. Mashayek and
Taulbee186 used the DNS data, generated for the homogeneous shear flow, to evaluate the
newly introduced constants in their models for pressure terms (equations (4.72) and (4.73)).
The magnitudes of the newly-introduced constants Cf 3 , Cf p1 , Cf p2 , and Cf p3 in these models
were determined by balancing the transport equations for all of the components of the fluid
Reynolds stress tensor (equation (4.69)) and the fluid-particle covariance tensor (equation
(4.71)). By considering all of the components of the stresses in all of the cases a large
number of data points was provided to determine the optimized values of these four model
constants. Two sample cases are shown in Fig. 3 for the energy budgets of hhu21 ii and hhu1 v1 ii
for a mass loading of Φm = 0.25 and a particle time constant of τp = 0.032s, where hh ii
indicates ensemble averaging over the number of particles. In this figure, the production,
the dissipation rate, and the contribution from drag are calculated from DNS. The values
of the pressure-strain and the pressure-dispersed phase velocity gradient are based on the
models as proposed. The term LHS indicates the derivative of the energy component (hhu21 ii
or hhu1 v1 ii) and is obtained from DNS. This value is compared with
P
RHS which is the
sum of the production, dissipation rate, and drag contribution (from DNS), and the pressure
term (from models). It is observed that the general agreement between the LHS and the
RHS is good, especially for St > 2 when the DNS results are considered for model validation.
Similar comparisons were performed for all the other components in all DNS cases. The final
optimized values of the empirical constants are given in table 1.
Mashayek and Taulbee207 used the DNS results of Ref. 186 to appraise their four-equation
92
model (see Section 4.1.2). A variety of cases were considered for model validation. The
Reynolds stress tensors for both the carrier and the dispersed phases, as well as the fluidparticle velocity covariance tensor were generated from the DNS results and were used to
appraise the model’s performance. In the assessment of the model, the values of k, ², kp ,
and kf p at St = 3 were taken from DNS as initial values. This time was chosen as the
initial time for model assessment in order to allow the flow and the particles to reach a
dynamic equilibrium, while the turbulence became well-developed and fell in the growth
region. Here, we present comparisons between the model predictions and DNS results for
one (two-way coupled) case only, with Φm = 0.25 and τp = 0.016s. The model predictions
(lines) are compared with DNS results (symbols) in Fig. 4 for all of the Reynolds stress
components. The overall agreement between the model predictions and DNS results is
promising. In particular, it is important to note that the shear components which are of
primary importance (in calculating the mean variables) are predicted very closely to DNS
data. It is also important to note that the model is capable of predicting the antisymmetry of
the shear components of the fluid-particle velocity covariance tensor. The model predictions
for the dissipation rate of the fluid turbulence kinetic energy were also compared with DNS
results, and good agreements were observed. The comparison showed that the model is
capable of capturing the trends of variations of the dissipation rate with the particle time
constant and the mass loading ratio as predicted by DNS.
Next, we consider the case of evaporating droplets in isotropic turbulent flow which has
been studied using DNS,53 stochastic,146 and pdf 246 approaches. The temporal variations of
1/2
1/2
hτd i, which is proportional to mean particle diameter, and standard deviation σ of τd
as obtained by these approaches are shown in Figs. 5 and 6, respectively. The droplet time
constant τd changes due to the evaporation and is governed by the Lagrangian equation
(4.192). At time t = 0 all the droplets have a time constant equal to τd0 . Figure 5 shows
temporal variation of h(τd /τd0 )1/2 i for one case with τd0 = 5τk , τec ≡ AτE /0.29 = 5τk , and
Scd = 5 (see Section 4.2.3 for definitions of these variables) as obtained by DNS, stochastic
(STH), and pdf (ANL1 and ANL2) solutions. Here τk and τE are Kolmogorov and Eulerian
93
time scales, respectively, for fluid turbulence. Also, ANL1 and ANL2 refer to the cases
in which the Lagrangian fluid integral time scale T̃L is given by 0.482k/² and 0.2826k/²,
respectively. In stochastic calculations, T̃L = 0.2826k/² was used. Figure 5 indicates that
STH curve is closer to the DNS as compared to ANL1 and ANL2. In Fig. 6, the temporal
1/2
1/2
variations of the normalized standard deviation, σ/τd0 , of τd
are shown for τd0 = 5τk ,
τec = 5τk for Scd = 1 and 5. The pdf (ANL1 and ANL2) predictions are better as compared
to the STH which predicts much narrower (thinner) droplet size distribution, for the reasons
discussed in Ref. 146, and gives smaller values for σ as compared to DNS results. Although
the pdf model predictions for σ are in good agreement with DNS, the pdf model is not
capable of fully capturing the minute details of temporal behavior of dσ/dt.
Lastly, we consider the case of homogeneous turbulent shear flow in which the particles
exchange heat with the surrounding fluid and their temperature is governed by the Lagrangian equation (4.151).247 DNS data were used to compute various Lagrangian integral
time scales T̃L , T̃ui t , T̃tt and T̃tui appearing in equations (4.162)-(4.164). These time scales
along with DNS data for other required correlations for fluid properties hu0i (x, t)u0j (x, t)i,
hu0i (x, t)t0 (x, t)i and ht0 (x, t)t0 (x, t)i were used to predict various statistics for particle phase
by solving the macroscopic equations. The predicted temporal evolutions of vi0 vj0 , vi0 θ0 , and
θ0 θ0 are shown in Figs. 7 and 8 and are compared with the DNS data. The temporal evolution of particle Reynolds stresses v10 v10 , v20 v20 , and v30 v30 are shown in Fig. 7(a) and of v10 v20
is shown in Fig. 7(b). The pdf model predictions are shown by solid line and referred to
as PDF I. This figure exhibits a good agreement between pdf model predictions and DNS
data for 0 ≤ αt ≤ 10 where α is the mean shear rate dhUb1 i/dx2 . The temporal evolution
of particle Reynolds stresses in homogeneous shear flow as predicted by pdf modeling were
also assessed by Hyland et al.219, 250 and Zaichik151 using LES data. The temporal evolution
of the temperature related statistical properties of the particle phase are shown in Fig. 8.
In this figure, PDF I refers to numerical solutions of macroscopic equations when the initial
condition effects of particle temperature and velocity fluctuations are taken into account.
PDF II refers to solutions without the initial conditions. Figure 8(a) shows the comparison
94
of PDF I and PDF II solutions for variance θ0 θ0 with DNS data. The inclusion of initial
conditions results in better agreement with DNS data and without the initial conditions, θ0 θ0
is under-predicted initially. Figure 8(b) shows the comparison of pdf model predictions, for
the flux vi0 θ0 with DNS data and suggests a good agreement.
The above mentioned cases show how the DNS data can be efficiently used to obtain
values for model constants and further used to assess the model predictions before the models
are ready to be implemented for more general situation of inhomogeneous flows.
6
Concluding Remarks
The accurate description of turbulent dispersed phase still remains an unfinished task. This
is, firstly, due to unavailability of accurate first-principle Lagrangian equations governing
particle’s motion, temperature, size variation due to evaporation, and its other properties
of interest in the presence of other particles in turbulent flows. At present, we rely on the
approximate Lagrangian equations which are strictly valid for single or lonely particle in
certain range of flow parameters. Even if these approximate equations are considered as a
set of first-principle equations, the task remains unfinished, secondly, due to our limitations
in extracting accurate and useful information from them.
Various approaches (DNS, LES, SM, RANS, pdf) developed in the past few decades for
extracting the information have certain merits and limitations at present. DNS being the
most accurate approach allows us to obtain details of fluctuations present in turbulent fluid
and particle flow properties but at present is restricted to flows in simple geometry and at
low velocity. The restriction is mainly imposed by the limitations of present day computing
resources and most of the DNS studies are designed accordingly. For example, we have
seen that the feasibility of DNS to obtain statistical properties of particle phase in simple
anisotropic homogeneous flows is achieved under the condition when only the Stokes drag
term is considered for the particle motion. Under this condition, the homogeneity of particle
95
phase has been achieved and utilized for accurate calculation of its statistical properties.
Also, with the advancement in the numerical schemes for DNS of turbulence, much suited
for parallel computing, new particle tracking algorithms in the framework of these schemes
need to be developed.
In contrast to DNS, the merit of LES allows the simulation of turbulent flows with higher
Reynolds numbers but at the cost of sacrificing fluctuations detail of small scales. The effects
of small or subgrid scales are present in the form of statistical average. The accurate modeling
of the subgrid scales effect in the presence of particles and the interactions between these
scales and particles remains as a main issue of LES research on two-phase flow. We have
seen that a limited amount of work has been done in this direction and much has yet to be
accomplished to make the LES as a reliable approach for inhomogeneous two-phase flows.
