Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Direct numerical simulation of particle dispersion in swirling jets
Nan Gui, Jianren Fan and Song Chen
Zhejiang University, State Key Laboratory of Clean Energy Utilization
Zheda Road 38#, Hangzhou, 310027, People's Republic of China
Email: fanir@iziu.edu.cn, ziuguinan(@gmail.com
Keywords: swirling jets; particle dispersion; direct numerical simulation; vortex breakdown
Abstract
In the present study, we carried out a direct numerical simulation of particle dispersion in swirling jets. The swirling jet is
issued into a rectangular container through a round nozzle with a diameter of 0.4mm. The gasphase flow is directly simulated
through solving the NavierStokes equations by the finite volume method and the fractionalstep projection technique on a set
of 384*128*128 grids. A critical value of swirl number of S=1.42 is used when the bubble vortex breakdown takes place. Five
types of particles with Stokes numbers of St=0.01, 0.1, 1, 10 and 100 are studied respectively on their dispersion
characteristics in the jets with the same flow rate of number concentration. Additionally and comparatively, three types of
particles with respectively St=0.5, 1 and 10 are also simulated with the same mass loading.
Based on the simulation, we found the configuration of the container induces an important modification to the basic
structures of vortex breakdown, and these effects influence the particle dispersion characteristics in a predominant manner. A
nonuniform dispersion of particles in the peripheral region of the vortex breakdown is observed, which is mainly dominated
by the modification of coherent structure of vortex breakdown. A quantitative analysis of the nonuniform spatial particle
dispersion is carried out. The differences in the spatial dispersion for different particle Stokes numbers as well as different
mass ladings and number flow rates are compared and explored analytically.
Introduction
Gassolid dispersed flow is of great importance in both
scientific researches and engineering applications, e.g. the
dispersion characteristics of coal particle is important for
transportation in ducts and combustion efficiency in
combustion devices, etc. It exhibits a variety of interesting
phenomena, e.g. preferential accumulations (Elghobashi &
Truesdell 1992, Squires & Eaton 1991) etc. Many
researches have shown that the preferential particle
concentration is associated with largescale structures which
disperse particles effectively and dominate particle motions
(Crowe, et al., 1988; Ling, et al., 1998; Longmire & Eaton,
1994).
The particle dispersion in swirling flows is important for
swirling combustion systems and gassolid cyclone
separators. With regard to particle concentration in swirling
flows, Wicker & Eaton (2001) showed the presence of large
vortex structures which have similar effects on particle
distribution. Apte et al. (2003) and Gui, et al., (2010) carried
out respectively a largeeddy simulation and a direct
numerical simulation of swirling particleladen flows in a
coaxialjet combustor corresponding to a previous
experiment by Sommerfeld & qiu (1991, 1993), focusing on
the particle dispersion characteristics and distribution
patterns, etc. In these studies, particle dispersion is shown to
be related closely to the largescales vortex structure as well
as the particle response characteristics. The particle
response property is appropriately characterized by the
Stokes number, which plays an important role in
preferential concentration. Heavy particles tend to
accumulate in regions where the strain rate dominates over
vorticity, whereas light particles tend to accumulate in
regions of intense vorticity (Balachandar & Eaton, 2010).
For the numerical simulation of dispersed flow, the
Lagrangian pointforce/particle method has been used for a
long history, which uses either a oneway coupling
approach (Elghobashi 1991, Maxey 1987, Squires & Eaton
1991) or a twoway coupling approach (Elghobashi &
Truesdell 1993). Under this approach, the dispersed phase is
considered as discrete points and depicted under the
Lagrangian framework through solving the equations of
motion, without taking into account the microeffects of
particle volumes, such as wakes after particles. Alternatively,
the carrier phase is simulated under the Eulerian approach
by some types of CFD technique, such as RANS, LES or
DNS, etc.
The present study focuses on the particle dispersion
characteristics in a gassolid swirling flow with a relatively
large swirl number but low and moderate Reynolds numbers
partially corresponding to a previous experiment by Billant
et al., (1998). In addition, two conditions with the either
particle number flow rate fixed or the particle mass loading
fixed are simulated to explore the multiple effects of particle
property, mass loading, and large scales of fluid vortices on
the behavior of particle dispersion. The EulerianLagrangian
description is adopted, in which direct numerical simulation
of the continuum phase and Lagrangian tracking of the
Paper No
discrete phase are performed, respectively.
