River Flow 2014 – Schleiss et al. (Eds)© 2014 Taylor & Francis Group, London, ISBN 978-1-138-02674-2

Effects of particle collisions on motions of mixed-size particles
and bed structures
T. Fukuda & S. Fukuoka

Research and Development Initiative, Chuo University, Bunkyo-ku, Tokyo, Japan

ABSTRACT: Gravel-bed rivers display a large variety of particle sizes and shapes. However, effects ofcollisions among particles in different sizes on particle motions and bed structures are hardly clarified.
This paper presents results of a numerical simulation with a numerical movable-bed channel capable ofsimulating three-dimensional motions of flows and particles in different sizes and shapes. The numericalsimulation revealed differences in action directions of contact forces of particles in different sizes andeffects of particle-volume concentrations on particle collisions. The simulation also demonstrated thatcollisions among particles in different sizes strongly affected particle motions and vertical structures ofthe movable bed.
1

Introduction

Sediment transports in gravel-bed rivers depend on
particle characteristics such as sizes, shapes and bed
structures. Large gravel particles exposed on bed
surfaces were considered to resist mainly against
hydrodynamic forces during floods and reduce
entrainment of small particles from the river bed(Osada & Fukuoka 2010, Fukuoka & Abe 2007).
Large-particle clusters were found to increased­river-bed stabilities by observations in gravel-bed
rivers (Hendrick et al. 2010). Many researchers
have investigated sediment-transport mechanism ingravel-bed rivers. However, their mechanisms havenot been clarified sufficiently because motions ofparticles could be hardly measured in flood flows.
The progress in computer processors made possible to investigate to sediment-transport mechanism
by numerical approaches in which particle motions
were simulated by a Lagrangian method (Truong
et al. 2007, Harada et al. 2013). Fluid motions
were simulated by the governing equations of
one-fluid model for solid-liquid multiphase flows
in an Eulerian method and particle motions were
simulated by the Distinct Element Method. This
numerical method can predict fluid forces acting
on particles directly by simulating fluid motions
around particles in detail using smaller computational cells than particle sizes. Most studies in the
Lagrangian method have modeled particles as asingle sphere. Therefore, it was unable to discuss
important effects of sizes and shapes of particles
on vertical structures of movable beds such as sorting and imbrications seen commonly in gravel-bed
rivers (Millane et al. 2006).

We developed a numerical model (numerical movable-bed channel) capable of simulating motionsof flows and particles in different sizes and shapes
which were made by the superposition of several
small spheres, as shown in Figure 1 (Fukuda et al.
2012, Fukuda et al. 2013). ­Sediment-transport
equations are generally derived by using hydrodynamic forces. However, the numerical movablebed experiment revealed that particles frequently
started to shift by collisions with particles moving down from upstream. These facts told us thatappropriate considerations of particle collisions aswell as hydrodynamic forces were essential for prediction methods of sediment transports in gravelbed rivers.
The present paper first explains a development
process of bed topographies of antidunes causedby sorting of particles and forming large-particle
clusters in the numerical channel, and then demonstrates dynamic features of collisions of large particles and small particles and effects of collisions
on particle motions and bed structures.
Eulerian equations of particle motions arepractical for simulating sediment transports over a

Figure 1.
particles.

799

Shapes of model particles and real gravellarge area. However, the Eulerian approach seems
to be difficult to incorporate properly dynamics
of particle collisions at present. Thus, establishing Eulerian equations of particle motions basedon results of Lagrangian particle motions in the
numerical movable-bed channel is required forpredictions of bed variations in gravel-bed rivers.
Stresses of particle collisions must be estimated forthis purpose. Time and space averaging of particle
collisions and averaging collision forces acting ona unit area are necessary for estimation of stressesof particle collisions. However, studies estimatingstresses of particle collisions based on simulated
Lagrangian particle motions showed to be small
variations of particle motions in space and time
(Harada et al. 2013). It was caused by numericalcomputations of small simulation regions for large
computational load.
The present numerical channel was long (15 m)
to discuss relationships among particle motions,particle collisions, and bed structures under a condition where particle sorting and bed waves wereformed. Estimating stresses of particle collisions
is difficult under these nonuniform conditions.
Therefore, we focus on not stresses of particle collisions but forces acting on each particle directly
and investigate contact forces on particles in different sizes. Then, relationships among particle
collisions, Lagrangian particle motions and bed
structures are discussed in the numerical movablebed channel.

and ∆: computational cell size. Physical propertyφ (density ρ, dynamic viscosity µ) and averaged
velocity ui were computed as volume-averaged values and mass-averaged values, respectively.
Water surfaces were simulated by computing
advection of water volume fractions in fluid computational cells with the Donor Cell Method.

