7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
DNS of Particles Sedimentation with PhaseChange Effect
F. Dierich t, S. Ananiev and P. A. Nikrityuk t
SCIC Virtuhcon, TU Bergakademie Freiberg, Freiberg, 09599, Germany
SInstitute of Aerospace Engineering, Technische Universitit Dresden, Dresden, 01062, Germany
Frank.Dierich@vtc.tufreiberg.de and Petr.Nikrityuk@vtc.tufreiberg.de
Keywords: EulerLagrange, Melting, Ice, Particles, Fluid Flow
Abstract
The main background of this work is the study of the behaviour of particulate flows by taking into account the
interfacial heat transfer on the particle surface. As a benchmark test we consider ice particles moving up in water
due to the gravity force in a twodimensional channel. The Boussinesq approximation was employed to take into
account the change of the water density due to the heat transfer between ice and water. An implicit fictitious boundary
method (FBM) over a fixed Cartesian grid is extended to model the heat transfer and the phase change in particulate
flows in two dimensions. The hydrodynamic forces acting on the particles were calculated directly through the
surface integrals without the use of any semiempirical correlations. The particles collisions were modeled directly
by use of the hard sphere approach, taking into account the inelastic collisions. The interface velocity of the melting
(solidliquid) was calculated by means of the Stefan condition for each particle. The code was validated against
experimental and analytical data published in the literature. The computations were performed for different number
of particles with initial diameter of 0.5 103 m. The base cases correspond to the 'sedimentation' of 16 and
32 ice particles during their dissolution in the water. The initial temperature of the water and ice particles was
set to 10 C and 0 "C, respectively. To illustrate the impact of particle rotation caused by the viscous torques, a
set of simulations with and without rotational effect was performed. The comparative analysis of results showed
that in the case of taking into account the viscous moment, the melting time of particles is reduced significantly.
This fact is explained by the significant impact of particle rotation on the interfacial heat transfer and on the
particle trajectories. Additionally, based on the analysis of the time history of the volumeaveraged velocity in the
entire domain, three regimes were found. The first one is so called acceleration regime defined by speed up of parti
cles, the second one is the transitional and the last regime is the passive one, where the melted particles follow the flow.
Nomenclature
Roman symbols
c, heat capacity (J kg 1 K 1)
g gravitational constant (m s 2)
hf latent heat offusion(J kg 1)
n surface normal ()
p pressure (N m 2)
t time (s)
T temperature (K)
iu velocity vector (m s 1)
V particle volume (m3)
TM viscous torque (N m)
Greek symbols
liquid volume fraction ()
A heat conductivity (W K 1 m 1)
p dynamic viscosity (kg m 1 s 1)
p density (kg m 3)
Subscripts
1 liquid
s solid
i particle number
Introduction
The problem of modeling solidliquid twophase flows
including physical and chemical transformations is the
subject of intense research in many engineering sci
ences, e.g. in chemistry, metallurgy and geophysics.
The basic applications, where solidliquid flows play the
most important role, are fluidized beds, sedimentation
columns, slurries and mining. These applications have
motivated increasing theoretical and experimental stud
ies of solidliquid two phase flows. For a detailed review
of existing works devoted to the multiscale modeling of
twophase flow, in particular fluidized beds, we refer to
the works of Deen et al. (2007) and Van der Hoef et al.
(2008).
Due to the microscale character of the phasechange
process occurring on the interface between solid and
fluid phases, the most appropriate model for this project
is the resolved discrete particle model (RDPM), see
Deen et al. (2007), or direct numerical simulations
model (DNS). The main idea of all DNS models is to
embed an irregular solid particle/particles into a larger
simple domain and to specify noslip boundary condi
tions on the particle boundaries. Thus the fluid flow is
computed only between solid particles. In spite of nu
merous work devoted to the direct numerical simulation
of the isothermal particulate flows, e.g. see the represen
tative works Pan et al. (2002); Wang & Turek (2007),
there are only few works concentrated on modeling the
heat and mass transfer process in solidfluid twophase
flows, e.g. see Gan, Feng & Hu (2003); Koynov, Khi
nast & Tryggvason (2005); Feng & Michaelides (2008).
