Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 4.5.1 - A numerical study of finite size particles in homogeneous turbulent flow
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 Material Information
Title: 4.5.1 - A numerical study of finite size particles in homogeneous turbulent flow Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Doychev, T.
Uhlmann, M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: particulate flow
homogeneous turbulence
resolved particles
DNS
immersed boundary method
 Notes
Abstract: The present study addresses finite size and finite Reynolds number effects in spatially homogeneous turbulent flow seeded with solid particles. The ratio between particle diameter and Kolmogorov length scale is chosen in the interval 5-25; the Reynolds number based upon the particle diameter and the terminal velocity in ambient flow is of order O(100). Under these conditions, the point-particle approach loses its validity, and we resort to fully-resolved simulations, realized with the aid of an immersed boundary method. The volume fraction of the solid phase is set to 5 10􀀀3, whereby dominant effects of inter-particle collisions are avoided. Two flow configurations are investigated: pure sedimentation in ambient fluid and sedimentation with decaying turbulent background flow, both without mean velocity gradients and with periodicity in all spatial directions. It is found that turbulence has a strong effect upon the settling velocity, initially causing a significantly slower average vertical particle velocity, and at later times (when the turbulence intensity has decayed to a weaker level) a slightly more rapid settling. At the same time turbulence is found to affect the inter-particle distances, bringing particles closer together than pure sedimentation. Further effects of settling particles upon the turbulent kinetic energy, the dissipation rate and anisotropy are discussed, as well as probability density functions of particle velocities and forces.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


A numerical study of finite size particles in homogeneous turbulent flow


T. Doychev and M. Uhlmann

Institute for Hydromechanics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
{todor.doychev, markus.uhlmann} @kit.edu
Keywords: particulate flow, homogeneous turbulence, resolved particles, DNS, immersed boundary method




Abstract

The present study addresses finite size and finite Reynolds number effects in spatially homogeneous turbulent flow
seeded with solid particles. The ratio between particle diameter and Kolmogorov length scale is chosen in the
interval 5-25; the Reynolds number based upon the particle diameter and the terminal velocity in ambient flow is of
order 0(100). Under these conditions, the point-particle approach loses its validity, and we resort to fully-resolved
simulations, realized with the aid of an immersed boundary method. The volume fraction of the solid phase is set to
5 10 3, whereby dominant effects of inter-particle collisions are avoided. Two flow configurations are investigated:
pure sedimentation in ambient fluid and sedimentation with decaying turbulent background flow, both without mean
velocity gradients and with periodicity in all spatial directions. It is found that turbulence has a strong effect upon
the settling velocity, initially causing a significantly slower average vertical particle velocity, and at later times (when
the turbulence intensity has decayed to a weaker level) a slightly more rapid settling. At the same time turbulence is
found to affect the inter-particle distances, bringing particles closer together than pure sedimentation. Further effects
of settling particles upon the turbulent kinetic energy, the dissipation rate and anisotropy are discussed, as well as
probability density functions of particle velocities and forces.


Introduction

The interaction between finite-size heavy particles and a
surrounding flow field is a fundamental hydrodynamic
problem with applications to various real-world con-
figurations, such as raindrop formation in clouds, flu-
idized bed reactors and combustion devices. In past
studies the details of the near-field around the particles
have often been neglected, thereby excluding a descrip-
tion of wake effects (cf. Balachandar and Eaton 2010,
for a recent review). Here we consider sedimentation
problems involving particles with diameters between 10
and 25 Kolmogorov length scales and average particle
Reynolds numbers (based upon diameter and settling ve-
locity) of 0(100). For this purpose we have carried out
direct numerical simulations with interface resolution.
Today these computations are still costly and limited to
a narrow parameter range. However, numerical experi-
ments are able to complement laboratory measurements,
in particular by providing data at high spatial and tem-
poral resolution.
The problem of particles settling in a turbulent back-
ground flow is characterized by a large set of non-
dimensional parameters. Let us first consider the case
of sedimentation of a set of mono-disperse spheres in