2 Numerical method

rg = Mg + F f + Fc
M

2.1

ω r = I r −1 R −1 N f + N c − ω r × I rω r

In the simulation of fluid motions, liquid and solid
were simulated as an incompressible multiphaseflow where regions of gravel particles were considered as fluids having densities different from water(Ushijima et al. 2008). Smagorinsky model wasused as the subgrid turbulence model:
∂ui
=0
∂xi

(1)

{

ν t = (CS ∆ )22Sij Sij

Particles of various sizes and shapes were made by
the superposition of several small spheres (Matsushima et al. 2009) to detect contact and collision
points between particles, as seen in Figure 1. It wasdifficult to compute geometrical shapes of lapped
part of spheres, therefore, computations of rigid
body properties such as mass, gravity centers, andtensor of momentum inertia become important.
We computed rigid body properties of particles
by using sufficiently small cells and integrating
numerically volumes and points of cells included
in individual particles. Rigid body properties ofparticles do not change in motion, therefore, they
were computed only once at the beginning of thesimulation for every particles.
2.3

Governing equations of particle
motions in flows

Particle motions were simulated by the momentum
equations and the angular momentum equationsof rigid bodies:{ (Governing equations of fluid motions

∂u
∂ui
∂
1 ∂P
+
+ u j i = gi −2 (ν + ν t ) Sij∂x j∂tρ ∂xi ∂x j

2.2 Model particles with different sizes and shapes

}

(2)
(3)

where ui: i-th component of averaged velocity of a
fluid computational cell including gravel particles,
ρ: density, P: sum of pressure and isotropic component of SGS stresses, g: gravitational acceleration,
Sij: strain rate tensors, ν: kinematic viscosity, νt:
SGS turbulent viscosity, Cs: Smagorinsky ­constant,

)

}

(4)
(5)

where bold face letters indicate vector, tensor, andmatrix, M: mass of particle, rg: position of centerof gravity, F: force acting on a particle surface, N:
torque acting on particles, ω: angular velocity, R:
transformation matrix (from the local coordinates
system to the global coordinates system), and I:
tensor of momentum inertia. Suffixes f and c are
fluid force and contact force, respectively, and suffix r is components in local coordinates systemsof each particle. In solving angular momentum
equations, next time step angular velocities inlocal coordinates system ωr were first solved witheq. (5). Then, angular velocities in global coordinates system ω were computed with transformation matrices R. Next time step transformation
matrices R were set in consideration of rotations
in time step ∆t with the angular velocities ω. Next
time step small sphere positions were set with nexttime step transformation matrices R and small
sphere positions in local coordinates system. In
these coordinates transformation between global

800

coordinates system and local coordinates system,
quaternions were used instead of transformation
matrix R (Ushijima et al. 2008).
2.4

Fluid force

Fluid forces acting on a particle are estimated withintegrating forces acting on a particle region in themultiphase flow (Ushijima et al. 2008):

 ∂P
∂
F f ,i = ∫  −+ρ
2 (ν + ν t ) Sij  d ΩΩs∂x j ∂xi


{

}

(6)


∂
 ∂P
N f ,i = ∫ ε ijk rf , j −+ρ
2 (ν + ν t ) Skl  d Ω
Ωs
∂xl

 ∂xk

{



}

(7)

where Ff,i: i-th component of fluid force, Nf,i:i-th component of torque caused by fluid force,
rf,i: position vector from a center of gravity of aparticle to a fluid computational cel, Ωs: region
occupied by a particle, and εijk: Levi-Civita
symbol.
2.5

3numerical movable-bed
experiment

The numerical movable-bed channel and the definition of coordinates system used in the analysis
are shown in Figure 2. The numerical channel was
15 m long, 1 m wide with a bed slope of 1:20 andlarge enough to transport gravel particles, sufficiently long for sand waves to form, and accommodated to computation load. Five different sizes

Contact forces

Contact forces acting among particles are computed by contact detections of each small spherecomposing a particle. Contact forces and torques
act on centers of gravity of particles. They arecomputed by the summation of computed contact
forces concerning each small sphere:
Fc = ∑ Fcp,n , N c = ∑ rcp,n × Fcp,nnFigure 2. Numerical movable-bed channel and definition of coordinates system.

