Effects of particle shapes and sizes on fundamental movement
processes of particles and sediment transport rates
in gravel streams
Hazumu Tadokoro
TOKEN C.E.E Consultants Co., Ltd.
1-8-63, Temmabashi, Osaka Kita-ku, Osaka 530-0042
Tel. +81 (06) 7636 1152, Fax: +81 (06) 7636 1153,
e-mail : tadokoro-h@tokencon.co.jp

Shoji Fukuoka
Professor, Research and Development Initiative, Chuo University
1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551:
Tel. +81 (03) 3817 1625, Fax: +81 (03) 3817 1625,
e-mail : sfuku@tamacc.chuo-u.ac.jp

Tomoo Fukuda
Associate Professor, Research and Development Initiative, Chuo University
1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551:
Tel. +81 (03) 3817 1625, Fax: +81 (03) 3817 1625,e-mail : t-fukuda@tamacc.chuo-u.ac.jp

Gravel-bed rivers are composed of particles with a large variety of sizes and shapes. Largeparticles would resist against flood flows, but are picked up and move intermittently. We estimate
fundamental movement processes of particles with different sizes and shapes and the sedimenttransport rate based on fundamental movement processes by numerical movable-bed simulations. The
effect of particle shapes on movement processes of large particles is greater than that of small
particles. Therefore, the effect of different particle shapes on sediment transport rate appears
conspicuously in large particles.
Key Words: gravel bed river, particle shape, particle size, fundamental movement process,sediment transport rate, numerical movable-bed channel.

1. Introduction
Gravel-bed rivers consist of a large variety of particle sizes and shapes．Large particles at the bed
resist against flood flows, but are picked up and move intermittently. These particle motions aredifferent from those in sandy rivers where most of particles tend to move continuously. The sedimenttransport rate of mixed particles is given by Eq.(1) based on fundamental processes such as pick-up
rate and step length of particle with diameter dij.
3Ad(1)q  f (d ) 3 ij    ps
Bij

ij

A2 d ij

2ij

ij

Where i: particle size , j particle shape, ps： pick-up rate, Λ： step length, A3d3：particle volume,
A2d2：particle area projected from above, f(dij): area ratio of dij.
To estimate sediment transport rates in gravel-bed rivers, fundamental movement processes ofparticles with different sizes and shapes have to be investigated. Fukuoka et al. (2014) developed anumerical movable-bed channel which could simulate three-dimensional motions of flows and gravel

通過重量100分率(%)

Cumulative percentage

flow
x=8m

x=10m

Fig.1 Numerical movable-bed channel.

100

8060
40

20010100

粒径(mm)
GrainSize (mm)1000

Fig.2 Particle size distribution.

Table 1. Shape factor and length of gravel particles.
SphereShape No.
a: Longest
b: Intermediatec: Shortest
Shape Factor

4

Shape 1
23Fig.3 Shape of gravels particles.

Sphere Shape 1 Shape 2 Shape 3 Shape 4
1
1.26
1.29
1.36
1.49
10.98
1.06
0.99
0.89
1
0.88
0.81
0.78
0.76
10.79
0.69
0.67
0.66

particles with different shapes and sizes (see Fig.1). In this paper, numerical movable-bed simulations
were conducted under three conditions of spheres, gravel particles and mixed particles, and sedimenttransport rates and fundamental processes were measured to estimate respective effects of the own
particle shape and surrounding particles on sediment transport.

2. Numerical computation method
Particle motions were simulated in the Lagrangian method as the rigid-body and fluid motions werein the Eulerian method. To take into account the effect of the solid phase on the liquid phase, fluid
motions were simulated by the governing equations of one-fluid model for solid-liquid flows. Particle
contacts were computed by distinct element method (DEM). Fluid dynamic forces on particles werecomputed directly by integrating the forces on a particle. Gravel particles with different shapes andsizes were made by the superposition of small spheres. The validation of the numerical computation
method was checked by distributions of velocities and concentrations of real gravel particles measured
in a large open channel flow(Fukuoka et al., 2014).

