Document [original]

Analysis of Mobility Effects in Particle-Gas Flows

by Particle-Resolved LBM-DEM Simulations

Tony Rosemann*, Simon R. Reinecke, and Harald Kruggel-Emden

DOI: 10.1002/cite.202000204

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any

medium, provided the original work is properly cited.

Dedicated to Prof. Dr.-Ing. Matthias Kraume on the occasion of his 65th birthday

Particle-resolved direct numerical simulations are performed to simulate the flow through particle assemblies that are

either static or freely moving to demonstrate the influence of particle mobility. To obtain a comprehensive understanding

for this influence essential parameters such as the Reynolds number, solids volume fraction, particle-fluid density ratio,

collision parameters and particle shape are varied. The influence of particle mobility is assessed by evaluating the particle-

fluid forces, the particle ensemble structure and particle velocities. It is found that the ability of existing correlations for

static particle systems to predict drag and lift forces correctly in dynamic particle-gas flows is limited and that drift forces

perpendicular to the drag force play an important role.

Keywords: CFD (computational fluid dynamics), Direct numerical simulation, Drag forces, Multiphase flow,

Non-spherical particles

Received: September 16, 2020; revised: November 30, 2020; accepted: December 01, 2020

1 Introduction

Fluidized systems occur in a wide range of industrial appli-

cation such as drying, coating, granulation, gasification, or

combustion. In gas-solid fluidization the fluidized bed is

characterized as a gas-solid mixture that behaves like a fluid

due to the introduction of an upward gas flow that counter-

acts the solid particles’ gravity. Numerical simulations of

fluidization processes play an increasingly important role in

the design and optimization thereof and can be divided into

three different classes: microscopic particle-resolved direct

numerical simulations (PR-DNS), mesoscopic Euler-

Lagrange DEM-CFD simulations and macroscopic two-

fluid model (TFM) simulations [1]. PR-DNS resolve the

fluid flow around individual Lagrangian particles and allow

a direct calculation of the particle-fluid forces acting upon

the particles whose motion can be predicted by the discrete

element method (DEM). Unresolved DEM-CFD simula-

tions also consider the motion of individual particles, but

rely on using a computational fluid dynamics (CFD) meth-

od to solve the volume-averaged Navier-Stokes equations in

which the assumed fluid cell size is usually larger than the

particle diameter. In TFM simulations both, the fluid and

the solid phase, are solved using volume-averaged equations

and the motion of single particles is not tracked.

While DEM-CFD and TFM simulations are generally

computationally cheaper than PR-DNS, their drawback is

that they rely on constitutive models for the interphase drag

forces, which have a large influence on the simulation’s

accuracy. In the past many different drag force correlations

have been proposed for the flow through random monodis-

perse assemblies of spheres for varying solids volume frac-

tions and Reynolds numbers based on PR-DNS where the

particles are assumed to be rigid [2–7]. Only recently it has

been questioned whether these static correlations are also

accurate for freely moving particles, which are ubiquitous in

industrial applications. By performing PR-DNS Tang et al.

[8] and Huang et al. [9] have shown that particle velocity

fluctuations increase the interphase drag at intermediate

Reynolds numbers and proposed drag correlations based on

the systems’ granular temperature, which is a measure for

the ensemble-averaged particle velocity fluctuations. Rubin-

stein et al. [10], on the other hand, concluded from particle-

resolved lattice Boltzmann (LB) simulations that in low

Reynolds number flows drag forces are lowered by the abil-

ity of mobile particles to adjust to the surrounding fluid

flow and proposed a Stokes number dependent drag corre-

lation for the Stokes regime that takes into account the sol-

id-fluid density ratio. Tavanashad et al. [11] also used LB

simulations to study the effect of particle velocity fluctua-

tions for varying density ratios at Re = 20 and found that

Chem. Ing. Tech. 2021,93, No. 1–2, 223–236 ª2020 The Authors. Chemie Ingenieur Technik published by Wiley-VCH GmbH www.cit-journal.com

–

Tony Rosemann, Simon R. Reinecke,

Prof. Dr.-Ing. Harald Kruggel-Emden

[email protected]

Technische Universita¨t Berlin, Chair of Mechanical Process Engi-

neering and Solids Processing, Ernst-Reuter-Platz 1, 10587 Berlin,

Germany.

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the density ratio of mobile particles has a negligible effect

on the mean drag force. From these studies it can be con-

cluded that the mobility of spherical particles has a different

impact on the drag forces depending on the flow regime.

However, a generalized theory or model that explains the

observed phenomena over a wide range of flow regimes is

still missing.

While many previous studies focused on interphase drag

forces, only few have addressed lift forces in fluidization

simulations. Esteghamatian et al. [12] compared PR-DNS

and DEM-CFD simulations of bubbling gas-solid fluidized

beds and found that available analytical models for lift force

contributions such as the Saffman force and the Magnus

force can neither predict the high magnitude nor the direc-

tion of the lift force computed by DNS correctly. Moreover,

Esteghamatian et al. [12] found that the particles’ transverse

slip velocities and fluctuations in the transverse force com-

ponent are both underpredicted by DEM-CFD simulations.

A contrary observation was made by Liu and Wachem [13],

who compared fluidization experiments in the bubbling

regime with corresponding DEM-CFD simulations and

reported an overestimation of time-averaged horizontal

particle velocities and horizontal velocity fluctuations in

their simulations. Thus, the correct implementation of later-

al forces in DEM-CFD simulations and their impact on the

simulation accuracy are still an open question.

Until now, lift force models based on DNS results have

mainly been developed in the context of non-spherical par-

ticles. However, while drag and lift force correlations with

variable angle of attack have been proposed for isolated

non-spherical particles [14–17], there is generally a lack of

correlations for dense non-spherical particle systems. A

rigorous drag and lift force correlation for spherocylinders

with aspect ratio 4 that is valid for variable Reynolds num-

bers and solids volume fractions has been recently devel-

oped from particle-resolved LB simulations of flow through

static spherocylinder assemblies by Sanjeevi and Padding

[18]. The authors of the latter study highlighted the impor-

tance of lift force, since their magnitude can especially at

higher Reynolds numbers attain values as high as the drag

force [18]. The influence of mobility effects, i.e., effects

related to the ability of particles to change their position

and velocity in a flow as a result of particle-particle and par-

ticle-fluid interactions, on the accuracy of static interphase

drag or lift correlations for non-spherical particles has not

been thoroughly addressed yet. Therefore, it remains un-

known if the recent findings for mobile spherical particles

are also transferable to particles of complex shapes.