(8)nwhere, Fc and Nc are contact force and torqueacting on center of gravity of a particle, respectively. Fcp,n is contact force acting on each small
sphere, and rcp,n is position vector from particle’s
center of gravity to a contact point. Contact
forces among particles were simulated by the Distinct Elements Method (Cundall & Strack 1979).
Spring constants k and coefficients of dashpots cwere computed by equations (9)–(11) (Tsuji et al.
1992):1
3

 4  r r   E  2 
kn =   1 2  en 2 9  r1 + r2   1 − pos s0 =where e: spring force, E: Elastic modulus, pos: Poisson’s ratio, h: dashpot coefficient, r1 and r2: radiiof two contacting spheres, m1 and m2: mass ofcontacting spheres, and suffixes n and s: components of directions from center of a sphere to thecontact point and orthogonal two directions on thetangential plane, respectively.

ks1=
kn 2 (1 + pos )m1m2
cn = 2 h
kn , cs = cn s0
m1 + m2

(9)
(10)
(11)

Figure 3.

Particle size distribution.

Table 1. Parameters used in the simulation.
∆x,∆y,∆z: Fluid-computational
cell size∆t: Time step for fluid simulation∆t’: Time step for particle simulation
ρw: Density of waterρs: Density of particles
μw: Viscosity coefficient of water
μs: Viscosity coefficient of particles
CS: Smagorinsky constant
E: Elastic modulus
pos: Poisson’s ratio
h: Dashpot coefficient

801

0.01

m

1.0 × 10-4
1.0 × 10-6
1,000
2,650
8.9 × 10-4
8.9 × 10-4
0.173
5.0 × 1010
0.33
0.11

s
s
kg/m3
kg/m3
Pa ⋅ sPa ⋅ sPa

of particles (40 mm, 50 mm, 70 mm, 90 mm, and
120 mm) with four different shapes of gravel particles as shown in Figure 1 were made by varying
sizes of small spheres and by superimposing themproperly. Diameters of different shape particles
were estimated as diameters of spheres with thesame volume. In this paper we defined particles
more than 90 mm as larger particles and less than
70 mm as smaller particles. The particle size distribution of the numerical channel bed was givenby that as shown in Figure 3. Then the channel
was made by dropping a set of particles with thesize distribution randomly and repeatedly. A discharge of 0.5 m3/s was supplied at the upstream
end of the channel and zero-pressure was given
at the downstream end. The numerical movablebed ­experiment was lasted for 400 s. It took about
3 months by the computation using shared memory
systems (64 core: Intel Xeon E7-8837 (2.67 GHz)
8 core × 8). The parameters used in the simulation
are shown in Table 1.
4 results of numericalmovable-bed experiments
4.1

Variations of sediment-transport rates
and hydrodynamic forces with developmentof antidune

Temporal variations in longitudinal profiles oflaterally averaged-bed levels and water-surface
levels are shown in Figure 4. The values wereaveraged for two seconds because flow fields frequently changed in a second due to turbulence.
The flow was supercritical and water surface profiles varied in the same phase as bed waves. Temporal variations of the bed surface are shown in
Figure 5. Beds at the beginning of the numericalexperiment were plane and changed into antidunes after t = 300 s. Antidunes moved upstream
between t = 300 s and 400 s. Variations in hydraulic conditions every 100 seconds are shown in
Table 2. The table shows values of maximum,
minimum, and cross-sectional average in termsof depth, velocity, and Froude number measured
from x = 6 m to x = 15 m. Differences between
the maximum and minimum values of depths,v­ elocities, and Froude numbers became greaterwith the elapsed time.
Sediment-transport rates of particles in different sizes and in total measured at the downstream
end of the channel are shown in Figure 6. The­figure shows that the total sediment-transport rate
decreased with time. Sediment-transport rates oflarger particles were more than twice as large asthat of smaller particles before the development of
antidunes, however, they became almost the sameafter the development of antidunes (t = 300 s).
Depositions of sediment on upstream slopes ofantidunes (in t = 300 s – 400 s at x = 3 m, 6 m, and13.5 m, as shown in Fig. 4) decreased sedimenttransport rates, and scourings at downstream
slopes of antidunes (in t = 300 s–400 s at x = 4.5 m,
9 m, and 12 m) increased them. Sediment depositions and scourings did not necessarily occur atthe same time. Therefore, sediment-transport
rates greatly varied with time after developmentof antidunes.
Hydrodynamic forces acting on the bed werecomputed to investigate effects of antidunes on
the motions of flows and particles. Figure 7 shows
distributions of hydrodynamic forces acting on
movable-bed in a unit area at t = 100 s and 400 s.
These values were evaluated by converting hydrodynamic forces to shear stresses τf,zx. The converted
forces τf,zx were calculated by eq. (12). The regionof integrations is from the channel bottom to the
top of moving particles in 0.01 m × 0.01 m area on
the x–y plane.
 ∂Pτ f , zx =∂∫ ∫ ∫ α − ∂x1 + ρ ∂x j {2 (ν + νt ) S1 j} dxdydz
∫ ∫ dxdy(12)

