Settling of finite-size particles in an ambient fluid: A ......8th International Conference on...
Transcript of Settling of finite-size particles in an ambient fluid: A ......8th International Conference on...
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
1
Settling of finite-size particles in an ambient fluid: A Numerical Study
Todor Doychev
1, Markus Uhlmann
1
1Institute for Hydromechanics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
Keywords: particulate flow, resolved particles, DNS, immersed boundary method, clusters, Voronoї analysis
Abstract
We have investigated the gravity induced settling of finite-size particles in an ambient fluid by means of direct numerical
simulation (DNS). Such configurations are relevant to a large number of applications such as meteorology, mechanical and
environmental engineering. For single particle a variety of motion patterns exist, from straight vertical to fully chaotic paths,
for which the fluid motion in the near field around the particles and the particle wake play a dominant role. Here, the interface
between the dispersed- and carrier-phase was fully resolved by means of an immersed boundary method. Particular care has
been taken to meet the respective resolution requirements. We have performed simulations of particles settling with two
different Galileo numbers, 𝐺𝑎 = 121 and 𝐺𝑎 = 178. The settling regimes for a single particle settling with this Galileo numbers are: (a) steady axisymmetric regime and (b) steady oblique regime. We observed, that in the steady oblique regime
(𝐺𝑎 = 178) the particles exhibit wake induced clustering, while in the steady axisymmetric regime (𝐺𝑎 = 121) this was not observed. Furthermore, the mean settling velocity of the particles with 𝐺𝑎 = 178 was strongly enhanced compared to the velocity of a single settling particle.
1. Introduction
Particle-laden flows are found in a large number of
environmental natural and technical processes. Examples
include pollution dispersion in the atmosphere, raindrop
formation in clouds, sediment transport, fluidized bed
reactors and combustion devices. Thus, the accurate
prediction of such flows is of great importance. This
naturally leads to the necessity of reliable theoretical
description of the mechanisms that take place in such flows.
Despite the progress made in the past, there is still a large
scatter of the available data. A recent review of the subject
can be found in (Balachandar and Eaton 2010).
Here we consider the sedimentation of finite-size particles
in an (initially) ambient fluid under the influence of gravity.
Under such conditions, the system is characterized by a set
of non-dimensional parameters. Given the fluid density 𝜌𝑓,
the kinematic fluid viscosity 𝜈, the vector of gravitational acceleration 𝒈 on the one hand, and the particle diameter 𝐷, particle density 𝜌𝑝 and solid volume fraction 𝛷𝑠 on the
other hand, dimensional analysis shows that the problem is
determined by three non-dimensional parameters. One has
already been mentioned, the solid volume fraction 𝛷𝑠. The other two can be taken as the density ratio 𝜌𝑝/𝜌𝑓 and the
Galileo number defined as the ratio between the
gravity-buoyancy and the viscosity forces
𝐺𝑎 = (|𝒈|𝐷3|𝜌𝑝/𝜌𝑓 − 1| )1/2/𝜈.
The particles under consideration in the present work have a
particle Galileo (Reynolds) number of the order of 𝑂(100). For single particle, the Galileo number and the
particle-to-fluid density ratio characterize the regime of
particle settling and in particular the particle wake (Jenny et
al. 2004). A variety of motion patterns exist, from straight
vertical to fully chaotic paths, for which the fluid motion in
the near field around the particles play a dominant role (Ern
et al. 2012). Therefore, the proper resolution of the flow
field in vicinity of the particles is crucial. In the present
work the motion of the fluid and the dispersed phase were
fully resolved by means of direct numerical simulation and
the interface between the dispersed- and carrier-phase was
fully resolved by means of an immersed boundary method
(Uhlmann 2005).
The interaction between the (turbulent) carrier flow and the
solid particles can lead to a number of hydro-dynamical
coupling phenomena which are most prominently
manifested by the following open questions: (a) Do
finite-size particles exhibit clustering or not? (b) How does
the flow field and/or particle clustering affect the settling
rate of the particles? (c) What are the characteristics of the
wake-induced turbulence?
Particle clustering has been investigated for long time.
Majority of the previous studies have been concentrated on
the so-called “preferential concentration” of small particles
in turbulent flows (Squires and Eaton 1991; Fessler et al.
1994). Few studies have been devoted to the question,
whether finite-size particles exhibit clustering, e.g. (Qureshi
et al. 2008; Fiabane et al. 2012). Furthermore, the
interaction between the flow filed and the solid particles
have been from great interest in the scientific community.
Especially the question, whether the particles enhance or
attenuate the flow turbulence. Lucci et al. 2010 have
investigated the interaction of finite-size heavy particles
with a decaying homogeneous-isotropic turbulence in the
absence of gravity.
In the present study the analysis will focus primary on: (i)
the mean apparent velocity lag; (ii) the flow field
fluctuations induced by the particles as they settle through
the computational domain as well as the velocity
fluctuations of the particles; (iii) the spatial structure of the
dispersed phase, i.e. do the particles form cluster and what
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
2
is the extension and shape of the clusters in case clustering
takes place.
