Effects of particle volume and shape on saltation motion of single particle
in a turbulent flow over roughened bed
Yuya Takakuwa1*, Shoji Fukuoka1
1

Research and Development Initiative, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan.
*Corresponding author. Tel.: +81 3 3817-1617. E-mail address: ytakakuwa099@g.chuo-u.ac.jp

Key Words: Non-spherical particle, Movement manner, Instantaneous particle velocity, Fluid forces
and collision forces, the APM method
1. Introduction
Motions of particles in a turbulent flow over a roughened bed are controlled by hydraulic conditions,particle properties (physical properties, volume, and shape), bed roughness, particle-particle
interactions, and fluid-particle interactions. We have assumed in previous sediment transport studies
that the particle shape is spherical. However, effects of the particle shape cannot be neglected when
dealing with large size particles, such as in bed variations in gravel-bed rivers. Effects of the particle
shape on saltation motions have been investigated by focusing on the time-averaged particle velocity
and hop length and height [1],[2], and there are few investigations about instantaneous particle
velocity and forces acting on a moving particle. The Arbitrary Particle Multiphase method (the APM
method [3]), a kind of Resolved CFD-DEM, is an effective method to solve these problems.
Takakuwa and Fukuoka [4] performed the APM numerical simulations on saltation motion
experiments [5] of single natural gravel with different volumes and shapes, and showed that the APM
method can explain the transport process and the instantaneous velocity of non-spherical particles.
In this paper, effects of volume and shape on moving particle are clarified from time-averaged andinstantaneous particle velocity, and fluid and collision forces of saltation motions of 12 particles withdifferent volumes and shapes.
Natural gravel: shape A, B and C. Sphere: D
2. Computational setup and45 m
d=75, 105
Qin =Particle
numerical schemeand 135mm
0.5 m3/szx=2.5 myAs shown in Fig.1, a steady flowx1.08 m
discharge 0.5 m3/s is supplied fromFree fall
the upstream end (x = 0.0 m) of ax=1.5 m→P=0 [Pa]Cyclicstraight channel with 45 m length,Measurement section for flow (13 ≤ x ≤ 33 m)
1.08 m width and 1/20 bed slope,and a single particle is gently
Abrasion bed
dropped from a height of 40 cm at
Figure.
1Overview
ofnumericalsimulations.
x = 2.5 m, which follows theShigemura’s experiment [5]. The
Table. 1 Particle shapes. “a, b, and c” denotecross-sectional shape of the channel is designed to be the longest, intermediate, and shortest axes,
a parabola, where particles tend to concentrate in the respectively.
Dimensionlessmaximum
center of the channel. The channel bed was uneven projected area PA*max is the maximum
due to the abrasion because the main aim of the projected area of the particle divided by that
Shigemura’s experiment was understanding of of the sphere with equivalent volume.
Shapeabrasion mechanism of the channel bed by moving Shape a/d b/d c/d Factor = Zingg's PA*max
nameshape typec/(ab)^0.5
particles. The roughness at the bed is set by trial andDisk
1.61
error by superposing small spheres to reproduce the A [6] 1.50 1.00 0.60 0.49
Massive𝐸𝑥𝑝.
̃
𝐸𝑥𝑝.
depth ℎ̅and the water surface velocity 𝑢̅̃𝑠
B [6] 1.39
1.01
0.79
0.67
and
1.21
symmetricmeasured by the Shigemura’s experiment. The
Massivevelocity distribution at x = 1.5 m is given at the C [6] 1.34 0.90 0.83 0.76
and
1.25
asymmetric
upstream end in every time step to make the boundary
D: Sphere 1.01