Recently, Feng & Michaelides (2008) reported about
direct numerical simulations of the nonisothermal par
ticulate flows for different Grashof numbers. The
Prandtl number of the fluid was set to unity. Simu
lation of two particles demonstrated that heat convec
tion around the particles modifies the draftingkissing
tumbling (DKT) scenario of their motion. In particular,
it was observed that the buoyancy currents induced by
the hotter particles reverse the DTK motion of the parti
cles or suppress it altogether.
Comprehensive DNS modeling of singe cylindrical
particles settling and melting in a hot fluid with Prandtl
number of Pr 0.7 has been reported in work Gan,
Feng & Hu (2003). It was shown that the melting rate of
each particle has a local character and strongly depends
on the sedimentation velocity of the particle. However,
these authors considered only two particles. Thus, the
influence of collisions on the particle dynamics has not
been investigated. To sum up, our understanding of the
complex interaction between melting particles and the
surrounding fluid is far from being complete. Motivated
by these facts, this work presents direct numerical simu
lations of ice particles moving up in the water. The im
plicit fictitious boundary method in finite volume formu
lation will be used to solve the problem Ananiev, Nikri
tyuk & Meyer (2009).
The present DNS method solves directly the mass
conservation, momentum and energy conservation equa
tions on a fixed Eulerian grid for the whole domain in
cluding particles. At the same time the particle dynamics
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
and their collisions are solved on a Lagrangian grid for
each particle. The particlefluid interaction are treated
by use of both grids. In this work we consider a particu
late flow system consisting of several circular ice parti
cles moving up due to the density difference in the two
dimensional channel filled with water. The Boussinesq
approximation is used for the fluid field to account the
variation of fluid properties due to the temperature gra
dients caused by remelting of ice particles.
Problem and Model Formulation
To proceed with the governing equations the following
basic assumptions have been done:
The fluid is incompressible and Newtonian. The
Eulerian grid is used to model the entire fluid flow.
The local thermodynamic equilibrium is satisfied
on the particle interface. Due to macroscopic size
of particles the GibbsThomson effect on the melt
ing temperature is negligible.
The shape of particle is cylindrical.
The heat transfer inside the particle is not consid
ered. Thus, the heat transfer equation is solved only
on the Eulerian grid.
The rate of particle melting is defined by the surface
averaged heat flux on the particle surface.
The particle collisions are inelastic and are modeled
by use of the hard sphere model.
Taking into account the assumptions done above the
conservation equations describing the behavior of ice in
water have the following form:
the continuity equation
V 0
the momentum equation
P t+po V = Vp+V ( (Vj + (Vg) +FB
(2)
where FB are the vector of buoyancy force resulting from
the Boussinesq approximation and
the heat transfer equation
Tpc
p C + pr, V (T)= AV2T.
at8
The hydrodynamic force and moment on the particle sur
face is calculated by the integrals
f, pjiidA + l A (V x d) x ldA (4)
J s. Jsi
and
TM, j v ( i) x ((V x d) x i) dA. (5)
Si
where denotes the coordinates of the surface and ri are
the coordinates of the mass centre.
The motion of the ice particles are modeled by use
of the Lagrangian approach where for each particle an
impulse conservation equation is solved. The equations
that govern the particle translation and rotation are the
following ones:
dvi
mi
dt
i i + tcols
J. '? r collis
1it =Tm i+ t (7)
where m, and 1, are the mass and the moment of the
particle, vi and Qi are its velocity and angular velocity
and Gi, Fi and T, are the force of gravity, the hydrody
namic force and moment, respectively. Here FEo~"i and
Tfcl" are the force and the torque due to the collision.
The coupling between the bulk flow and the parti
cles is done via the hydrodynamic force F, and the
torque TM, calculated directly for each particle by use
of Eqs. (4) and (5), respectively.
A phase change may involves volume change with a
jump in the normal velocity at the surface. In general
case due to the particle movement and its continuous
melting the particle shape depends strongly on the flow
Gan, Feng & Hu (2003). Thus, the location of the mass
center of the particle is time dependent, which can lead
to the additional rotational effect. The literature analysis
shows, that up to now there are no works which solved
this problem. Thus, as a first step in this work we use a
simplified 1D model to track the particle interface dur
ing its melting. The temperature inside the ice particle
is set to the melting temperature and the shape of the ice
particle is fixed to a cylinder which leads to a simplified
Stefan condition:
pshf dV A T dA, (8)
Sdt Is, On
where Vi is the volume of the particle.