where α is a volume fraction of particles in a fluid
computational cell.
Figure 7 shows that large hydrodynamic forces
acted sparsely on large particles exposed on the
bed surface at t = 100 s before the development of
antidunes. On the other hand, large hydrodynamic
forces after the development of antidunes (t = 400 s)

Figure 4. Longitudinal profiles of laterally averaged water-surface levels and bed levels.

802were seen on the bed of x = 3–4 m, x = 7–8 m, andx = 11–12 m. They were concentrated on antidunes.
The mean velocities in the channel center averaged
in t = 390 s∼t = 400 s are shown in Figure 8. Largehydrodynamic forces concentrated on downstream
slopes of antidunes where flows near the bed wereaccelerated as shown in Figures 7 and 8. We averaged hydrodynamic forces acting on particles in aunit area over x = 6 m to x = 15 m. A temporal variation of averaged hydrodynamic forces on particles
in a unit area is shown in Figure 9. ­Hydrodynamic
forces acting on the bed after development of antidunes (after t = 300 s) were larger than those of theinitial stage. The comparison between sedimenttransport rates (Fig. 6) and hydrodynamic forces(Fig. 9) shows that sediment-transport rates in the
final stage were smaller than those of the initialstage while hydrodynamic forces in the final stage
were greater. This demonstrates that the sedimenttransport rate was controlled by not only hydrodynamic forces acting on bed surfaces but also

bed structures. A temporal variation of particle
size distributions on the bed surface is shown in
Figure 10. When hydrodynamic forces were a minimum at t = 200 s (Fig. 9), the particle size distribution on the bed surface was a little smaller than thatof the initial but almost the same. When antidunes
developed (t = 400 s), the rate of large particles on
the bed surface were not necessarily great. Thus,
the reason why hydrodynamic forces increased at
t = 400 s was not due to increasing the percentageof large particles on the bed surface. It was because
antidune formation increased the resistance of the
bed and decreased sediment-transport rates.
4.2 Vertical structures of antidune beds
A distribution of particle-volume concentrationsin the center of the channel in t = 400 s is shownin Figure 11. Particle-volume concentrations
were computed by counting the number of small
cells (0.0053 m3) occupied by particles in a region(0.4 m square and 0.01 m height) as illustrated in
Figure 12 and dividing the number by the total
number of cells in the region. We set a region withthe height of 0.01 m to evaluate detailed distributions changing greatly in the vertical direction.
The figure indicates particle-volume concentrationin the x–z plane along the center of the channel.
A distribution of average particle sizes in the x–z
plane is similarly shown in Figure 13. These values
are computed as (∑dk)/Nkmax, where dk is a diameterof a particle occupying a cell k in the region, andNkmax is the total number of cells occupied by particles in the region.
Vertical distributions of particle-volume concentrations of each size and average particle sizes areshown in Figure 14. These figures depicted vertical
bed structures of antidune beds at x = 7 m wherelarge particles formed clusters on the bed and at
x = 9 m where they did not. Figures 11, 13, and14 show that particle-volume concentrations were
nearly 0.8 in the lower part of the bed. At the crownof antidunes (x = 7 m), particle-volume concentrations gradually decreased from about 0.15 m tothe vertical upward direction. Porosities of the bed
surface were considerably large there. In contrast,

Figure 5.
surfaces.

Temporal and spatial variations of bed

Table 2.