The experiments of (Parthasarathy and Faeth 1990a,b)
provide one of the most relevant datasets for the present
study (𝑅𝑒𝑝=[65,147,262]). They performed measurements
of heavy particles settling in an ambient fluid. Their
experiments provide measurements of the velocity fields for
both phases. Considering the wake-induced clustering of the
particles, our parameters are matched most closely by the
numerical simulations of Kajishima and Takiguchi (2002)
and the experiments of Nishino and Matsuchita (2004).
2. Numerical Method
The numerical code used for the present simulation utilizes
an efficient and precise formulation of the immersed
boundary method for the simulation of particulate flows
(Uhlmann 2005). This method employs direct forcing
approach by adding a localized volume force term to the
momentum equations. The basic idea of the immersed
boundary method is to solve the Navier-Stokes equations in
the entire computational domain, including the space
occupied by the particles. The force term is formulated in
such way, as to impose a rigid body motion upon the fluid at
the locations of the solid particles and is explicitly computed
at each time step as a function of the desired particle
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 values from Eulerian
grid positions to particle-related Lagrangian positions and
vice versa are performed by means of the regularized delta
function approach of (Peskin 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.
The solution of the Navier-Stokes equations is realized in
the framework of a standard fractional-step method for
incompressible 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
procedure for the non-linear part (Verzicco and Orlandi
1996). The spatial operators are discretized by means of
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 coupled to the fluid
equations. The hydrodynamic forces acting upon a particle
are readily obtained by summing the additional volume
forcing term over all discrete forcing points. Thereby, the
exchange of momentum between the two phases cancels out
identically and no spurious contributions are generated. The
analogue procedure is applied for the computation of the
hydrodynamic torque driving the angular particle motion. In
the case of periodic boundary conditions, the spatial average
of the force term needs to be subtracted from the momentum
equation for compatibility reasons (Fogelson and Peskin
1988; Höfler and Schwarzer 2000).
Since particles are free to visit any point in the
computational domain and in order to ensure that the
regularized delta function verifies important identities (such
as the conservation of the total force and torque during
interpolation and spreading), a Cartesian grid with uniform
isotropic mesh width is used. For reasons of efficiency,
forcing is only applied to the surface of the spheres, leaving
the flow field inside the particles to develop freely.
During the course of the simulation, particles can approach
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 lead to possible partial “overlap” of the particle
position. In order to prevent such non-physical situations,
we use the artificial repulsion potential of (Glowinski at al.
1999), relying upon a short-range repulsion force.
For detailed description of the method and for information
on the validation tests and grid convergence please refer to
(Uhlmann 2005, 2008; García-Villalba et al. 2012;
Kidanemariam 2013) and further references therein.
3. Setup of the Simulations
The sedimentation of multiple heavy spherical particles in
an otherwise ambient fluid under the influence of gravity
was studied. We carried out numerical experiments with two
different Galileo numbers, 𝐺𝑎 = 121 and 𝐺𝑎 = 178 . In both cases the particle-to-fluid density ratio and the solid
volume fraction were kept constant to 𝜌𝑝/𝜌𝑓 = 1.5
and 𝛷𝑠 = 0.5 % . The corresponding particle Reynolds number based on the balance between drag and immersed
weight, using the standard drag formula (Clift 1978, p.112)
𝑅𝑒𝑝∞ = 𝑤𝑝∞𝑐𝑙𝑖𝑓𝑡
𝐷/𝜈 , was calculated to 𝑅𝑒𝑝∞ = 141 and
𝑅𝑒𝑝∞ = 245. Under this conditions the flow is considered
to be dilute, thus dominant effects of inter-particle collisions
are avoided. Here and in the following, we will refer to the
particles settling with 𝐺𝑎 = 121 as case M120 and to the particles settling with 𝐺𝑎 = 178 as case M180. Additionally we performed simulations of the settling of a
single particle with the exact physical parameters as in case
M120 and case M180. The corresponding simulations are
denoted by S120 and S180.
The computational domain Ω for both cases M120 and
M180 is a box elongated in the vertical direction. The
domain extends in terms of the particle diameter 𝐷 to: 68𝐷 × 68𝐷 × 341 for case M120 and 85𝐷 × 85𝐷 ×170 for case M180. In all three directions periodic boundary conditions are applied. The selected solid volume
fraction corresponds then to a total of 15190 particles in case M120 and 11867 particles in case M180. The particles are initially placed randomly in the computational
domain. The initial particle position and the extensions of
the computational domains are depicted in figure 1. The
flow is resolved by 1024 × 1024 × 5120 grid points in case M120 and 2048 × 2048 × 4096 grid points in case M180. Consequently the particle resolution in case M120
results in 𝐷/𝛥𝑥 = 15 and in case M180 in 𝐷/𝛥𝑥 = 24. The physical and numerical parameters of the performed
simulations are summarized in table 1 and table 2.
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
3
Figure 1: Dimensions of the computational domain with the
initial particle distribution for (a) case M120 and (b) case M180 . D denotes the particle diameter. Gravity acts in the negative z-direction. Periodic boundary conditions are applied in all three directions in both cases.
