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

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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.

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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

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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

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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 .

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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.

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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).

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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.

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