In the present study we aim at extending the understand-

ing for mobility effects in dense gas-solid fluidized systems

by performing particle-resolved lattice Boltzmann (PR-LB)

simulations of flow through both, static and freely moving

assemblies of particles, which are coupled with discrete ele-

ment modelling to account for particle collisions that fre-

quently occur when particles are allowed to freely move.

PR-LB simulations as a subgroup of PR-DNS are in this

case a very promising approach, because they allow for

analyzing the directly computed particle-fluid forces of in-

dividual particles that are not accessible in unresolved

DEM-CFD simulations. Note that in the following to the

performed PR-LB simulations will be referred as PR-DNS.

The following parameters that are expected to have an in-

fluence on the mobility effects are varied in the present

study: Reynolds number, solids volume fraction, particle-

fluid density ratio, particle shape and collision behavior, i.e.,

restitution coefficient and friction coefficient. Especially the

effects of particle shape and collision parameters have not

been addressed in DNS studies on mobility effects yet, but

could reveal and explain further limitations of static correla-

tions. This work is divided as follows. In Sect. 2 the applied

coupling between the lattice Boltzmann method and the

discrete element method are briefly outlined and the per-

formed simulations are explained. In Sect. 3 the observed

mobility effects are addressed by considering three aspects:

deviation from static drag as well as lift force correlations,

particle ensemble structure (relative position of particles),

and particle velocity correlations. Finally, in Sect. 4 the con-

clusion, that motivates further investigations of mobility

effects in future works, is drawn.

2 Methodology

2.1 LBM-DEM Coupling

The particle-resolved direct numerical simulations of the

fluidized systems investigated in this work were carried out

using a coupling of a lattice Boltzmann flow solver and the

discrete element method. The coupling of used lattice Boltz-

mann flow solver and DEM solver has been previously vali-

dated for suspension flows with moving particles in Rose-

mann et al. [19] and Kravets et al. [20]. The LB algorithm is

based on the collision of velocity distribution functions in

fluid nodes separated by the grid spacing Dxand the con-

secutive streaming of post-collision distribution function to

neighboring fluid nodes. This fundamental process can be

written as

firþeiDt;tþDtðÞ¼fir;tðÞWifðÞþFiDt;i¼1;2; :::N

(1)

where the distribution functions f

are streamed in one time

step Dtfrom the node at position ralong the discrete lattice

velocities e

to the neighboring nodes at position r+e

Dtby

taking into account the collision operator W(f) and external

forces F

. Here, a three-dimensional D3Q19 lattice model is

used that features N= 19 discrete lattice velocities and the

multiple relaxation time (MRT) collision operator presented

in d’Humie

`rs et al. [21], which allows using different relaxa-

tion times for different moments of the Navier-Stokes equa-

tion besides the viscosity-linked relaxation time t

enhance the stability of the method. The no-slip boundary

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224 Research Article

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condition for particles is established by the lin-

early interpolated bounce-back method by Bou-

zidi et al. [22] that computes the ‘‘bounce-back’’

of distribution functions at the particle surface

during a streaming step. The particle-fluid force

p–f

and resulting torque T

p–f

are computed

using the momentum exchange method by Wen

et al. [23]. Solid nodes that are turned into fluid

nodes in the wake of moving particles are initial-

ized with the simple equilibrium refilling meth-

od which performs equally well as more sophis-

ticated refilling methods in dynamic particulate

flows [19]. Further details about the LBM imple-

mentation can be found in Rosemann et al. [19].

The particle trajectories are calculated by the

discrete element method that tracks the particle

positions x

, translational velocities u

and rota-

tional velocities _

wpaccording to Newton’s equa-

tions of motion given by

d2xp

dt2¼FcþFpf(2)

and

dt¼TcþTpf(3)

where the particle-particle collision force F

, the

particle-fluid force F

p–f

and the corresponding

torques T

p–f

are acting upon the particles with

mass m

and inertia tensor I

. The collision force includes a

normal and a tangential component. The normal compo-

nent is modelled as a linear spring damper model which

depends on the normal spring stiffness and the normal

damping coefficient. These two coefficients are calculated

based on the particle mass, collision duration and restitu-

tion coefficient eas outlined in Scha¨fer et al. [24]. A colli-

sion duration is chosen that is 25 times the DEM integra-

tion time step Dt

DEM

, which is necessary to resolve particle

collision with sufficient accuracy [25]. For the tangential

contact force, a spring model with a tangential spring stiff-

ness that is 0.9 times the normal spring stiffness is used

which is additionally constrained by the Coulomb friction

based on a friction coefficient m. Rolling friction is neglected

in the collision model.

2.2 Simulation Setup

To investigate mobility effects in gas solid-flows the flow

through particle systems is considered as depicted in Fig. 1,

where particles are either assumed to be static with zero ve-

locity or able to freely move due to the forces and moments

acting on them. All of the analyzed systems include

= 100 particles in a cubic domain with length Land fully

periodic boundaries. The solids volume fraction jis

adjusted by the domain size to be either 0.2 or 0.4. The

number of particles in the present work is comparable with

the number of particles used in the above mentioned DNS

studies investigating particle-fluid forces in assemblies of

spherical [2–11] and non-spherical [18] particles, where it

varies between 27 and 200 depending on the study.

In this study two different particle shapes were consid-

ered: spheres and spherocylinders with aspect ratio 4 that

have the same volume as the considered spheres. These

spherocylinders were also used in the previous study of San-

jeevi and Padding [18], but instead of an exact spherocylin-

der shape representation the multi-sphere approach is used

with which the spherocylinders are modelled as 11 aligned

and equispaced spheres as can be seen in Fig. 1. The simula-

tions start with a random configuration being created by

two consecutive steps: initialization at overlap-free position

and randomization of particle positions. These two steps

are carried out for the two investigated particle shapes as

follows.

The spheres are first inserted one after another at random

positions in the cubic domain that do not lead to overlaps

with already inserted particles. When all spheres are

inserted a Monte-Carlo simulation follows in which the

particle positions are varied by a small displacement in each

time step that is only accepted if the new position does not

lead to an overlap with an already inserted particle. The

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Figure 1. Exemplary particle systems with 100 particles consisting of spheres

(left) and spherocylinders (right) that are used for the simulations in this work

and the decomposition of the particle-fluid forces F

p–f

into their contributions

drag

(aligned with slip velocity u

), F

drift

and F

lift

. The angle of attack qfor spher-

ocylinders is defined as the angle between the longitudinal cylinder axis and

the direction of the slip velocity.

