River Flow 2020 – Uijttewaal et al (eds)© 2020 Taylor & Francis Group, London, ISBN 978-0-367-62773-7

Effects of irregular particle shapes on the sediment movement andtransport rate in gravel-bed channels
Takatoshi Atsumi
Graduate School of Science and Engineering, Chuo University, Tokyo, Japan

Shoji Fukuoka
Research and Development Initiative Chuo University, Tokyo, Japan

ABSTRACT: The numerical movable-bed experiments were conducted to investigate theeffect of particle shapes on the transport of gravel particles. The river-bed materials used in
the numerical experiment were three kinds of particles with the same nominal diameter(70 mm) and different shapes (Thin, Rod, Sphere). The analysis of the fundamental movement
process of sediment transport revealed the importance of particle shape on the pick-up process
in comparison with the particle velocity. Non-spherical particles form the imbrication andclusters which make the river-bed stabilize, and reduce the pick-up volume compared to thatof spherical particles. In particular, the difference in the pick-up rate between thin and rod
particles is considerable under high tractive force.

1 INTRODUCTION
The gravel-bed river is composed of particles of various shapes and sizes and takes the imbrication form which makes the river-bed stable. The shape and size of gravel particles are considered
to be significant on river-bed structure and sediment transport rate. For example, Gomez (1994)
pointed out that the increasing roughness of a stable armor changed due to the particle shape.
However, quantitative effects of shapes on bed variation and sediment transport have not beenclarified because it is difficult to measure the mechanism of particle motions during floods. In
addition, because particle size and shape that could be used in hydraulic experiments are limited,
the results of hydraulic experiments could not provide sufficient knowledge on the effect of particle shapes. Fukuoka et al. (2014) constructed a numerical movable-bed channel that takes
account of dynamic interactions between flow and particles using gravel particles in different
sizes and shape. They conducted numerical movable-bed experiments with sphere particles andgravel particles and clarified effects of particle sizes and shapes on sediment movement and sediment transport rates. Fukuda & Fukuoka (2017) applied their model to the debris flow experiment. The investigation by Fukuda & Fukuoka (2017) showed that this model could reproduce
distribution of the particle velocities and discharge rates of water and particles in hyperconcentrated flows. The investigation of Fukuoka et al. (2014) was difficult to consider the influence ofeach particle shape on particle motion because numerical movable-bed experiments with fivedifferent shapes and sizes particles. Therefore, the present numerical movable-bed experiments
are conducted using three kinds of gravels with uniform particle sizes of different shapes, whichfacilitate to understand the influence of particle shape.

2 NUMERICAL METHODSThe numerical simulation model of Fukuoka et al. (2014) is used. This model can simulate
three-dimensional motions of flows and gravel particles in different shapes and sizes. Fluid1

motions were simulated using the governing equations of a single-fluid model for the solidliquid multiphase flows in the Eulerian approach. Particle motions were computed as rigid
bodies with the Lagrangian approach using DEM method.
2.1 Fluid motions
The fluid motions are calculated using the governing equations used in a single-fluid model
for solid-liquid multiphase flows in the Eulerian approach. The Smagorinsky model was usedas the subgrid scale turbulence model.
∂ui
¼0
∂xi

Dui
1 ∂P
∂ 
¼ gi þ
2ð þ t ÞSijDt
ρ ∂xi ∂xi
1 ∂ui ∂ujþ
Sij ¼
2 ∂xj ∂xi
 ¼ μ=ρ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
t ¼ ðCs ΔÞ2 2Skl Skl

ð1Þ
ð2Þ
ð3Þ
ð4Þ
ð5Þ

where ui = i-th component of the weighted average velocity within a fluid cell; p = sum of
the pressure and isotropic component of SGS stress; ρ = volume-averaged density; μ =volume-averaged viscosity; gi = gravitational acceleration; νi = SGS turbulent viscosity; Sij =strain rate tensor; Cs = Smagorinsky constant; Δ = computational grid size.
The free-surface variation was simulated using the continuity equation of the fluid volume
fraction f based on the volume-of-fluid (VOF) method.
∂f ∂fui¼0
þ
∂t ∂xið6Þ

2.2 Particle motions
Gravel particles were modeled by connecting several small spheres without gaps. The particle
motion was simulated using the following momentum and angular momentum equations as
a rigid body:M€ri ¼ Mgi þ Fif þ Ficn 
o
Rj0 i Tif þ Tic  εj0 k0 l0 ωk0 Il0 m0 ωm0
ω_ i0 ¼ Ii1
0 j0