Water Properties
The properties of water are taken from the VDI
Warmeatlas (2006). We derived a polynomial for
the viscosity of water (valid for 273 K to 288 K) in
kg m1 s1
S= 9.5989 10 6.3085 104 T (9)
+1.047 10 6T2
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
In Eq. (2) we use the Boussinesq approximation in a
quadratic form to consider the density maximum of wa
ter, FB can be expressed as follows:
FB = po [,3(T To) + 2(T To)2]
with the parameters (valid for 273 K to 288 K)
To 273.15 K
Po = 999.843 kg/rm3
31 = 0.57776710 104 K 1
32 0.72210337 .105 K 2
(10)
also derived from the VDI Warmeatlas (2006). The
thermal conductivity of the water was assumed to be a
constant 0.5723 W K 1 m 1. The thermal capacity cp
of the fluid was entered as a constant 4205 J kg 1 K 1
The density of the ice ps = 916.7 kg m 3 and the la
tent heat hf = 333.6 103 J kg 1 are taken from Lide
(2002). Thus, the Prandtl number of the water is about
9.6.
Numerics and Code Validation
The set of transport equations has been discretized by a
finitevolume and finitedifference based method. A grid
with 200 x 1000 control volumes (CV) was used. The
SIMPLE algorithm with collocatedvariables arrange
ment was used to calculate the pressure and the veloc
ities, for details see Ferziger & Peric (2002). The Rhie
and C ho"\ stabilization scheme was used for the stabi
lization of pressurevelocity coupling, see Rhie & C ho"\
(1983). The system of linear equations was solved by us
ing Stone strongly implicit procedure, see Stone (1968).
The time derivatives were discretized by a threetime
level scheme. The convection terms were treated by
the deferred correction scheme, which converges to the
secondorder central difference scheme. For details de
scribing the full algorithm we refer to Ferziger & Peric
(2002). Time marching with a fixed time step was used.
The time step was equal to 5 104 and 1 10 4 for the
cases of particulate flows without and with rotational ef
fect due to viscous torque, respectively. For every time
step the outer iterations were stopped if the normalized
maximal residual of all equations is less than 1 104.
To set up the noslip and Dirichlet boundary condi
tion for the momentum and energy conservation equa
tions on the particle surface, respectively, we use orig
inal implicit fictitious boundary method (FBM) intro
duced by Zienkiewicz (1971); Turek, Wan & Rivkind
(2003). The finite volume formulation was developed
by Ananiev, Nikrityuk & Meyer (2009). The main idea
of this method is the modification of the matrix coeffi
cients in such a way that the boundary conditions in the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
interface control volumes are satisfied automatically. In
particular, to illustrate the basic principle of FBM, we
consider the system of algebraic equations arising from
the use of the fivepoint difference scheme, e.g. for the
equation Eq. (3) given by:
[A] T QT
where [A] is the sparse matrix of known coefficients, T
is the column vector of unknowns, and QT is a right
hand side vector. In order to fix the temperature in
side the immersed region, we set the elements of ma
trix [A] equal to unity in the solid region and subtract
corresponding values from QT, details can be found in
Zienkiewicz (1971).
To validate the interface tracking algorithm a diffu
sion controlled solidification of a cylindrical ice particle
was modeled. This problem has a similar solution in
the case of semiinfinite domain filled with undercooled
melt, Carslaw & Jaeger (1960). This problem is often
used as a benchmark test for the validation of programs
dealing with phase change problems, e.g. see Gan, Feng
& Hu (2003); Li et al. (2003). In particular, in the
following analytical solution the radius of the cylinder
growth obeys the following equation, Carslaw & Jaeger
(1960):
Re(t) 2rat (12)
where a is the thermal diffusivity of the fluid and r is
the root of the following relation:
r2 e72 Ei (_2) + St = 0 (13)
where St is the Stefan number and Ei is the exponential
integral function defined by:
ej T
Ei (x) El (x) 
Ji T
dr for x > 0. (14)
00016
00015
00014
% 00013
00012
00011
0 001

analytic
o numeric
40 50 60 70 80 90
Figure 1: Validation Case 1: Time history of the radius
of the icecylinder in undercooled water. The
undercooling temperature was set to 10 OC.