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spherocylinders are first inserted at random overlap-free

positions with random orientations in a larger domain with

size L·L·Hsurrounded by walls, where the height His

larger than Lsuch that each particle can be inserted with a

minimum midpoint-to-midpoint distance of one cylinder

length. A downward gravity force is then assigned to the

particles to make them settle to the bottom of the enclosure.

During the settling process friction-free particle-particle

and particle-wall collisions are assumed using the discrete

element method. Once the particles are settled, they fit into

the cubic domain with size L·L·L. With the final particle

positions of the settling process another DEM simulation

with fully periodic boundaries in the desired cubic domain

is initialized, where random translational and rotational

velocities are assigned to the particles and fully elastic and

friction-free collisions are assumed during the simulation.

After a sufficiently long simulation period this results in the

desired random particle configuration in the cubic domain.

To achieve different flow conditions, the superficial

Reynolds number are adjusted.

Re ¼Dp1fðÞus

n(4)

Eq. (4) depends on the particle diameter D

(volume-

equivalent diameter of a sphere in the case of spherocylin-

ders), the fluid’s kinematic viscosity vand on the slip-veloc-

ity between fluid and particles given by

us¼uf<up>

 (5)

Where <u

> denotes the ensemble-averaged particle ve-

locity and u

the fluid velocity averaged over all fluid nodes

in the domain. The desired particle Reynolds numbers are

established by applying an appropriate external fluid force

Fi in Eq. (1). The total external fluid force is obtained by

summing up this force contribution over all fluid nodes. In

the case of freely moving particles, the total fluid force

divided by the number of particles is imposed into opposite

direction on each particle to keep the system’s net momen-

tum zero and achieve a constant time-averaged Reynolds

number.

The LB simulation parameters are listed in Tab. 1. The

resolution D

/Dxand relaxation times t

are chosen to be

very close to the recent study of Sanjeevi and Padding [18],

who performed a resolution-dependency study for the ob-

tained drag forces acting upon static ensembles of spherocy-

linders under various flow conditions that are similar to the

flow conditions considered in this study, i.e. similar solids

volume fractions and Reynolds numbers. For reasons of

consistency the same simulations parameters for spheres

and spherocylinders are chosen. Since 100 particles are used

in each simulation the resulting domain lengths Lare 6.4D

for the solids volume fraction j= 0.2 and 5.1D

for j= 0.4.

The LBM time step has been chosen such that the Mach

number of the flow is kept below 0.3 during the simulation,

which is a stability criterion of the underlying incompressi-

ble lattice Boltzmann scheme. The DEM time step has to be

sufficiently small to keep the overlap between particles dur-

ing a collision below 1 % and obtain a correct collision

behavior. Since the LBM time steps at lower Reynolds num-

bers (Re 1 and Re = 20) would be too large to ensure the

required small particle overlaps, DEM time steps smaller

than the LBM time steps have to be chosen and the ratio

LBM

/Dt

DEM

is therefore larger than one for lower

Reynolds numbers. This means that the particle-fluid force

p–f

in Eq. (2) is assumed to be constant for Dt

LBM

/Dt

DEM

DEM time steps.

In this work three different simulation cases were consid-

ered to probe the influence of particle density and contact

parameters on the correlations’ ability to accurately predict

the drag force for freely moving particles, which are given

in Tab. 2. Case I and II assume a fully elastic and friction-

free collision behavior but differ in the particle-fluid density

ratio (case I: r

= 1000, case II: r

= 100). The density

ratio in case III is the same as in case II but in case III an

inelastic collision behavior with restitution coefficient

e= 0.5 and friction coefficient m= 0.3 is simulated. These

values are exemplarily chosen here to demonstrate the pos-

sible effect of elasticity and friction. However, we note that

the chosen values for the friction coefficient and the restitu-

tion coefficient are close to the corresponding values of bio-

materials such as corn grains [26].

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Table 1. LB simulation parameters of the particle-fluid simula-

tions.

Re jD

/DxL/D

LBM

/Dt

DEM

<< 1 0.2 38.8 6.4 1.8 100

<< 1 0.4 48.8 5.1 1.8 100

20 0.2 69.8 6.4 0.8 10

20 0.4 87.9 5.1 0.8 10

1000 0.2 93.0 6.4 0.51 1

1000 0.4 117.2 5.1 0.51 1

Table 2. Simulation parameters of the three simulation cases

involving freely moving particles: particle-fluid density ratio r

, friction coefficient mand restitution coefficient e.

Case r

I 1000 0.0 1.0

II 100 0.0 1.0

III 100 0.3 0.5

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3 Results and Discussion

3.1 Particle-Fluid Forces

The main contribution of particle-fluid forces in gas-solid

flows is usually assumed to be the drag force which is

aligned with the slip-velocity. In the present work the parti-

cle-fluid force is decomposed into the drag force and the

perpendicular drift force as follows (see Fig. 1):

Fpf¼Fdrag þFdrift (6)

Note that the particle-fluid force F

p–f

does only include

the viscous contribution and the contribution from the fluc-

tuating pressure field. The contribution of the external body

force that is driving the flow is not included. Note also that

especially in studies on interphase drag of gas-solid flows

considering spherical particles F

p–f

and F

drag

are often

regarded as identical since the drift force averaged over all

particles is generally assumed to be zero and only the

ensemble-averaged forces were of interest in these studies.

As can be seen in Fig. 1, the drift force contains in the case

of spherocylinders also the contribution from the lift force

based on the particle orientation that cannot be defined for

the spherical particle. The drag force vector can be calcu-

lated as

Fdrag ¼Fdrag^

edrag ¼Fpf^

edrag



edrag (7)

Where F

drag

denotes the scalar projection of the particle-

fluid force onto the unit vector e

drag

being aligned with the

slip-velocity (Eq. (5)). The drift force is computed as

Fdrift ¼Fdrift^

edrift ¼Fpf^

edrift



edrift (8)

with the unit vector

edrift ¼FpfFdrag

FpfFdrag



(9)

Analogously, the lift force for the spherocylinders is de-

fined as

Flift ¼Flift^

elift ¼Fpf^

elift



elift (10)

and the definition of e

lift

given in Mema et al. [27] is used,

which calculates the expected lift force direction based on

the particle’s orientation with respect to the slip-velocity.

