A TwoWay Coupled Reynolds Stress Model for Suspensions with
Comparison to Experiment
O. Skjaeraasen* and R. Skartlien and G. Zarruk
Institute for Energy Technology, P.O. Box 40, N2027 Kjeller, Norway
*Email: olafs@ife.no
Keywords: Particleladen flow, suspension modelling, turbulence modelling, PIV
Abstract
We derive a 2waycoupled Reynolds stress model for dispersions, and use it to sim
ulate particleladen turbulent channel flow. We incorporate closure relations for the
interphase coupling based on turbulencekinetic theory (Reeks (1992); Skartlien et al.
(2009)) and include the added mass effect to handle solid particles, droplets or bubbles
in a carrier liquid. Simple scaling formulae are obtained for the turbulent kinetic energy
of the fluid, from which one can identify two primary mechanisms for the modulation
of turbulent kinetic energy by the particles: 1) production/loss due to momentum
transfer (and work exchange) between the particles and the fluid, and 2) modified pro
duction by the mean velocity gradients due to the added mass effect. The predictions
are compared qualitatively with experimental data, including new Particle Imaging
Velocimetry (PIV) measurements using silvercoated polystyrene spherical particles
with a diameter of 230 micron, a density of 1.15 g/cc, and a flow Reynolds number
between 10,000 and 30,000. Even at a very low mean mass loading ratio ( 10 4)),
the experiments reveal significant particle feedback on the fluid.
Introduction
It is known from experiments that a tur
bulent flow involving suspended particles
can deviate significantly from its single
phase counterpart, in terms of its statis
tical mean properties. This holds true
even at very low particle volume fractions.
Previous Eulerian modelling efforts for 2
way coupled, turbulent suspensions have
almost exclusively been based on the ke
framework (Kataoka and Serizawa (1989);
Elghobashi and AbouArab (1983)). An
exception to this is the twoway coupled
Reynolds stress model of Simonin et al.
(Squires and Simonin (2002)), developed
for solid particles suspended in gas. An
important observation in recent experi
mental data is that the Reynolds stress
components can be modified differently
and in an opposite sense. A model for the
turbulent kinetic energy only may there
fore be insufficient.
In view of the above, we have re
cently developed a twoway coupled fluid
particle Reynolds stress model intended
for stationary turbulent flow in channels
or pipes. In the limit of zero volume
fraction of particles, the model reduces
to a standard singlephase stresscw model
(Wilcox (2006)). Our work is a step
forward for such modelling, since we in
corporate 1) more rigorous closure rela
tions for the interphase momentum trans
fer based on turbulencekinetic theory, 2)
include the added mass effect to handle
droplets and bubbles in a carrier liquid,
and 3) we treat all the Reynolds stresses
of the fluid to account for the individual
modulations of these in the presence of
particles. So far, we have not included
the interparticle interactions that char
acterize dense suspensions, but the model
closures are good approximations in the
regime where particleparticle interactions
can be ignored. The model is applicable
for dense solid particles, and sufficiently
small bubbles and droplets that can be de
scribed as unique particles with negligible
interfacial deformations. In this paper, we
discuss the implications of the model in
terms of the fluid and particle momentum
and kequations.
For light particles with a density com
parable to or smaller than the carried
fluid, it is well known that the particles
respond to the fluid acceleration (Drew
and Passmann (1999)). This "added mass
effect" is required when one wants to
model droplets or bubbles in the carrier
fluid; in fact, it can only be safely ne
glected for particles whose intrinsic den
sity is much larger than that of the carrier
fluid. Skartlien et al. (2009) implemented
a Reynolds stress type model based on
the kinetic framework of Reeks(1992),
with account for drag and added mass.
The model was tested against data for
polystyrene particles suspended in water
in pipe flow, and the model closure rela
tions were evaluated and tested using PIV.
In the current work, there is a new set of
closure relations needed for the momen
tum transfer terms in the Reynolds stress
and momentum equations. We follow the
same kinetic framework to develop these
closure relations.