In the stochastic modeling we loose all the details of ‘real’ fluid turbulence fluctuations
in the sense that these were never calculated in SM. Instead, a synthetic turbulence has
been created from certain known statistical properties, mainly obtained from single point
turbulence closure schemes, of real turbulence. This reduces the computational efforts significantly. The accuracy of predictions is governed by the accuracy of predicted single point
statistical properties and then how well the generated synthetic turbulence represents the
real turbulence. Even when the accurate statistical properties are known, the difficulty arises
due to the fact that more than one synthetic turbulence can have identical values for certain
statistical properties. We have also seen that the stochastic scheme having better prediction
for non-evaporating particles has failed to predict correctly the size distribution of evaporating droplets in isotropic case. Much efforts have been devoted in devising various stochastic
schemes and their assessment in a variety of flow cases. Further improvement would continue
to remain as an active area of research because of the reduction in computational efforts and
advantage of Lagrangian framework for particle in stochastic modeling approach.
Despite the suitability of Lagrangian framework for particle phase in DNS, LES and
SM, one has to abandon these approaches in many cases of practical applications where the
96
number of particles are substantially large. Then other alternate techniques of RANS and
pdf modeling become useful in which the resulting equations for particle phase statistical
properties are in the form of ‘fluid’ equations in Eulerian framework. In the RANS, instantaneous fluid equations for the dispersed phase have been the starting set of equations. The
ensemble average of these equations have resulted in closure problems due to the appearance
of the various unknown correlations. These problems have been tackled using the singlepoint, second-order closure schemes available for fluid turbulence closure. We have seen that
a detail analysis by taking the proper account of two-way coupling and the effect of particles
on the pressure-strain correlations has been done recently. The number of these averaged
partial differential equations are large and which have been reduced by incorporating certain
approximations leading to algebraic models for second order correlations. As we are aware
that there is no universal single-point model available for fluid turbulence, we expect the
same in case of dispersed phase. And in view of this, RANS type models which involve various non-universal constants would require improvement and adjustment in different cases of
flows which are not similar in nature.
The most recent method is the pdf modeling, in which the transition from Lagrangian to
Eulerian frame is achieved by first using the Liouville theorem, solving the closure problems
to obtain closed pdf equation and then taking various moments of the pdf equation. The
various pdf models can differ in their selection for phase space variables. The single-point
pdf modeling, having variables describing the particle properties, has been discussed in this
review. The pdf transport equation, if solved numerically, would contain more information
as compared to RANS models. The accuracy of the closed pdf equation depends on the
employed closure scheme and our ability in predicting or ‘seeing’ the statistical properties
of the fluid as ‘seen’ by the particle. We have seen that all the three closure methods,
namely, LHDIA, functional and Van Kampen’s method, have resulted in an identical pdf or
kinetic equation. The statistical properties of the fluid (correlations) along the particle path
have been approximated using the exponential form with fluid integral time scales. Predictions of these time scales and assessment of ‘fluid’ equations obtained from pdf equation in
97
nonhomogeneous flows remain as one of the important future tasks.
Acknowledgment
The support for this work was provided by the U.S. Office of Naval Research under Grant
N00014-01-1-0122 with Dr. G.D. Roy as Program Officer, and by the National Science Foundation under Grant CTS-0096349 with Dr. C.K. Aidun as Program Director. FM would
like to thank Professors Peyman Givi and Dale B. Taulbee of State University of New York
at Buffalo for introducing him to the subject of two-phase turbulent flow. RVRP dedicates his contribution to his guru (late) Professor Emeritus Subir Kar of Indian Institute of
Technology, Bombay, India.
References
[1] Friedlander, S. K., Smoke, Dust and Haze. Fundamental of Aerosol Dynamics, Oxford
University Press, New York, NY, 2000.
[2] Williams, F. A., Combustion Theory, The Benjamin/Cummings Publishing Company,
Menlo Park, CA, 2nd edition, 1985.
[3] Sirignano, W., Fluid Dynamics and Transport of Droplets and Sprays, Cambridge
University Press, Cambridge U.K., 1999.
[4] Gidaspow, D., Multiphase Flow and Fluidization: Continuum and Kinetic Theory
Descriptions, Academic Press, Boston, 1994.
[5] Crowe, C., Sommerfeld, M., and Tsuji, Y., Multiphase Flows with Droplets and Particles, CRC Press, Boston, 1998.
[6] Tsuji, Y., Activities in Discrete Particle Simulation in Japan, Powder Technology,
113:278–286 (2000).
[7] Wilcox, D. C., Turbulence Modeling for CFD, DCW Industries, Inc., La Cañada, CA,
1993.
[8] White, F. M., Viscous Fluid Flow, McGraw-Hill, Inc., New York, 1991.
[9] Schlichting, H., Boundary-Layer Theory, McGraw-Hill Book Co., 7th edition, 1979.
98
[10] Kim, I., Elghobashi, S., and Sirignano, W. A., On the Equation for Spherical-Particle
Motion: Effect of Reynolds and Acceleration Numbers, J. Fluid Mech., 367:221–253
(1998).
[11] Michaelides, E. E., Review–The Transient Equation of Motion for Particles, Bubbles,
and Droplets, Journal of Fluid Engineering, 119:233–247 (1997).
[12] Michaelides, E. E. and Feng, Z.-G., Analogies between the Transient Momentum and
Energy Equations of Particles, Prog. Energy Combust. Sci, 22:147–162 (1996).
[13] Gouesbet, G. and Berlemont, A., Eulerian and Lagrangian Approaches for Predicting
the Behaviour of Discrete Particles in Turbulent Flows, Prog. Energy Combust. Sci.,
25:133–159 (1999).
[14] Odar, F. and Hamilton, W. S., Forces on a Sphere Accelerating in a Viscous Fluid, J.
Fluid Mech., 18:302–314 (1964).
[15] Odar, F., Verification of the Proposed Equation for Calculation of the Forces on a
Sphere accelerating in a viscous fluid, J. Fluid Mech., 25:591–592 (1966).
[16] Odar, F., Unsteady Motion of a Sphere Along a Circular Path in a Viscous Fluid, J.
Appl. Mech., 86:652–654 (1968).
[17] Tchen, C. M., Mean Value and Correlation Problems Connected with the Motion of
Small Particles Suspended in a Turbulent Fluid, Ph.D. Thesis, Delft University. The
Hague, 1947.
[18] Corrsin, S. and Lumley, J., On the Equation of Motion for a Particle in Turbulent
Fluid, Appl. Sci. Res., 6:114–116 (1956).
[19] Buevich, Y. A., Motion Resistance of a Particle Suspended in a Turbulent Medium,
Fluid Dynamics, 1:119 (1966).
[20] 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).
[21] Clift, R., Grace, J. R., and Weber, M. E., Bubbles, Drops, and Particles, Academic
Press, New York, NY, 1978.
[22] Saffman, P. G., The Lift on a Small Sphere in a Slow Shear Flow, J. Fluid Mech.,
22:385 (1965).
[23] Saffman, P. G., Corrigendum to “The Lift on a Small Sphere in a Slow Shear Flow”,
J. Fluid Mech., 31:624 (1968).
[24] Michaelides, E. E. and Feng, Z.-G., Heat Transfer from a Rigid Sphere in a Nonuniform
Flow and Temperature Field, Int. J. Heat Mass Transfer, 37:2069–2076 (1994).
[25] Ranz, W. E. and Marshall, W. R., Evaporation from Drops, Chem. Engng Prog.,
48:141–173 (1952).
99
[26] Loth, E., Numerical Approaches for Motion of Dispersed Particles, Droplets and
Bubbles, Prog. Energy Combust. Sci., 26:161–223 (2000).
[27] Feng, Z.-G. and Michaelides, E. E., Heat and Mass Transfer Coefficients of Viscous
Spheres, Int. J. Heat Mass Transfer, 44:4445–4454 (2001).
[28] Magnaudet, J. and Eames, I., The Motion of High-Reynolds-Number Bubbles in
Inhomogeneous Flows, Annu. Rev. Fluid Mech., 32:659–708 (2000).
[29] Batchelor, G. K., The Theory of Homogeneous Turbulence, Cambridge University
Press, New York, 1982.
[30] Hinze, J. O., Turbulence, McGraw Hill Book Company, New York, NY, 1975.
[31] Tennekes, H. and Lumley, J. L., A First Course in Turbulence, MIT Press, Cambridge,
MA, 1972.
[32] Druzhinin, O. A. and Elghobashi, S., Direct Numerical Simulations of Bubble-Laden
Turbulent Flows Using the Two-Fluid Formulation, Phys. Fluids, 10(3):685–697
(1998).