Note that the values F

drag

and F

lift

can be negative in case

the drag force acts into the opposite direction of the

slip-velocity or the lift force acts into the opposite direction

of e

lift

, respectively.

In the following the particle-fluid forces obtained from

the DNS F

DNS

are compared to the predicted force F

cor

given

by reference correlations from literature. The predicted

force is not based on the resolved flow around the particles

but only on the superficial Reynolds number as defined in

Eq. (4). The relative deviation between the two forces is

computed for each particle as

drel Fcor

ðÞ¼

Fcor FDNS

FDNS

jjhi (11)

where the ensemble-averaged mean of the force magnitude

is used as the denominator to avoid near zero divisions

which would distort the deviation computations. The rela-

tive deviation of the force magnitude is calculated as fol-

lows:

drel Fcor

ðÞ¼

Fcor

FDNS

FDNS

jjhi (12)

Tab. 3 lists the drag force deviations for the flow through

a static ensemble of spheres and the three different moving

cases (see Tab. 2). Note that the deviations are time-aver-

aged for the simulations with freely moving particles and al-

so for the static simulations at the highest Reynolds number

(Re = 1000).

In the Stokes regime (Re 1) the static correlation by

van der Hoef et al. [28] for rigid particle systems can ac-

curately predict the drag force for the static simulation

case but overpredicts the drag force for the moving cases,

especially for low particle-gas density ratios. The Wen and

Yu [29] correlation was obtained from sedimentation ex-

periments and thereby inherently considers the particle

mobility. A lower deviation is obtained with this correla-

tion for the low density-ratio case II, but not for case I.

The reason for this observation is that the density ratio

has a notable influence on the interphase drag but is not

included in the Wen and Yu [29] correlation. The recently

developed correlation of Rubinstein et al. [10], on the

other hand, explicitly takes into account the density ratio

in form of the Stokes number St =Re/18 ·r

and can

reduce the maximum deviation observed in these cases.

There is not a significant influence of friction and inelas-

ticity during the collisions (case III) in the Stokes regime

for the low solids volume fraction but a notable reduction

of interphase drag for j= 0.4, which can be attributed to

the fact that collisions happen more often in dense parti-

cle systems. The inelasticity and friction promote the

clustering of particles and the drag reduction since the

relative particle velocities are reduced during a collision

event. The deviation of the correlation by Rubinstein

et al. [10] is also significantly higher in case III since it

was obtained solely from simulations with friction-free

and perfectly elastic collisions.

At elevated Reynolds numbers (Re = 20 and Re = 1000)

the static correlation of Tang et al. [7] is applicable for the

simulations with static spheres and its modified version [8]

for freely moving spheres is applicable to moving case I, but

is outside the applicability range of case II and III because

particle-fluid density ratios below 500 were not considered

in the study of Tang et al. [8]. The modified version does

not only take into account the slip Reynolds number

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(Eq. (4)) and solids volume fraction, but also the Reynolds

number based on the granular temperature defined as

ReT¼ﬃﬃﬃﬃ

pDp

n(13)

with the granular temperature

Q¼QxþQyþQz

3(14)

and its components Q

that are derived from the velocity

components of the particle velocities u

in each spatial

direction kby the following sum over all particles nas fol-

lows:

Qk¼1

NpP

n¼1

p;n;k<up;k>2;k˛x;y;z

fg (15)

The granular temperature can be understood as a mea-

sure for the ensemble-averaged particle velocity fluctuations

that occur when particles are allowed to continuously

change their velocities due to the forces acting upon them.

From their simulations Tang et al. [8] derived a correlation

for Re

that is dependent on the particle-fluid density ratio

and the Reynolds number. There is already a fair deviation

between the DNS results and the correlation for the static

simulation case which cannot only be attributed to different

underlying numerical methods (lattice Boltzmann method

vs. immersed boundary method) but also to the fact that

Tang et al. [7] performed the simulations with a relatively

small resolution (D

/Dx~12) and a grid-size effect correc-

tion. The number of particles cannot be the origin of the

deviation since the number particles used in the study of

Tang et al. [7] and the present study is approximately the

same (108 vs. 100) and periodic boundary conditions are

also used in both of the studies. In our previous work [6],

we have extensively compared results of the used LB solver

for flow through static assemblies of spheres for a broader

range of intermediate Reynolds numbers and solids volume

fraction with available references and found that our results

agree on average better with DNS studies such as Bogner

et al. [4] and Tenneti et al. [5] where higher resolutions than

in the study of Tang et al. [7]. are inherently used. It has

also to be noted that there is generally a lack of highly

resolved simulation data for gas-solid flows at elevated

Reynolds numbers, even for spherical particles.

In contrast to the Stokes regime, Tang et al. [8] found in

PR-DNS that the interphase drag in dynamic gas-solid

flows increases significantly with increasing Reynolds num-

bers and decreasing particle-fluid density ratios compared

to the flow through a static particle system. However, the

DNS results of the present study do not support that the ef-

fect is as large as proposed by the correction by Tang et al.

[8]. The static correlation [7] predicts the drag force equally

well and in most systems even much better than the

dynamic correlation [8] for the moving case I. This is even

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Table 3. Time- and ensemble averaged relative deviation of drag force correlations as defined in Eq. (11) for the simulations with spheri-

cal particles being either static or freely moving. Positive values indicate an overprediction and negative values an underprediction of

the DNS results by the correlation.

Re jCorrelation Correlation parameter Relative deviation d

rel

drag

cor

) [%]

static moving I

moving II

moving III

<<1 0.2 Van der Hoef et al. [28] j8.4 23.9 66.6 65.1

Wen and Yu [29] j– –33.0 –7.2 –8.4

Rubinstein et al. [10] j,St(Re,r

) – 9.8 24.5 23.0

0.4 Van der Hoef [28] j6.9 16.7 53.3 70.1

Wen and Yu [29] j– –36.1 –14.4 –5.4

Rubinstein et al. [10] j,St(Re,r

) – 9.7 24.6 38.8

20 0.2 Tang et al. [7] j,Re 15.0 8.0 5.8 1.5

Tang et al. [8] j, Re, Re

,Re) – 21.0 45.7 39.1

0.4 Tang et al. [7] j,Re 15.4 11.9 7.3 –25.0

Tang et al. [8] j, Re, Re

,Re) – 30.8 63.1 5.7

1000 0.2 Tang et al. [7] j,Re –17.8 –22.0 –21.4 –55.6

Tang et al. [8] j, Re, Re

,Re) – 14.8 94.0 2.4

0.4 Tang et al. [7] j,Re –23.2 –30.2 –30.8 –65.5

Tang et al. [8] j, Re, Re

,Re) – 28.5 133.6 –0.2

a) r

= 1000, e= 1.0, m= 0.0; b) r

= 100, e= 1.0, m= 0.0 b; c) r

= 100, e= 0.5, m= 0.3.