A model for the particle distribution
is also required in a twoway model. It
has been recognized that the full Reynolds
stresses tensor for the fluid is required to
account for the particle diffusion tensor,
which is anisotropic in the general case
(Reeks (1992)). This is accompanied by
a Reynolds stress type model for the par
ticles as well (expressed in terms of the
turbulent particle velocity). Therefore, a
Reynolds stress model for both fluid and
particles is necessary to accurately pre
dict the turbulent diffusion of the par
ticles. Various Reynolds stress Eulerian
formulations for the particles are due to
Reeks (1992); Swailes et al. (1998); Young
and Leeming (1997); Squires and Simonin
(2002); Zaichik and Alipchenkov (2005).
Recent experimental work using PIV
has shed new light on turbulence mod
ulation in particle laden flow. In par
ticular, the augmentation of turbulence
seems to be characteristic for many of
these flows, even for low mass loadings
on the order of 10 3. Kiger and Pan
(2000) used glass beads (with diameter
195pm) suspended in turbulent channel
flow with water, and Wu et al. (2006)
used polythene spheres in air with diam
eter 60 and 110pm, to demonstrate tur
bulence augmentation. Gore and Crowe
(1989) discussed the effect of particle size
on turbulence modulation, while Hetsroni
(1989) discussed the dependency on par
ticle Reynolds number.
A general trend in the available exper
imental data is that the larger particles
tend to augment the turbulence kinetic en
ergy, while smaller particles tend to sup
press it. Although we may not expect
a I,,, I I 1 .v of turbulence modifica
tion in relation to particle diameter, it
seems that the particle diameter is one
of the prime parameters in twoway cou
pling. The concentration of particles will
obviously also play a role. The diameter of
the particles affects the particle Reynolds
number Rep, and the particle relaxation
time Tp. Hetsroni (1989) discussed the
influence of Re,. The local Stokes num
ber (the ratio between the particle relax
ation time and the characteristic turbu
lence timescale or turnover time) is also
expected to be a central parameter.
In the current paper, we apply a stress
omega model to turbulent, statistically
steady channel flow. Particular empha
sis is put on the kequation, while a de
tailed discussion of all the components of
the turbulent stress tensor is deferred to a
forthcoming work. Model predictions are
compared with new PIV experimental re
sults.
PhasicAveraged
Momentum Equations
First, we briefly discuss the general form
of the averaged momentum equations.
The phasicaveraged momentum equa
tions for the fluid and the particles is,
Otpaui + ilpauiu, + 0 ,.. .." =
atip + a17 Fi + pag + F,, (1)
dtp'' + dadvi, + pdadv;'v
adapp + adal d + pddg + ,d, (2)
where p is the intrincic density of the fluid,
a is the fluid volume fraction, gi is the
gravitational acceleration, F, = Fi is
the interphase mean force, particle quan
tities are labelled with a superscripted d,
ensemble averages are labelled by an over
bar, and socalled phasic averaged quan
tities are labelled with a tilde. Phasic
averaging is defined in the appendix, and
consists of a volume averaging and then
an ensemble averaging. We have used a
standard Reynolds decomposition for the
scalar pressure p, together with a phasic
decomposition for the viscous stress ten
sor Tij and the velocity fields, in consis
tence with previous works by Laux (1998)
and Wilcox (2006) (Laux; Wilcox (2006)).
In the limit ad 0 and a 1, the
fluid momentum equation converges to the
standard Reynolds averaged singlephase
NavierStokes equation.
For the force between the two phases,
we use a MaxeyRiley type equation with
both added mass and drag forces in
cluded. As discussed by Drew and Pass
mann (1999), the volumeaveraged force
on the particle phase from the fluid phase
is given by
Drag
Fd pPpad (u v)
Added Mass I
pa d (9OU
+ Y +u Vu
Added Mass II
2 8'at
Compression
+ (v u) (aV u) (3)
where 3p 1/Tp is the inverse of the
particle relaxation time, and interparticle
collisions, lift forces and forces of order
O(vul) with n > 1 have been omitted.