Temporal variations of hydrodynamic conditions.
Depth (m)Average velocity (m/s)

Fr

Time

Minimum

Average

Maximum

Minimum

Average

Maximum

Minimum

Average

Maximumt = 100 s
t = 200 s
t = 300 s
t = 400 s

0.208
0.219
0.212
0.200

0.236
0.258
0.258
0.268

0.259
0.307
0.370
0.414

1.93
1.63
1.35
1.21

2.12
1.95
1.97
1.93

2.41
2.29
2.36
2.50

1.22
0.94
0.71
0.60

1.40
1.24
1.26
1.22

1.69
1.56
1.64
1.78

803

particle-volume concentrations sharply decreased
from about 0.1 m to the vertical upward direction
at the trough of antidunes (x = 9 m). Porositiesof the bed surface are relatively small. Therefore,
particle-volume concentrations of troughs at antidunes were high where few large particles locatedon the bed surface.

Figure 6. Temporal variations of sediment-transport
rates.

5

Effect of contact forceson particle motions and bed
structures

5.1 Effect of contact forces on vertical structuresof beds
We understood that the bed came to be stablethrough particle sorting on the bed surface causedby collisions among particles in different sizes.
­Figure 15 shows relationships between particle
contact forces in the x-direction and the z-direction, which clarifies directions of contact forces in­collisions. Particle-collision velocities which werelarger than 0.1 m/s were used as sampling data of the
figure. If these criteria of samples were the momentum, particle-velocity criteria of the smallest particles were 27 times larger than the largest particles,
because the mass of the smallest particles was 1/27

Figure 7. Hydrodynamic forces acting on a unit area in the x–y plane.

Figure 8. Flow in the center of the channel (Values are averaged in t = 390 s ∼ t = 400 s).

Figure 9. Temporal variations of averaged hydrodynamic forces acting on particles in a unit area.

Figure 10. Temporal variations of the particle size distribution on the bed surface.

Figure 11. Distribution of particle-volume concentrations in the center of the channel at t = 400 s.

804

times (0.043/0.123) as much as that of the largest particles. However, small particles would not
move so fast (Fukuda et al. 2013). Therefore, we
selected a particle velocity as the criteria of samples to investigate contact forces among particles
in motions.
Contact forces were nondimentionalized bysubmerged weights of particles. Contact forces inthe z-direction of larger particles were great compared to those in the x-direction. The reasons are
explained as follows: Small particles in a cloudof moving particles drop into gaps among particles easily. In contrast, large particles do not into
gaps because of being supported by surrounding
particles. Therefore, the larger particles were, thegreater vertical components of contact forces were.
There are differences between action directions ofcontact forces of particles in different sizes, andlarger particles can move in upper levels of the bed
surface. This is the mechanism of particle sortingon the movable bed.
Effects of particle-volume concentrations onparticle motions were investigated. Relationships
between particle-volume concentrations and ratiosof velocities before collisions to after collisions(v2/v1) are shown in Figure 16 where v1 and v2 arevelocities in the x-direction before and after collisions, respectively. The ratios (v2/v1) decreased with
the increase in particle-volume concentrations.

Figure 12. Region for measurement of particle-volumeconcentration and average particle size.

This means that particles velocities decreased byparticle collisions.
Relationships between stabilization of the
bed and particle sorting on the bed surface are­considered. We found that large particles movingon the bed surface received contact forces in the
upward direction more easily than small particles
in collisions and moved upper positions on the
bed surface. Large particles moving in the upper
position of the bed surface are not so many, andparticle-volume concentrations are relatively small.
Protruding large particles are easy to be transported
by flows and to stop by collisions with stationary
large particles on the bed surface. Then, larger particles form clusters and increase particles volumeconcentrations at the bed. In this manner, all particles on the bed surface come to be in relatively
high volume concentrations and to resist against
particle collisions and hydrodynamic forces. Thismechanism leads to stable bed structures in gravelbed rivers.
5.2 Contact forces at the time of incipientand settling motions
The x-direction particle velocities, contact forces,and hydrodynamic forces nondimentionalized bysubmerged weight of particles at the time of incipient and settling motions are shown in F
­ igure 17.
There are positive and negative contact forces. Positive and negative contact forces are forces which
particles receive in collision with upstream particles
and receive from downstream particles supporting them, respectively. On positive contact forces,
small particles received larger contact forces thanhydrodynamic forces and large particles did largerhydrodynamic forces than contact forces. The difference is explained as follows: Particle-volumeconcentrations were large in the lower part of the

Figure 13. Distribution of average particle size in the center of the channel at t = 400 s.

Figure 14. Vertical distributions of particle-volume concentrations of each size and average particle size.

805

Figure 15. Directions of contact forces acting on particles in motion.

Figure 16. Relationships between particle-volume concentrations and ratios of particle velocities before collisions to after collisions.