ð7Þ
ð8Þ

where index i = the components of the global coordinate system fixed on the space; theindices i'-m' = the components of the local coordinate system fixed on a rigid body; M =mass of a particle; gi = gravitational acceleration; ri = position of the center of gravity of therigid; Fi = force acting on the surface of the particle; suffix f, c = components of the fluid
forces and contact forces between particles, respectively; ωi = angular velocity; Ti = torqueon the center of the gravity generated by the force; Ii1
0 j 0 = inverse of the matrix consisting of
the components of the moment of inertia tensor in the local coordinate system; Rj'i = equal
to ej'ei (ej' and eiare unit basis vectors); εi'j'k' = Levi-Civita symbol. The basis vectors were
simulated by using the angular velocity for the next time step solved by Eq.(9) and the time
step Δtp. The positions of the small spheres in the next time step with respect to the global2

coordinate system were computed using the initial components of the positions of the small
spheres in the local coordinate system, the basis vectors that were computed at every time
step, and the center of gravity of the rigid body that was estimated on based on Eq.(8). Quaternion was used for the rotation of the basis vector, and the local coordinate systems using
Rj'i (Ushijima et al., 2008).
The parameters used in the simulations are shown in Table 1.

Table 1. Simulations parameters.
Δx,Δy,Δz: the size of the ﬂuid computation cellΔtf: Time step for the ﬂuid simulation
Δtp: Time step for the particle simulation(Case1/Case2,Case4/Case3)
ρw: Density of waterρp: Density of particles
μw: Fluid viscosity
Cs: Smagorinsky constant
E: Elastic modulus
pos: Poisson’s ratio

0.01
5.0 × 10-4
2.5 × 10-6/3.3 × 10-6/2.5 × 10-6

m
s
s

1000
2650
8.9 × 10-4
0.173
5.0 × 1010
0.33

kg/m3
kg/m3
Pa・s
Pa
-

3 NUMERICAL MOVABLE-BED EXPERIMENT
3.1 Experimental conditions
Particle shapes used in numerical experiments imitate particles in a real gravel-bed river. On thesand bar of the Asa river 2.3 km, 50 gravels were collected by the line grid method, and measured the longest axis a, the intermediate axis b and the shortest axis c are shown in Figure 1.
The solid line indicates the Shape Factor (c/(ab)0.5). The gravels are in the range of 0.4 ~ 0.6.
Figure 2 shows Thin particle and Rod particle modeled by small spheres, which represent particles having a shape factor in the range of 0.5 ~ 0.6. Thin particle in Figure 2a have a rugged
surface, and rod particle in Figure 2b have a smooth surface. The two types of particles have thesame volume, and the nominal diameter estimated as a sphere of the equal volume is 70 mm.
The two model particles and the gravels collected in the Asa river are plotted in Figure 1.
The numerical movable-bed channel and the coordinate system are shown in Figure 3. The
numerical channel is 15 m long, 1 m wide with a bed slope of 1:20. The particles were packed
randomly to make a river-bed. Table 2 shows the experimental conditions. Four numericalmovable-bed experiments were conducted, and the thin particle was used in Case1, the rod
particle in Case2, mixture of these two particles in Case3 and sphere particle in Case4. The

Figure 1.

Characteristics of particle shapes.

Figure 2. Particle shape viewed from different
directions.

3

Figure 3.

Numerical movable-bed channel and the coordinate system.

Table 2. Experimental conditions (Case1 ~ Case4).

Water discharge [m3/s]
Water depth [m]Froudeanumber(t=100s)[-]Particle shape
Norminal diameter [mm]Packing depth [m]

Case1

Case2

Case3

Case4

0.5
0.24
1.27
Thin
70
0.23

0.5
0.23
1.35
Rod
70
0.21

0.5
0.24
1.31
2Shapes (Thin,Rod)
70
0.22

0.5
0.25
1.25
Sphere70
0.22

volume of packed particles was equal in all cases. In Case3, the same amount of thin and rod
particles was mixed. The water discharge of 0.5 m3/s was supplied to the channel, and the pressure was set to zero at the downstream end. The same number and volume of particles discharging out from the end of the channel were dropped back simultaneously to a section (x =1~3m) near the upstream end of the channel. The water depth and the Froude number inTable 2 are spatially averaged values at x = 5 ~ 11 m.
3.2 Transported volume of gravel particles
Figure 4 Shows the volume of gravel particles passed through the cross section at x = 11 m. The
measurement was performed at t = 0 ~ 100 s while the river-bed remained almost plane. In
Case4, the transported volume of spherical particles is extremely large compared to other particle
cases. Unlike the non-spherical particle, the spherical particle has only one contact point. Therefore, the sphere is generally considered to be the most mobile shape. Comparing the volume inCase1(Thin) and Case2(Rod), Case2 is about 1.85 times that of Case1. The effect of the particle
shape on the transport rate is large. The total transported volume in Case3(2 shapes) wasapproximate half of that in Case1 and Case2, and there is not much difference between thin androd particles. This suggests that the transport volume of mixed gravels is affected not only theshape of the particle itself but also that of the surrounding particles. It was confirmed that thedifference in particle shapes has a large effect on sediment transport rates. In order to clarify
the cause of the difference, the fundamental movement process of the particle is analyzed below.
3.3 Influence of particle shape on moving process of gravels
In order to analyze particles in saltation movement on the gravel bed, the streamwise particle
velocity in the x direction of 0.05 m/s or more were defined as moving particles. In addition,
particles that have a speed of 0.05 m/s less due to collision between particles and continue tomove further were counted as the moving particle. At t = 0 ~ 100 s and x = 10 ~ 11 m, these4