The simulations were initialized from a succession of
coarse-grained runs with randomly distributed fixed
particles. We start with a coarse simulation with fixed
randomly distributed particles and evolve the simulation in
time until stationary steady state is reached. The results of
the coarse simulations are then linearly interpolated on a
finer computational grid and the simulations are resumed.
This is repeated until the targeted resolution is reached.
During this procedure the particles were kept at fixed
positions. This ensures that any particle deviations from a
random distribution due to insufficient resolution are
avoided. Once the desired resolution is reached, the particles
are released to move freely and the actual recording of data
is started. Here and in the following we arbitrary set the
time at which the particles are released to zero, 𝑡 = 0. During the course of the present document the following
nomenclature is applied: the domain is discretized by a
Cartesian grid (x, y, z), where z is the vertical component in
direction of the gravity g; velocity vectors and their
components corresponding to the fluid and the particle
phases are distinguished by subscripts “f” and “p”
respectively, as in 𝒖𝑓 = (𝑢𝑓 , 𝑣𝑓 , 𝑤𝑓)𝑇 and 𝒖𝑝 =
(𝑢𝑝, 𝑣𝑝, 𝑤𝑝)𝑇 ; particle position vector is denoted as
𝐱𝑝 = (𝑥𝑝, 𝑦𝑝, 𝑧𝑝)𝑇 . The fluctuations of particle quantities
over time are denoted by a single prime, i.e. 𝑢𝑝′ and are
defined as the difference of the instantaneous values and the
averaged value at that same time e.g. 𝑢𝑝′ (𝑥𝑝 (𝑡), 𝑡) =
𝑢𝑝(𝑥𝑝 (𝑡), 𝑡) −< 𝑢𝑝 >𝑝 (𝑥𝑝 (𝑡), 𝑡) . Similarly, the
fluctuations of the fluid velocity field with respect to the
average over the volume occupied by the fluid are defined
as 𝑢𝑓′ (𝑥 (𝑡), 𝑡) = 𝑢𝑓(𝑥 (𝑡), 𝑡) −< 𝑢𝑓 >Ω𝑓 (𝑥 (𝑡), 𝑡) . The
time in the present work is scaled by the gravitational time
scale 𝜏𝑔, which is defined as 𝜏𝑔 = (|𝒈|𝐷|𝜌𝑝/𝜌𝑓 − 1| )1/2.
Table 1: Physical parameters for particulate flow with a
single and multiple settling particles in an ambient fluid.
Solid volume fraction 𝛷𝑠 , density ratio 𝜌𝑝/𝜌𝑓 , Galileo
number 𝐺𝑎 = (|𝒈|𝐷3|𝜌𝑝/𝜌𝑓 − 1| )1/2/𝜈 , Reynolds
number 𝑅𝑒𝑝∞ = 𝑤𝑝∞𝑐𝑙𝑖𝑓𝑡
𝐷/𝜈 based on the terminal
velocity of a single particle 𝑤𝑝∞𝑐𝑙𝑖𝑓𝑡
, particle diameter 𝐷
and fluid viscosity 𝜈 and number of particles 𝑁𝑝 . The
gravitational constant 𝒈 is applied against the vertical direction 𝒛.
𝛷𝑠 𝜌𝑝/𝜌𝑓 𝐺𝑎 𝑅𝑒𝑝∞ 𝑁𝑝
𝑀120 0.005 1.5 121 141 15190 𝑀180 0.005 1.5 178 245 11867 𝑆120 𝑂(10−5) 1.5 121 141 1 𝑆180 𝑂(10−5) 1.5 178 245 1
Table 2: Numerical parameters for particulate flow of
single and multiple settling particles in an ambient fluid.
Particle resolution 𝐷/𝛥𝑥, number of grid nodes 𝑁𝑖 in the 𝑖-th coordinate direction.
𝐷/𝛥𝑥 𝑁𝑥 × 𝑁𝑦 × 𝑁𝑧
𝑀120 15 1024 × 1024 × 5120 𝑀180 24 2048 × 2048 × 4096 𝑆120 15 128 × 128 × 2048 𝑆180 24 256 × 256 × 4096
4. Results and Discussion
4.1 Settling velocity
Figure 2 depicts the temporal evolution of the mean
apparent velocity lag 𝑤𝑝,𝑙𝑎𝑔 for both cases M120 and
M180. The velocity 𝑤𝑝,𝑙𝑎𝑔 is defined as the difference in
the mean streamwise velocity components of the two
phases:
𝑤𝑝,𝑙𝑎𝑔(𝑡) =< 𝑤𝑝(𝑡) >𝑝− < 𝑤𝑓(𝑡) >Ω𝑓 , (3.1)
where Ω𝑓 defines a spatial averaging operator over the
domain occupied by the fluid and 𝑝 denotes an
averaging operator over the particles. The velocities have
been scaled as a Reynolds number with the particle diameter
and the fluid kinematic viscosity, 𝑅𝑒𝑝,𝑙𝑎𝑔 = |𝑤𝑝,𝑙𝑎𝑔 |𝐷/𝜈.