an ambient fluid. Given the fluid density pf, the kine-
matic fluid viscosity vf, the vector of gravitational ac-
celeration g on the one hand, and the particle diame-
ter D, particle density pp and solid volume fraction sb
on the other hand, dimensional analysis shows that the
problem is determined by three non-dimensional param-
eters. One has already been mentioned (the solid vol-
ume fraction); the other two can be taken as the den-
sity ratio pp/pf and the ratio between gravitational and
viscous velocities, D g\D/.v. Now, when adding a
turbulent background flow to the picture, we introduce
at least two additional reference quantities to the prob-
lem, namely a fluid velocity scale ... f and a fluid flow
length scale C. Therefore, there will be five independent
non-dimensional parameters, the two additional ones be-
ing, e.g., a fluid flow Reynolds number ... fj/v and a
length scale ratio D/i. In the present work we con-
sider fully developed homogeneous-isotropic turbulence
as the background flow, implying that the scales of the
single-phase flow field can be represented e.g. by the
Taylor micro-scale A and the r.m.s. velocity u,,, m
2q/73 (where q2 = E(K)dK is twice the integral of
the turbulent kinetic energy, here as evaluated in spectral
space over radial wavenumbers K).











The problem of settling particles can be character-
ized by various time-scales: the gravitational scale T7
ID/.g, the purely viscous scale 7, D2/v and the
Stokes drag scale Tp ppD2/(pf18v). The back-
ground turbulence is often characterized by the Kol-
mogorov time-scale, T, ,/, and the large-eddy
turnover time TL k/E.
These dimensional considerations illustrate the ampli-
tude of the parameter space. It is therefore in general dif-
ficult to find reference studies matching a given param-
eter point. The experiment of Parthasarathy and Faeth
(1990) provides one of the most relevant datasets for the
present simulations. It provides measurements of fluid
and particle data of particles settling in a fluid which is
initially at rest, carried out for parameter values mostly
comparable to those investigated in the present study, al-
beit at a much lower solid volume fraction. Their value
is approximately = 10-4, whereas we are targeting
flows with a somewhat higher solid volume fraction of
# 5 10-3.
The purpose of the present work is to investigate the
influence of fluid turbulence upon the settling velocity
of solid spherical particles. Contrary to previous studies
dealing with single particles fixed in space (e.g. Bagchi
and Balachandar 2003; Zeng et al. 2010), our particles
are mobile, they are present in large numbers, and the
flow is turbulent a priori. In order to separate the tur-
bulence effect from the effects of mobility and collectiv-
ity, we consider configurations with and without back-
ground turbulence.

Numerical method

The present simulations have been carried out with the
aid of a variant of the immersed boundary technique (Pe-
skin 1972, 2002) proposed by Uhlmann (2005a). This
method employs a direct forcing approach, where a lo-
calized volume force term is added to the momentum
equations. The additional forcing term is explicitly com-
puted at each time step as a function of the desired parti-
cle positions and velocities, without recurring to a feed-
back procedure; thereby, the stability characteristics of
the underlying Navier-Stokes solver are maintained in
the presence of particles, allowing for relatively large
time steps. The necessary interpolation of variable val-
ues from Eulerian grid positions to particle-related La-
grangian positions (and the inverse operation of spread-
ing the computed force terms back to the Eulerian grid)
are performed by means of the regularized delta function
approach of Peskin (1972, 2002). This procedure yields
a smooth temporal variation of the hydrodynamic forces
acting on individual particles while these are in arbitrary
motion with respect to the fixed grid.
Since particles are free to visit any point in the com-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