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more visible for the moving case II which is, however, out-

side the correlation’s applicability range due to the low den-

sity ratio that has a very high impact on the correlation.

Since the deviations between our DNS results and correla-

tion [7] are almost identical for moving case I and II, the

respective ensemble-and time-averaged drag forces are not

significantly different despite the different density ratios.

These finding regarding the rather low influence of the den-

sity ratio on the ensemble-averaged drag forces acting on

spheres in dynamic systems are consistent with the recent

findings of Tavanashad et al. [11, 30], who performed simi-

lar PR-DNS for Reynolds numbers between 20 and 100 and

found only significant deviations for liquid-solid systems

£10). Regarding the moving case III, the DNS re-

sults of the present study further suggest that elasticity and

friction significantly increase the drag force at intermediate

and higher Reynolds numbers, which is a contrary observa-

tion to the Stokes regime where interphase drag is slightly

reduced by the introduction of this collision behavior. A

possible explanation for this effect could be that particles

experience a sudden loss of kinetic energy during an inelas-

tic collision with friction, which increases the slip velocity

with respect to the surrounding fluid and thereby the drag

force.

Tab. 4 shows the deviations between the drag and lift

forces computed in the PR-DNS of spherocylinders and the

respective forces obtained by the correlations of Sanjeevi

and Padding [18]. These correlations do not only depend

on the superficial Reynolds number (Eq. (4)) and the solids

volume fraction but also the angle of attack q, which is the

angle between the slip velocity u

and the longitudinal cylin-

der axis (see Fig. 1). First, the results for the static simula-

tion case are discussed. Since the lift forces are very sensitive

with respect to the particle orientations, the results for this

case are additionally averaged over three independent parti-

cle configurations. Very low deviations with respect to the

correlations are obtained regarding the values for F

drag

cor

and F

lift

cor

, which can be explained by the fact that the same

underlying numerical method and resolution is used as in

the study of Sanjeevi and Padding [18]. One difference is

the number of considered particles, which is 200 in Sanjeevi

and Padding [18] and 100 and in the present study. How-

ever, the good agreement suggests that the size of the system

has no significant influence when studying static particle

configurations. The second difference is that Sanjeevi and

Padding [18] derived the correlations based on particle con-

figurations where the particle orientation is only varied

around one axis such that particles are aligned with respect

to one plane parallel to the mean flow. The fact that par-

ticles have random orientations around all principal axes in

the present study proves that this difference does not trans-

late into different particle-fluid forces, which was also

exemplarily shown in Sanjeevi and Padding [18]. Finally, we

note that the particle shape representation by the multi-

sphere approach does apparently not lead to different drag

or lift forces compared to an exact shape representation

used in Sanjeevi and Padding [18], which indicates that the

number of used spheres for one spherocylinder (11 spheres)

is sufficient.

When comparing the deviations for the signed forces

with the force magnitudes in the static case, there is no dif-

ference between the values for the drag force (d

rel

drag

cor

rel

(⏐F

drag

cor

⏐)), but a notable difference between the values

for the lift force, since the correlation is significantly under-

predicting the lift force magnitude. The reason for this

observation is that we consider the ensemble-averaged force

deviations in Tab. 4 and can be explained as follows: Espe-

cially for angles of attack close to 0or 90the lift force is

expected to be close to zero. However, due to the deflection

of the flow at neighboring particles, this is rarely the case

and significant force contributions in the direction of the lift

force are also obtained for these angles. These force contri-

butions are zero on average since they have varying direc-

tions. However, when considering the force magnitudes,

Chem. Ing. Tech. 2021,93, No. 1–2, 223–236 ª2020 The Authors. Chemie Ingenieur Technik published by Wiley-VCH GmbH www.cit-journal.com

Table 4. Time- and ensemble averaged relative deviation of the static drag and lift force correlation of Sanjeevi and Padding [18]

as defined in Eq. (11) for the simulations with spherocylinder particles being either static or freely moving (moving case II). Correlation

parameters are Reynolds number Re, solids volume fraction j, and angle of attack q. Results for the static case are averaged over three

independent random particle configurations. Positive values indicate an overprediction and negative values an underprediction of the

DNS results by the correlation.

Re jRelative deviation d

rel

[%]

static moving II

rel

drag

cor

rel

lift

cor

rel

(⏐F

drag

cor

⏐)d

rel

(⏐F

lift

cor

⏐)d

rel

drag

cor

rel

lift

cor

rel

(⏐F

drag

cor

⏐)d

rel

(⏐F

lift

cor

⏐)

<<1 0.2 –3.7 –13.4 –3.7 –39.7 92.3 57.3 85.2 –19.2

0.4 –2.8 5.1 –2.8 –32.1 130.1 80.9 122.8 –37.8

20 0.2 –1.3 –11.8 –1.3 –29.5 –3.0 –4.0 –3.4 –33.9

0.4 3.0 –1.6 3.0 –24.2 1.6 –5.2 1.1 –43.6

1000 0.2 –0.6 0.3 –0.6 –18.9 –2.6 –2.7 –5.0 –24.2

0.4 –5.4 –1.7 –5.4 –7.8 –19.5 –13.5 –26.4 –50.2

a) r

= 100, e= 1.0, m= 0.0.

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these fluctuating force contributions do not cancel out in

the ensemble average. This effect becomes less pronounced

for increasing Reynolds numbers because the lift force con-

tribution to the total particle-fluid force becomes larger with

increasing Reynolds number [18]. Thus, the fluctuating

force contribution in the lift force decreases.

The results for moving case II in Tab. 4 reveal that both,

the lift forces and drag force, are significantly lower in the

Stokes regime when the cylinder particles are allowed to

move in the PR-DNS compared to the static case, which

leads to an overprediction of these forces by the correlation.