The ..I.. 1 1 ...... I" force in Eq. (3) is
related to particle motion; i.e., after local
volume averaging, this term arises from a
net flux of particles into a control volume.
A detailed discussion of the general
ized Reynolds stress equation that governs
the tensor au'i u is beyond the scope of
the present paper; instead, we discuss be
low the momentum equations and the k
equations for the two phases.
Modelling and Measuring Axial
Momentum Balance in Steady
Channel Flow
The xcomponent of the fluid momentum
can be obtained from Eq. (1);
adxp + ad y + s .S. ,
+p/p (ad (Ud U)
2 dt
2'
d dux
dt )
, i i ,
The first three terms in Eq. (4) repre
sent the usual pressure gradient and stress
terms also present in single phase flow.
The first drag term depends on the mean
velocity slip (Ud U). The second drag
term represents turbulent transport (ui)
of particles, and is nonzero because of the
turbulent fluctuations of the particle vol
ume fraction. We will show below that the
transport term is positive across the whole
channel, except in the viscous boundary
layer. The slipterm can play the opposite
role, depending on the sign of the veloc
ity slip between the two phases. This, in
turn, depends on the particle properties;
i.e., their size and intrinsic density, and
their volume fraction distribution.
The added mass term is proportional to
half the particle volume fraction, and the
acceleration slip between the two phases.
For zero velocity slip, the added mass ef
fect dominates drag, but in steady channel
flow with dense solid particles, drag gen
erally dominates in the mean momentum
equation. The last term in Eqs. (4) rep
resents momentum carried into or out of
a control volume by particles crossing its
border. For small particle volume frac
tions, this effect is weak, and it can be
neglected.
For large ( 950pm), close to neutrally
buoyant (pd/p 1.03) particles in turbu
lent pipe water flow at 4 x 104 < Re <
1.2 x 105, Drazen and Jensen (2007) found
that the particles slightly lag the fluid over
most of the cross section. Kaftori et al.
(1995) found their particles (pd/p 1.03)
to be slightly lagging the fluid in the core
regions of open channel flow for 5000 <
Re < 14000.
For dense (pd/p = 2.6) particles in open
channel flow at Re 14000, Righetti and
Romano (2004) found that the particles
lag the fluid in the core of the channel,
while the fluid lags the particles near the
boundaries. In their case, the net effect of
the drag from the particles is to steepen
the fluid velocity gradient near the chan
nel walls, and flatten the fluid velocity
profile in the core regions. Obviously, the
production terms in the equations for the
turbulent stress tensor of the fluid will be
affected by the particlemodified velocity
shear.
Recent PIV experiments using water,
silvercoated polystyrene particles with a
diameter of 230pm, a density of pd/p
1.15, channel dimensions (length x height
x width) of 800 x 5 x 30 cm, a mean
mass loading ratio of approximately 0.015
per cent, and Reynolds numbers Re ~
30000, qualitatively differ from the results
of Righetti & Romano. We find that in
the mean, the fluid lags the particles in
core channel regions very slightly; i.e., by
approximately 1 per cent. Near the chan
nel walls, the signaltonoise level is rather
high, making it difficult to draw firm con
clusions. The same qualitative behaviour
is seen for Re 2 x 104. At Re 104,
however, we find that the particles lag the
fluid in the bottom 1/3 of the channel,
while the opposite is the case above this
region. It should be stressed that further
analysis of the data remains before their
statistical significance can be inferred pre
cisely.