Figure 1 shows numerically simulated and analytical
results for the ice interface location in time dependence.
It canbe seen that the agreement between numerical pre
dictions and exact solution is good.
Next, we validated our FBM model and the code
against experimental results of [Kuehn (1976); Kuehn
& Goldstein (1978)]. The test case compares the nu
merical prediction of the temperature profiles along the
symmetry lines with experimental data. The experimen
tal set up included a heated inner cylinder, placed in the
center of another cold cylinder. Due to the gravity field
a buoyancyinduced flow occurs. The temperature con
tour plot and the flow pattern are shown in Fig. 2. The
comparison of the temperature profiles along the sym
metry line compared with the data of Kuehn (1976);
Kuehn & Goldstein (1978) are given in Fig. 3. It can
be seen that the agreement between our predictions and
the experimental data is very good.
For the numerical calculation of Ei, see also Press,
Teukolsky, Vetterling, Flannery (1992). The tempera
ture in the fluid phase is given by:
Tm To Ei r2
T(r) To + Ei (n2) Ei t (15)
To compare the analytic and numerical solutions, the
ice particle with initial diameter of do = 1mm was en
closed in a square box of the size 100 do x 100 do. The
initial conditions were derived from the Eq. (15) cor
responding to the time to, when the particle diameter
reached the initial diameter do. The number of control
volumes in X and Y directions was set to 900 x 900.
The temperature of the undercooled water was set to
To = 10 C and the melting temperature Tm equals
zero.
Results and Discussions
Before we proceed with the analysis of the results, let us
recall briefly the description of the investigated setup.
We consider a two dimensional enclosed cavity filled
with water. The height is Ho 50 do and the width
is Lo = 10 do, where do is the initial diameter of the
ice particle. In this work do equals 0.5mm. The initial
temperature of the water is 10 C. The temperature of
the ice particles equals the melting temperature Tm = 0
C. The cavity has noslip and adiabatic, impermeable
walls. Initially, the ice particles are placed on the bottom
of the cavity. Two basic cases are considered: 16 and 32
ice particles in order to investigate the influence of colli
sions on the particle dynamics. Additionally, in order to
investigate the impact of particle rotation caused by the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
370
360
H350
328.0 334.9 341.9 348.8 355.8 362.7 369.7
0.094
y, m
Figure 2: Validation Case 2: Spatial distributions of the
velocity vectors and the temperature isolines.
Here every 3rd vector is shown.
!
S experiment
present work, grid 50x50
 present work, grid 100x100
,6
7
i/e'
_Fj_E W4,144
0 0.01 0.02 0.03
x, m
370 
360 ~ "
350 experiment 0
I present work, grid 50x50
 present work, grid 100x100
340 
330
0.06 0.07 0.08 0.09
x, m
Figure 3: Validation Case 2: Axial profiles of the tem
perature at y = 0.5D1. Here D1 is the diam
eter of the outer cylinder.
t, sec
1   ^ ^
099
0 98 
097 316 particle, without viscous moment
S9 32 partcle, without viscous moment
16 particle, with viscous moment
32 particle, with viscous moment
0 96
095 '
0 2 4 6 8 1
t, sec
Figure 4: (a) Time history of the volumeaveraged ve
locity U,. and (b) the total volume fraction of
water ,, calculated for different scenarios.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
viscous moment, TM, (see Eq. (5)), on the flow pattern
we performed a set of simulations where this term was
excluded from Lagrangian equation for the calculation
of particle rotation Eq. (7).
To characterize the system we use the following di
mensionless parameters:
where
B 1 Hi P (To Tm)
Gr o
U Ido Po Ps
Uref = g P do
PO
(16)
(17)
(18)
Here po is the molecular viscosity of the water by the
initial temeparture To. After substitution of the water
properties and geometric parameters into Eq. (16) we
have U,,f 0.0202 m s1, Re 7.73 and Gr
51.93 103. The density ratio between the solid parti
cles and the liquid phase is PO = 1.091.