putational domain and in order to ensure that the regu-
larized delta function verifies important identities (such
as the conservation of the total force and torque dur-
ing interpolation and spreading), a Cartesian grid with
uniform isotropic mesh width Ax A y Az is em-
ployed. For reasons of efficiency, forcing is only applied
to the surface of the spheres, leaving the flow field inside
the particles to develop freely.
The immersed boundary technique is implemented
in a standard fractional-step method for incompress-
ible flow. The temporal discretization is semi-implicit,
based on the Crank-Nicholson scheme for the viscous
terms and a low-storage three-step Runge-Kutta proce-
dure for the non-linear part (Verzicco and Orlandi 1996).
The spatial operators are evaluated by central finite-
differences on a staggered grid. The temporal and spatial
accuracy of this scheme are of second order.
The particle motion is determined by the Runge-
Kutta-discretized Newton equations for translational and
rotational rigid-body motion, which are explicitly cou-
pled to the fluid equations. The hydrodynamic forces
acting upon a particle are readily obtained by summing
the additional volume forcing term over all discrete forc-
ing points. Thereby, the exchange of momentum be-
tween the two phases cancels out identically and no spu-
rious contributions are generated. The analogue proce-
dure is applied for the computation of the hydrodynamic
torque driving the angular particle motion.
During the course of a simulation, particles can ap-
proach each other closely. However, very thin inter-
particle films cannot be resolved by a typical grid and
therefore the correct build-up of repulsive pressure is not
captured which in turn can lead to possible partial 'over-
lap' of the particle positions in the numerical computa-
tion. In practice, we use the artificial repulsion potential
of Glowinski et al. (1999), relying upon a short-range
repulsion force (with a range of 2Ax), in order to pre-
vent such non-physical situations. Essentially the same
method is used for the treatment of particles approaching
solid walls.
The present numerical method has been submitted
to exhaustive validation tests (Uhlmann 2004, 2005a,b,
2006a), as well as grid convergence studies (Uhlmann
2006b). The computational code has been applied to the
case of vertical plane channel flow (Uhlmann 2008).

Setup of the simulations

We have performed two different types of numerical ex-
periments. In the first one, particles are released from
rest in ambient fluid; the cases realized with this con-
figuration of "pure a'ldillicill.lliln are denoted as "SA"
and "SB". The second type of configurations features a
turbulent background flow, which is homogeneous, (ini-












Table 1: Particle properties. In all cases considered
here, particles possess the same relative density with re-
spect to the fluid, viz. pp/pf = 1.5, as well as a global
solid volume fraction of bs = 5 10 3. Two differ-
ent values for the gravitational acceleration are consid-
ered, leading to the following non-dimensional parame-
ters. Note that the particle Reynolds number ReDo is
based upon the "nominal" settling velocity given by a
balance between drag and immersed weight, using the
standard drag formula (Clift et al. 1978).
cases D gD/v ReDoo
A, SA 93.9125 66.53
B, SB 141.4600 145.85


Table 2: Properties of the initial homogeneous-isotropic
flow field.
cases Rec kmaxrI Lint/Lbox
A, B 180.6 0.884 0.0995
SA, SB (fluid at rest)


tially) isotropic, and which decays in time. The simula-
tions of this type are denoted as cases "A" and "B". All
simulations are performed with the same particle/fluid
density ratio of pp/pf 1.5. Likewise, we chose
the same value for the global solid volume fraction of
t s 5 10-3 in all cases, corresponding to Np 3038
particles. Two different values for the gravitational ac-
celeration were chosen (cf. table 1) such that a single
particle in ambient fluid experiences a relative flow at
a diameter-based Reynolds number of 66.53 (145.85)
in cases A and SA (B and SB), according to a balance
between drag and immersed weight, using the standard
drag formula (Clift et al. 1978)
The initial turbulent flow field is generated through
spectral simulation of forced homogeneous-isotropic
turbulence, using the code of A. Wray (cf. Jim6nez and
Wray 1998). When the flow has reached a statistically
stationary state (verified by monitoring skewness and
normalized dissipation rate) a flow field is spectrally in-
terpolated upon the staggered finite-difference grid. The
turbulence properties of the single-phase flow are re-



Table 3: Two-phase flow properties: the following pa-
rameters apply to both cases with turbulence background
flow (based upon quantities at the initial time t 0).
cases D/r St, Stint TL/Tp
A, B 22.5 59.22 2.66 0.84


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


10


10-1


S10-2
ko
10-3


100 101
t/Tp


Figure 1: Turbulent kinetic energy k (normalized with
the initial value of cases A and B, ko) as a function of
time (normalized with the particle relaxation time Tp).
*, single-phase flow; case A; ---- case SA;
- case B; ---- case SB. The straight reference
line ( ) varies as 1-4.





ported in table 2. The actual simulation with the finite-
difference code is then started from this initial field (ar-
bitrarily setting the time to zero, t 0), adding par-
ticles at random positions and assigning them an ini-
tial translational velocity equal to the local fluid veloc-
ity. Further pertinent parameter values of the simula-
tions with turbulent background flow are given in ta-
ble 3. The ratio between the particle diameter and the
Kolmogorov length is initially 26.7 (0.85 with respect to
the initial Taylor microscale); this ratio then decreases
due to turbulence decay, as will be seen below. Con-
cerning the Stokes number, two definitions will be con-
sidered. In the first, the particle response time (based on
Stokes drag) is divided by the Kolmogorov time scale,
viz. St,7 = Tp/. In the second definition, the ratio be-
tween the same particle time scale and the integral fluid
time scale is formed, yielding Stit = TplTit. At the
time of particle injection, the value of St, is approxi-
mately 60, whereas the value of Stint is 2.66.


