Two-pHase modElling for Sediment dynamIcS in geophysical flows
THESIS-2016
September 12-14, 2016, Tokyo – Japan

Motions of single and a group of particles with different shapes
flowing down in fixed bed channels
Organizing committees
Yuya Takakuwa

Graduate School of Science and Engineering, Chuo University
1-13-27 Kasuga, Bunkyou-ku, Tokyo 112-8551, Japan
International Scientific Committee
Tel. +81 (03) 3817 1615
Email: takakuwa0715@civil.chuo-u.ac.jp

Balachandar, S. (University of Florida, USA)
Blondeaux, P. (University of Genova, Italy)
Shojidelle
Fukuoka
Brocchini, M. (Università PolitecnicaMarche, Italy)
Research and Development Initiative, Chuo University
Dong, P. (UniversityofDundee,U.K.)
1-13-27 Kasuga, Bunkyou-ku, Tokyo 112-8551, Japan+81 (03)Irvine,3817 1625
Elghobashi, S. (University of Tel.
CaliforniaUSA)
Guillou, S. (University of Caen, France)
Email: sfuku@tamacc.chuo-u.ac.jp
Hsu, T.J. (University of Delaware, USA)
Izumi,N. (HokkaidoUniversity,Japan)
Gravel-bed
riversare composedof particles
with various shapes and sizes. In this paper, the image
analysis ofJenkins,laboratoryof singleUSA)
gravel motions in flows and numerical simulations wereJ. T.experiments(Cornell University,conducted.Meiburg,Effects ofE.particle
shapessedimentUSA)
transport mechanism were discussed.
(Universityofon
California,Particles were transported in characteristic manners depending on their shapes. Spheres rolled, on
Nguyen,K.D. (Laboratoryfor moved
Hydraulics
Saint-Venant,France)Velocities of single particles
the other hand,non-sphericalparticlesby saltation
and rolling.
Pham Van
Bang, D.
(LaboratoryHydraulics
were influenced
by shapes
morethan sizesforand
became Saint-Venant,large as they France)
deviated from sphere shape.
Motions ofShimizu,non-sphericalparticles
varied inJapan)
the vertical direction, so the interaction among nonY. (HokkaidoUniversity,
spherical particles
smallerthan thatof spheres.de
Further,the France)group velocity of mixed particles
Simonin,becameO. (InstitutNational
Polytechnique
Toulouse,
was estimated by average velocities of single particles and average velocity deceleration rates of aSoldati, A. (University of Udine, Italy)
group of particles.
Toorman, E.A. (KU Leuven, Belgium)
Key Words:
gravel-bed
sediment transport,interaction among particles, particle shape
Uhlmann,M.rivers,(KIT Karlsruhe,Germany)
Wang, G.Q. (Tsing-Hua University, China)

1. Introduction

Understandingof sedimenttransport mechanism in gravel-bed rivers is insufficient because
Local Organizing
Committee
transportation experiments of gravels with various shapes and sizes are complicated in comparison withthose of sands. Therefore, effects of particle shapes and sizes on transport mechanism should be
examined Fukuoka,
in detail. S. (Chair) (Chuo University)
Gotoh,
H. the(KyotoUniversity)
In recent years,progressof computing performance has made possible to analyze threedimensional
flowN.
motionsand particle
motions by the Euler-Lagrange approach [1]. Fukuoka et al. [2]Izumi,(HokkaidoUniversity)
developedKajishima,
the numericalmovable-bed
channel expressing motions of flow and particles with various
T. (OsakaUniversity)
shapes and sizes, and emphasized the importance of considering particle shapes on transport mechanism
K. well(Chuoof a groupKashiyama,of particles asas University)a single particle.
In this paper,toclarify
fundamental
effectsof particle
shapeson motions of single and a group ofMizuyama, T. (National GraduateInstitutefor PolicyStudies)particles, following
investigations
wereconducted.
First,
Euler-Lagrange
simulations were applied toNakagawa, H. (Kyoto University)
motions of single gravels with different shapes and sizes in the laboratory experiments conducted by
M.Fukuoka(The UniversityofParticle
Tokyo) motions in the numerical channel were compared with
ShigemuraSakai,[3] andet al. [4].
S. laboratory(The Universityof Tokyo)
the resultsSato,of theexperiments.
Second, transportations of single particles (4 shapes and 3sizes) wereSekine,simulatedand effectsof particle shapes on motions of single particles were examined. Third,M. (WasedaUniversity)transportationsof aY.groupof particles
were simulated and effects of particle shapes and interactions
Shimizu,(HokkaidoUniversity)
among particles on motions of a group particles were discussed.
Ushijima, S. (Kyoto University)
Uchida, T.of(Secretary)(ChuoUniversity)results and numerical simulations
2. Comparisonslaboratory
experiment2.1. Conditions of experiments and numerical simulations
SecretariatTo clarify
transportof Initiative
single gravels
withdifferent
sizes, Shigemura [3] and Fukuoka etResearchand mechanism
Development(RDI),ChuoUniversity
al. [4] photographed motions of single gravels (see Table 1) in an open channel flow and analyzed them.
Figure 1 shows conditions of the experimental channel with 45 m long, 1 m width, 1:20 bed slope and0.5 m3/s discharge. The channel cross-section was a broad concave shape.
In the numerical simulations, model particles (see Table 1) were transported by water in the numericalchannel. The length of the numerical simulation channel was 38 m. The discharge 0.5 m3/s was supplied

89Table 1. Gravels used for the laboratory experiments
and model particles for the numerical simulations.