In order to analyze the temporal evolution of the
global flow field induced by the particles and the particle
melting rate the volumeaveraged velocity and the total
volume fraction of fluid in the domain are expressed as
follows:
1 [Ho 'LO
U. av= L L + , dx dy
HoLo JO Jo
1 Ho o 0 Lo
H"0L0 Jo Jo
(19)
: dx dy (20)
10 2 4 6 8 10
a b
Figure 5: Trajectories of the 16 melting particles calcu
lated (a) without and (b) with rotational effect.
The resulting histories for both Uav and Eav are depicted
in Fig. 4 for different number of particles. It can be
seen that the increase of the number of particles leads to
the higher velocity induced, which is somehow logical.
Each particle generates the fluid flow due to its upward
movement and no slip condition on its surface. Next
we deal with the question on the influence of viscous
torque on the particle behavior. For that purpose we plot
the evolution of the particles trajectories in Fig. 5. The
comparison of trajectories calculated with and without
taking into account rotational effect shows significant
differences in particles paths. To explain it in detail we
start with the classification of flow regimes, which can
be tracked by the time history of the volumeaveraged
velocity Uav.
The analysis of Fig. 4a shows the existence of three
phases. We identify the first phase as acceleration phase,
characterized by the speed up of particles from rest due
to the gravity. This phase is finished when Uav reaches
its first maximum, see Fig. 4a. The snapshots of the
nondimensional velocity vectors, u~ , and the contour
plot of the nondimensional temperature, T T, for this
To T'
2 4 6 8
poUrefdo
regime are shown in Figs. 6 13, second column from
the left. It can be seen, that independent on the number
of particles we have quite different flow pattern, if the
viscous moment is excluded from the model (no rota
tion). Without rotation the ice particle forms 'bubble'
like structure. This can be explained as follows. As the
particles start to rise, the central particles move faster,
and the near wall particles come up slower due to the
friction effect. However, in the case of including the vis
cous moment into the momentum balance equation 7,
rotation of ice particles occur due to the presence of a
velocity gradient such as the shear layer near the chan
nel wall. As a result, the rotation causes fluid entrain
ment, increasing velocity on the one side of the particle
and lowering the velocity on the other side. This phe
nomenon, known as the Magnus effect, tends to move
the particle toward the region of higher velocity, e.g. see
Soo (1967). In our case, particles begin to migrate to
the center of the channel. It can be seen, that at the first
and second phases the particles tend to concentrate at the
distance of about 0.3Lo and 0.7Lo from the left and right
side, respectively. Thus, the particles form the 'M'like
structure. These findings are in good correlation with
experimental data of Segre & Silberberg (1961).
The second phase is the transitional one. This phase
is characterized by progressive melting of ice particles
and by the local acceleration and braking of particles due
to the local DKT (drafting, kissing, and tumbling) effect
explained below. Comparison of Figs. 10 and 11 reveals,
that within this regime in the case of not including the
viscous moment, the particle trajectories are closed to
the so called DKT scenario, e.g. see Feng et al. (1994).
The main feature of drafting is the suction effect, which
is caused by the leading particle by creation of a wake
behind it. As a result the nearest particle will be caught
up by the leading particle. Afterwards, the so called
kissing occurs, which means that both particles move
close with almost the same velocity. However, due to
collisions and unsteadiness of particle motion the kiss
ing state is unstable. Thus, finally the so called tumbling
happens, leading to the separation of particles. The en
tire DKT scenario can be repeated again with other par
ticles.
The third phase is the passive regime. Due to the per
manent melting of the ice particles their size is decreas
ing. As a result, from active players particles become
passive flowers, which follow the flow induced in the
past. This can be seen in Figs. 6, 7, 10 and 11 fifth
column from the left.
It must be pointed out, that in this work we have not
found significant impact of buoyancy induced flow near
the particles on the particle trajectories in comparison to
the work Feng & Michaelides (2008).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Conclusions
The behavior of particulate flows by taking into account
the interfacial heat transfer on the particle surface was
studied numerically. As a benchmark test we considered
the ice particles moving up in the water due to the gravity
force in twodimensional channel. An implicit fictitious
boundary method (FBM) over a fixed Cartesian grid is
extended to model the heat transfer and the phase change
in particulate flows in two dimensions. The hydrody
namic forces acting on the particles were calculated di
rectly through the surface integrals without the use of
any semiempirical correlations. The particles collisions
were modeled directly by use of the hard sphere ap
proach taking into account the inelastic collisions. The
interface velocity of the melting (solidliquid) was cal
culated by means of Stefan condition for each particle.