The present simulations are realized with 5123 grid
points. This mesh provides a moderate resolution of
initially Ax/r = 0.28 (equivalently kmax,,r 11 I
which, however, improves quickly over the course of the
simulation. The particles are resolved with D/Ax
7.5, which is approximately the same resolution em-
ployed by Lucci et al. (2010). The time step is chosen
such that the CFL number remains below 0.5.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


10 20 30 40 50


Figure 2: Anisotropy of the vertical diagonal
component of the Reynolds stress tensor, b33
(w'w')f/(2k) 1/3. Linestyles as in figure 1.


Results


Figure 1 shows the temporal evolution of the turbulent
kinetic energy (TKE) defined as k = '.'..'f/2 for all
four cases. Here and in the following the notation () f is
used for spatial averages over the volume occupied ex-
clusively by the fluid. As can be observed from the fig-
ure, the overall energy of the fluctuating fluid velocity is
only affected at later times by the presence of the parti-
cles. In case B, the difference becomes appreciable after
approximately 10 particle time units (equivalent to 12
initial large-eddy turnover times TL), while in case A a
difference of only 10% with respect to single-phase flow
is measured at t/tp 20. As will be discussed below,
the initial turbulence intensity is very strong compared to
the fluctuations induced by the particle motion. There-
fore, the two-phase flow in cases A and B at early times
can be considered as nearly one-way coupled, at least as
far as kinetic energy is concerned. Conversely, at later
times the flow becomes more and more influenced by the
presence of particles, and finally the statistics of cases
A/SA and B/SB should converge. The figure shows that
the asymptotic limit of "pure s'diniimci,'l.l, i has not yet
been reached in cases A and B by the end of the present
runs.
The anisotropy of the fluctuating velocity field
can be measured by the anisotropy tensor bij
'.'..')/(2k) 6ij/3. Here only the diagonal ele-
ments are non-zero, and both horizontal components
are identical by symmetry. Therefore only component
b33 is shown in figure 2 (the other two being given by
bl b22 = -b33/2). In both cases without tur-
bulent background flow a value of approximately 0.5
is quickly reached and remains constant. The fluctua-


Figure 3: The ratio between the particle diameter and
the Kolmogorov length scale as a function of time.


I -80

-100
-120
-140
0 10 20 30 40 50



Figure 4: The temporal evolution of the average particle
settling velocity, w,l = (wp)p (w)f, scaled by the
viscous reference velocity.


tions are therefore principally concentrated in the verti-
cal direction (i.e. aligned with the settling velocity). The
experimental study of P.nili.s.n.Il.\ and Faeth (1990)
has likewise shown that sedimenting particles lead to an
anisotropic velocity field. In their cases (at low solid vol-
ume fractions of = 10 4), the anisotropy was found
to measure b33 A at various particle Reynolds num-
bers (including the present range). With background
turbulence, on the other hand, the anisotropy remains
initially close to zero and only slowly approaches the
strongly anisotropic value obtained in pure sedimenta-
tion. The temporal growth of b33 seems to be roughly
constant, at least for values of b33 < 1/3.
The ratio between the particle diameter and the Kol-
mogorov length scale, = (v3/)(1/4), in cases A, B is
shown in figure 3. While the background turbulence de-
cays the Kolmogorov scale grows, leading to a decrease


I - - - - -







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


tITp


Figure 5: The temporal evolution of the difference be-
tween the average particle settling velocity in turbulent
conditions, wre, and without background flow, w(,,
normalized by the value without background flow.