3. Simulation conditions
In the numerical simulations, we used five particle sizes(40, 50, 70, 90, 120 mm) (see Fig.2), andfour gravel particle shapes (see Fig.3). Particles of 40, 50 and 70 mm were categorized as small
particles, 90 and 120 mm as large particles. The diameter of gravels was defined by a diameter ofspheres having the same volume. This means that A3d3 of Eq (1) is the same value for every particle
sizes in this stady. It is common to use the shape factor (Eq.(2)) which consists of long, middle andshort axes of particles.
S .F . c(2)abWhere a: long axis, b: middle axis, c: short axis. The shape factors are close to 1 as particle shapes arealike to the sphere shape. Shape No.1 is the closest to the sphere, and other gravel shape factors arealmost similar values (see Table1). Particles were packed in numerical movable-bed simulation
channels (length is 15 m, width 1 m, depth 1 m and bed slope 1/20). A water discharge of 0.5 m3/s was
supplied at the upstream end of the channel, and the zero pressure condition was set at the downstream
end. The equal amount of water and particle volumes discharging out from the channel were suppliedat the upstream end of the channel. Sediment transport rate and movement processes of particles weremeasured in sections from x=2 m to x=12 m.

4. Transport rates of particles with different shapes and sizes
Figure 4 and 5 show average sediment transport rates measured in 2m and 60 seconds intervals withrespect to space and time. To compare sediment transport rates of every particle with that mixed
particles, transport rates of spherical particles and gravel particles were multiplied by 1/5, and by 4/5
respectively. Transport rates of large particles become greater than that of small particles. Moreover,sediment transport rate became large as particle shapes were close to the sphere. Sphere transport ratesin mixed particles were smaller than that of sphere particles only due to the increase in engagement
effects of surrounding particles. In other words, the particle movement was affected by not only own
particle shape, but also surrounding particle shapes.

5. Fundamental movement processes of particles
Fundamental movement processes (pick-up rate, step length and particle projected area) ofsediments with different sizes and shapes are estimated from results of the numerical movable-bedsimulation. Effects of particle sizes and shapes on fundamental movement processes and sedimenttransport rates were discussed below.
The A2d2 is defined as a projected area measured from above at the time just before particles arepicked up. Figure 6 shows dimensionless average areas of particle from above just before picking up aparticle. They were nondimensionalized by the projected area of the equivalent size sphere (πd2/ 4).
We assume that particles are picked up when the velocity of particles exceeds 0.05m/s. Figure 7 shows
the pick-up process of particles. The A2d2 becomes large as the particle shape deviates from a sphereshape, because particles tend to rest stably by directing flat plane toward the top and lowering a centerof gravity (see Fig.7(a)). The dimentionless A2d2 of large particles is larger than that of small particles.
It is reason why large particles were not affected much by surrounding particles compared with smallparticles.
Pick-up rates of particles in every shapes and sizes were evaluated by the ratio of area which someparticles picked up in a unit time had occupied on the bed surface to the area which the same particles
occupied on the bed surface. Figure 8 shows averaged pick-up rate of particles in every sizes andshapes. Pick-up rates of small particles became smaller as particle shapes deviate from a sphere, but
those of large particles took almost constant values. Pick-up rates did not greatly vary in comparison
with A2d2. In short, pick-up rates were mainly affected by the placement of the surrounding particles.
Therefore, it is hard to recognize conspicuously the effect of particle shape on pick-up rate.
The step length was defined as the distance that particles move from pick-up to deposition. Steplengths of large particles were unable to be measured within length of the numerical channel. Figure 9shows step lengths of small particles nondimensionalized by particle diameters. The step length
became larger as particle shapes differed much from a sphere shape. Rolling particles had large
projected area to the flow and hard to stop as particle shapes deviated from a sphere. However, No.3
and 4 particles do not show such a characteristic. Therefore, it is difficult to evaluate effects of particle
shapes on step lengths by only using the conventional shape factor (Eq.(2)). Figure 10 shows therelationship between the step length and the smallest projected area of particles during moving periods.
Dimensionless step lengths are small except spheres as particles are small and have smallest projection
areas. Then, rolling particles are easy to stop, because projected areas in the streamwise direction
Gravel particles

Mixed particles

2.5E-04

2.0E-04 2.0E-04

2.0E-04

1.5E-04 1.5E-04

1.5E-04

1.0E-04 1.0E-04

1.0E-04

5.0E-05 5.0E-05

5.0E-05

0.0E+00 0.0E+00
Sphere12.5E-04 2.5E-04

Sediment transport
3rate(m /s)

Sediment transport
3rate(m /s)

Spheres

2.5E-04 2.5E-04

0.0E+00
Shape4
1nshp=5 2
nshp=4
5

Shape1
Shape2
Shape3
nshp=1
nshp=2
nshp=32
3
4

Dimensionless area ofparticle from above(A2d2/(πd2/4))

Fig.4 Sediment transport rate in sphere and gravel.
Spheres
Mixed particles
conditions.
1.3

1.5E-04 1.5E-04

1.0E-04 1.0E-04
5.0E-05 5.0E-05
0.0E+00 0.0E+00
Sphere451
球形Shape112形状 1

Shape223
形状 2Shape334

形状 3Shape445
形状 440mm
50mm

1.1

70mm

flowflow(a) Stop(b) Pick-up

90mm

1120mmSphereSphereShape1

Shape2

Shape35

Fig.5 Sediment transport rate in mixed conditions.