For higher Reynolds numbers the mean drag and lift forces

do not differ significantly from the correlations. These find-

ings are comparable with the findings for spheres in Tab. 3,

but the increase in interphase drag for moving particles in

the Stokes regime is more pronounced for the spherocylin-

ders. This can be explained by the fact that particle cluster

formations causing the drag reduction are more persistent

when considering the elongated particles due to a higher

degree of entanglement which is not possible for spherical

particles. When it comes to the averaged magnitude of the

forces, the deviations in the drag forces are very close

rel

drag

cor

)~d

rel

(⏐F

drag

cor

⏐)) as in the static case. The

values for d

rel

(⏐F

drag

cor

⏐), however, show a different

behavior compared to the static case. In the moving case

the magnitude of lift forces increases with increasing

Reynolds number and with increasing solids volume frac-

tion. In the Stokes regime one can also observe that

⏐d

rel

(⏐F

drag

cor

⏐)⏐<⏐d

rel

drag

cor

)⏐for moving particles,

which is caused by a very low correlation between the pre-

dicted direction of the lift force and the real direction in the

PR-DNS (F

lift

DNS

is negative in Eq. (11) when the DNS lift

force acts into the opposite direction of the predicted lift

force direction).

Fig. 2 shows the ratio between the ensemble-averaged

magnitude of the drift force (see Fig. 1) and the ensemble-

averaged magnitude of the drag force. From Fig. 2a it can be

seen that the significance of the drift force for spherical par-

ticles increases with increasing solids volume fraction and is

especially high in the Stokes regime. Furthermore, the ratio

www.cit-journal.com ª2020 The Authors. Chemie Ingenieur Technik published by Wiley-VCH GmbH Chem. Ing. Tech. 2021,93, No. 1–2, 223–236

a) b)

Figure 2. Ratio of ensemble averaged drift force magnitude and drag force magnitude under different flow conditions: a) PR-

DNS values for spheres, b) comparison between DNS values and values predicted by the correlations of Sanjeevi and Padding

[18] for cylinders and c) ratio of the two drift force components for cylinders obtained in the DNS. Note that the ratios are also

time-averaged in the case of moving particles and that the correlation prediction only contains the lift force contribution in the

drift force.

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increases with increasing Reynolds number outside the

Stokes regime and increases with decreasing particle-fluid

density ratio. Note that the static case can be regarded as

the limit r

ﬁ¥. The ratio is only significantly influ-

enced by inelasticity and friction during the collisions at

higher Reynolds numbers (Re = 1000) and reduced by this

collision behavior.

Fig. 2b shows the ratio between the ensemble-averaged

magnitude of the drift force and the ensemble-averaged

magnitude of the drag force for simulations with static

spherocylinders and freely moving spherocylinders (moving

case II). This figure includes not only the PR-DNS values

but also the ratio that is obtained by computing the particle

drag forces and drift forces based on the correlations of

Sanjeevi and Padding [18]. Note that the correlation predic-

tion of the drift force only includes the lift force contribu-

tion and not the third contribution being perpendicular to

both, lift force and drag direction, since there is no available

correlation for this force contribution which purely arises

from the deflection of flow in the dense particle system. In

both, static and moving case, the ratio directly computed in

the PR-DNS is significantly larger than the ratio predicted

by the correlation and the discrepancy is more pronounced

for the moving case than for the static case. The large ratios

suggest that the lift force is not only underpredicted by the

correlation, but also that the third force component has,

especially in the moving case, a magnitude that is compara-

ble with the lift force magnitude. When comparing the

results shown in Fig. 2a and Fig. 2b, it can be seen that the

ratios obtained from the PR-DNS for static case and

moving case II do not differ notably. This indicates that the

magnitude of the drift force relative to the drag force mag-

nitude is not strongly shape-dependent regarding the two

considered particle shapes.

The significance of the inclusion of a lift force model in

unresolved DEM-CFD fluidization simulations with the

spherocylinders considered in the present work has been re-

cently studied by Mema et al. [31]. The authors found that

the inclusion of lift forces has a notable effect on the flow

velocity in mean flow direction (opposite direction of grav-

ity). The results of the present study further underline the

significance of drift forces in general, but suggest that lift

force models based on the particle orientation alone may be

insufficient to predict these force contributions correctly,

especially when it comes to dynamics gas-solid flows. This

assessment regarding the significance also holds for spheri-

cal particles which do not feature a specific orientation. A

special attention needs to be paid to the transition from the

Stokes regime to higher Reynolds numbers, because the in-

fluence of particle mobility and the magnitude of drift

forces are remarkably different depending on the flow

regime.

The observation that the lift force model for non-spheri-

cal particles alone is insufficient to fully describe the forces

acting perpendicular to the drag force is further supported

by Fig. 2c, where the ratio of the force being perpendicular

to both, lift force and drag force, and the lift force is shown.

It is even slightly higher than the lift force for the Stokes

regime and at Re = 20, and increases significantly at

Re = 1000 when particles are allowed to be mobile. This fur-

ther explains the discrepancies in Fig. 2b and highlights the

importance of forces that arise from the deflection of flow

at neighboring particles.

3.2 Particle Ensemble Structure

To further analyze the effects of particle mobility, the under-

lying structure in the particle system is of interest. While

static correlations usually assume a random distribution of

particles, this randomness does not necessarily need to be

present in a dynamic fluidized system. One tool that has

been recently used to find structures in the particle system

are radial distribution functions [8, 11]. The radial distribu-

tion function g(⏐x

p,i

–x

p,j

⏐) is a measure for the probability

of a particle iwith position x

p,i

to be separated by another

particle jwith position x

p,j

by the distance ⏐x

p,i

–x

p,j

⏐.If

g(⏐x

p,i

–x

p,j

⏐) = 1, the particle separation ⏐x

p,i

–x

p,j

⏐in

the particle system is as likely as finding the particle separa-

tion in a fully random particle distribution, since the distri-

bution function is normalized by the domain-averaged

particle density. Significantly larger values than unity for

g(⏐x

p,i

–x

p,j

⏐) indicate that particles are preferably found

with separation distance ⏐x

p,i

–x

p,j

⏐in the particle system.

When particle clustering occurs and many particles are in

contact, one would expect peaks near ⏐x

p,i

–x

p,j

⏐/D

=1,

since particles are in contact at this separation distance.