moving particles were sampled at 0.1s intervals, and the moving velocity and the projectedarea in the x direction were analyzed. Figure 5 shows the particle moving speed in x direction
divided by the friction velocity of the flow in Case1 ~ Case4. In each case, the mean velocity
and the variation pattern of the moving particles show almost the same value. Figure 6 shows
the frequency distribution of the projected area in the x direction of moving particles in Case1(Thin) and Case2(Rod). The projected area in the x direction does not take into account the
influence of shelter of other moving particles. From Figure 6, Case1(Thin) and Case2(Rod)take the peak near the mean value, which are roughly symmetrical distributions. In Case1(Thin) and Case2(Rod), the mean value is approximately equal to the projected areaof a sphere with a diameter of d = 0.07 m. Since the saltation movement are in irregularrotational motion, the projected area in the x direction on the average seems to be regulatedby the particle volume. Figure 7 shows the unevenness distribution of the river-bed in Case1 ~Case4. The unevenness distribution is close to the normal distribution, and the difference dueto the particle shape is small. In addition, the existence probability of the unevenness is in
the range around the mean value to the nominal diameter (0.07m). It is considered that tofind the effect of the shape of the moving particle itself on the particle moving speed was difficult because the scale of the unevenness was almost the same as that of the moving particle.
Figure 8 shows the frequency distribution of the dimensionless step length in Case1 ~ Case4.
The number of samples in Case1 ~ Case4 was 322, 560, 367 and 707, respectively. In eachcase, it could be confirmed that the distribution almost follows the exponential distribution,
which is the same as the previous study (e.g. Hill et al. 2010). Also, the mean step length ofCase1 ~ Case4 is 2.07, 2.20, 1.95 and 1.76 m, respectively, which are almost the same.
3.4 Influence of particle shape on pick-up of particles from the river-bed
Figure 9 shows the pick-up volume at t = 0 ~ 100 s, x = 10 ~ 11 m in Case1 ~ Case4. The ratioof the pick-up volume in each case is almost the same distributions as the transported volumeof gravel particles shown in Figure 4, the particle shape is a prominent effect in the pick-up
process and river-bed structure. One of the hydro-geological phenomena characterizing nonspherical particles is the imbrication of the gravel-bed surface. Figure 10 shows the arrangement of particles on the river-bed surface before and after experiment. The imbrication isoften seen at the bed surface in gravel rivers after floods where gravel’s longest axis is directed
toward the streamwise direction and slightly vertically upward. The imbrication takes a stableposture by directing the contact force to the upstream direction (Figure 11). In Case1 ~ Case3,
the imbrication was clearly formed on the gravel bed. In addition, in Case1 ~ Case3, a groupof clusters composed of several gravel particles are formed. Figure 12 shows the clusters
formed in Case1. A group of gravel enclosed by circles is clusters. The cluster has a stablestructure in a group against stream forces. Unless particles moving from the upstream collidewith clusters, the cluster kept stable.

Figure 4. Volume of gravel particles passing
through the cross section at x = 11 m.

5

Figure 5.

Particle moving speed.

Figure 6. Frequency distribution of the projectedarea in x direction.

Figure 7. Frequency distribution of unevenness of the river-bed.

Figure 8.

Frequency distribution of dimensionless step length.

Figure 9.

Pick-up volume at x = 10 ~ 11 m in the moveable-bed experiments.

Figure 10. Formation of the imbrication in Case2(Rod).

6

Figure 11. Posture of particles forming theimbrication.

Figure 12. Formation of clusters in Case1(Thin).