1.2

0.9

3

混合

2.0E-04 2.0E-04

Shape4

Fig.6 Average area of particle measured from above,
just before picking up particles.

Fig.7 An example of picked up particle.

50mm

25

D=40mm

20

D=50mm

15

D=70mm

70mm
90mm

0.01
030120mm
SphereSphereShape1

Shape2

Shape3

Shape4step length(Λ/d)

0.0240mmDimensionless

Pick-up rate (1/s)

0.03

Fig.8 Pick-up rate with respect to particle
shapes and sizes.
D=70mm
3D=50mm22520 Shape 4

1Sphere shape

31510
0.70

0.80
0.90
1.00
無次元最小投影面積
Dimensionlesssmallest projected area

Fig.10 Comparison between step length andsmallest projected area.

SphereSphereShape1

Shape2

Shape3

Shape4

Fig.9 Step length with respect to particle
shapes and sizes.

3/s）transport rate(Sedimentm /s)
流砂量（mstep length(Λ/d)
step length

Dimensionless

D=40mm301040mm（数値実験で計測した流砂量の素過程から算定）
40mm（計測区間において2m間隔で計測した流砂量の時空間平均）
50mm（数値実験で計測した流砂量の素過程から算定）
50mm（計測区間において2m間隔で計測した流砂量の時空間平均）: Sediment transport rate calculated by Eq.(1)70mm（数値実験で計測した流砂量の素過程から算定）: Sediment transport rate measured directly
70mm（計測区間において2m間隔で計測した流砂量の時空間平均）
1.0E-04
3.0E-05

7.5E-05

2.0E-05
5.0E-05
2.5E-05

1.0E-05
0.0E+00
123Sphere
1Sphere2

Shape13456

0.0E+00
Shape2
4

Shape3
5

Shape4

6Fig.11 Comparison between Eq.(1) andsediment transport rate.

become small.
Sediment transport rates of Eq.(1) calculated using results of fundamental movement processes
were compared with those of numerical movable-bed simulations(see Fig.11). It was confirmed thatEq.(1) reproduced well sediment transport rates of each particle sizes and shapes measured directly inthe numerical channel.
Finally, we investigated further the relationship between sediment transport rate and fundamental
movement processes (step length, pick-up rate and A2d2). In both small and large particles, A2d2
become large as particle shapes varied from the sphere, but the pick-up rate did not greatly change incomparison with A2d2. In short, the number of picked up particles per unit time (ps/ A2d2) become
small as particle shapes differ from the sphere. The step length becomes large as particle shapes aredifferent from the sphere. The product ((Ps /A2d2)×Λ) of the number of picked up particles and step
length is affected by the number of picked up particles (ps/ A2d2) than step length (Λ) , because of theeffect of particle shapes. Therefore, sediment transport rates are small as particle shapes vary from thesphere. In regard to large particles, effects of the bottom roughness on particle motions were relatively
small compared to own particle shapes. Therefore, the difference in particle shapes on sedimenttransport rates appears markedly on large particles. Sediment transport rates of small spheres and largespheres were maximum 1.3 times larger and 2 times larger than those of gravel particles, respectively(see Fig.5). Fundamental movement processes are different in every sizes and shapes. Therefore, toassess direct effects of particle shapes and sizes on fundamental movement processes are important for
estimations of transport rates of mixed sediments.

6. Conclusions
The applicability of sediment transport equation(1) was checked with the fundamental movement
processes measured in the numerical channel. Eq.(1) reproduced well sediment transport rates of eachparticle sizes and shapes measured directly in the numerical channel. However, they are different inevery sizes and shapes. From the present investigations, it was found that the assessments of direct
effects of particle shapes and sizes on fundamental movement processes are important for moreaccurate estimations of mixed sediments transport rates.
References[1]

Fukuoka, S., Fukuda, T. and Uchida, T. (2014). Effects of sizes and shapes of gravel particles on
sediment transports and bed variations in a numerical movable-bed channel., Advances in Water
Resources, Volume 72, p. 84-96.