Neglecting the compression term, the
mean momentum equation for the parti
cle phase is
ad adOy'Ty + ,) s .dsd
+p (ad(U Ud)+
p (a du>
2 [d
Sdvx
dt )
which exhibits a strong degree of sym
metry with Eq. (4), since momentum
balance between the two phases requires
Fx = Fx. It should be noted, however,
that the intrinsic stress and pressure of the
particles does not play any role in the cur
rent work, and Eq. (5) can be solved ex
plicitly in the limit where drag dominates
added mass,
P!3pdUd p3dU ad9p+ ad9yFxY
+,9 .dSy +, (6)
The u'term will be discussed below.
Turbulent Kinetic Energy
Equation for the Fluid
An equation for the turbulent kinetic en
ergy of the carrier phase can be obtained
by taking the trace of the equation for the
turbulent stress tensor,
.. U 3*pakw
+a, [( + r) adk] + F .." = 0,(7)
where cw is the specific dissipation ~ e/k
and pT is the eddy viscosity. The single
phase value of the constant 3* is given in
Wilcox (2006). The turbulent stress ten
sor and the turbulent kinetic energy are
defined in terms of phasic averages,
aui Uj
a
auia.,
2 a
The first term in (7) is production due
to the mean fluid velocity gradient, the
second term is dissipation in the fluid it
self, and the third term is due to diffusion
of turbulent kinetic energy. The source of
turbulent kinetic energy due to forces be
tween the particles and the fluid is
F .," = (adSxyOy
2k
+p[p (U U)W + "] ,
with the mean axial particle velocity given
by
The first source term is due to the dif
ference in mean velocity. Since the mean
velocities are nonzero only in the axial di
rection for channel flow, only i = x con
tributes in this term.
The turbulent volume flux of particles
gives a force
Ti = ,',,, (9)
./a, (10)
in the fluid momentum equation (and an
equal, but opposite force in the particle
momentum equation). The fluid moves
in the opposite direction of the turbulent
particle flux to conserve the total volume
fraction a + ad = 1. Alternatively, one
can think of a net turbulent drift veloc
ity .." that does work in combination with
the average drag force on the fluid. The
corresponding work term ppp(Ud U)u'a
may generate or absorb turbulent kinetic
energy. We will show that it is positive in
the core regions of the flow if the particles
lag the fluid. In the boundary layer, we
may expect the opposite situation, if the
fluid lags the particles.
The second source term in (8) is the
work done on the fluid by the fluctuating
part of the drag force (added mass forcing
enters here via the fluctuation in particle
velocity). The closure relations for these
source terms will be given below. The
third source term in (8) is dissipation of
kinetic energy due to the fluctuating drag
force, and is always negative. To close this
term, we assume
2adk.
The final term in (8) is due to the added
mass effect. At low particle Reynolds
number, the added mass term is propor
tional to (one half of) the liquid mass ex
pelled by the particle, and to the (mean)
local difference between the particle accel
eration and the fluid acceleration.
In the equations for the turbulent stress
tensors of both the liquid phase and the
dispersed phase, the added mass effect
gives an extra production term, which
mathematically arises from the increased
effective inertia. It should be noted, how
ever, that the total fluid turbulence pro
duction given by the velocity shear has
a coefficient a + ad/2 1 ad/2; i.e.,
it decreases with increasing particle vol
ume fraction. This reflects the fact that
with increasing ad, the increased produc
tion from the particle coupling is domi
nated by a reduction in the fluid's own
(volumeaveraged) inertia.
It should be noted that equations which
are structurally similar to Eq. (7) have
been frequently discussed in the litera
ture, although most previous works on liq
uid dispersions have neglected the added
mass effect, and have been based on strict
ensembleaveraging (Rogers and Eaton
(1991)).
A key novelty of the current work is the
construction of new closure formulae for
the interphase terms in the kequation
(7). This is discussed below.
Closure for the work term
For the second source term in (8), we
adopt
ad ap (11)
2
where (..)' is the fluctuation relative to the
straight ensemble average,
V' = Vi Ti,
Ui = Ui Ui.