[33] Yeung, P. K. and Pope, S. B., An Algorithm for Tracking Fluid Particles in Numerical
Simulations of Homogeneous Turbulence, J. Comp. Phys., 79(2):373–416 (1988).
[34] Elghobashi, S. E., On Predicting Particle-Laden Turbulent Flows, Appl. Sci. Res.,
52:309–329 (1994).
[35] Maxey, M. R., Patel, B. K., Chang, E. J., and Wang, L. P., Simulations of Dispersed
Turbulent Multiphase Flow, Fluid Dynamics Research, 20:143–156 (1997).
[36] Maxey, M. R. and Patel, B. K., Forced-Coupled Simulations of Particle Suspensions at
Zero and Finite Reynolds Numbers, Center for Fluid Mechanics Report #97-2, Brown
University, Providence, RI, 1997.
[37] Riley, J. J. and Patterson, G. S., Diffusion Experiments with Numerically Integrated
Isotropic Turbulence, Phys. Fluids, 17:292–297 (1974).
[38] Squires, K. D. and Eaton, J. K., Measurements of Particle Dispersion Obtained from
Direct Numerical Simulations of Isotropic Turbulence, J. Fluid Mech., 226:1–35 (1991).
[39] Yudine, M. I., Physical Considerations on Heavy-Particle Diffusion, Adv. Geophys.,
6:185–191 (1959).
[40] Csanady, G. T., Turbulent Diffusion of Heavy Particles in the Atmosphere, J. Atmos.
Sci., 20:201–208 (1963).
[41] Elghobashi, S. and Truesdell, G. C., Direct Simulation of Particle Dispersion in a
Decaying Isotropic Turbulence, J. Fluid Mech., 242:655–700 (1992).
100
[42] Snyder, W. H. and Lumley, J. L., Some Measurements of Particle Velocity Autocorrelation Functions in a Turbulent Flow, J. Fluid Mech., 48:41–47 (1971).
[43] Elghobashi, S. and Truesdell, G. C., On the Two-Way Interaction Between Homogeneous Turbulence and Dispersed Solid Particles. I: Turbulence Modification, Phys.
Fluids, 5(7):1790–1801 (1993).
[44] Truesdell, G. C. and Elghobashi, S., On the Two-Way Interaction Between Homogeneous Turbulence and Dispersed Solid Particles. II: Particle dispersion, Phys. Fluids,
6(3):1790–1801 (1994).
[45] Boivin, M., Simonin, O., and Squires, K. D., Direct Numerical Simulation of Turbulence Modulation by Particles in Isotropic Turbulence, J. Fluid Mech., 375:235–263
(1998).
[46] Squires, K. D. and Eaton, J. K., Particle Response and Turbulence Modification in
Isotropic Turbulence, Phys. Fluids, 2:1191–1203 (1990).
[47] McMurtry, P. A. and Givi, P., Direct Numerical Simulations of Mixing and Reaction
in a Nonpremixed Homogeneous Turbulent Flow, Combust. Flame, 77:171–185 (1989).
[48] Eswaran, V. and Pope, S. B., Direct Numerical Simulations of the Turbulent Mixing
of a Passive Scalar, Phys. Fluids, 31(3):506–520 (1988).
[49] Squires, K. D. and Eaton, J. K., Preferential Concentration of Particles by Turbulence,
Phys. Fluids, 3(5):1169–1178 (1991).
[50] Squires, K. D. and Eaton, J. K., Effect of Selective Modification of Turbulence on
Two-Equation Models for Particle-Laden Turbulent Flows, J. Fluids Eng., 116:778–
784 (1994).
[51] Eaton, J. K., Experiments and Simulations on Turbulence Modification by Dispersed
Particles, Appl. Mech. Rev., 47(6):544–548 (1994).
[52] Wang, L.-P. and Maxey, M. R., Settling Velocity and Concentration Distribution of
Heavy Particles in Isotropic Turbulence, J. Fluid Mech., 256:27–68 (1993).
[53] Mashayek, F., Jaberi, F. A., Miller, R. S., and Givi, P., Dispersion and Polydispersity
of Droplets in Stationary Isotropic Turbulence, Int. J. Multiphase Flow, 23(2):337–355
(1997).
[54] Mashayek, F., Taulbee, D. B., and Givi, P., Modeling and Simulation of Two-Phase
Turbulent Flow, in Roy, G. D., editor, Propulsion Combustion: Fuels to Emissions,
chapter 8, pp. 241–280, Taylor & Francis, Washington, D.C., 1998.
[55] Jaberi, F. A. and Mashayek, F., Temperature Decay in Two-Phase Turbulent Flows,
Int. J. Heat Mass Transfer, 43(6):993–1005 (2000).
[56] Mashayek, F. and Jaberi, F. A., Particle Dispersion in Forced Isotropic Low-MachNumber Turbulence, Int. J. Heat Mass Transfer, 42(15):2823–2836 (1999).
101
[57] Mashayek, F., Direct Numerical Simulations of Evaporating Droplet Dispersion in
Forced Low Mach Number Turbulence, Int. J. Heat Mass Transfer, 41(17):2601–2617
(1998).
[58] Mashayek, F., Simulations of Reacting Droplets Dispersed in Isotropic Turbulence,
AIAA J., 37(11):1420–1425 (1999).
[59] Kida, S. and Orszag, S., Energy and Spectral Dynamics in Forced Compressible Turbulence, Journal of Scientific Computing, 5(2):85–125 (1990).
[60] Mashayek, F., Droplet-Turbulence Interactions in Low-Mach-Number Homogeneous
Shear Two-Phase Flows, J. Fluid Mech., 376:163–203 (1998).
[61] Mashayek, F., Numerical Investigation of Reacting Droplets in Homogeneous Shear
Turbulence, J. Fluid Mech., 405:1–36 (2000).
[62] Taulbee, D. B., Mashayek, F., and Barré, C., Simulation and Reynolds Stress Modeling
of Particle-Laden Turbulent Shear Flows, Int. J. Heat Fluid Flow, 20(4):368–373
(1999).
[63] Rogallo, R. S., Numerical Experiments in Homogeneous Turbulence, NASA TM 81315,
1981.
[64] Jackson, R., Locally Averaged Equations of Motion for a Mixture of Identical Spherical
Particles and a Newtonian Fluid, Chem. Eng. Sci., 52(15):2457–2469 (1997).
[65] Barré, C., Mashayek, F., and Taulbee, D. B., Statistics in Particle-Laden Plane Strain
Turbulence by Direct Numerical Simulation, Int. J. Multiphase Flow, 27(2):347–378
(2001).
[66] Liao, S., Mashayek, F., and Guo, D., Numerical Simulations of Particle-Laden Axisymmetric Turbulent Flows, Numer. Heat Transfer, Part A, (2001), in press.
[67] Mashayek, F., Velocity and Temperature Statistics in Reacting Droplet-Laden Homogeneous Shear Turbulence, J. Propulsion Power, 17(1):197–202 (2001), also appeared
as AIAA Paper 2000-0183.
[68] Mashayek, F. and Jacobs, G. B., Temperature-Dependent Reaction in Droplet-Laden
Homogeneous Turbulence, Numer. Heat Transfer, Part A, 39:101–121 (2001).
[69] McLaughlin, J. B., Numerical Computation of Particles-Turbulence Interaction, Int.
J. Multiphase Flow Suppl., 20:211–232 (1994).
[70] Kim, J., Moin, P., and Moser, R. D., Turbulent Statistics in Fully Developed Turbulent
Channel Flow at Low Reynolds Number, J. Fluid Mech., 177:133–166 (1987).
[71] McLaughlin, J. B., Aerosol Particle Deposition in Numerically Simulated Channel
Flow, Phys. Fluids, 1:1211–1224 (1989).
102
[72] Brooke, J. W., Kontomaris, K., Hanratty, T. J., and McLaughlin, J. B., Turbulent
Deposition and Trapping of Aerosols at a Wall, Phys. Fluids, 4(4):825–834 (1992).
[73] Brooke, J. W., Hanratty, T. J., and McLaughlin, J. B., Free-Flight Mixing and Deposition of Aerosols, Phys. Fluids, 6(10):3404–3415 (1994).
[74] Ounis, H., Ahmadi, G., and McLaughlin, J. B., Dispersion and Deposition of Brownian
Particles from Point Sources in a Simulated Turbulent Channel Flow, J. Colloid and
Interface Science, 147(1):233–250 (1991).
[75] Ounis, H., Ahmadi, G., and McLaughlin, J. B., Brownian Particle Deposition in a
Directly Simulated Turbulent Channel Flow, Phys. Fluids, 5(6):1427–1432 (1993).