0.99

0.99

0.99

Massive

1.02

layer develop in the short distance. At the downstream end, a zero pressure is given for the free-fallflow in the experiment. 12 particles are used, consisting of 3 sizes (d = 75, 105, and 135 mm) and 4shapes (see Fig.1 and Tab. 1. Natural gravel [5]: A Disk type, B Massive and symmetric type, C
Massive and asymmetric type. D Sphere type). In this study, the particles are named by denoting theirshapes and sizes, such as “A105”.
The APM method [3] is a one of a Euler-Lagrange Table. 2 Physical and numericalparameters.
solvers. The fluid forces acting on an arbitrary shaped
Density of fluid
1000kg/m3
particle are directly calculated by detailed flows Density of particle
2500kg/m3
0.0010kg/(m s)evaluated using a computational grid that is sufficiently Fluid viscosity
modules
5.0E+10
Pasmaller than a particle. Point collisions and areal contacts Elastic
Poisson's ratio
0.23
0.30
between particles and channel boundaries are evaluated Coefficient of restitution
0.20
Coefficient of friction
by applying the Distinct Element Method (the Voigt Grid size
0.008m
5.7E+07
model) to each small sphere making up a particle and a Number of grids (Nx*Ny*Nz)(=5629*147*69)
channel bed. Physical and numerical parameters are Time step for fluid calculation
2.0E-04s
Time step for paritcle calculation
2.0E-06s
shown in Tab. 2.
Tab. 3 shows flow and channel characteristics. Fig.2
demonstrates time-averaged flow structures at x = 22.5 Table. 3 Flow and channel characteristics
at the channel center (y = 0.0 m).
m. A high velocity of about 3.7 m/s is calculated at theEquivalent roughness
0.023
mchannel center (y = 0.0 m), 0.05 m above the bottom, andDepth0.205m
Depth-averaged velocity
3.88m/ssecondary flows of Prandtl’s second kind are also
Friction velocity
0.32m/s
calculated in the corner between the bottom and theFroude number
2.73
Bulk Reynolds number
7.96E+05
sidewall.
Friction Reynolds number
Shields number(d = 0.075, 0.105, 0.135)
d/ks(d = 0.075, 0.105, 0.135)

6.53E+04
0.092, 0.066,0.051
3.26, 4.57,
5.87-sidewall

3. Results and Discussions
First, characteristics of movement manners of eachnon-spherical particle are examined. As shown in Fig.
3(a), A105 moves in saltation with the longest orz[m]4.50.20intermediate axis as the rotation axis and in rolling and
4.05
0.15
saltation with the shortest axis as the rotation axis.
3.6
3.15
0.10
Massive typed particles (B075 and C135) basically move
2.7
2.25
0.05
by rolling or saltation with the longest axis as the rotation
1.8
0.00
1.35
axis (Figs. 3(b) and (c)). However, C135, which has a
Secondary flows
0.9
-0.05
-0.4
0.45
-0.2
strongly asymmetrical shape, has an unstable rotation,0.0
y[m]0and sometimes leaps greatly after the angular point hits Figure. 2 Time-averaged flow structures
the roughened channel bed. The movement manners of at x = 22.5 m.
these 3 particles found in the numerical simulations are
similar to those of each particle in the Shigemura’s (a) A105
x=10mexperiment. The spherical particle (D135) simply rolls
and jumps repeatedly (Fig. 3(d)). Under the conditions x=5m
x=40mof these simulations, each particle is transported in themanner corresponding to its shape regardless of the x=35m(b) B075
particle size.
x=20mSecond, the posture of non-spherical particles is1minvestigated when particles are transported in their x=15m(c) C135
characteristic movement manners. Probability density
x=30m
distributions of the projected area of the non-sphericalparticles in the flow direction are compared (see Fig. 4). x=25m(d) D135
When disk typed (A) particles roll or jump with theshortest axis as the rotation axis, probability density
x=25m
distributions are different from the others, but theprojected area of the other non-spherical particles in Figure. 3 Trajectories of A105, B075,
C135 and D135 every 0.1 second.

(b): d=135mm(a): d=75mm

Figure.
4
Probability
density
distributions of the longitudinalcomponent of the projected area of eachnon-spherical particle (d = 135 mm).

Figure. 5 Probability density distributions of thelongitudinal component of the instantaneous particle
velocity (Vpx(t)).
(a): d=75mm

Table. 4velocity.