The closure tensor ;i,. that enter in the
fluid Reynolds stress equations has the fol
lowing contributions,
dd da dd da,(12)
1 = i 12
where pairs in "a" and "d" refer to correla
tions between added mass and drag forces.
Formulae for these tensors are be obtained
from particle path statistics, and Green's
function for the equation of motion.
Without added mass forcing, and a
negligible mean fluid shear, we obtain
I ,,
U aui u
i a
the simplest linear relation in the fluid
Reynolds stresses,
Tpjidi (13)
 [ 4pTdd7 (14)
1 + PpTdd_
(15)
2k [iSt ,
where St 1 /(f3pTdd) is the particle
Stokes number, and 7dd is the local cor
relation time of the fluid velocity seen by
the particles. This is also the form used by
many authors previously (cited in the in
troduction), by adopting a suitable corre
lation timescale. For large Stokes number
(heavy particles), '..' is small and little
work is done on the fluid. For vanishing
Stokes number (light, small particles) the
particles follow the fluid, and we recover
The more general case with shear
and added mass gives a dependency on
more Reynolds stress components via the
anisotropic components of the particle dif
fusion tensor. These general closure for
mulae are obtained via
Tp [(Xii)dd/Tdd+ (+ii)da/Tda]
(16)
that depend on the Atensor from the the
ory of Reeks (1992). The dispersion ten
sor Xji is the correlation between the lo
cal force fi(x,t) on the particle in the i
direction, and the total displacement Axj
in the jdirection of the particle before it
passes through x at time t,
Aji = (fi(x,t)Axj(x,t)).
These components are functions of the
force correlation functions and Greens
function of the equation of motion,
ji (fi(x,t)fk(xp(s), s))Gkj(ts)ds.
(17)
where xp(s) is the particle trajectory.
Closure for the turbulent volume
flux
Since a+ad 1 and therefore a'+a' = 0,
we have with respect to the first source
term in (4) and (8),
7 a'u a' .
a a
The closure for the axial turbulent vol
ume flux of particles, a' ..' follows di
rectly from the diffusion current of Reeks
(1992),
(ad)' ,
T[O (adAki) + adid, (19)
where [..]d means inclusion of only the
drag component of the dispersion tensors.
The 7 vector represents a drift flux that
is nonzero in inhomogeneous turbulence.
In channel flow,
(ad?: Tp[y(addy) +a ]d. (20)
We will assume zero axial drift, = 0,
since there is no variation of the turbu
lence in this direction (Reeks 1992). With
the explicit Atensor formulae derived in
Skartlien et al. (2009), we obtain
(ad: T p[y(ad AX )d (21)
Sad Qdd T ],(22)
[' TPOy P 1 + p T QCa T2j
with Tr 7dd. Here, a = 3/2p+/(l + pt/2)
with p/p p/rhod, and C is the sec
ond derivative of the wall normal stress in
the fluid, and Q~ = ; is propor
tional to the turbulent fluid shear stress.
An estimate in the channel core regions
is then (using characteristic timescales for
the core flow),
length scale of the turbulence. Experi
ments suggest C,, 0.09. To account for
particles, we assume
ka/2
/3*pak paC, (25)
3 T2 O (ad' ..'..') where lh < 1 is a smaller "1l I.... length
u1  C" 2 scale, depending on the mean particle sep
Saration A. The viscous dissipation is more
0 (23) ifflinnt ,nrhn nortirlne orn intrnoidurnA
with '.,, ~ ad, .' ..' > 0. We thus
have turbulence production (Ud U)u >
0 in the core flow if U > Ud. In the bound
ary layers, we may have (U < Ud), and
turbulence suppression (given the same
shear stress '..' ..' ). As discussed previ
ously, the sign of the mean velocity slip
between the two phases depends on the
fundamental particle properties.
New experimental results qualitatively
confirm these results, but a careful assess
ment of the data remains before conclu
sions can be drawn more firmly.
Turbulence Modulation in
the kequation
We will estimate k in the bulk flow suffi
ciently far away from the walls where gra
dients are small, such that we can neglect
the diffusion of turbulent kinetic energy.