The terminal settling velocity of the single settling particles,
case S120 and case S180, is shown as well. As can be seen
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
4
the mean apparent lag in case M120 remains over the entire
course of the simulation at approximate value of 𝑅𝑒𝑝,𝑙𝑎𝑔 ≈
141. Moreover, the velocity 𝑤𝑝,𝑙𝑎𝑔 in case M120 is well
represented by the settling velocity of the single particle
from case S120. This implies that, the presence of many
freely moving particles in case M120 does not have strong
influence upon the mean apparent velocity lag of the
particles. In case M180 on the other hand, after the particles
are released they begin to accelerate for approximately two
hundred gravitational time units and the velocity 𝑤𝑝,𝑙𝑎𝑔
reaches a maximum of approximately 𝑅𝑒𝑝,𝑙𝑎𝑔 = 271. After
reaching maximum, the mean settling velocity decelerates
and levels up, still exhibiting some fluctuations, at an
approximate value of 𝑅𝑒𝑝,𝑙𝑎𝑔 = 260. In contrast to case
M120, we found out that the velocity 𝑤𝑝,𝑙𝑎𝑔 in case M180
deviates significantly from the one for single particle in case
S180. As will be discussed below, the increase of the mean
apparent velocity lag in case M180 in comparison to the
settling velocity of a single particle is a direct result of the
clustering of the dispersed phase in case M180.
Figure 2: Average particle settling velocity for case M120
(black solid line) and case M180 (red solid line)
(normalized with the particle diameter and kinematic
viscosity) as function of time. The time is normalized with
the gravitational time scale 𝜏𝑔 = (|𝒈|𝐷|𝜌𝑝/𝜌𝑓 − 1| )1/2 .
Terminal settling velocity of a single particle in ambient
fluid for case S120 (black dashed line) and case S180 (red
dashed line).
4.2 Velocity fluctuations
As mentioned above, the particles are settling in an
(initially) ambient fluid. Therefore, any fluctuations of the
fluid are induced by the settling of the particles. The
relevant question in this context is aimed at determining the
intensity of the self-induced fluid motion. This analysis is
different (but related to) the study of the modulation of
existing (background) turbulence due to the addition of
particles. Figure 3a shows the temporal evolution of the root
mean square (r.m.s.) values of the fluid velocity components
for both cases, M120 and M180. The r.m.s. values are
normalized with the terminal settling velocity of the
particles from case 𝑆120 and case 𝑆180 respectively.
(a)
(b)
Figure 3: (a) R.m.s. of the fluid velocity fluctuations for
case M120 (black lines), and case M180 (red lines) as
function of time. Solid lines show the horizontal
components, dashed lines show vertical component. (b) As
in (a), but for the particle velocity. The reference velocity
𝑤𝑟𝑒𝑓 denotes the terminal settling velocity 𝑤𝑝,𝑙𝑎𝑔 of case
S120 and case S180 respectively.
As can be immediately seen, the fluid velocity components
in case M180 show higher fluctuations than in case M120. It
can be observed, that in both cases the fluid velocity
fluctuations of the vertical component are highly dominant.
This can be attributed to the wake-induced character of the
fluid motion, since fluid motion is primarily caused by the
particles moving in streamwise direction. This high level of
anisotropy also suggests strong effects of the particle wakes,
where the mean particle wake contributes to the fluctuating
velocity field. Similar observations were made in the work
of Parthasarathy and Faeth (1990a,b). In case M180 the
r.m.s. values increase after releasing the particles for about
200𝜏𝑔 time units after which they oscillate around a mean
value of 0.27 for the streamwise component and around
0.08 for the lateral component. On the other hand the
fluctuations in case M120 do not experience significant
increase after particles are released. The r.m.s. values in
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
5
streamwise direction seem to have a large period oscillating
behaviour. In the following we assume the flow to be in
statistically steady state after 200𝜏𝑔. For case M180, after
reaching statistically stationary state, the fluid velocity
fluctuations exhibit somehow more prominent peaks in the
r.m.s. values than in case M120 and they are more
distinctive for the streamwise component than for the lateral
velocity component.
The corresponding fluctuations of the velocity components
of the dispersed phase are depicted in figure 3b. The
fluctuations of the dispersed phase experience similar
evolution over time as for the fluid velocity: the fluctuations
in case M180 are larger then in case M120 and the
streamwise component is approximately 2.5 times larger
then the cross-stream component of the particle velocity.
The intensity of the fluctuations of the vertical velocity
component can be attributed entirely to the collective effects,
since a single particle at both Galileo numbers is in steady
motion (Jenny et al. 2004).
By comparison of the velocity fluctuations of both phases,
we found out that, both phases experience comparable
values. Similar values for the cross-stream velocity
component indicate that turbulent dispersion plays an
important role in the present cases. As mentioned earlier the
flow is dominated by the wakes generated from the particles
as they settle through the domain. The particles do not settle
on straight vertical trajectories, rather they settle on curved
or oblique paths. As result the particle wakes are not
oriented exactly in vertical direction, which causes
momentum along the wake axis to be deposited into the
lateral direction.