in time of the relative diameter D/r. On the figure
we can observe that the value is approximately halved
over the first five time units. Asymptotically, values of
7 (10) are obtained in cases A (B). This comparison of
scales shows that the relative particle size (with respect
to the spectrum of turbulent flow scales) varies in time.
It should be noted, however, that due to the developing
strong anisotropy (favoring the vertical velocity compo-
nent) the scales will also be highly anisotropic. Their
analysis should be performed by means of spatial corre-
lation functions, separately by horizontal/vertical coor-
dinate direction.
The average particle settling velocities wrel are shown
(in viscous scaling) in figure 4. This quantity is de-
fined as the difference between the mean velocities of
the two phases, viz. wrl (wp)p (w)f (wp being the
vertical component of the particle velocity, w the corre-
sponding fluid velocity component, and the operator (-)p
denotes averaging over the number of particles), and it
is therefore in general different from the average over
the relative velocities seen by each particle. Fundamen-
tal differences between cases A, B on the one hand and
cases SA, SB on the other hand can be observed from
the figure. Firstly, the initial mean acceleration due to
gravity is much smaller in the turbulent cases. The dif-
ference diminishes with time and after t/tp 7 (12) in
cases A/SA (B/SB) a cross-over takes place, after which
the particles settle on average faster in the turbulent en-
vironment. The relative difference in settling velocity
between cases A and SA is approximately 4% at times
larger than 20 particle time units; for cases B, SB the rel-
ative difference amounts to 1% (cf. figure 5). Although
the long-time difference is relatively small, it is system-


Figure 6: The temporal evolution of the relative turbu-
lence intensity I = us/Wrel.


atic, and we therefore consider it as physical, especially
between cases A and SA.
Let us first discuss the different short time behavior.
One possible mechanism acting to decrease the mean
settling velocity is the well-known non-linear drag ef-
fect (Crowe et al. 1998). Although experiments show
considerable scatter, the general view is that turbulent
fluctuations of the velocity seen by the particles together
with the non-linearity of the drag law lead to an increase
of the mean drag coefficient, and therefore a decrease
in settling velocity. Since the turbulence intensity is ini-
tially very strong, the non-linearity of the drag can be ex-
pected to play a, ig.ilik.ii role at short times. However,
the argument hinges on the knowledge of (i) a drag law
with instantaneous validity, and (ii) the precise definition
of a relative velocity (cf. discussion in Bagchi and Bal-
achandar 2004). In fact, Bagchi and Balachandar (2004)
did not observe any systematic effect of turbulence upon
the mean drag.
Concerning the long-time behavior, it should be re-
marked that the apparent persistence of the effect of the
initial background turbulence is somewhat surprising. It
seems that the small residual turbulent agitation is caus-
ing the particles to fall faster than in the pure sedimen-
tation cases. The principal mechanism known to lead
to faster settling velocities under turbulent conditions is
the preferential sweeping mechanism (Wang and Maxey
1993), which, however, has thus far only been confirmed
for very small particles. In order to test the possibility
of preferential particle trajectories in the present cases,
conditional sampling of fluid quantities along particle
trajectories needs to be performed.
Figure 6 depicts the ratio between the square root of
the turbulence intensity (ums = vk) and the instan-
taneous average settling velocity, viz. I = urm,/Wrel.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


(a)


10-2


10-
10


100 101 102


Figure 7: The temporal evolution of the average dis-
sipation rate normalized by the initial value 0o, shown
for case A (-- ) and case B (- ). The result for
single phase flow is shown in blue.



The quantity I is a measure of the relative intensity
of particle-induced flow structures (i.e. wakes) as com-
pared to velocity fluctuations attributed to the turbulent
background flow. Its value in the present cases A and
B is initially very large (in fact it has a singularity at
the origin since particles are at rest), dropping to val-
ues below unity by t/t,,f 4 (2) in case A (B). The
difference between the curves corresponding to the two
cases A and B reflects the fact that the respective settling
velocities differ by roughly a factor of two. The tempo-
ral decay of the parameter I in cases A and B implies
that these simulations encompass the whole spectrum of
relative turbulence intensities from very strong agitation
(early times) to the limit of purely particle-induced fluc-
tuations (long times). In fact, figure 6 shows that the
asymptotic value of the parameter I in both pure sedi-
mentation cases SA, SB is approximately 0.08, indepen-
dent of the particle Reynolds number. It can also be seen
that at the end of the observation interval discussed here,
cases A, B have not yet reached the regime of pure sedi-
mentation, with values of I still between 1.5 and 2 times
larger than the counterparts without initial background
turbulence.
The dissipation rate of turbulent kinetic energy is
shown in figure 7. Visibly, the addition of particles is
causing an increase of energy dissipation. This is ex-
pected since each particle introduces additional gradi-
ents near its surface, thereby dissipating kinetic energy.
In the present cases, however, the additional dissipation
is offset by additional generation of kinetic energy due
to the work the particles exert on the fluid, as evident
from figure 1 discussed above. Concerning the differ-
ence in dissipation between cases A and B, an approxi-