Nomenclature
CD Drag coefficient ()
d Jet diameter at the inlet (mm)
f, Drag force of particles ()
f Drag factor ()
mi Mass loading of particles ()
mp Particle mass (kg)
np Number flow rate of particles, (/step)
p Fluid pressure (Pa)
Re Reynolds number ()
Rep Particle Reynolds number ()
r(x, 0) Integral radial dispersion function
r,(x, 0, t) Radial distance from the jet axis at (x, 0, t)
S Swirl number ()
S, Fluid strain rate tensor
St Stokes number ()
u,u Fluid velocity (vector) ()
Uo Inflow mean velocity of fluid (m/s)
vp Particle velocity ()
V,, Max azimuthal velocity of fluid at the inlet ()
Xp Particle position vector ()
Greek letters
At Simulation time step
e Turbulent dissipation rate
1r Kolmogorov length scale (m)
p Kinetic viscosity of fluid (Pa's)
v Kinematic viscosity of fluid (m2/s)
p Fluid density
Subsripts
f flow
1 loading
max maximum
p particle
Governing equation of fluids
The governing equation for the carrier phase is the three
dimensional, time dependent, incompressible NavierStokes
equations
u 0 (1)
=0 (1)
ax
t 9 +
at 8x
at
ap 1 a a
, Rex fp
8x Re ax x '
where u,, p are fluid velocity and pressure respectively, and
f, is the drag force of particles. When the particle density is
far larger than the fluid, the drag force is of the leading
order compared to other hydrodynamic forces. Thus, the
other forces, such as virtual mass force, Saffman force,
Magnus force, pressure gradient force and buoyancy forces
are omitted. As a result, only the fp is considered here as the
backward force from particle to fluid.
The dimensionless flow domain is 30dx10dx10d (Fig. 1),
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
where d is the inlet diameter of the jet. The inlet axial and
azimuthal velocity profiles are given in Fig.2. The outlet is
the nonreflecting boundary (Orlanski, 1976). Otherwise,
the side walls are set as nonslip wall boundaries.
The Reynolds number is Re=U., =.;in; and 3000 for
case 1 and case 2 respectively (Table 1), where Uo is the
dimensionless inflow mean streamwise velocity. The swirl
number is defined as the ratio of maximum azimuthal
velocity to the mean streamwise velocity at the inlet
S=2Vm /Uo, which is fixed as S=1.42 here.
To solve the above equations, a total number of
384x128x128 grids are used, which can resolve the scales
of turbulence as fine as about 0.075d. The Kolmogorov
length scale rf ( (v3/E)1/4) is estimated as 7 = 0.065d when
Re=606. Thus, the spatial resolution requirement for direct
numerical simulation is met. The simulation time step is
JAt0.005, and a total 100 dimensionless time is computed.
To perform numerical solution, the finite volume method
and the fractionalstep projection technique (Chorin, 1968)
are applied. An explicit lowstorage, thirdorder Runge
Kutta scheme (Williamson, 1980) is used for time
integration. A direct fast elliptic solver is used to solve the
Poisson equation.
r x AXI outlet: nonreflecting
Figure 1: Sketch of the simulation setup.
a b
0.75s
0.5
o0.25 
2 1 0 1 2 2 1 0 1 2
r/R r/R
Figure 2: Given profiles of inflow axial velocity (a) and
azimuthal velocity (b).
Motion equation of particles
In this study, focusing on the dispersion characteristics of
particles, the discrete phase is assumed to be: 1) a dilute
flow regime, where the particleparticle collisions are
omitted; b). spherical particles with uniform diameters and
densities; c). as demonstrated by Gui, et al., (2010), the
Saffman force and Magnus force are of the secondary
importance when compared to the drag force, they are
omitted here as well as other types of hydrodynamic forces.
Hence, only the drag force is computed.
Based on these assumptions, the motion equations of any
discrete particle are solved in a deterministic way. For any
j/
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particle, the particle motion equation is
dV ;r dp2
MP " P CD p u p) (3) (3)
dt 8
where CD=24 I'F is the drag coefficient. f 1+0.15Rep,687
(Clift et al., 1978) is the drag factor and Rep=luvpd,/v is
the particle Reynolds number. By simple deductions, Eq.3
is reduced to
dvp f
= (u Vp) (4)
dt 7p
Where rp=pdp2/(18p) is the particle aerodynamic response
time. Finally, Eq.4 is nondimensionalized
dt S(U (5)
dX=
dt P (6)
where St=r, /r=rp /(d/Uo) is the Stokes number.
Initially, the particles are generated in the crosssectional
area at the jet inlet with a random and uniform distribution.
To study the characteristics of particle response behaviour, a
difference between the inlet velocities of particle and fluid is
required. We set the inlet velocity of particles vp(0)=0.59U0
here.