A measure of the influence of the fluid turbulence on the
particles is the relative turbulence intensity, defined as the
ratio between the intensity of the incoming fluid flow
fluctuations and the apparent slip velocity. In homogeneous
flows the definition of relative turbulence intensity is often
based upon the three-component turbulent intensities, viz.
𝐼𝑟 = (< 𝑢𝑓,𝑖′ >/3)1/2/𝑤𝑝,𝑙𝑎𝑔 . In cases with unidirectional
mean flow (z-coordinate direction) the following definition
is commonly employed, 𝐼𝑟𝑉 = (< 𝑤𝑓′ >)1/2/𝑤𝑝,𝑙𝑎𝑔.
The temporal evolution of the relative turbulence intensity
𝐼𝑟 (𝐼𝑟𝑉) is depicted in figure 4. Although, the flow in the present simulations is not considered to be turbulent in the
general sense, the relative turbulence intensities in the
present simulations are showed to be comparable to the
turbulence levels often considered in studies of the influence
of background turbulence upon the particle motion, e.g. (Wu
and Faeth 1994; Bagchi and Balachandar 2004; Yang and
Shy 2005; Legendre et al 2006; Poelma et al. 2007; Snyder
et al 2008; Amoura et al. 2010). For the present cases we
calculated the relative turbulence intensity to 𝐼𝑟𝑉 =0.2 (0.24) for case M120 (M180). This indicates that the particles in the present cases generated substantial fluid
“turbulence” as they settle through the domain.
4.3 Spatial structure of the dispersed phase
As aforementioned, an interesting feature of particulate
flows is the ability of the particles to form clusters. As
already mentioned all the fluctuations of the flow field are
induced by the settling particles. Moreover, we saw that the
particles react to the flow field fluctuations. This reaction
was manifested in the fluctuations of the particle velocity
field. For the considered Galileo (Reynolds) numbers in this
study, the wakes are important even at considerably large
distances behind the particles. Thus, the particles can
interact with each other over large distances through the
particle wakes. The most prominent example of the particle
wake interaction is the “drafting-kissing-tumbling” motion
of pairs of trailing particles (Fortes et al. 1987; Wu and
Manasseh 1998). Since the flow field is homogeneous in all
three directions any deviation of the particles from the
random distribution can be attributed to the wake character
of the flow.
The particle distribution for the present simulations can be
visually examined in figure 5 (case M120) and figure 6
(case M180) where the position of the particles is projected
on the 2 -D horizontal plane. While in case M120 no significant difference in the distribution of the particles at
the beginning and at later time of the simulation can be
observed, it is clearly observable that the particle
distribution in case M180 at later time deviates significantly
from the random distribution at the beginning of the
simulation. The distribution of the particles shows regions
with high number of particles (high particle concentration)
and regions with small number of particles (low particle
concentration), or even void regions with complete absence
of particles. Hereafter, regions with high particle
concentration are referred as clusters and regions with low
particle concentration as voids.
The clusters and the voids in case M180 (figure 6b) appear
to extend throughout the entire height of the computational
domain. This is in line with previous experimental and
numerical findings (Kajishima and Takiguchi 2002,
Kajishima 2004a) where similar “columnar particle
accumulation” (Nishino and Matsushita 2004) was observed.
Time sequences of such visualization (not shown here)
shows that these structures are quite robust and they persist
over long time intervals.
More quantitative information on the particle clustering can
be obtained by performing a Voronoї analysis (Monchaux et
al. 2010). The Voronoї tessellation is a decomposition of the
space into independent cells, which have the property that
Figure 4: Temporal evolution of the relative turbulence
intensity for case M120 (black lines) and case M180 (red
lines). Solid lines: 𝐼𝑟 = (< 𝑢𝑓,𝑖′ >/3)1/2/𝑤𝑝,𝑙𝑎𝑔 . Dashed
lines 𝐼𝑟𝑉 = (< 𝑤𝑓′ >)1/2/𝑤𝑝,𝑙𝑎𝑔 .
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
6
each point in the cell is closer to the cell's site than to any
other cell's site. As a consequence, the inverse of a Voronoї
cell's volume is proportional to the local particle
concentration. In order to quantify preferential concentration
we compared the probability distribution function (p.d.f.) of
the normalized Voronoї volumes with the p.d.f. of randomly
distributed particles (Monchaux et al. 2010, 2012; Fiabane
et al. 2012). The Voronoї cell volumes are normalized to be
of unit mean. This normalization allows a qualitative
comparison of the p.d.f.s for different particle number
densities, since the so normalized p.d.f.s are independent of
the particle number density (Ferenc and Neda 2007).
The distribution of particles experiencing clustering is
expected to be more intermittent than the distribution of
randomly distributed particles. This implies that regions
with higher particle concentration are more probable than
for random distribution. Respectively, void regions with low
particle concentration are also more probable.
Figure 5: (a) Top view of the particle position at the begin
of the simulation 𝑡/𝜏𝑔 = 0 for case M120. (b) The same,
but at 𝑡/𝜏𝑔 = 1200.