Singleparticle

LengthLaboratory experiments: 45m
Numerical simulations: 38m
Concave cross-section shape

FLOW

Boundary condition of
the upstream end：Q=0.5m3/s

Width：1m
Gradient：1/20

Boundary condition of
the downstream end：
Free flow condition

40.15
0.1
0.05

0-4 -3 -2 -1 0 1 2 3Lateral distance / dDisk-type, d=105mm

0.2
Left bank sideExp.
Cal.

Existence probabilities

0-4 -3 -2 -1 0 1 2 3Lateral distance / d
Ellipsoid-type, d=75mm

Left bank side

0.1
0.05

0.2

0.15

Exp.
Cal.

0.1
0.05

0-4 -3 -2 -1 0 1 2 3Lateral distance / dPear-type, d=135mm

4Right bank side

0.15

Existence probabilities

Exp.
Cal.

Right bank side

Left bank side

0.2

Right bank side

Existence probabilities

Fig. 1. Conditions for the experimental channel andthe numerical simulation channel.

4Fig. 2. Lateral distributions of existence probability density of single particles.
Table 2. Average velocities of single particles.
Average velocity
Exp. (m/s)
Cal. (m/s)

Ellipsoid-type
d=75mm3.30
3.03

Disk-typed=105mm3.81
3.49

Pear-typed=135mm
3.54
3.24

at the upstream end and the condition of the zero pressure at the downstream end was given.

2.2. Numerical method
The numerical method of Fukuoka et al. [2] was used for numerical simulations. In the numericalsimulations, flow and particle motions were calculated by the Euler-Lagrange approach. Fluid motions
are simulated by governing equations (1)-(4) of one-fluid model of the incompressible fluids:∂ui=0∂xi(1)υt  C S Δ 

22 S ij S ij

(3)

∂ui
∂ui
1 ∂P∂2υ  υt Sij uj gi ∂t∂x jρ ∂xi ∂x j
Sij

1  ∂ui ∂u j 
2  ∂x j ∂xi (2)(4)

where ui: i-direction mass averaged velocity within a fluid cell including particles, ρ: volume averaged
density, gi: i-direction gravitational acceleration, P: sum of the pressure and isotropic component of SGS
stress, ν: kinematic viscosity, νt: SGS turbulent viscosity, Sij: strain rate tensor, Cs: Smagorinsky constant(0.173) and Δ: calculation grid size.
Particle motions are expressed by translational and rotational equations of rigid bodies. Fluid forces
acting on particles are calculated by the second term and the third term in the right side of equation (2)and integrated them on the region of a particle. Contact forces and torques are evaluated by Distinct
Element Method [5]. Model particles are made by superposing small spheres [6] as shown in Table 1.

2.3. Comparisons of the result of experiments and numerical simulations
Figure 2 shows probability distributions in the lateral direction of experimental and simulated singlegravels. Simulated probability distributions almost agreed with the result of laboratory experiments ineach case. Table 2 shows average velocities of the experiments and numerical simulations of single
particles. Although average velocities of numerical simulations of single particles were about 8 %smaller than those of the experiment in all cases, the relative magnitude of average velocities ofnumerical simulations was almost the same as that of the experiments. We confirmed that the numericalsimulations could reproduce well motions of single particles flowing down in the fixed-bed channel.

3. Effects of particle shapes on motions of single particles
Transportation simulations of single particles (4 shapes shown in Table 3: sphere, ellipsoid-type, peartype and disk-type, 3 sizes: d = 75, 105, 135 mm) were conducted under the conditions described in
Figure1.
Figure 3 describes consecutive motions of single particles with different shapes plotted at every 0.1

90Table 3. Characteristics of particle shapes.

43.5
32.5
21.5
10.50average velocity of flow (3.86m/s)
43.5
32.5
2Sphere1.5
Ellipsoid-type particle
1Pear-type particle
0.5
Disk-type particle
d=75mmd=105mmd=135mm
01.00 1.20 1.40 1.60 1.80
90110
130150DimensionlessParticle size (mm)maximum projection area
Average velocity
of single particle (m/s)

Average velocity
of single particle (m/s)

Fig. 3. Consecutive motions of single particles
with different kinds of shapes.