Production and dissipation in the
fluid
In single phase fluids one may adopt
the usual scaling law for the dissipation
(Wilcox 2006),
k*/2 C3/2
L I
since these will enhance the local fluid ve
locity gradients. Crowe (2000) assumes
1 1 1
Sh + (26)
lh As I
For the production term, we follow
Crowe (2000) and adopt a similar scaling
to the same order of magnitude as the dis
sipation,
k.3/2
,.,. .i fT paCO, (27)
1
where ko is the single phase turbulence ki
netic energy.
Scaling of k with respect to
particle diameter and Stokes
number
With the scaling formulae for the fluid dis
sipation and production, and with the ne
glect of diffusion, the kequation reduces
pf3p [Sb + Sf] + Pp (a ) o
(2
lk, I3/2 k/2 )
with lh < 1 and Sf < 0. Here,
S St
8+ 2k
1 + St
>k3 2 [ T
C1 1 (1 + St)
where L is a pseudo dissipation length
scale, and I is the characteristic integral
aTad
1 + p Tad]
(29)
and co Ck 2/1 gives the singlephase
dissipation rate.
Sb K_ ; T T2
1 + 3p aC" Y2
I d ~ (.' .' )
x (Ud U)ad ) (30)
a
We recover k ko and lh = 1 for ad = 0.
For the ratio k/ko, we obtain
(k >3/2
f1
1h 3/ [Sb + adSf]
C, k/2 a
"/G'0
+ i1 ad
+1)
For sufficient volume fraction of particles,
we may use lh ~ As with a mean parti
cle separation As ~ D(a) 13. Note that
Pp ~ 1/(ppD2), and St ~ ppD2.
Added mass effect.
If the added mass term dominates for the
larger particles, and the dispersion is rel
atively dense,
( \)3/2 (D ( d 3/3
For a volume fraction of 0.5, we get sig
nificant augmentation for D ~ 1, with
(k/ko)3/2 3/2. For a dilute suspen
sion, k/ko 1 for lh 1. The classi
cal empirical result is that k/ko increases
with D/L, with k/ko ~ 1 for D/L ~ 0.1
(Kenning and Crowe (1997)). This is con
sistent with D/1 ~ 1, since 1 C,L =
0.09L. The literature reports a clear cor
respondence between turbulence enhance
ment and D, indicating the importance of
the above relation.
Phase interaction effect.
The sum of the drag related interaction
terms may be written
3p [Sb + dSf]
ad T ((2/3)k 2k
1 + St a L T
S o + aTad(l1 + St)2) (32)
St(1+St) + 1+ 3pTad )]
where we have introduced a "slip param
eter" (or mean acceleration) ,
(U Ud) Tp.
For small St, we confirm that the interac
tion suppresses the turbulence intensity,
3p [Sb + adSf] 3p [adSf] (34)
a ad 1 [ + (1d) < 0. (35)
St 1 +,3pTad
For large St, both Sb and Sf contributes,
+ [ adSf d [T ( (2/3)k) 2k]
1p [Sb ]f f
St a R T
and the interaction may provide turbu
lence enhancement for sufficient slip pa
rameter > 0 in the Sb term.
The Particle kEquation
After a number of algebraic steps involv
ing the MaxeyRiley equation of motion
presented above, we obtain an equation
for the turbulent kinetic energy associated
with the particles;
1
d d d pad AM
" a 2adkd +p ^
p Ud
S + 2 ' U
+ da dsd ayud 6d
near the core of the channel, where the
spatial gradients are small. Here, kd
aui'." /2 and e is the dissipation rate (a
discussion of the latter is deferred to a
forthcoming paper). The first drag term
is given by
1
2
with
ad ( dd +da), (37)
( i T.