Figures 7a and 8a depict the p.d.f. of the normalized
Voronoї volumes for case M120 and case M180. As can be
observed, the p.d.f. for both cases at the time when the
particles were released in the computational domain is well
represented by the theoretical Gamma distribution for
random particle fields (Ferenc and Neda 2007), confirming
that the initial particle distribution was indeed random. At
later times the p.d.f.s in case M120 (figure 7a) show that the
extremes for the present data are less probable than in the
case of randomly distributed particles. This indicates that
the particles in case M120 become more ordered than
randomly distributed particles. On the other hand, the p.d.f.s
in case M180 (figure 8a) deviate significantly from the
distribution function for the random distributed particles.
The distribution function of the present data exhibits tails
with higher probability of finding Voronoї cells with very
large or very small volumes than in the random case with
uniform probability, indicating that cluster formation takes
place in case M180.
Figure 6: (a) Top view of the particle position at the begin
of the simulation 𝑡/𝜏𝑔 = 0 for case M180. (b) The
same, but at 𝑡/𝜏𝑔 = 820.
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
7
Further, we have analysed the shape of the regions with high
particle concentration by computing the aspect ratio of each
Voronoї cell, defined as the ratio of the largest cross-stream
extension 𝑙𝑥,𝑉𝑖 to the largest streamwise extension 𝑙𝑧,𝑉𝑖 of
the Voronoї cell, 𝐴𝑉 = 𝑙𝑥,𝑉𝑖/𝑙𝑧,𝑉𝑖 . The aspect ratio 𝐴𝑣
provides a qualitative measure of the anisotropy of the
particle clusters and can be seen as measure for the
stretching of the cells. Figures 7b and 8b show the p.d.f.s of
the Voronoї cells aspect ratio 𝐴𝑉 for case M120 and case M180. It can be immediately seen that the p.d.f.s in case
M120 do not show any deviation from the p.d.f. of the
randomly distributed particles, while in case M180 an
appreciable difference is observed. This indicates, that the
majority of the Voronoї cells are squeezed/stretched in the
vertical/horizontal direction and that the particle structures
in case M180 are more likely to be aligned in vertical
direction. This confirms the observations made in figure 6b.
Figure 7: Case M120: Probability density function of (a)
the normalized Voronoї cell volumes and (b) the aspect
ratios 𝐴𝑉. Different lines represent data assembled over different time intervals. The initial random distribution is
represented by black solid line. 𝑡/𝜏𝑔 = [279,307] (red).
𝑡/𝜏𝑔 = [559,587] (blue); 𝑡/𝜏𝑔 = [1216,1244] (green);
𝑡/𝜏𝑔 = [1496,1524] (magenta). The dashed black line
represents an analytical Gamma function fit (Ferenc and
Neda 2007).
An alternative way of characterizing the spatial structure of
the dispersed phase is by performing nearest-neighbour
analysis (Kajishima 2004b). Figure 9 depicts the time
evolution of the average distance to the nearest particle
neighbour 𝑑𝑚𝑖𝑛. The distance 𝑑𝑚𝑖𝑛 is calculated as:
𝑑𝑚𝑖𝑛 =1
𝑁𝑝 ∑ min
𝑗=1,𝑁𝑝𝑗≠𝑖
𝑑𝑖,𝑗
𝑁𝑝
𝑖=1
, (4.2)
where 𝑑𝑖,𝑗 = |𝑥𝑝,𝑖 − 𝑥𝑝,𝑗| is the distance between the centers of particles 𝑖 and 𝑗. The distance 𝑑𝑚𝑖𝑛 has been normalized by its value for a homogeneous distribution with
the same solid volume fraction, 𝑑𝑚𝑖𝑛ℎ𝑜𝑚 = (|Ω|/𝑁𝑝)1/3. The
lower limit of 𝑑𝑚𝑖𝑛ℎ𝑜𝑚 corresponds to the minimum value of
the function, when all particles are in contact with a
neighbouring particle.
Figure 8: Case M180: Probability density function of (a)
the normalized Voronoї cell volumes and (b) the aspect
ratios 𝐴𝑉. Different lines represent data assembled over different time intervals. Initial random distribution is
represented by black solid line. 𝑡/𝜏𝑔 = [199,213] (red).
𝑡/𝜏𝑔 = [345,359] (blue); 𝑡/𝜏𝑔 = [572,576] (green);
𝑡/𝜏𝑔 = [794,810] (magenta). The dashed black line
represents an analytical Gamma function fit (Ferenc and
Neda 2007).