[76] Chen, M. and McLaughlin, J. B., A New Correlation for the Aerosol Deposition Rate
in Vertical Ducts, J. Colloid and Interface Science, 169:437–455 (1995).
[77] Chen, M., Kontomaris, K., and McLaughlin, J., Dispersion, Growth, and Deposition
of Coalescing Aerosols in a Direct Numerical Simulation of Turbulent Channel Flow,
pp. 27–32, ASME, FED-Vol. 228, 1995.
[78] Rouson, D. W. I. and Eaton, J. K., On the Preferential Concentration of Solid Particles
in Turbulent Channel Flow, J. Fluid Mech., 428:149–169 (2001).
[79] Rouson, D. W. I., Eaton, J. K., and Abrahamson, S. D., A Direct Numerical Simulation
of a Particle-Laden Turbulent Channel Flow, Department of Mechanical Engineering
Report TSD-101, Stanford University, Stanford, CA, 1997.
[80] Eaton, J. K., Turbulence Modification by Particles in Shear Flows, in Gas-Solid Flows,
pp. 273–278, ASME, FED-Vol. 228, 1995.
[81] Pan, Y. and Banerjee, S., Numerical Simulation of Particle Interactions with Wall
Turbulence, Phys. Fluids, 8(10):2733–2755 (1996).
[82] Pedinotti, S., Mariotti, G., and Banerjee, S., Direct Numerical Simulation of Particle Behavior in the Wall Region of Turbulent Flows in Horizontal Channels, Int. J.
Multiphase Flow, 18(6):927–941 (1992).
[83] Ling, W., Chung, J. N., Troutt, T. R., and Crowe, C. T., Direct Numerical Simulation
of a Three-Dimensional Temporal Mixing Layer with Particle Dispersion, J. Fluid
Mech., 358:61–85 (1998).
[84] Ling, W., Chung, J. N., , and Crowe, C. T., Direct Numerical Simulation of the TwoWay Coupled Interaction Between Particles and Mixing Layer, P. Roy. Soc. Lond. A,
456:2931–2955 (2000).
[85] Ling, W., Chung, J. N., and Crowe, C. T., Direct Numerical Simulation of a Two-Way
Thermally Coupled Droplet-Laden Mixing Layer, J. Fluid Mech., 437:45–68 (2001).
103
[86] Miller, R. S. and Bellan, J., Direct Numerical Simulation of a Confined ThreeDimensional Gas Mixing Layer with One Evaporating Hydrocarbon-Droplet Laden
Stream, J. Fluid Mech., 384:293–338 (1999).
[87] Miller, R. S. and Bellan, J., Direct Numerical Simulation and Subgrid Analysis of a
Transitional Droplet-Laden Mixing Layer, Phys. Fluids, 12(3):650–671 (2000).
[88] Okong’o, N. and Bellan, J., A Priori Subgrid Analysis of Temporal Mixing Layers
with Evaporating Droplets, Phys. Fluids, 12(6):1573–1591 (2000).
[89] Miller, R. S., Effects of Nonreacting Solid Particle and Liquid Droplet Loading on an
Exothermic Reacting Mixing Layer, Phys. Fluids, 13(11) (2001).
[90] Karniadakis, G. E. and Sherwin, S. J., Spectral/hp Element Methods for CFD, Oxford
University Press, New York, NY, 1999.
[91] Jacobs, G. B., Kopriva, D. A., and Mashayek, F., A Particle Tracking Algorithm for
the Multidomain Staggered-Grid Spectral Method, AIAA Paper 2001-0630, 2001.
[92] Jacobs, G. B., Kopriva, D. A., and Mashayek, F., Outflow Boundary Conditions for
the Multidomain Staggered-Grid Spectral Method, AIAA Paper 2002-0903, 2002.
[93] Kopriva, D. A., A Staggered-Grid Multidomain Spectral Method for the Compressible
Navier-Stokes Equations., J. Comp. Phys., 244:142–158 (1998).
[94] Lesieur, M. and Métais, O., New Trends in Large-Eddy Simulations of Turbulence,
Ann. Rev. Fluid Mech., 28:45–82 (1996).
[95] Meneveau, C. and Katz, J., Scale-Invariance and Turbulence Models for Large-Eddy
Simulation, Ann. Rev. Fluid Mech., 32:1–32 (2000).
[96] Yeh, F. and Lei, U., On the Motion of Small Particles in a Homogeneous Isotropic
Turbulent Flow, Phys. Fluids, 3(11):2571–2586 (1991).
[97] Yeh, F. and Lei, U., On the Motion of Small Particles in a Homogeneous Turbulent
Shear Flow, Phys. Fluids, 3(11):2758–2776 (1991).
[98] Armenio, V., Piomelli, U., and Fiorotto, V., Effect of the Subgrid Scales on Particles
Motion, Phys. Fluids, 11:3030–3042 (1999).
[99] Boivin, M., Simonin, O., and Squires, K. D., On the Prediction of Gas-Solid Flows
with Two-Way Coupling Using Large Eddy Simulation, Phys. Fluids, 12:2080–2090
(2000).
[100] Yuu, S., Ueno, T., and Umekage, T., Numerical Simulation of the High Reynolds Number Slit Nozzle Gas-Particle Jet Using Subgrid-Scale Coupling Large Eddy Simulation,
Chemical Engineering Science, 56:4293–4307 (2001).
[101] Wells, M. R. and Stock, D. E., The Effects of Crossing Trajectories on the Dispersion
of Particles in a Turbulent Flow, J. Fluid Mech., 136:31–62 (1983).
104
[102] Deutsch, E. and Simonin, O., Large Eddy Simulation Applied to the Motion of Particles in Stationary Homogeneous Fluid Turbulence, in Turbulence Modification in
Multiphase Flows, pp. 255–260, ASME, FED-Vol. 110, 1991.
[103] Simonin, O., Deutsch, E., and Minier, J. P., Eulerian Prediction of the Fluid/Particle
Correlated Motion in Turbulent Two-Phase Flows, Appl. Sci. Res., 51:275–283 (1993).
[104] Pozorski, J., Minier, J. P., and Simonin, O., Analysis and New Propositions for the
Crossing-Trajectory Effect in Lagrangian Turbulent Dispersion Models, pp. 63–71,
ASME, FED-Vol. 166, 1993.
[105] Champagne, F. H., Harris, V. G., and Corrsin, S., Experiments on nearly homogeneous
shear flow, J. Fluid Mech., 41:81–139 (1970).
[106] Harris, V. G., Graham, A. H., and Corrsin, S., Further experiments in nearly homogeneous turbulent shear flow, J. Fluid Mech., 81:657–687 (1977).
[107] Simonin, O., Deutsch, E., and Boivin, M., Large Eddy Simulation and Second-Moment
Closure Model of Particle Fluctuating Motion in Two-Phase Turbulent Shear Flows, in
Durst, F., Kasagi, N., Launder, B., Schmidt, F., and Whitelaw, J., editors, Turbulent
Shear Flows 9, pp. 85–115, Springer-Verlag, 1995.
[108] Wang, Q. and Squires, K., Large Eddy Simulation of Particle-Laden Turbulent Channel
Flow, Phys. Fluids, 8(5):1207–1223 (1996).
[109] Wang, Q. and Squires, K., Large Eddy Simulation of Particle Deposition in a Vertical
Turbulent Channel Flow, Int. J. Multiphase Flow, 22(4):667–683 (1996).
[110] Germano, M., Piomelli, U., Moin, P., and Cabot, W. H., A Dynamic Subgrid-Scale
Eddy Viscosity Model, Phys. Fluids A, 3(7):1760–1765 (1991).
[111] Rouson, D. W. I. and Eaton, J. K., Direct Numerical Simulation of Particles Interacting
with a Turbulent Channel Flow, in Sommerfeld, M., editor, Proceedings of the 7th
Workshop on Two-Phase Flow Predictions, Erlangen, Germany, 1994.
[112] Kulick, J. D., Fessler, J. R., and Eaton, J. K., Particle Response and Turbulence
Modification in Fully Developed Channel Flow, J. Fluid Mech., 277:109–134 (1994).
[113] Gosman, A. D. and Ioannides, E., Aspects of Computer Simulation of Liquid-Fuelled
Combustors, AIAA Paper 81-0323, 1981.
[114] Shuen, J.-S., Solomon, A. S. P., Zhang, Q.-F., and Faeth, G. M., A Theoretical and
Experimental Study of Turbulent Particle-Laden Jets, NASA CR 168293, 1983.
[115] Solomon, A. S. P., Shuen, J.-S., Zhang, Q.-F., and Faeth, G. M., A Theoretical and
Experimental Study of Turbulent Evaporating Sprays, NASA CR 174760, 1984.