Fig. 3 depicts the radial distribution functions for the mov-

ing cases I–III simulated with spheres (a–c) and the moving

case II simulated with spherocylinders (d). The lower parti-

cle-fluid density ratio promotes particle clustering only in

the Stokes regime and not higher Reynolds numbers outside

the Stokes regime (compare Fig. 3a and 3b). The inclusion

of particle elasticity and friction, on the other hand, only

promotes clustering for the higher Reynolds numbers, while

it is not a significant effect in the Stokes regime (compare

Fig. 3b and 3c). These observations regarding the influence

of the density ratio and the collision behavior depending on

the flow regime are consistent with the observations made

with respect to the drag forces in Sect. 3.1. Thus, the ten-

dency for particle clustering is one possible explanation for

the observed decrease (Stokes regime) or increase (higher

Reynolds numbers) of the drag forces. One final observa-

tion regarding the simulations of moving spheres can be

made in Fig. 3a where the simulations performed at Re =20

have the lowest value for g(⏐x

p,i

–x

p,j

⏐) = 1 compared to

the simulations with the other two Reynolds numbers and

thereby the lowest tendency for particle clustering. This can

be explained by the fact that two phenomena can cause par-

ticle clustering: First, long-term particle cluster formations

and void regions occur in the Stokes regime due to the par-

ticle’s ability to adjust to the flow [10]. Second, at elevated

Chem. Ing. Tech. 2021,93, No. 1–2, 223–236 ª2020 The Authors. Chemie Ingenieur Technik published by Wiley-VCH GmbH www.cit-journal.com

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Reynolds numbers strong particle velocity fluctuations

introduce a more chaotic behavior of the particle system

that favors frequent particle collisions. Since the particle

velocity fluctuations increase with Reynolds number [8],

particles can be found more often in close vicinity to each

other at Re = 1000 than at Re = 20.

Interestingly, the spherocylinders do not have a peak

close to ⏐x

p,i

–x

p,j

⏐/D

= 1 but a drop in the radial distribu-

tion function (Fig. 3d). A possible explanation for this phe-

nomenon is that the distance between the cylinder centers

and not the closest distance between two cylinders is con-

sidered. It is more likely that the cylinders’ touch each with

outer particle surface regions than with surface regions

close to the center. Especially for non-spherical particles,

better tools for an analysis of the particle system structure

would be desirable.

Another aspect concerning the structure in the particle

system is the preferred orientation when it comes to non-

spherical particles. Fig. 4 shows the probability of finding a

spherocylinder with angle of attack qin the simulation with

freely moving particles (moving case II). For the derivation

of the static correlation of Sanjeevi and Padding [18] parti-

cle configurations with an even distribution of attack angles

were used. In a dynamic system, however, the particle

orientations can generally not be assumed to be evenly dis-

tributed. As the Reynolds number increases, particles are

increasingly oriented perpendicular to the mean flow veloc-

ity. This observation is consistent with the results of the un-

resolved DEM-CFD fluidization simulations of Mema et al.

[31], who found that spherocylinders are preferably aligned

horizontally in the upper region of the vertical fluidization

column, where particles are more mobile and no longer

densely packed. This flow situation is similar to the flow sit-

uation in the simulations of the present study. Since these

DEM-CFD simulations also considered gas-solid fluidiza-

tion, the Reynolds numbers can be expected outside the

Stokes regime and in the range of Re = 1000, where also an

orientation perpendicular to the mean flow direction is

obtained. The frequent occurrence of particles with attack

angles between 60and 90supports the idea that lift forces

are not strongly correlated with the angle of attack (relative

to the mean flow velocity) in dynamic gas-solid flows,

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Figure 3. Ensemble- and time-averaged radial distribution functions for the fluidized system simulations with moving parti-

cles under different flow conditions: a) moving case I – high particle-fluid density ratio with spheres, b) moving case II – low

particle-fluid density ratio with spheres, c) moving case III – low particle-fluid density with additional particle elasticity and

friction with spheres, d) moving case II – low particle-fluid density ratio with spherocylinders.

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because a low lift force would be predicted by a static lift

force correlation for such angles [18]. However, as can be

seen in Fig. 2b, the drift forces (and thereby also lift forces)

have a significant influence in the simulations with freely

moving particles at Re = 1000 despite high angles of attack.

3.3 Particle Velocities

Besides the particle ensemble structure, an analysis of parti-

cle velocities can also shed light on the mobility effects in

dynamic gas-solid flows. Fig. 5 shows the normalized corre-

lation between the particle velocities of two particles iand j

against their separation distance ⏐x

p,i

–x

p,j

⏐for moving

cases I–III simulated with spheres (a–c) and moving case II

simulated with spherocylinders (d). Values near zero indi-

cate that the particle velocities are uncorrelated and values

significantly larger than zero indicate that the particle veloc-

ities are positively correlated. When a fully elastic and

friction-free collision behavior is assumed, the particle

velocities of neighboring spherical particles are only posi-

tively correlated in the Stokes regime and this correlation

increases with decreasing particle-fluid density ratio (com-

pare Fig. 5a and 5b). Inelastic collisions with additional fric-

tion lead to a significant correlation of particle velocities at

higher Reynolds numbers (Re = 20 and Re = 1000) but do

not change the correlation in the Stokes regime (compare

Fig. 5b and 5c). This flow regime dependent influence of

density ratio and collision parameters was already found for

the density distribution functions and the drag forces (see

Tab. 3 and Fig. 3). The corresponding results obtained for

the spherocylinders (Fig. 5d) also show strong particle

velocity correlations in the Stokes regime, which is even

more pronounced compared to spheres. Especially, for the

solids volume fraction j= 0.4 remarkably far-ranging cor-

relations are obtained. This phenomenon can be explained

by the entanglement of the cylindrical particles, which

forces them to move with the same velocity in a dense parti-

cle system. The strong far-ranging correlation of particle

velocities may also explain that the reduction of interphase

drag is stronger for the spherocylinders than for the spheres

when comparing the static to the moving case (see

Sect. 3.1).

Another phenomenon in Fig. 5 that should be addressed

is the occurrence of slightly negative particle velocity corre-

lations at larger particle separation distances in the cases

where velocity correlations are strongly positive at small

separation distances. The simulation system is designed by

construction to keep the net momentum zero. Once a larger

particle cluster starts to move collectively in a distinct direc-

tion with the surrounding flow, this will therefore excite

remote particles to move in the opposite direction with the

surrounding flow and result in a negative particle velocity

correlation. Finally, the small but noticeable positive

velocity correlation at small separation distances in Fig. 5a

is attributed for the case Re = 1000 with j= 0.2 to the fact

that particles have a high mobility (high granular tempera-

ture due to high Reynolds number), which makes it easier

to reach other particles and form a small cluster that can

persist for a certain amount of time due to high weight of

the particles (r

= 1000) and the fact that the formation

of isolated clusters is facilitated by the low solids volume

fraction. This can lead to the observed correlated particle

motion, which is however not strongly pronounced.