The numerical simulations revealed the existence of
three regimes. The first one is the so called acceleration
regime defined by speed up of particles, the second one
is the transitional and the last regime is the passive one,
where the melted particles follow the flow.
Comparative analysis of different models showed the
importance of taking into account the viscous torques
by calculation of the particle trajectories. In particu
lar, the results demonstrated that inclusion of the vis
cous moment into the Lagrangian equation for the par
ticle movement leads to the 'M'like particulate pattern.
More over, the rotational effect reduces the melting time
of particles significantly. This fact is attributed to the
enhancement of the heat transfer due to the rotation of
particles.
Acknowledgements
Authors appreciate the financial support of the Govern
ment of Saxony and the Federal Ministry of Education
and Science of Federal Republic of Germany as a part of
Centre of Innovation Competence VIRTUHCON.
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0.5
50
40
30
20
10
0
0 5 10
X/do
0.00 s
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.5
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; **>
0 5 10
X/do
2.00 s
0 5 10
X/do
3.50 s
0.5
: :
0 5 10
X/do
6.00 s
50 .
40
30
20
10
0 5
0 5 10
X/do
7.40 s
Figure 6: Snapshots of the nondimensional velocity vectors, , calculated for 16 particles without taking into
account the rotational effect caused by the viscous torque.
50
40
* * * * I s r.l : . :
0 O I, 0 0C 0 0 ,
0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10
X/do X/do X/do X/do X/do X/do
0.00 s 0.50s 1.05 s 2.00 s 3.00s 4.50s
Figure 7: Snapshots of the nondimensional velocity vectors, ', calculated for 16 particles by taking into account
the rotational effect due to the viscous torque TM,.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0 5 10 0 5 10
X/do X/do
0.00 s
0.80 s
10
0 5 10 0 5 10
X/do X/do
2.00 s
3.50 s
6.00 s 7.40 s
Figure 8: Snapshots of the nondimensional temeprature, contour plots calculated for 16 particles without
taking into account the rotational effect.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
50r 50 50
11 40 40 40
1 30 30 30
11 20 20 20
10 10 10
I00 0L 0
0 5 10 0 5 10 0 5 10 0 5 10
X/do X/do X/do X/do
0.00 s
0.50 s
1.05 s
2.00 s
X/do X/do
3.00 s
4.50 s
Figure 9: Snapshots of the nondimensional temeprature, T T, contour plots calculated for
into account the rotational effect due to the viscous torque.
16 particles by taking
0.5
50
40
30
20
10
0 5 10
X/do
0.00 s
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
X/do X/do
0.80 s 2.00 s
X/do
3.50 s
*i
0 5 10
X/do
6.00s
63
5 0 . 5
30
20
10
1 0 . ..: :.
0 5 10
X/do
7.40 s
Figure 10: Snapshots of the nondimensional
account the rotational effect.
) 5 10
X/do
0.00 s
0 5 10
X/d0
0.50 s
velocity vectors, u , calculated for 32 particles without taking into
50
40
0 5 10
X/do
0.80 s
Figure 11: Snapshots of the nondimensional velocity vectors,
the rotational effect due to the viscous torque.
1.80s
I' '
0 5 10
X/do
3.40s
0 5 10
X/do
5.50 s
", calculated for 32 particles by taking into account
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0 5 10 0 5 10
X/do X/do
0.00 s
0.80 s
50
40
30
20 4
.0
10
0 5 10
X/do
2.00s
Figure 12: Snapshots of the nondimensional temeprature,
taking into account the rotational effect.
T T contour plots calculated for 32 particles without
To T_
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0 5 10 0 5 10
X/do X/do
0.00 s
0.50 s
0 5 10 0 5 10
X/do X/d,
0.80 s
1.80s
Figure 13: Snapshots of the nondimensional temeprature, T T contour plots calculated for 32 particles by taking
into account the rotational effect due to the viscous torque.
3.50 s
6.00 s
7.40 s
3.40 s
X/do
5.50 s