4 6 8 10 12


R/(D/2)
(b)


2 4 6 8 10 12


R/(D/2)

Figure 8: The dissipation rate averaged over a spherical
shell between an outer radius R (measured from the par-
ticle center) and the particle surface (R D /2), addi-
tionally averaged over all particles. Different lines corre-
spond to different instants in time: t/7p 0.1;
- t/p 4.6; t/Tp = 16.6. (a) case A,
(b) case B.


mate factor of 8 separates the two curves at large times
(t/Tp > 30). When considering that the average particle
settling velocities of these two cases differ by a factor of
2.2, it appears that the average dissipation rate (when the
background turbulence has already sufficiently decayed)
is roughly proportional to the third power of the settling
velocity.
In order to quantify the location of regions where ad-
ditional dissipation is acting, we have averaged the dis-
sipation rate over spherical shells between a radial dis-
tance p D /2 (the particle surface) and an outer radius
p = R, viz.


/R 27r fr
D, /2 J O J O


dydydp/V,,







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010




09
08
07
06
05
04


Figure 9: Vorticity magnitude in case A, showing a ver-
tical slice at three different instants: t/Tp 0.14, 1.92,
10.35 (from top to bottom).



where Vs = J /2 J Jd dydqydp is the volume of the
shell. In figure 8 the quantity (E)s is shown as a func-
tion of the outer shell radius R for different instants in
time in cases A and B. It can be seen that dissipation is
indeed much larger than average near the particle sur-
faces, decreasing towards unity with distance from the
particle. The local values increase in time relative to the
box-average dissipation since turbulence decays in time,
i.e. the dissipation rate becomes more and more local-
ized around the particles. When plotted in log-log scale
(not shown), it becomes obvious that the dissipation rate
decays as a power-law R" for short distances. Specifi-
cally, the measured radial decay rates for the three curves
in figure 8 are n 0.4, 2.2 and 2.5 (with increasing
time).
Snapshots of the flow field of case B at different in-
stants are shown in figure 9. The visualization of the
vorticity magnitude in vertical slices demonstrates how
vortical structures in the neighborhood of the particles


03I

0 10 20 30 40 50



Figure 10: The temporal evolution of the average dis-
tance to the nearest neighbor, normalized by the value
for a homogeneous distribution on a regular cubical lat-
tice. The straight line ( ) indicates the value for a
random distribution with the same solid volume fraction.
The lower limit of the plot corresponds to the minimum
value of the function (when all particles are in contact
with a neighboring particle).


(located above the particle location) are nearly invisible
at early times, and then progressively start to occupy the
stage. In the final state (t/Tp 10.35) the vorticity field
is visibly dominated by wakes. A further analysis of
the properties of wakes behind collectively sedimenting
particles as well as the implications for the turbulence
spectra would be desirable.
In order to describe the spatial distribution of the dis-
persed phase, we determine the average distance be-
tween nearest neighbors, defined as follows:


1 Np
Pdmin = in1 i),
N i J=1i


where dij Ixi) xj) is the distance between the
centers of particles i and j. In figure 10 the quantity
dmin is normalized by the value of particles distributed
on a cubical lattice d, = (VQ/N,)1/3 (where VQ
is the volume of the computational domain). It can be
seen that without background turbulence the distance to
the nearest neighbor is somewhat larger than with tur-
bulence (approximately 0.65 compared to 0.58). The
turbulent cases A, B exhibit values which are initially
very close to a random distribution, similar to findings
in vertical turbulent channel flow (Uhlmann 2008). At
later times the values of dmin in cases A, B tend slowly
towards the asymptotic value reached in pure sedimenta-
tion. This result implies that the initial strong turbulence
maintains an approximately random spatial distribution


dmin
o rr.








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


10
-10 -5 0 5 10

(b)
100


101

10,



10


10 . ..