70

Fig. 4. Relations between average velocities versus sizes and shapes of single particles.

second. Naturally single particles were transported in characteristic manners depending on their shapes.
Sphere (a) moved by rolling. Ellipsoid-type particle (b) and pear-type particle (c) moved by saltation
and rolling taking the long-axis to a rotating-axis. Disk-type particle moved in two typical manners(gliding just above the bed (d) or rolling taking the short-axis to rotating-axis) and in a compound manner.
Figure 4 shows relations between particle velocities versus sizes and shapes. Average velocities ofsingle particles varied between 2 ~ 4 m/s, and effects of sizes on particle velocities were not appreciatedclearly. On the other hands, average velocities were considerably different by particle shapes. Averagevelocities of a disk-type particle became about one and a half as much as those of a sphere. A shapeindex of particles should be selected depending on the sedimentation subject. We chose thedimensionless maximum projection area as a shape index shown in Table 3 because many particles
moved taking a large projection area against the flow. The maximum projection area controls particle
motions as Sneed et al. [7] pointed out. Average velocities of single particles were influenced by shapes
more than sizes and became large as the dimensionless maximum projection area increased.

4. Effects of particle shapes and interactions among particles on motions of a group ofparticles
Transportation simulations of single particles and 4 types of 24 particles (spheres, pear-type particles,
disk-type particles and mixed particles of three different types, size: d = 105 mm) were performed in therectangular channel of 21 m long, 2.2 m width and 1:100 bed slope shown in Figure 5. The 3.0 m3/s
discharge was given at the upstream end and the zero pressure was imposed at the downstream end. Thewater depth was 0.44m and the Froud number was 1.4. Particles were given momentarily in the regionof 4 < x < 5 m, - 0.7 < y < 0.7 m, 0 < z < 0.35 m with the initial velocity of 2.5 m/s.
At first we discuss vertical distributions of existence probability density of single and a group ofparticles (see Figure 6). It was found that vertical distributions of existence probability density of agroup of particles were almost equal to those of single particles. The moving height of particles wasdetermined by the center of gravity of particles. Heights were different from shapes and became large
as particle shapes deviated from the spherical shape.
Table 4 shows average velocities of single and a group of particles and average velocity deceleration
rates (ri: see equation (5)) of particles group. Average group velocities to average velocities of single
particles were less for spheres 13%, pear-type 11% and disk-type 8%. This demonstrated interactions
between particle-particle and flow-particles. The reason was because individual particles composing agroup of particles were affected by wakes of other particles and fluid forces acting on particles became
small. The relation between particle velocities and vertical motions and interaction among particles wasdescribed as follows. As particles deviated from the spherical shape, they jumped to higher levels andthen interactions among particles (average velocity deceleration rates: ri) were reduced. Finally, theaverage velocity of mixed particles with three kinds of shapes was estimated by equation (6) using91

2Velocity(m/s)
204z/dz/d0

Velocity(m/s)
24SpherePear-type

1.5

1.5
1

Disk-type
Mixed particles

1Velocity

0.5

0.5
02

00.2 0.4 0.6 0.8 1
Probability density

00Single particle

0.2 0.4 0.6 0.8 1
Probability density
A group of particles

Fig. 5. Motions of a group of particles
Fig. 6. Vertical distributions of existence probability density.
with different kinds of shapes.
Table 4. Average velocity of single and a group of particles.
Average velocity
AverageSingle particle: A group of particles:�� � ��
deceleration rate: ri�� �Velocity
Vs (m/s)Vm (m/s)SpherePear-typeDisk-type
Mixed particles

1.59
2.28
1.96
-

1.39
2.03
1.81
1.74

V�
0.13
0.11
0.08
-

ri Vs  VmVs

(5)

average velocities of respective single particles and average velocity deceleration rates expressing
effects of interaction among particles(6)

where Vmix: average group velocity of mixed particles, Vsi: average velocity of i shape single particle, n:number of different particle shapes, αi: mass rate of i shape particles and ri: average velocity deceleration
rate of i shape group of particles. The simulated velocity was 1.74 m/s and the velocity estimated byequation (6) was1.73 m/s.

5. Conclusions
Main conclusions were described as follows:
1. Particles moved taking characteristic manner on their shapes in a flow. Spheres, for instance, moved
by rolling, ellipsoid-type particles and pear-type particles moved by saltation or rolling taking thelong-axis to a rotating-axis and disk-type particles moved by gliding and rolling taking the shortaxis to a rotating-axis.
2. Average velocities of single particles were influenced by their shapes compared to their sizes.
Dimensionless maximum projection area was used as the shape index because particles mainly
moved taking a large projection area against the flow. Average velocities became large as
dimensionless maximum projection area increased.
3. Average velocity deceleration rates of non-spherical particles expressing the interaction among them
became smaller than that of spheres because non-spherical particles jumped to high levels. Averagevelocity of mixed particles with different kinds of shapes was estimated using average velocities ofsingle particles and average velocity deceleration rates of respective shape of a group of particles.
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