4 Conclusions

Particle-resolved direct numerical simulations help to gain

a deeper understanding for complex flow characteristics in

particle-fluid flows. While PR-DNS are commonly used to

study the flow through static arrangements of particles, the

question of the transferability to dynamic gas-solid flows

occurring in a wide range of applications is of high impor-

tance. In the present study, it is demonstrated that this

transferability is not always given. Correlations for particle-

fluid forces obtained from the consideration of flow through

static particle configurations should therefore be applied in

dynamic unresolved particle-fluid simulations with caution.

The key findings of this study are as follows:

– The mobility of fully elastic and friction-free particles has

a significant influence on interphase drag in the Stokes

regime, but not at elevated Reynolds numbers. The

strong reduction of interphase drag in the Stokes regime

obtained for mobile particles compared to static particles

is stronger for cylindrical particles than for spherical par-

ticles, indicating a strong particle shape dependency. A

lower particle-fluid density ratio further decreases the

interphase drag, when considering spheres.

Chem. Ing. Tech. 2021,93, No. 1–2, 223–236 ª2020 The Authors. Chemie Ingenieur Technik published by Wiley-VCH GmbH www.cit-journal.com

Figure 4. Ensemble- and time-averaged spherocylinder orienta-

tions with respect to the mean flow velocity under different

flow conditions in the simulations with freely moving particles

(case II). The longitudinal cylinder axis is aligned with the mean

flow velocity for an angle of 0and perpendicular to it for an

angle of 90(see Fig. 1).

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– Outside the Stokes regime, the collision parameters that

model elasticity and friction play an important role.

Inelastic collisions with friction lead to an increase of

interphase drag when considering spheres.

– The Reynolds number dependent influence of particle

density ratio and collision parameters can also be found

in the tendency for particle clustering and the correlation

between the velocities of neighboring particles.

– The magnitude of drift forces relative to the magnitude

of drag forces is very high for both of the analyzed parti-

cle shapes, especially in dynamic systems. A static lift

force correlation based on the particle orientation with

respect to the slip velocity cannot predict the drift force

sufficiently accurate.

Since we have focused on time- and ensemble-averaged

forces, structures and correlations in the present work, a fol-

lowing PR-DNS study has to shed light on the individual

time-dependent particle forces and their relationship with

the position and velocity of neighboring particles. Particu-

larly, the origin of drift forces has to be studied more closely

in order to derive rigorous drift force models. Moreover, the

influence of collision parameters and particle shape was

only shown exemplarily here and deserves a detailed inves-

tigation by considering a wider range of collision parame-

ters and particle shapes.

Generally, the particle-resolved simulations can be as-

sumed to be more accurate than particle-unresolved simula-

tions, since they do not rely on constitutive models for the

particle-fluid forces but inherently allow a direct calculation

of the forces and thereby a derivation of constitutive mod-

els. This is also a major advantage over state-of-the-art

experimental methods for particle tracking [32–36] that

have been recently developed to track positions, velocities

and orientations of non-spherical particles and result in

macroscopic distributions of these quantities, but do not

allow a measurement of individual particle-fluid forces. Due

to the high computational demand of resolved simulations,

unresolved simulations will nevertheless still be the only

option for large-scale models in the foreseeable future.

Thus, future particle-resolved DEM-CFD analyses should

be accompanied by particle-unresolved DEM-CFD simula-

tions to detect discrepancies between the two modelling

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Figure 5. Ensemble- and time-averaged correlation between the particle velocities of neighboring particles as a function of

particle center distances under different flow conditions: a) moving case I – high particle-fluid density ratio with spheres,

b) moving case II – low particle-fluid density ratio with spheres, c) moving case III – low particle-fluid density ratio with addi-

tional particle elasticity and friction with spheres, d) moving case II – low particle-fluid density ratio with spherocylinders.

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stages. The development of accurate constitutive models for

particle-unresolved simulation will therefore also remain an

important research area that should – according to the find-

ings of the present study – pay special attention to mobility

effects and particle shape effects. Finally, as heat transfer

plays a decisive role in various applications, it should also

be examined how mobility effects and particle shape effects

influence the heat transfer in thermal particle-laden flows

and the validity of constitutive heat transfer models.

Financial support by the DFG KR3446/13-1, KR3446/

14-1 and KR3446/14-2 is greatly acknowledged. The

original form of the DEM-code ‘‘DEM-Calc’’ applied is

based on a development of LEAT, Ruhr-Universita¨t

Bochum, Germany. The used DEM code ‘‘DEM-Calc’’

has then been continuously extended both at Ruhr-

Universita¨t Bochum and Technische Universita¨t Berlin,

Germany. We thank all who have contributed. The

developed LBM code bases on SUSP3D developed by

Anthony Ladd. We thank him for providing it. Open

access funding enabled and organized by Projekt DEAL.

Symbols used

D[m] diameter

e[–] restitution coefficient

e[–] lattice velocity vector

ˆ[–] unit vector

f[–] velocity distribution function

F[N] scalar force

F[N] force vector

g[–] radial distribution function

H[m] height

I[kg m

] inertia tensor

L[m] length

m[kg] mass

N[–] number

r[–] lattice node position vector

Re [–] Reynolds number based on the

slip-velcocity

[–] Reynolds number based on the

granular temperature

St [–] Stokes number

t[s] time

T[N m] torque

u[m s

–1

] particle velocity vector

x[–] particle position vector

Greek letters

d[–] deviation

D[–] difference operator

q[] angle of attack

Q[m

–2

] granular temperature

m[–] friction coefficient

n[m

s] kinematic viscosity

r[kg m

–3

] density

t[–] relaxation parameter

j[–] solids volume fraction

w[s

–1

] angular velocity vector

W[–] collision operator

Sub- and superscripts

c contact

cor correlation

DNS direct numerical simulation

drag acting in the drag force direction

drift acting in the drift force direction

f fluid

i particle index or lattice direction

j particle index

k spatial dimension index

lift acting in the lift force direction

p particle

rel relative

s slip

x spatial direction x

y spatial direction y

z spatial direction z

Abbreviations

CFD computational fluid dynamics

DEM discrete element method

DNS direct numerical simulation

LB lattice Boltzmann

LBM lattice Boltzmann method

PR-DNSparticle-resolved direct numerical simulation

PR-LB particle-resolved lattice Boltzmann

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