-10 -5 0 5 10


Figure 11: Normalized probability density function of
(a) particle velocity, and (b) hydrodynamic forces act-
ing on the particles, both for case B. The histogram was
evaluated over the temporal interval t/Tp E [22.2, 26.6].
F; F; -- F. For the purpose
of comparison a Gaussian is shown as a dashed line
( ).



in cases A, B, while pure sedimentation is characterized
by a structure with values closer to a homogeneous pat-
tern (i.e. with particles spread further apart on average).
Probability density functions (pdfs) of particle veloc-
ity and hydrodynamical force components acting on the
particles are shown in figure 11. The pdfs are normal-
ized in order to study their shape independently of the
mean value and the standard deviation. The figure shows
data for case B at t/Tp z 24, averaged over a short
interval of AT/Tp 4.4. We observe that all veloc-
ity components behave nearly Gaussian (both horizontal
components should be equal by symmetry, which pro-
vides a measure of the quality of the statistics). On the


. . . . . . . . . . . . . . .


.. .. .. .. .. .. .. ..


.. .. .. .. .. .. .. ..


.. .. .. .. .. .. .. ..


.. .. .. .. .. .. .. ..


. . . . . . . . . . . . . . . . .



. . . . . . . . . . . . . . . .


other hand, the pdfs of the particle forces, which under
this normalization are identical to the particle accelera-
tion statistics, exhibit a clear deviation from Gaussianity.
Firstly, the horizontal components are still symmetric (as
they should be due to symmetry), but have longer tails
than the Gaussian reference curve. Secondly, the verti-
cal force component is markedly asymmetric with a vis-
ible positive skewness. It should be noted that the force
statistics have been filtered such that data from particles
which are instantaneously "in contact" (i.e. those which
are within the range of the artificial repulsion force) have
been removed from the dataset. Now, a positive fluctu-
ation F' implies an instantaneous fluctuation which is
directed upward. A positive skewness of F' could be
generated by a non-linear drag effect. If one assumes
that the drag is quasi-steady (i.e. instantaneously given
by the standard drag law), and the characteristic relative
flow velocity has a symmetric pdf, then the non-linearity
of the velocity/drag-force relation would indeed lead to a
positively skewed pdf for the force. A positive skewness
of the pdf for the vertical force has also been observed in
the case of vertical channel flow (unpublished data from
the simulation of Uhlmann 2008). This persistent result
merits further investigation in the future.


Conclusions

We have simulated the settling of spherical particles at
moderate Reynolds numbers and low solid volume frac-
tions. The solid/fluid interfaces were fully resolved by
means of an immersed boundary method. Two spatially
homogeneous configurations were investigated: pure
sedimentation in ambient fluid and sedimentation with
decaying turbulent background flow.
The results show that turbulence initially leads to a
,ig2ilk.ili decrease in the mean settling velocity. For
large times, however, particles settle on average slightly
more rapidly in turbulent flow. Our analysis of the spa-
tial distribution of particles has revealed that the parti-
cles are more evenly distributed in the cases of pure sed-
imentation. This effect is small, but clearly noticeable
from the average distance to the nearest neighbor.
It was found that particle velocity pdfs are close to
a Gaussian function, but hydrodynamic particle forces
possess wider tails. Interestingly, the pdf of the force
component in the vertical direction exhibits a positive
skewness.
The effect of particles upon the fluid turbulence is
only felt at later times, when the initial background
turbulence has already sufficiently decayed. The two-
way coupling effect is such that turbulent kinetic en-
ergy and average dissipation rate are both enhanced. The
later quantity was shown to be more and more localized
around the particles. Moreover, anisotropy is generated











such that the vertical fluctuations are strongly enhanced.
In the future the current results will first be verified at
a higher spatial resolution. It is planned to investigate
in more detail the effect of the size of the computational
domain upon the results, especially in the vertical di-
rection. We will then turn to higher particle Reynolds
numbers, with the purpose of studying possible cluster
formation through wake attraction.

Acknowledgements

The authors thankfully acknowledge the computer re-
sources, technical expertise and assistance provided by
the Barcelona Supercomputing Center Centro Nacional
de Supercomputaci6n as well as by the Steinbuch Centre
for Computing (SCC) at KIT. We thank A. Wray for pro-
viding the spectral simulation code. A. Dejoan has pro-
vided assistance in the adaptation of the spectral code.

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