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
8
The upper limit of the function 𝑑𝑚𝑖𝑛/𝑑𝑚𝑖𝑛ℎ𝑜𝑚 has a value of
unity and arises for homogeneously distributed particles, i.e.
particles are positioned on a regular lattice. It can be
observed, that the average distance to the nearest neighbour
at initial time is very close to the value for randomly
distributed particles. As time evolves, the value of
𝑑𝑚𝑖𝑛/𝑑𝑚𝑖𝑛ℎ𝑜𝑚 in case M120 increases quickly and reaches a
statistically steady state value of approximately 0.61. This
finding is in line with the findings of the Voronoї analysis
that the particles in case M120 are more ordered than
randomly distributed particles. Contrarily, the value of
𝑑𝑚𝑖𝑛/𝑑𝑚𝑖𝑛ℎ𝑜𝑚 in case M180 initially decreases for
approximately 200𝜏𝑔 and undulates around approximate
value of 0.5. This again corroborates the results obtained by
the Voronoї analysis that the particles in case M180 tend to
form agglomerations.
Figure 9: Temporal evolution of the average distance to
the nearest neighbor for cases M120 and M180, normalized
by the value for a homogeneous distribution on a regular
cubical lattice. Case M120 (black solid line). Case M180
(red solid line). Random particle distribution with the same
volume fraction (black dashed line).
5. Conclusions
We have simulated the settling of spherical particles at
moderate Reynolds numbers and low solid volume fractions.
The solid/fluid interfaces were fully resolved by means of
an immersed boundary method. The particles were released
to move freely in an initially ambient fluid. The settling in
two different single particle regimes was investigated:
steady axisymmetric regime with Galileo number
𝐺𝑎 = 121 (𝑅𝑒𝑝,𝑙𝑎𝑔 = 141) and steady oblique regime
with 𝐺𝑎 = 178 (𝑅𝑒𝑝,𝑙𝑎𝑔 = 250).
Voronoї analysis of the particle spatial distribution was
performed. Our analysis revealed that the particles in the
steady oblique regime exhibit significant clustering, while in
the steady axisymmetric regime no clustering of the
dispersed phase was observed. It was observed that, the
clustering of the particles led to a significant increase of the
mean apparent velocity lag. Furthermore, the shape of the
clusters was investigated and it was found that the cluster
structures have strong anisotropic shape, where particles
were aggregated in a column like structures which extended
throughout the entire height of the computational domain.
It was found that the flow field was considerably anisotropic
with the vertical direction being the dominant direction. The
results show that, independent of clustering, considerable
turbulence levels were induced by the settling particles, with
relative turbulence intensities reaching values of 0.2 to 0.24.
Moreover, the velocity fluctuations of both phases were
comparable indicating that the particles respond strongly to
the flow field.
In the future more detailed analysis of the flow will be
performed, e.g.: (i) The effect of clustering upon the flow
statistics will be more deeply analysed; (ii) Lagrangian
analysis of the data; (iii) Statistics of conditionally averaged
data. The existence of such strong differences in the spatial
distribution of the particles between the two different
settling regimes is quite interesting and the role of the
settling regimes on the spatial distribution of the particles
will be further analysed. Next step will be the simulation of
the interaction between finite size particles and forced
homogeneous turbulence.
Acknowledgements
Support through a research grant from DFG (UH 242/1-1) is
thankfully acknowledged. The authors also want to also
acknowledge the computer resources, technical expertise
and assistance provided by the Leibniz Supercomputing
Center (LRZ), Jülich Supercomputing Center (JSC) as well
as by the Steinbruch Center for Computing (SCC) at KIT.
References
Amoura, Z., Roig, V., Risso, F., Billet, A.M., 2010.
Attenuation of the wake of a sphere in an intense incident
turbulence with large length scales. Physics of Fluids 22,
055105.
Bagchi, P., Balachandar, S., 2004. Response of the wake of
an isolated particle to an isotropic turbulent flow. Journal
of Fluid Mechanics 518, 95–123.
Balachandar, S., Eaton, J.K., 2010. Turbulent Dispersed
Multiphase Flow. Annual Review of Fluid Mechanics
42, 111–133.
Clift, R., Grace, J., Weber, M., 1978. Bubbles, drops,
and particles. Academic Press.
Ern, P., Risso, F., Fabre, D., Magnaudet, J., 2012.
Wake-Induced Oscillatory Paths of Bodies Freely Rising
or Falling in Fluids. Annual Review of Fluid Mechanics 44,
97–121.
Ferenc, J.S., Néda, Z., 2007. On the size distribution of
Poisson Voronoi cells. Physica A: Statistical Mechanics
and its Applications 385, 518–526.
Fessler, J.R., Kulick, J.D., Eaton, J.K., 1994. Preferential
concentration of heavy particles in a turbulent channel flow.
Physics of Fluids 6, 3742.
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
9
Fiabane, L., Volk, R., Pinton, J.F., Monchaux, R., Cartellier,
A., Bourgoin, M.M., 2012. Do finite size neutrally buoyant
particles cluster? arXiv preprint , 1–6.
Fogelson, A., Peskin, C.S., 1988. A fast numerical method
for solving the three-dimensional Stokes’ equations in the
presence of suspended particles. Journal of Computational
Physics 79, 50–69.
Fortes, A., Joseph, D., Lundgren, T., 1987. Nonlinear
mechanics of fluidization of beds of spherical particles.
Journal of Fluid Mechanics 177, 467–483.