In this study, two cases are simulated (Table 1): At first,
we kept the number flow rate of particles and study the
dispersion characteristics of particles under different Stokes
numbers at a low Reynolds number. Alternatively, we kept
the mass loading of particles and study the dispersion of
particles under different number flow rates at a moderate
Reynolds number.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
covers, concentrating around the central region of four
large scale vortices. It is considered as the coherent structure
dominated pattern of dispersion of particles.
y/d
Figure 3: Threedimensional largescale fluid vortex
structure (a) and its crosssectional visualization (b).
Table 1: Two simulation cases
Case 1: (Re=606) n,=10/step, St
Case 2: np=268/step
(Re=3000 np=95/step
ml=0.134) n,=3/step
0.01, 0.1, 1, 10, 100
St=0.5
St=l
St=10
Results and Discussion
1. Large scale vortex structure
At first, Fig.3 shows the structure of largescale vortices
with regard to the vortex breakdown for Case 1 (at t=5,
x=10d). It is shown that there exists an evident recirculation
zone enclosed by the largescale vortices (Fig.3a). It is
socalled as the bubble vortex breakdown. Moreover, it is
found that the development of large scale coherent structure
of vortices is confined by rectangular configuration of
container. Four large vortices are observed in the covers of
the container symmetrically where they have more space to
develop, whereas they are greatly restrained by the side
walls of the flow domain (Fig.3b). Thus, it is concluded that
the configuration of the container makes an important
modification to the basic structures of vortex breakdown,
inducing a breaking of symmetry from axisymmetric
distribution of vortices into four main large scale vortices in
the covers.
Correspondingly, the dispersion of particles is dominated
by the symmetrical modification of large scale structure of
vortices, especially for small particles (St=0.01, Fig.4). It is
observed that they are dispersed dominantly by the large
scale vortices from the jet central region to the peripheral
.
3 2 1 0 1 2 33
y/d
Figure 4: Particle dispersion for St=0.01 dominated by the
large scale vortices in Fig.3.
2. Case 1: Keeping number flow rates of particles
To evaluate the dispersion of particles in the swirling jets,
especially the particle dispersion characteristics under the
influence of the fluid vortex structure, we used a integral
dispersion function (IDF) r(x,O) or r(x) defined as follows
r(x,O)= fJmr(x,O,t)dt/Jmdt (7)
r(x)= fr,(x,0)d (8)
where r,(x,O,t) is the distance from particle at (x,O,t) to the
jet axis. The IDFs indicate the mass averaged dispersion of
particles in the radial direction. Similarly, we defined a mass
averaged dispersion velocity of particles as follows:
,(x, 0)= \m,v,(x,O,t)dtfmpdt (9)
I
Paper No
v(x)= Jv,(x,0)dO (10)
The typical results for the integral dispersion function are
illustrated in Fig.5. It is observed that for St=0.01 (Fig.5a)
the particles are welldispersed by the fluid vortex. At
0 0.257E, 0.757E, 1.257E and 1.757E, corresponding to the four
covers of the container respectively, the maximum of IDFs
is over 4d (from the comer point to the container center is
5 J2 d); whereas at At 0=0.57n, 7E, 1.57E and 27E, which
corresponds to the four sidewalls of the container
respectively, the IDFs are about 3d. Referring to Fig.4, it is
concluded that the particles are welldistributed in and
around the vortex breakdown region. The dispersion is
mainly dominated by the fluid vortex structure, presenting a
symmetrical distribution of particles within the largescale
vortices enclosing the central recirculation zone. In contrast,
for St= (Fig.5b), the particle dispersion is mainly restrained
within the vortex breakdown region, with outside regions
almost empty as well as the four comers. Thus, the IDF
doesn't have symmetrical four peaks as before. In addition,
for St=100 (Fig.5c), the particle dispersion seems to be
fairly independent of largescale vortex structure. No radial
dispersion is observed, quite like a particulate direct jet
without any dispersion. The difference of particle dispersion
characteristics is due to the response characteristics. As
wellknown, the Stokes number is the ratio of particle
response time to the characteristic time scale of fluid. Small
particles respond to fluid vortex motion quickly and follow
the fluid motion well whereas large particles respond slowly
and follow the vortex motion badly.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
is fully developed and the IDFs reach the maximum peaks.
Moreover, at about x=25d, the streamwise dispersion of
particles is throughly prohibited even in the centre of the jet.