[116] Shuen, J.-S., Solomon, A. S. P., Zhang, Q.-F., and Faeth, G. M., Structure of ParticleLaden Jets: Measurements and Predictions, AIAA J., 23(23):396–404 (1985).
105
[117] Sargianos, N. P., Anagnostopoulos, J., and Bergeles, G., Turbulence Modulation of
Particles, Downstream of a Two-Phase, Particle-Laden, Round Jet, in Rodi, W. and
Ganić, editors, Engineering Turbulence Modeling and Experiments, pp. 897–906, Elsevier Science Publishers Co., Inc., 1990.
[118] Sommerfeld, M., Qiu, H. H., Ruger, M., Kohnen, G., Spies, P. J., and Muller, D., Spray
Evaporation in Turbulent Flow, in Rodi, W. and Martelli, F., editors, Engineering
Turbulence Modeling and Experiments 2, pp. 935–945, Elsevier Science Publishers B.
V., 1993.
[119] Wang, L.-P. and Stock, D. E., Stochastic Trajectory Models for Turbulent Diffusion:
Monte Carlo Process versus Markov Chains, Atmospheric Environment, 26(9):1599–
1607 (1992).
[120] Etasse, E., Meneveau, C., and Poinsot, T., Simple Stochastic Model for Particle Dispersion Including Inertia, Trajectory-Crossing, and Continuity Effects, ASME Journal
of Fluid Engineering, 120:186–192 (1998).
[121] Mei, R., Adrian, R. J., and Hanratty, T. J., Particle Dispersion in Isotropic Turbulence
under Stokes drag and Basset Force with Gravitational Settling, J. Fluid Mech.,
225:481–495 (1991).
[122] Graham, D. and James, P., Turbulent Dispersion of Particles using Eddy Interaction
Models, Int. J. Multiphase Flow, 22(1):157–175 (1996).
[123] Graham, D. I., On the Inertia Effect in Eddy Interaction Models, Int. J. Multiphase
Flow, 22(1):177–184 (1996).
[124] Graham, D. I., Dispersion of Initially-Static and Initially-Excited Particles in a Turbulent Fluid Flow, Int. J. Multiphase Flow, 22(5):1005–1021 (1996).
[125] Graham, D. I., An Improved Eddy Interaction Model for Numerical Simulation of
Turbulent Particle Dispersion, J. Fluids Eng., 118:819–823 (1996).
[126] Graham, D. I., Spectral Characterestics of Eddy Interaction Models, Int. J. Multiphase
Flow, 27(6):1065–1077 (2001).
[127] Wang, Y. and James, P. W., On the Effect of Anisotropy on the Turbulent Dispersion
and Deposition of Small Particles, Int. J. Multiphase Flow, 25(3):551–558 (1999).
[128] Chen, X.-Q., Heavy Particle Dispersion in Inhomogeneous Anisotropic Turbulent
Flows, Int. J. Multiphase Flow, 26(4):635–661 (2000).
[129] MacInnes, J. M. and Bracco, F. V., Stochastic Particle Dispersion and the TracerParticle Limit, Phys. Fluids, 4:2809–2824 (1992).
[130] Ormancey, A. and Martinon, J., Prediction of Particle Dispersion in Turbulent Flows,
PhysicoChemical Hydrodynamics, 5(3):229–244 (1984).
106
[131] Parthasarathy, R. N. and Faeth, G. M., Turbulent Dispersion of Particles in SelfGenerated Homogeneous Turbulence, J. Fluid Mech., 220:515–537 (1990).
[132] Box, G. E. P. and Jenkins, G. M., Time Series Analysis, Holden-Day, Oakland, CA,
1976.
[133] Berlemont, A., Desjonqueres, P., and Gouesbet, G., Particle Lagrangian Simulation in
Turbulent Flows, Int. J. Multiphase Flow, 16(1):19–34 (1990).
[134] Berlemont, A., Grancher, M.-S., and Gouesbet, G., On the Lagrangian Simulation of Turbulence Influence on Droplet Evaporation, Int. J. Heat Mass Transfer,
34(11):2805–2812 (1991).
[135] Chauvin, G., Gouesbet, G., and Berlemont, A., Influence of Particle Interactions on
Particle Dispersion in Low Turbulence Field, in Rodi, W. and Martelli, F., editors,
Engineering Turbulence Modeling and Experiments 2, pp. 915–924, Elsevier Science
Publishers B. V., 1993.
[136] Berlemont, A., Simonin, O., and Sommerfeld, M., Validation of Inter-Particle Collision
Models Based on Large Eddy Simulation, in Stock, D. E., Reeks, M. W., Tsuji, Y.,
Michaelides, E. E., and Gautam, M., editors, Gas-Particle Flows, pp. 359–369, ASME,
FED-Vol. 228, 1995.
[137] Blumcke, E., Brandt, M., Eickhoff, H., and Hassa, C., Particle Dispersion in Highly
Swirling Turbulent Flows, Part. Part. Syst. Char., 10(4):182–190 (1993).
[138] Zhou, Q. and Leschziner, M. A., A Time-Correlated Stochastic Model for Particle
Dispersion in Anisotropic Turbulence, in 8th Turbulent Shear Flows Symp., Munich,
1991.
[139] Lu, Q. Q., Fontaine, J. R., and Aubertin, G., Particle Motion in Two-Dimensional
Confined Turbulent Flows, Aerosol Sci. Technol., 17:169–185 (1992).
[140] Lu, Q. Q., Fontaine, J. R., and Aubertin, G., A Lagranian Model for Solid Particles
in Turbulent Flows, Int. J. Multiphase Flow, 19(2):347–367 (1993).
[141] Lu, Q. Q., Fontaine, J. R., and Aubertin, G., Numerical Study of the Solid Particle
Motion in Grid-Generated Turbulent Flows, Int. J. Heat Mass Transfer, 36(1):79–87
(1993).
[142] Lu, Q. Q., Fontaine, J. R., and Aubertin, G., Particle Dispersion in Shear Turbulent
Flows, Aerosol Sci. Technol., 18:85–99 (1993).
[143] Lu, Q. Q., An Approach to Modeling Particle Motion in Turbulent Flows–I. Homogeneous, Isotropic Turbulence, Atmospheric Environment, 29(3):423–436 (1995).
[144] Chen, X.-Q. and Pereira, J. C. F., Prediction of Evaporating Spray in Anisotropically
Turbulent Gas Flow, Numerical Heat Transfer, Part A, 27:143–162 (1995).
107
[145] Chen, X.-Q. and Pereira, J. C. F., Computation of Turbulent Evaporating Sprays with
Well-Specified Measurements: a Sensitivity Study on Droplet Properties, Int. J. Heat
Mass Transfer, 39(3):441–454 (1996).
[146] Mashayek, F., Stochastic Simulations of Particle-Laden Isotropic Turbulent Flow, Int.
J. Multiphase Flow, 25(8):1575–1599 (1999).
[147] Pismen, L. M. and Nir, A., On the Motion of Suspended Particles in Stationary
Homogeneous Turbulence, J. Fluid Mech., 84:193–206 (1978).
[148] Pascal, P. and Oesterle, B., On the Dispersion of Discrete Particles Moving in a
Turbulent Shear Flow, Int. J. Multiphase Flow, 26(2):293–325 (2000).
[149] Burry, D. and Bergeles, G., Dispersion of Particles in Anisotrpic Turbulent Flows, Int.
J. Multiphase Flow, 19:651–664 (1993).
[150] Moissette, S., Oesterle, B., and Boulet, P., Temperature Fluctuations of Discrete
Particles in a Homogeneous Turbulent Flow: A Lagrangian Model, Int. J. Heat Fluid
Flow, 22(3):220–226 (2001).
[151] Zaichik, L. I., A Statistical Model of Particle Transport and Heat Transfer in Turbulent
Shear Flows, Phys. Fluids, 11(6):1521–1534 (1999).
[152] Zhou, Q. and Yao, S. C., Group Modeling of Impacting Spray Dynamics, Int. J. Heat
Mass Transfer, 35(1):121–129 (1992).
[153] Chen, X.-Q. and Pereira, J. C. F., Efficient Computation of Particle Dispersion in
Turbulent Flows with a Stochastic-Probabilistic Model, Int. J. Heat Mass Transfer,
40(8):1727–1741 (1997).
[154] Chen, X.-Q. and Pereira, J. C. F., Computation of Particle Dispersion in Turbulent
Liquid Flows using an Efficient Lagrangian Trajectory Model, International Journal
for Numerical Methods in Fluids, 26:345–364 (1998).