+ Tda/Tp
and a (3/2)p/(pd + p/2). By ignoring
gradients in the channel core, we have
dd 2k
Pi r,[1+St) (39)
and the added mass term is
AM 2 (Tad 3p Q d
+ ( Taa)Qn a (40)
(i ,Pp aa
where Q7d dT. For dense particles,
p/1p > 1, drag dominates added mass,
and hence T7Q[ < k; i.e., the turbu
lent fluid velocity field has a stronger self
correlation that its acceleration field. For
light particles, the opposite is the case.
Between these two extremes, the relative
roles of added mass and drag are con
trolled by the particle volume fraction, the
particlefluid density contrast, as well as
Re, Rep, and St.
Conclusions
We have presented a 2way coupled stress
omega model for turbulent dispersions,
and applied it to steady channel flow
involving suspended, solid particles. It
demonstrated that the current model
framework is consistent with turbulence
modulation as reported in the literature,
where large particles can provide turbu
lence augmentation, and small particles
can provide turbulence suppression. The
larger particles inject turbulence due to
the average slip between the particles in
the core flow, and smaller particles pro
vide turbulence suppression due to drag
related dissipation. A new ingredient in
the current work is that turbulence aug
mentation may be induced by the added
mass effect via an extra production term
that is proportional to the mean fluid ve
locity gradient.
We obtain well founded closure approx
imations which, in their general tensor
form, can be used in a Reynolds stress
model with twoway coupling (in the limit
of negligible particleparticle interaction).
From the kequation, we identify four ba
sic mechanisms for the modulation of tur
bulent kinetic energy by the particles: 1)
production/loss due to momentum trans
fer (and work exchange) between the par
ticles and the fluid, 2) modified dissipation
due to the introduction of new particle
induced length scales, 3) modified produc
tion by the mean velocity gradients, due
to the added mass effect, and 4) modified
production due to a change in the mean
velocity gradient.
In the near future, we will use new
PIV experimental data to tune the Clo
sure Relations, and make a detailed, quan
titative comparison between the measured
and modelled profiles for the various com
ponents of the turbulent stress tensor.
Acknowledgements
This work was performed by the FACE
center a research cooperation between
IFE, NTNU, and SINTEF. The center
is funded by The Research Council
of Norway, and by the following in
dustrial partners: StatoilHydro ASA,
Norske ConocoPhillips AS, Vetco Gray
Scandinavia AS, Scandpower Petroleum
Technology AS, FMC, CDAdapco, ENI
Norge AS, and Shell Technology Norway
AS. O.S. thanks C'li. Lawrence and Olav
Sendstad for useful input.
Phasic Decomposition
For dispersed multiphase systems, it is ad
vantageous to express any physical quan
tity Ok(x, t), where k denotes the phase
(e.g., fluid or particles), as the sum of the
phasic mean 9k and a fluctuating compo
nent k where the former is defined to be
the ensemble average of the local volume
average;
a (r, t)k (r, t))
(41)
E (ak (r, t)),
1 k (a (r, t) (r, t)) ,(42)
where a Vk/V and Eqs. (41)(42)
involve a sum over realizations, each de
noted by s. Note that in Eq. (42), the
quantity 9k under the summation sign
on the righthand side has already been
volumeaveraged; i.e.,
9 (r,t)
V1 d3r'9(r',t), (43)
vk V
where y(r, t) is the exact value of y(r, t)
at the point (r, t), and Vk is a control vol
ume centered on r and covering only the
phase k. To simplify the notation in this
paper, we have omitted the curly brackets
({}) used in Eq. (43) to denote a volume
averaged quantity.
By its definition, the phasic average is
welldefined only over a control volume
which is sufficiently large; i.e., substan
tially larger than the characteristic inter
particle separation.
The phasic decomposition scheme satis
0 =
,k akok
ak
akoc" 0,
a
akI' k"
ak' k' ,
where overbars denote ensemble averag
ing, and single primes denote a fluctuating
quantity according to standard Reynolds
decomposition. The reader should note
that underlying Eq. (48) is a decomposi
tion of the volume fraction ak according to
the latter scheme, which is standard pro
cedure.