It indicates the existence of a limit region for dispersion of
small particles, which depends greatly on the structure and
motion of large scale fluid vortices. It is well known that the
vortex breakdown takes place and a recirculation zone is
established (Fig.6b) when the swirl number is large enough.
The streamwise motion of fluid is greatly attenuated after
the recirculation zone (Fig.6b). As a result, for the particles
which moves following the fluid motion well, the
streamwise dispersion is limited due to the attenuation of
the jet.
x/d
 t=o.o1
recirculation zone St=O.1
with vortex breakdown St=l
. St=10
St=100
I
3 6 9 12
x/d
Figure 5: Integral particle dispersion function in the
streamwise and azimuthal plane (x, 0) for St=0.01 (a), 1 (b)
and 100 (c), respectively.
Then, by integrating the r(x,0) or vp(x,0) in the 0 direction,
we obtained the results of r(x) or v(x), respectively. The
results are showed in Fig.6. It is observed from Fig.6a that
at the immediate outlet of the nozzle (x=l.25d, Fig.1), small
particles (St=0.01 and 0.1) are dispersed suddenly and
immediately from the core of the jet to the peripheral
regions, jumping from 0.5d to about 2.53d within a very
short axial distance. At about x=12d, the radial distribution
x/d
Figure 6: Integral particle dispersion function (a) and the
streamwise (b) and radial components (c) of the mass
averaged particle dispersion velocity.
Quite on the contrary, for large particles (St=10 and 1000),
they are almost not dispersed by the expansion of fluid
vortices (Fig.6c, v, 0) even within a very long axial
distance (010d). Moreover, the velocities are not affected
much by the fluid motion too (Fig.6a, b and c). Additionally,
the intermediate Stokes are of transitional characteristics of
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dispersion (St=1).
In addition, from Fig.6c, it is found that the radial
dispersion velocities for small particles are varied. It can be
divided mainly into five subzones, i.e. ZO, zi, z2, Z3, z4.
Subzone zo is within the nozzle before the vortex breakdown,
whereas the others are after the vortex breakdown. These
subzones are separated by locations with nearly zero or
locally minimum radial dispersion velocities. The existence
of clearly separated subzones indicates an evident coherent
motion of largescales vortices (Fig.3a and b), which moves
downstream regularly in a spiral type. It provides another
possible validation for the coherent dominated dispersion
characteristics of particles.
3. Case 2: Keeping mass loadings of particles
In the above sections, the results are for the same number
flow rate of particles but with different mass loadings,
which increase as ~St3/2. However, the results with the same
mass loadings are important and necessary to be studied,
since under the same mass loadings, the momentum
exchanges between the fluid and particle phase would be
approximately of the same order.
Fig. 7 shows the dispersion results for case 2. Compared
Fig.7a to Fig.6a, for the small Stokes numbers (St=0.5 and
1), it is found that the maximum dispersion functions are
decreased. Referring to Table 1, the number flow rate of
particles of small Stokes numbers for case 2 is far larger
than that of case 1. Thus, the mass loading for case 2 is far
heavier than that of case 1. Given the same flow condition,
the energy for large scales of vortices of fluid to disperse
particles is not increased, whereas the particle loadings are
increased greatly. For this reason, the obtained dispersion
energy from fluid for each particle is attenuated which
causes a consequential decrease of the dispersion functions
of Fig.7a.
Quite on the contrary, for large particles (St=10), the
number flow rate for case 2 is less than case 1, which
indicates the mass loading for case 2 are smaller than that of
case 1. Thus, as indicated by Fig.7a with reference to Fig.6a,
the dispersion energy for each particle obtained from the
large scales of fluid vortices should be increased, which
results in an increase of the dispersion function.
The above analysis on the effects of the balance of energy
or momentum exchange between the carrier phase and the
dispersion phase is also indicated by Fig.7b and c.
Compared Fig.7b to Fig.6b, for St=0.5 and 1, the
streamwise component of dispersion velocity doesn't get
decreased to zero within the recirculation zone. It is because
the loadings are increased and the exchange of momentum
between the gas and particle phases is not sufficient to
decrease particle velocities to zero. Meanwhile, in Fig.7c,
the subzones separated by nearly zero or locally minimum
radial dispersion velocities are not as clear as that in Fig.6c.
In contrast, for large particles (St=10), as the mass loadings
decreased, the attenuation of particle streamwise velocity is
more rapid and evident than before, which is caused by
sufficient exchange of momentum for each particle to follow
the fluid motion. The similar conclusion is also indicated by
Fig.7c.