[155] Rizk, M. A. and Elghobashi, S. E., Two-equation Turbulence Model for Dispersed
Dilute Confined Two-phase Flows, Int. J. Multiphase Flow, 15(1):119–133 (1989).
[156] Zhang, D. Z. and Prosperetti, A., Averaged Equations for Inviscid Disperse Two-Phase
Flow, J. Fluid Mech., 267:185–219 (1994).
[157] Zhang, D. Z. and Prosperetti, A., Momentum and Energy Equations for Disperse
Two-Phase Flows and Their Closure for Dilute Suspensions, Int. J. Multiphase Flow,
23(3):425–453 (1997).
[158] Pourahmadi, F. and Humphrey, J. A. C., Modelling Solid-Fluid Turbulent Flows with
Application to Predicting Erosive Wear, PhysicoChemical Hydrodynamics, 4(3):191–
219 (1983).
[159] Elghobashi, S. E. and Abou-Arab, T. W., A Two-equation Turbulence Model for
Two-phase Flows, Phys. Fluids, 26(4):931–938 (1983).
108
[160] Elghobashi, S., Abou-Arab, T., Rizk, M., and Mostafa, A., Prediction of the ParticleLaden Jet with a Two-Equation Turbulence Model, Int. J. Multiphase Flow, 10(6):697–
710 (1984).
[161] Mostafa, A. A. and Elghobashi, S. E., A Two-equation Turbulence Model for Jet Flows
Laden with Vaporizing Droplets, Int. J. Multiphase Flow, 11(4):515–533 (1985).
[162] Chen, C. P. and Wood, P. E., Turbulence Closure Modeling of the Dilute Gas-Particle
Axisymmetric jet, AIChE J., 32(1):163–166 (1986).
[163] Marble, F. E., Dynamics of a Gas Containing Small Solid Particles, in Proc. 5th
AGARD Combustion and Propulsion Symp., New York, NY, 1963, Pergamon.
[164] Issa, R. I. and Oliveira, P. J., Modelling of Turbulent Dispersion in Two Phase Flow
Jets, in Rodi, W. and Martelli, F., editors, Engineering Turbulence Modeling and
Experiments 2, pp. 947–957, Elsevier Science Publishers B. V., 1993.
[165] Picart, A., Berlemont, A., and Gouesbet, G., Modeling and Predicting Turbulence
Fields and the Dispersion of Discrete Particles Transported by Turbulent Flows, Int.
J. Multiphase Flow, 12(2):237–261 (1986).
[166] Rodi, W., Turbulence Models for Environmental Problems, in Prediction Methods for
Turbulent Flows, von Karman Institute for Fluid Dynamics, Lect. series 2979-2, 1979.
[167] Abou-Arab, T. W. and Roco, M. C., Solid Phase Contribution in the Two-Phase
Turbulence Kinetic Energy Equation, J. Fluids Eng., 112:351–361 (1990).
[168] Kashiwa, B. A. and Gore, R. A., A Four Equation Model for Multiphase Turbulent Flow, Technical Report LA-UR-90-2273, Los Alamos National Laboratory, Los
Alamos, NM, 1990.
[169] Viollet, P. L., Simonin, O., Olive, J., and Minier, J. P., Modelling Turbulent TwoPhase Flows in Industrial Equipments, in Hirsch, C., editor, Computational Methods
in Applied Sciences, Elsevier Science Publishers B. V., 1992.
[170] Modarress, D., Tan, H., and Elghobashi, S., Two-Component LDA Measurement in a
Two-Phase Turbulent Jet, AIAA J., 22(5):624–630 (1983).
[171] Launder, B. E. and Spalding, D. B., Lectures in Mathematical Modeling of Turbulence,
Academic Press, London, UK, 1972.
[172] Launder, B. E. and Spalding, D. B., The Numerical Computation of Turbulent Flows,
Comput. Methods Appl. Mech. Engng, 3:269–289 (1974).
[173] Reynolds, W. C., Computation of Turbulent Flows, Annu. Rev. Fluid Mech., 8:183–
208 (1976).
[174] Lumley, J. L., Computational Modeling of Turbulent Flows, Adv. Appl. Mech., 18:123–
176 (1978).
109
[175] Taulbee, D. B., Engineering Turbulence Models, in George, W. K. and Arndt, R.,
editors, Advances in Turbulence, pp. 75–125, Hemisphere Publishing Co., New York,
NY., 1989.
[176] Launder, B. E., Current Capabilities for Modelling Turbulence in Industrial Flows,
Appl. Sci. Res., 48:247–269 (1991).
[177] Simonin, O., Prediction of the Dispersed Phase Turbulence in Particle-Laden Jets, in
Proc. 4th Int. Symp. on Gas-Solid Flows, pp. 197–206, ASME, FED-Vol. 121, 1991.
[178] He, J. and Simonin, O., Non-Equilibrium Prediction of the Particle-Phase Stress
Tensor in Vertical Pneumatic Conveying, in Proc. 5th Int. Symp. on Gas-Solid Flows,
pp. 253–263, ASME, FED-Vol. 166, 1993.
[179] Simonin, O., Continuum Modelling of Dispersed Two-Phase Flows, in Combustion
and Turbulence in Two-Phase Flows, von Karman Institute for Fluid Dynamics, Lect.
series 1996-02, 1996.
[180] Grad, H., On the Kinetic Theory of Rarefied Gases, Communications on Pure and
Applied Mathematics, 2(4):331–407 (1949).
[181] Drew, D. A., Mathematical Modeling of Two-Phase Flow, Ann. Rev. Fluid Mech.,
15:261–291 (1983).
[182] Haworth, D. and Pope, S. B., A Second-Order Monte Carlo Method for the Solution of the Ito Stochastic Differential Equation, Stochastic Analysis and Applications,
4(2):151–186 (1986).
[183] Shih, T.-H. and Lumley, J. L., Second-Order Modelling of Particle Dispersion in a
Turbulent Flow, J. Fluid Mech., 163:349–363 (1986).
[184] Crowe, C., Troutt, T., and Chung, J., Numerical Models for Two-Phase Turbulent
Flows, Ann. Rev. Fluid Mech., 28:11–43 (1996).
[185] Hinze, J., Turbulent Fluid and Particle Interaction, , Vol. 6, pp. 433–452, Pergamon
Press, Oxford, 1972.
[186] Mashayek, F. and Taulbee, D. B., Turbulent Gas-Solid Flows. Part I: Direct Simulations and Reynolds Stress Closures, Numer. Heat Transfer, Part B, 41(1):1–29
(2001).
[187] Launder, B. E., Reece, G. J., and Rodi, W., Progress in the Development of a ReynoldsStress Turbulence Closure, J. Fluid Mech., 68:537–566 (1975).
[188] Elperin, T., Kleeorin, N., and Rogachevskii, I., Turbulent Thermal Diffusion of Small
Inertial Particles, Phys. Rev. Lett., 76:224–227 (1996).
[189] Elperin, T., Kleeorin, N., and Rogachevskii, I., Anomalous Scalings for Fluctuations
of Inertial Particles Concentration and Large-Scale Dynamics, Phys. Rev. E, 58:3113–
3124 (1998).
110
[190] Elperin, T., Kleeorin, N., and Rogachevskii, I., Dynamics of the Passive Scalar in Compressible Turbulent Flow: Large-Scale Patterns and Small-Scale Fluctuations, Phys.
Rev. E, 52:2617–2634 (1995).
[191] Elperin, T., Kleeorin, N., and Rogachevskii, I., Turbulent Barodiffusion, Turbulent
Thermal Diffusion, and Large-Scale Instability in Gases, Phys. Rev. E, 55:2713–2721
(1997).
[192] Elperin, T., Kleeorin, N., Rogachevskii, I., and Sokoloff, D., Passive Scalar Transport
in Random Flow with a Finite Renewal Time: Mean-Field Equations, Phys. Rev. E,
61:2617–2625 (2000).
[193] Pandya, R. V. R. and Mashayek, F., Turbulent Thermal Diffusion and Barodiffusion
of Passive Scalar and Dispersed (Particle) Phase in Turbulent Flows, Phys. Rev. Lett.,
(2002), in press.
[194] Pope, S. B., A More General Effective-Viscosity Hypothesis, J. Fluid Mech., 72:331–
340 (1975).
[195] Rodi, W., A New Algebraic Relation for Calculating the Reynolds Stresses, ZAMM,
56:T219–T221 (1976).
[196] Launder, B. E., On the Effects of a Gravitational Field on the Turbulent Transport of
Heat and Momentum, J. Fluid Mech., 67:569–581 (1975).
[197] Speziale, C. G., Analytical Methods for the Development of Reynolds Stress Closures
in Turbulence, Ann. Rev. Fluid Mech., 23:107–157 (1991).