García-Villalba, M., Kidanemariam, A.G., Uhlmann, M.,
2012. DNS of vertical plane channel flow with finite-size
particles: Voronoi analysis, acceleration statistics and
particle-conditioned averaging. International Journal of
Multiphase Flow 46, 54–74.
Glowinski, R., Pan, T., Hesla, T., Joseph, D.D., Periaux, J.,
1999. A distributed Lagrange multiplier/fictitious domain
method for flows around moving rigid bodies: application
to particulate flow. Int. J. Numer. Meth. Fluids 1066,
1043–1066.
Höfler, K., Schwarzer, S., 2000. Navier-Stokes simulation
with constraint forces: finite-difference method for
particle-laden flows and complex geometries. Physical
review. E, Statistical physics, plasmas, fluids, and related
interdisciplinary topics 61, 7146–60.
Jenny, M., Dušek, J., Bouchet, G., 2004. Instabilities and
transition of a sphere falling or ascending freely in a
Newtonian fluid. Journal of Fluid Mechanics 508, 201–
239.
Kajishima, T., 2004a. Influence of particle rotation on the
interaction between particle clusters and particle induced
turbulence. International Journal of Heat and Fluid Flow
25, 721–728.
Kajishima, T., 2004b. Numerical investigation of collective
behavior of gravitationally settling particles in a
homogeneous field, in: Proc. ICMF, pp. 1–10.
Kajishima, T., Takiguchi, S., 2002. Interaction between
particle clusters and particle-induced turbulence.
International Journal of Heat and Fluid Flow 23, 639–646.
Kidanemariam, A.G., Chan-Braun, C., Doychev, T.,
Uhlmann, M., 2013. DNS of horizontal open channel flow
with finite-size, heavy particles at low solid volume
fraction. arXiv preprint , 1–44.
Legendre, D., Merle, A., Magnaudet, J., 2006. Wake of a
spherical bubble or a solid sphere set fixed in a turbulent
environment. Physics of Fluids 18, 048102.
Lucci, F., Ferrante, A., Elghobashi, S., 2010. Modulation of
isotropic turbulence by particles of Taylor length-scale size.
Journal of Fluid Mechanics 650, 5.
Monchaux, R., Bourgoin, M., Cartellier, A., 2010.
Preferential concentration of heavy particles: A Voronoї
analysis. Physics of Fluids 22, 103304.
Monchaux, R., Bourgoin, M., Cartellier, A., 2012.
Analyzing preferential concentration and clustering of
inertial particles in turbulence. International Journal of
Multiphase Flow 40, 1–18.
Nishino, K., Matsushita, H., 2004. Columnar particle
accumulation in homogeneous turbulence. Proc. ICMF04
(5th Int. Conf. Multiphase Flow) , 1–12.
Parthasarathy, R., Faeth, G.M., 1990a. Turbulence
modulation in homogeneous dilute particle-laden flows.
Journal of Fluid Mechanics 220, 485–514.
Parthasarathy, R., Faeth, G.M., 1990b. Turbulent
dispersion of particles in self-generated homogeneous
turbulence. Journal of Fluid Mechanics 220, 515–537.
Peskin, C.S., 2002. The immersed boundary method. Acta
Numerica 11, 1–39.
Poelma, C., Westerweel, J., Ooms, G., 2007. Particlefluid
interactions in grid-generated turbulence. Journal of Fluid
Mechanics 589, 315–351.
Qureshi, M., Arrieta, U., Baudet, C., Cartellier, A., Gagne,
Y., Bourgoin, M., 2008. Acceleration statistics of inertial
particles in turbulent flow. The European Physical Journal
B 66, 531–536.
Snyder, M.R., Knio, O.M., Katz, J., Le Maitre, O.P., 2008.
Numerical study on the motion of microscopic oil droplets
in high intensity isotropic turbulence. Physics of Fluids 20,
073301.
Squires, K.., Eaton, J.K., 1991. Preferential concentration
of particles by turbulence. Physics of Fluids A: Fluid
Dynamics 3, 1169.
Uhlmann, M., 2005. An immersed boundary method with
direct forcing for the simulation of particulate flows.
Journal of Computational Physics 209, 448–476.
Uhlmann, M., 2008. Interface-resolved direct numerical
simulation of vertical particulate channel flow in the
turbulent regime. Physics of Fluids 20, 053305.
Verzicco, R., Orlandi, P., 1996. A finite-difference scheme
for three-dimensional incompressible flows in cylindrical
coordinates. Journal of Computational Physics 123, 402–
414.
Wu, J., Faeth, G.M., 1994. Sphere wakes at moderate
Reynolds numbers in a turbulent environment. AIAA
Journal 32, 535–541.
Wu, J., Manasseh, R., 1998. Dynamics of dual-particles
settling under gravity. International Journal of Multiphase
Flow 24, 1343–1358.
-
8th
International Conference on Multiphase Flow ICMF 2013, Jeju, Korea, May 26 - 31, 2013
10
Yang, T.S., Shy, S.S., 2005. Two-way interaction between
solid particles and homogeneous air turbulence: particle
settling rate and turbulence modification measurements.
Journal of Fluid Mechanics 526, 171–216.