Figure 17. Time variations of particle velocities, hydrodynamic forces and contact forces at incipient and settling
motions.

bed surface where many small particles rested inbeing sandwiched by other particles. Small particles
received larger positive contact forces compared tolarge particles at the time of incipient motions. Thisis because small particles often collide with moving
large particles. Thus, contact forces from relatively
large particles are essential for incipient motionsof small particles. In contrast, large particles rarely
received contact forces from larger particles. Largeparticles started to move by receiving larger hydrodynamic forces than contact forces.
Hydrodynamic forces acting on large particles
and small particles were considerably small at thetime of settling compared to that at incipient motion
because flows decelerated on upstream slopes ofantidunes and a large number of particles settled
there. These results show that proper estimations of
momentum exchanges and collisions among particles in different sizes are required for predictions of
sediment transport in gravel-bed rivers.
6 conclusions
Numerical movable-bed experiments were conducted by using numerical movable-bed channel

with particles in different sizes and shapes. The following main conclusions were obtained.
1. Effects of hydrodynamic forces and contact
forces of particle in different sizes were clarified
by focusing on particle motions and verticalstructures of movable-beds.
2. Contact forces in collisions make large particles
move vertical upward compared to small particles and large particles move over small particles
because small particles drop into gaps among
particles.
3. Particle collisions bring a great impact on
incipient motions of particles. Ratios of contact forces to hydrodynamic forces at incipient
motions of small particle are higher than thoseof large particles.
4. Large particles moving in upper positions of
the bed surface are not so many and particlevolume concentrations are relatively small.
Protruding large particles are easy to be transported by flows and to stop by collisions withstationary large particles on the bed surface.
5. Mechanisms of sorting and stabilization of
the bed surface were clarified on the basis of
motions of particles in different sizes.

806

references
Cundall, P.A. & Strack, O.D.L. 1979. A discrete numerical model for granular assemblies. Geotechnique 29 (1):47–65.
Fukuoka, S. & Abe, T. 2007. Mechanism of low water
channel formation and the role of grain-size distribution in gravel-bed rivers, 10th International Symposium
on River Sedimentation, Moscow, Russia.
Fukuda, T. Fukuoka, S. & Uchida, T. 2012. Threedimensional numerical modeling for the motion ofstones with different sizes and shapes in streams. 10th
International Conference on Hydroscience & Engineering, Orlando, USA.
Fukuda, T. Fukuoka, S. & Uchida, T. 2013. Three-dimensional motions in numerical movable bed channel
with particles of various shapes and sizes. 12th International Symposium on River Sedimentation, Kyoto,
Japan.
Harada, E. Gotoh, H. & Tsuruta, N. 2013. Distribution of shear stress in oscillatory sheet-flow layer by
LES. Journal of JSCE, B2 (Costal Engineering) 69(2):I_556-I_560. in Japanese.
Hendrick, R.R. Ely, L.L. & Papanicolaou, A.N. 2010.
The role of hydrologic processes and geomorphology
on the morphology and evolution of sediment clusters
in gravel-bed rivers. Geomorphology 114: 483–496.

Matsushima, T. Katagiri, J. Uesugi, K. Tsuchiyama, A. &Nakano, T. 2009. 3D shape characterization and imagebased DEM simulation of the lunar soil simulant
FJS-1. J.Aerospace Engineering 22(1): 15–23.
Millane, R.P. Weir, M.I. & Smart, G.M. 2006. Automatedanalysis of imbrication and flow direction in alluvial
sediments using laser-scan data. Journal of Sedimentary Research 76: 1049–1055.
Osada, K. & Fukuoka, S. 2010. Analysis of two-dimensional riverbed variation in channels with stony beds.
11th International Symposium on River Sedimentation
Stellenbosch South Africa.
Truong, H. Nguyen, A. Sugimoto, H. & Wells, J. 2007.
Fictitious-domain simulations of bedload transport;
free-surface flow in a rotating tumbler. Annual Journalof Hydraulic Engineering, JSCE 51: 157–162.
Tsuji, Y. Tanaka, T. & Ishida, T. 1992. Lagrangian numerical simulation of plug flow of cohesionless particles
in a horizontal pipe. Powder Technology 71; 239–250.
Ushijima, S., Fukutani, A. & Makino, O. 2008. ­Prediction
method for movements with collisions of arbitraryshaped objects in 3D free-surface flows. JSCE Journal
B 64(2): 128–138. in Japanese.

807