Thus, by comparing the results of case 2 and case 1, it is
reasonable to conclude that the response characteristics of
particle to fluid motion are not only related to the intrinsic
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
characteristics of particle property and fluid flow conditions,
but also related closely to the loadings of particles. To say
specifically, the heavy loadings of particles can make the
insufficient momentum exchange between the large scale
fluid vortices and any dispersed particle, and cause the less
rapid response of particle to fluid. Quite on the contrary,
light loadings can cause large particles to respond to fluid
motion well due to the sufficient exchange of momentum or
energy between the particles and the energetic fluid vortices.
b 0.8
0.6
' 0.4
0.2
0.0
5 10 15 20 25 3
0 5 10 15 20 25 30
x/d
Figure 7: Integral particle dispersion function (a) and the
streamwise (b) and radial components (c) of the mass
averaged particle dispersion velocity.
4. Probability density functions
Finally, we would like to show some interesting results on
the dispersion of particles related to the invariants of fluid
strain rate tensor. As wellknown, the fluid strain rate tensor
is S,=l 21" i'o 1' +i,1. 1', I The first and second invariants of
the tensor are I=S11+S22+S33 and I2=811S22S12S21+S22S33
S23S32+S11S33S13S31, respectively (11=0 for any instantaneous
incompressible fluid field). We think to use the time
averaged fluid field here instead of instantaneous flow filed
is more meaningful for engineering consideration. Thus, the
 St=0.5
 St=l
St=10
Paper No
invariants are calculated based on the mean fluid fields and
they are called the invariants of the timeaveraged strain rate
tensor. In this way, I1 is not necessary to always be zero.
Thus, we calculated the probability density functions
(PDFs) of integral dispersion of particles on the invariants
of I1 and 12, i.e. to integrate the number of dispersed
particles over all time on any given possible value of the
invariants, and then normalize it. The results are shown in
Fig.8a and b for PDFs of particle dispersion on invariants I1
and I2 respectively.
From Fig.8, it is seen that the PDFs on I1 is much
narrower than those on 12. It is reasonable since the mean
value of I1 is close to zero. However, it is still of large
probability for the extreme events to occur, i.e. for the
dispersed particles to occur on large I1 (Fig.8a). It means the
particles are possible to occur in the regions with large line
strains. Moreover, the PDFs for large Stokes number are
narrower than those for small Stokes numbers. It means that
small particles are more possible than large particles to
occur at points with extremely large line strains.
In addition, it is seen that the PDFs on 12 have long tails
in the negative direction (Fig.8b). It means that particles are
more likely to occur in the regions where the shear strains
dominate over linear strains. Moreover, the PDFs for small
Stokes numbers are wider than those for large Stokes
numbers too. Thus, it seems that the small particles are more
likely than large particles to occur in the regions with
extremely large shear strains.
st=.01
10 St=0.1
st=1
st=10
10
10
S VI
... . .
10 ..frt 0
10 ,
300 250 200 10 10 0
0 500
Figure 8: Probability density functions on invariants I 1(a)
and 12 (b) respectively (Case 1).
Conclusions
The present study carried out a direct numerical
simulation of gassolid swirling jets under the Eulerian
Lagrangian deterministic approach, focusing mainly on the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
dispersion characteristics of particles under the effects of
large scale fluid vortices. The main results show that:
1).The configuration of the flow domain does a basic
modification on the bubble vortex breakdown and induces a
symmetry breaking from axisymmetric bubbles into four
main large scale fluid vortices in the comers.
2). The dispersion of particles is mainly dominated by the
large scales fluid vortices, especially for small Stokes
number. Thus, the modification of bubble vortex breakdown
does a consequential modification of particle dispersion, i.e.
small particles are dispersed by the energetic large scale
fluid vortices immediately from the jet core region to the
covers of the container. Thus, the dispersion functions r(x,O)
have four clear peaks. However, the dispersion of large
particles especially extremely large particles are not greatly
influenced by the fluid vortices.
3). The particle dispersion characteristics are not only
determined by the intrinsic particle response property and
the structure of large scale fluid vortices, but also influenced
greatly by the mass loadings. Heavy mass loadings can
cause insufficient momentum exchange for any individual
particle and make light particles follow fluid motion badly.
On the contrary, light mass loadings can improve the
response characteristics of large particles due to sufficient
momentum exchange between the particle and the energetic
large scale vortices.
4). Particles tend to occur in the regions with large strains,
especially the regions where the shear strains dominate over
linear strains. Small particles are more likely than large
particles to occur at points with extremely large strains.
Acknowledgements
The authors are grateful for the support of this research
by the National Natural Science Foundations of China
(Grant No. 50736006 and 50976101)
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