References
C. T. Crowe. On models for turbulence
modulation in fluidparticle flows. In
ternational Journal of Multiphase Flow,
26:719727, 2000.
D. Drazen and A. Jensen. TimeResolved
Combined PIV/PTV Measurements of
TwoPhase Turbulent Pipe Flow. 6th
Int. Conf. on Multiphase Flow, Leipzig,
July 913, 2007.
D.A. Drew and S.L. Passmann. Theory of
Multicomponent Fluids. Springer, New
York, 1999.
S. E. Elghobashi and T. W. AbouArab. A
twoequation turbulence model for two
phase flows. Physics of Fluids, 26(4):
931938, 1983.
R. A. Gore and C. T. Crowe. Effect of par
ticle size on modulating turbulent in
tensity. International Journal of Multi
phase Flow, 15:279285, 1989.
G. Hetsroni. ParticleTurbulence Inter
action. International Journal of Mul
tiphase Flow, 15(5):735746, SEPOCT
1989. ISSN 03019322.
D. Kaftori, G. Hetsroni, and S. Baner
jee. Particle Behavior in the Turbulent
Boundary Layer II. Velocity and Distri
bution Pi..ll. Phys. Fluids, 7:1107
1121, 1995.
I. Kataoka and A. Serizawa. Basic equa
tions of turbulence in gasliquid two
phase flow. International Journal of
Multiphase Flow, 15(5):843 855, 1989.
V.M. Kenning and C. T. Crowe. On the
effects of particles on carrier phase tur
bulence in gasparticle flows. Interna
tional Journal of Multiphase Flow, 23:
403408, 1997.
K.T. Kiger and C. Pan. Piv technique for
the simultaneous measurement of dilute
twophase flows. Journal of Fluids En
gineering, 122:811818, 2000.
H. Laux.
M. W. Reeks. "on the continuum equa
tions for dispersed particles in nonuni
form flows.". Physics of Fluids A Fluid
Dynamics, 4(6):12901303, 1992.
M. Righetti and G.P. Romano. Particle
Fluid Interactions in a Plane NearWall
Turbulent Flow. Journal of Fluid Me
chanics, 505:93121, 2004.
C. B. Rogers and J. K. Eaton. The effect
of small particles on fluid turbulence in
a flatplate, turbulent boundary layer in
air. Physics of Fluids, 3:928936, 1991.
R. Skartlien, D. Drazen, D.C. Swailes, and
A. Jensen. Suspension in Turbulent
Liquid Pipe Flow: Kinetic Modelling
and Added Mass Effects. International
Journal of Multiphase Flow, 2009.
K. D. Squires and O. Simonin. An analy
sis of twoway coupling in gassolid tur
bulent flows. APS Meeting Abstracts,
page K4, November 2002.
D. C. Swailes, Y. A. Sergeev, and
A. Parker. Chapmanenskog closure ap
proximation in the kinetic theory of di
lute turbulent gasparticulate suspen
sions. Physica A: Statistical and The
oretical Physics, 254(34):517 547,
1998.
D.C. Wilcox. Turbulence Modelling for
CFD, 3rd edition. DCW Industries, La
Canada, California, May 2006.
Y. Wu, H. Wang, Z. Liu, J. Li, L. Zhang,
and C. Zheng. Experimental investiga
tion on turbulence modification in hor
izontal channel flow at relatively low
mass loading. Acta Mechanica Sinica,
22:99108, 2006.
J. Young and A. Leeming. A theory of par
ticle deposition in turbulent pipe flow.
Journal of Fluid Mechanics, 340:129
159, 1997.
L.I. Zaichik and V.M. Alipchenkov. Statis
tical models for predicting particle dis
persion and preferential concentration
in turbulent flows. International Jour
nal of Heat and Fluid Flow, 26(3):416
430, 2005.