[198] Taulbee, D. B., An Improved Algebraic Reynolds Stress Model and Corresponding
Nonlinear Stress Model, Phys. Fluids A, 4(11):2555–2561 (1992).
[199] Gatski, T. B. and Speziale, C. G., On Explicit Algebraic Stress Models for Complex
Turbulent Flows, J. Fluid Mech., 254:59–78 (1993).
[200] Taulbee, D. B., Sonnenmeier, J. R., and Wall, K. W., Stress Relation for ThreeDimensional Turbulent Flows, Phys. Fluids, 6(3):1399–1401 (1993).
[201] Yakhot, V. and Orszag, S. A., Renormalization Group Analysis of Turbulence. I. Basic
Theory, J. Sci. Comput., 1(1):3–52 (1986).
[202] Horiuti, K., Higher Order Terms in the Anisotropic Representation of Reynolds
Stresses, Phys. Fluids, 2(10):1708–1711 (1969).
[203] Yoshizawa, A., Statistical Analysis of the Deviation of the Reynolds Stress from its
Eddy-Viscosity Representation, Phys. Fluids, 27:1377–1387 (1984).
[204] Yoshizawa, A., Statistical Modelling of Passive-Scalar Diffusion in Turbulent Shear
Flows, J. Fluid Mech., 195:541–555 (1988).
111
[205] Zhou, L., Theory and Numerical Modeling of Turbulent Gas-Particle Flows and Combustion, CRC Press, Boca Raton, FL, 1993.
[206] Zhou, L. X., Liao, C. M., and Chen, T., A Unified Second-Order-Moment Two-Phase
Turbulent Model for Simulating Gas-Particle Flows, in Crowe, C. T., Johnson, R.,
Prosperetti, A., Sommerfeld, M., and Y., T., editors, Numerical Methods in Multiphase
Flows, pp. 307–313, ASME, FED-Vol. 185, New York, NY, 1994.
[207] Mashayek, F. and Taulbee, D. B., Turbulent Gas-Solid Flows. Part II: Explicit Algebraic Closures, Numer. Heat Transfer, Part B, 41(1):31–52 (2001).
[208] Rivlin, R. S., Further Remarks on the Stress-Deformation Relations for Isotropic
Materials, J. Rat. Mech. Anal., 4:681–702 (1955).
[209] Spencer, A. J. M., Theory of Invariants, , Vol. 1, pp. 240–352, Academic, 1971.
[210] Mashayek, F. and Taulbee, D. B., A Four-Equation Model for Prediction of Gas-Solid
Turbulent Flows, Numer. Heat Transfer, Part A, 41.
[211] Hanjalić, K. and Launder, B. E., A Reynolds Stress Model of Turbulence and Its
Application to Thin Shear Flows, J. Fluid Mech., 52:609–638 (1972).
[212] Boltzmann, L., Lectures on Gas Theory, University of California Press, Berkeley and
Los Angeles, 1964, Translated by Stephen G. Brush.
[213] Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases,
Cambridge University Press, Cambridge, London, 1970.
[214] Chandrasekhar, S., Stochastic Problems in Physics and Astronomy, in Wax, N., editor,
Selected Papers on Noise and Stochastic Processes, Dover Publications, New York, NY,
1954.
[215] Gardiner, C. W., Handbook of Stochastic Methods, Springer-Verlag, New York, NY,
1985.
[216] Pozorski, J. and Minier, J. P., Probability Density Function Modeling of Dispersed
Two-Phase Turbulent Flows, Phys. Rev. E, 59:855–863 (1999).
[217] Minier, J.-P. and Peirano, E., The PDF Approach to Turbulent Polydispersed TwoPhase Flows, Physics Reports, 352:1–214 (2001).
[218] Pope, S. B., PDF Methods for Turbulent Reactive Flows, Prog. Energy Combust. Sci.,
11:119–192 (1985).
[219] Hyland, K. E., McKee, S., and Reeks, M. W., Derivation of a PDF Kinetic Equation for
the Transport of Particles in Turbulent Flows, J. Phys. A: Math. Gen., 32:6169–6190
(1999).
[220] Van Kampen, N. G., Stochastic Processes in Physics and Chemistry, Elsevier Science
Publishers, North Holland, Amsterdam, 1992.
112
[221] Kraichnan, R. H., Irreversible Statistical Mechanics of Incompressible Hydromagnetic
Turbulence, Phys. Rev., 109:1407–1422 (1958).
[222] Kraichnan, R. H., The Structure of Isotropic Turbulence at Very High Reynolds
Numbers, J. Fluid Mech., 5:497–543 (1959).
[223] Leslie, D. C., Developments in the Theory of Turbulence, Clarendon Press, Oxford,
1973.
[224] McComb, W. D., The Physics of Fluid Turbulence, Oxford University Press, New
York, NY, 1990.
[225] Lesieur, M., Turbulence in Fluids, Kluwer, Dordrecht, 3rd edition, 1997.
[226] Kraichnan, R., Dynamics of Nonlinear Stochastic Systems, J. Math. Phys., 2:124–148
(1961).
[227] Kolmogorov, A. N., The Local Structure of Turbulence in Incompressible Viscous
Fluid for Large Reynolds Numbers, Dokl. Akad. Nauk SSSR, 30(11):301–305 (1941).
[228] Sreenivasan, K. R., On the Universality of the Kolmogorov Constant, Phys. Fluids,
7:2778–2784 (1995).
[229] Kraichnan, R. H., Lagrangian-History Closure Approximation for Turbulence, Phys.
Fluids, 8:575–598 (1965).
[230] Orszag, S. A. and Kraichnan, R. H., Model Equations for Strong Turbulence in a
Vlasov Plasma, Phys. Fluids, 10:1720–1736 (1967).
[231] Orszag, S. A., Stochastic Acceleration by Strong Electric Fields, in Proceedings of
the Symposium on Turbulence of Fluids and Plasmas, pp. 17–28, Microwave Research
Institute Symposia Series, Polytechnic Press, Brooklyn, NY, New York, 1968.
[232] Reeks, M. W., Eulerian Direct Interaction Applied to the Statistical Motion of Particles
in a Turbulent Fluid, J. Fluid Mech., 97:569–590 (1980).
[233] Reeks, M. W., The Transport of Discrete Particles in Inhomogeneous Turbulence, J.
Aerosol Sci., 14(6):729–739 (1983).
[234] Reeks, M. W., On a Kinetic Equation for the Transport of Particles in Turbulent
Flows, Phys. Fluids, 3(3):446–456 (1991).
[235] Reeks, M. W., On the Continuum Equations for Dispersed Particles in Nonuniform
Flows, Phys. Fluids, 4(6):1290–1303 (1992).
[236] Reeks, M. W., On the Constitutive Relations for Dispersed Particles in Nonuniform
Flows. I: Dispersion in a simple Shear Flow, Phys. Fluids, 5(3):750–761 (1993).
[237] Novikov, E. A., Functionals and the Random-Force Method in Turbulence Theory,
Sov. Phys. - J. Exp. Theoret. Phys., 20(5):1290–1294 (1965).
113
[238] Furutsu, K., On the Statistical Theory of Electromagnetic Waves in a Fluctuating
media, J. Res. Natl. Bur. Stand. D, 67:303 (1963).
[239] Donsker, M. D., On Function Space Integrals, in Martin, W. A. and Segal, I., editors,
Analysis in Function Space, MIT, Cambridge, MA, 1964.
[240] Derevich, I. V. and Zaichik, L. I., Particle Deposition from a Turbulent Flow, Fluid
Dynamics, (5):722–729 (1989).
[241] Derevich, I. V. and Zaichik, L. I., An Equation for the Probability Density, Velocity,
and Temperature of Particles in a Turbulent Flow Modelled by a Random Gaussian
Field, Journal of Applied Mathematics and Mechanics (Prikl. Mat. Mekh.), 54(5):631–
637 (1990).
[242] Zaichik, L. I., Pershukov, V. A., Kozelev, M. V., and Vinberg, A. A., Modeling of Dynamics, Heat Transfer, and Combustion in Two-Phase Turbulent Flows: 1. Isothermal
Flows, Experimental Thermal and Fluid Science, 15:291–310 (1997).
[243] Zaichik, L. I., Pershukov, V. A., Kozelev, M. V., and Vinberg, A. A., Modeling of
Dynamics, Heat Transfer, and Combustion in Two-Phase Turbulent Flows: 2. Flows
with Heat Transfer and Combustion, Experimental Thermal and Fluid Science, 15:291–
310 (1997).