Additional numerical experiments were conducted to investigate the pick-up rate of thin,
rod and spherical particles. The channel of the numerical experiment was 10 m, the width was
1 m, the bed slope was 1:20, and gravel particles are laid at a thickness of about 20 cm in therange of x = 6 ~ 8 m. In other sections of the channel, moving particles are laid only the surface layer. The discharge was increased from 0.1 ~ 0.5 m3/s every 0.1 m3/s and the pick-up rate
was measured. Table 3 shows the friction velocity versus the discharge. The pick-up rate wascalculated by examining the residual rate of tracer particles in the same way as Nakagawa &Tsujimoto (1976). Of the particles exposed on the moving bed surface x = 6 ~ 8 m and y = -0.4
~ 0.4 m, particles that are higher level on the bed were selected as tracer particles. Similarly,
for Case1 ~ Case4, tracer particles were selected at x = 9 ~ 11 m and y = -0.4 ~ 0.4 m at t = 20
s. Figure 13 shows that the bright color particles are tracer particles in CaseA(Thin). The
number of tracer particles in each experiment is from 67 ~ 105.
Figure 14 shows the measured results of the pick-up rate. The friction velocity wascalculated from (gRI)0.5. The solid lines are CaseA(Thin), CaseB(Rod), and CaseC(Sphere),
and the plots show the values measured in Case1 ~ Case4 of the movable-bed experiments. In
case C (sphere), the pick-up rate increases at about u* = 0.230 m/s, whereas the pick-up rate
for non-spherical particles (CaseA and CaseB) increases around u* = 0.260 m/s. Therefore, it

Table 3.

Friction velocity [m/s] (CaseA ~ CaseC).

Water discharge [m3/s]
CaseA(Thin)
CaseB(Rod)CaseC(Sphere)

0.1
0.194
0.202
0.196

Figure 13. Bright color tracer particles in
CaseA (Thin,0.5 m3/s).

0.2
0.220
0.228
0.230

0.3
0.254
0.252
0.248

0.4
0.266
0.273
0.255

0.5
0.283
0.298
-

Figure 14. Pick-up rate versus friction velocity.

7was confirmed that the critical friction velocity changes depending on the particle shape. Also,
comparing CaseA(Thin) and CaseB(Rod), there is almost no difference at low tractive force
near the critical tractive force. It is probable for both cases that there are a certain number ofparticles in unstable positions and they can easily dislodge from river-bed by the flow. On theother hand, in the high tractive force state, pick-up rates of CaseA(Thin) and CaseB(Rod) areclearly different. The thin particle has a large flat area and irregular shape, so the thin particle
is easy to form clusters and is easier to transmit force to surrounding particles than rod particles. It was observed that particles just downstream of the cluster are picked up along withthe collapse of the cluster.
The future study is to clarify the existence ratio of the cluster on the river-bed surface andcluster’s roles to evaluate the pick-up volume.

4 CONCLUSIONS
Numerical movable-bed experiments with the same nominal diameter of three gravel particles
were conducted in order to examine the effect of particle shapes on sediment transport. Mainconclusions drawn are described as follows:
1. Signiﬁcant differences in sediment transport mechanism and transported volume werecaused by the difference in particle shape. Case4(sphere) has the largest volume of sedimenttransportation compared to the other cases, and the sphere is considered as the mostmobile particle shape. The sediment transported volume in Case3(2 shapes) showed thatthe particle shape affects not only the motion of the particle itself, but also the motion ofsurrounding particles.
2. The moving speed of the saltating particles showed little difference depends on the particle
shape and was almost the same mean value and variation pattern. Because the mean valueof the projected area in each Case1(Thin) and Case2(Rod) do not differ much from the
projected area of a sphere of the same volume. The mean step length was almost the samefor four cases. The distribution of dimensionless step length (λ/d) was exponential.
3. It was found that the effect of particle shape clearly appears in the pick-up process from riverbed. Non-spherical particles in Case1 ~ Case3 formed the imbrication and clusters whichmake river-bed stable. The experiments to analyze the pick-up rate of thin, rod and sphericalparticles showed that the particles shape had a remarkable effect on not only the critical shear
stress but also the pick-up rate under high tractive force of the ﬂow. It is necessary to analyze
the role of clusters on the river-bed in detail in order to evaluate the pick-up volume.

REFERENCES
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, Vol. 72, pp. 84–96.
Fukuda, T. and Fukuoka, S. (2017). Interface-resolved large eddy simulations of hyperconcentrated
ﬂows using spheres and gravel particles., Advances in Water Resources, Vol. 129, pp. 297–310.
Gomez, B. (1994). Effects of particle shape and mobility on stable armor development. Water Resources
Research, Vol.30, No.7, pp. 2229–2239.
Hill, K. M., L. DellAngelo, and M. M. Meerschaert (2010), Heavy-tailed travel distance in gravel bed
transport: An exploratory enquiry, Journal of Geophysical Research, Vol. 115, F00A14, doi: 10.1029/
2009JF001276.
Nakagawa, H. and Tsujimoto, T. (1976). On probabilistic characteristics of motion of individual sediment particles on stream beds. Proceedings of the 2nd International Symposium on Stochastic Hydraulics, International Association of Hydraulic Research, pp. 293–316.
Ushijima, S., Fukutani, A. and Makino, O. (2008). Prediction method for movements with collisions ofarbitrary-shaped objects in 3D free-surface ﬂows, JSCE Journal B, Vol.64, No.2, pp. 128–138, inJapanese.

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