[244] Derevich, I. V., Statistical Modelling of Mass Transfer in Turbulent Two-Phase Dispersed Flows - 1. Model Development, Int. J. Heat Mass Transfer, 43:3709–3723
(2000).
[245] Derevich, I. V., Statistical Modelling of Mass Transfer in Turbulent Two-Phase Dispersed Flows - 2. Calculation Results, Int. J. Heat Mass Transfer, 43:3725–3734
(2000).
[246] Pandya, R. V. R. and Mashayek, F., Probability Density Function Modeling of Evaporating Droplets Dispersed in Isotropic Turbulence, AIAA Journal, 39:1909–1915
(2001).
[247] Pandya, R. V. R. and Mashayek, F., Kinetic Equation for Particle Transport and
Heat Transfer in Non-Isothermal Turbulent Flows, in 40th AIAA Aerospace Sciences
Meeting & Exhibit, Reno, NV, 2002, AIAA Paper Number 2002-0337.
[248] Kraichnan, R. H., Eulerian and Lagrangian Renormalization in Turbulence Theory, J.
Fluid Mech., 83:349–374 (1977).
[249] Minier, J., Closure Proposals for the Langevin Equation Model in Lagrangian TwoPhase Flow Modelling, in Proceedings of the 3rd ASME/JSME Joint Fluids Engineering Conference, San Francisco, CA, 1999.
[250] Hyland, K. E., McKee, S., and Reeks, M. W., Exact analytic solutions to turbulent
particle flow equations, Phys. Fluids, 11(5):1249–1261 (1999).
114
[251] Van Kampen, N. G., A Cumulant Expansion for Stochastic Linear Differential Equations. I, Physica, 74:215–238 (1974).
[252] Van Kampen, N. G., A Cumulant Expansion for Stochastic Linear Differential Equations. II, Physica, 74:239–247 (1974).
[253] Caporaloni, M., Tampieri, F., Trombetti, F., and Vittori, O., Transfer of Particles in
Nonisotropic Air Turbulence, J. Atmos. Sci., 32:565–568 (1975).
[254] Swailes, D. C., Sergeev, Y. A., and Parker, A., Chapman-Enskog Closure Approximation in the Kinetic Theory of Dilute Turbulent Gas-Particulate Suspensions, Physica
A, 254:513–547 (1998).
[255] Peirano, E. and Leckner, B., Fundamentals of Turbulent Gas-Solid Flows Applied
to Circulating Fluidized Bed Combustion, Prog. Energy Combust. Sci., 24:259–296
(1998).
115
Constant Magnitude Basis for choice
Cf 1
1.75
LRR187
Cf 2
-0.159
LRR187
Cf 3
0.5
Budget of ui uj
Cf p1
2.5
Budget of ui vj
Cf p2
0.5
Budget of ui vj
Cf p3
0.2
Budget of ui vj
C²1
1.45
Standard k-²
C²2
1.85
Standard k-²
C²3
0.8
Overall model performance
Table 1: Empirical constants.
116
Figure Captions
Figure 1. Correspondence between simple homogeneous flows and various regions in a
backward-facing step flow.
Figure 2. Instantaneous particle distribution in a homogeneous plane strain turbulence at
various times for (a) τp = 0.1τpcr , and (b) τp = 0.5τpcr . The prediction of the theory for the
particle-containing box is also shown with dashed line.
Figure 3. Energy budgets of << u21 >> and << u1 v1 >> in homogeneous shear flow for a
case with Φm = 0.25 and τp = 0.032s.
Figure 4. Comparisons between the algebraic model predictions and DNS results for various
components of (a) fluid Reynolds stress, (b) particle Reynolds stress, and (c) fluid-particle
velocity covariance, in homogeneous shear flow for a case with Φm = 0.25 and τp = 0.016s.
Figure 5. Temporal variations of h(τd /τd0 )1/2 i for evaporating droplets in isotropic turbulence for τd0 = 5τk , τec = 5τk and Scd = 5.
1/2
Figure 6. Temporal variations of the standard deviation of τd , for evaporating droplets
in isotropic turbulence, at different droplet Schmidt numbers for τd0 = 5τk , τec = 5τk .
Figure 7. Temporal evolution of particle Reynolds stresses in homogeneous shear turbulence
with constant
b1 i
∂hU
.
∂x2
Figure 8. Temporal evolution of particle temperature-temperature and velocity-temperature
correlations in homogeneous shear turbulence with constant
117
b1 i
∂hU
∂x2
and constant
∂hT i
.
∂x2
plane strain
shear
isotropic
Figure 1: Correspondence between simple homogeneous flows and various regions in a
backward-facing step flow.
118
4
4
St=0
St=0
3
3
x2/π 2
x2/π 2
1
1
1
2
x1/π
3
4
1
4
4
St=0.693
3
x2/π 2
x2/π 2
1
1
2
x1/π
3
4
1
4
3
3
x2/π 2
x2/π 2
1
1
2
x1/π
2
x1/π
3
4
St=1.386
1
4
3
4
3
St=0.693
1
2
x1/π
3
4
St=1.386
1
2
x1/π
3
4
(a)
(b)
Figure 2: Instantaneous particle distribution in a homogeneous plane strain turbulence at
various times for (a) τp = 0.1τpcr , and (b) τp = 0.5τpcr . The prediction of the theory for the
particle-containing box is also shown with dashed line.
119
1.0
Budget of <<u1u1>>
0.5
0.0
−0.5
Production (DNS)
Dissipation (DNS)
Drag (DNS)
Pressure−strain (model)
RHS (DNS)
ΣLHS (DNS+model)
−1.0
(a)
−1.5
0
2
4
6
St
8
10
12
1.0
Budget of <<u1v1>>
0.5
0.0
−0.5
−1.0
−1.5
(b)
−2.0
0
2
4
6
St
Production (DNS)
Drag (DNS)
Pressure−strain (model)
RHS (DNS)
ΣLHS (DNS+model)
8
10
12
Figure 3: Energy budgets of << u21 >> and << u1 v1 >> in homogeneous shear flow for a case
with Φm = 0.25 and τp = 0.032s.
120
1.0
(a)
0.8
<<uiuj>>
0.6
0.4
0.2
0.0
−0.2
−0.4
0
2
4
6
8
10
12
1.0
(b)
0.8
<<vivj>>
0.6
0.4
0.2
0.0
−0.2
−0.4
0
2
4
6
8
10
12
1.0
0.8
<<uivj>>
0.6
0.4
i=1, j=1
i=2, j=2
i=3, j=3
i=1, j=2
i=2, j=1
(c)
0.2
0.0
−0.2
−0.4
0
2
4
6
St
8
10
12
Figure 4: Comparisons between the algebraic model predictions and DNS results for various
components of (a) fluid Reynolds stress, (b) particle Reynolds stress, and (c) fluid-particle
velocity covariance, in homogeneous shear flow for a case with Φm = 0.25 and τp = 0.016s.
121
1.0
1/2
<(τd/τd0) >
0.8
0.6
0.4
DNS
STH
ANL1
ANL2
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
t/τE
Figure 5: Temporal variations of h(τd /τd0 )1/2 i for evaporating droplets in isotropic turbulence
for τd0 = 5τk , τec = 5τk and Scd = 5.
122
0.06
Scd=5.0, DNS
Scd=1.0, DNS
Scd=5.0, STH
Scd=1.0, STH
Scd=5.0, ANL1
Scd=1.0, ANL1
Scd=1.0, ANL2
Scd=5.0, ANL2
0.05
σ/τd0
1/2
0.04
0.03
0.02
0.01
0.00
0
1
2
3
t/τE
1/2
Figure 6: Temporal variations of the standard deviation of τd , for evaporating droplets in
isotropic turbulence, at different droplet Schmidt numbers for τd0 = 5τk , τec = 5τk .
123
2
0.2
(b)
(a)
PDF I τd=0.3
i=j=1, DNS
i=j=2, DNS
i=j=3, DNS
1
0
v1’v2’
vi’vj’
1.5
−0.2
0.5
0
DNS
−0.4
−0.6
0
5
10
15
−0.8
αt
0
5
10
15
αt
Figure 7: Temporal evolution of particle Reynolds stresses in homogeneous shear turbulence
b
with constant ∂h∂xU21 i .
124
2
3
(a)
PDF I τd=0.3
PDF II
DNS
1
vi’θ’
θ’θ’
2
(b)
i=1, DNS
i=2, DNS
1.5
0.5
0
1
−0.5
0
0
5
10
15
−1
αt
0
5
10
15
αt
Figure 8: Temporal evolution of particle temperature-temperature and velocity-temperature
b
i
correlations in homogeneous shear turbulence with constant ∂h∂xU21 i and constant ∂hT
.
∂x2
125