Time-averaged𝑉̅𝑝𝑥(b): d=135mm

particle
[m/s]

75mm

105mm 135mm

A: Disk type

3.07

3.13

2.78

B: Massive and symmetric

3.15

3.18

3.16

C: Massive and asymmetric3.193.18

3.18

D: Sphere

3.12

3.22

3.10

The depth averaged velocity (Uc ) is 3.88 m/s.

A particle moves
straight ahead.

A particle moves
to the left side.

Figure. 6 Probability density distributions of thetransverse component of the instantaneous particle
velocity (Vpy(t)).

motion are basically larger than that of the sphere with the equivalent volume. This is different fromthe numerical result of Jain et al. [6] about bedload motions with various shapes over the fixed roughbed. In their numerical simulations, the particle-roughness ratio 𝑑 ⁄𝑘𝑠 and the Shields number 𝑆ℎare small about 1.0 and 0.047, respectively, and non-spherical particles roll by interlocking with bedroughness (so called the engagement effect). On the other hand, the particle-roughness ratio and theShields number in the present simulations are 𝑑 ⁄𝑘𝑠 (=3.26 ~ 5.87) and 𝑆ℎ (=0.051 ~ 0.092), andthe saltation motion is dominant. It is considered that the particle movement postures are affected bythese two parameters (𝑑⁄𝑘𝑠 and 𝑆ℎ) whose values are greatly different in Jain’s and our simulations.
Third, effects of particle volume and shape on the particle velocity are investigated. Except forA135, which frequently jumps higher than the water surface, the variation of the longitudinalcomponent of the time-averaged particle velocity (𝑉̅𝑝𝑥 ) is very small, at most 4.7 %, as shown in
Tab.4. It is also found that there are almost no effect of particle shape on the time-averaged velocityof a particle transported by flows with large Froud number, as shown in the results of previous studies[1] and [2].
In contrast, probability density distributions of the longitudinal component of the instantaneousparticle velocity (𝑉𝑝𝑥 (𝑡)) vary greatly depending on the particle volume and shape (see Fig. 5). Whenthe diameter is 75 mm, regardless of the particle shape, the 𝑉𝑝𝑥 (𝑡) is about 0.60 ~ 1.00 times 𝑈𝑐 ,and shapes of probability density distributions are almost similar. This is because the ratio of theparticle diameter to bed roughness 𝑑⁄𝑘𝑠 (= 3.26) is relatively small, and particles that collide headon with unevenness of the bed increase in number. However, as the particle diameter increases,
probability density distributions of the 𝑉𝑝𝑥 (𝑡) clearly differ depending on the particle shape. Whenthe particle shape deviates from the sphere, the peak value of the distribution and the width are small
and large, respectively.
Focused on the transverse component of the instantaneous particle velocity 𝑉𝑝𝑦 (𝑡) (see Fig. 6),
the 𝑉𝑝𝑦 (𝑡) of spherical particle is at most about ± 0.4 m/s (10 % of 𝑈𝑐 ), while that of non-sphericalparticles is at most about ± 0.8 m/s (20 % of 𝑈𝑐 ), regardless of the particle size. Non-sphericalparticles move more three-dimensionally than sphericalparticles. These analytical results indicate that the motion ofarbitrary shaped particles changes in the Shields number, theratio 𝑑 ⁄𝑘𝑠 and the particle shape.
As described above, effects of particle shape on thedistribution of the instantaneous particle velocity differ
depending on the particle size. This mechanism is discussed
on the fluid forces and collision forces acting on a moving
particle. Fig. 7 shows the longitudinal component of timeaveraged fluid force 𝐹̅𝑓𝑥 acting on each particle dominant Figure. 7 The longitudinal component
in the accelerated motion. 𝐹̅𝑓𝑥 acting on non-spherical of the time-averaged fluid force acting
on each particle. Fluid forces are
particles such as A075 and C075 are about 1.5 times larger averaged over the time that eachthan 𝐹̅𝑓𝑥 acting on the spherical particle (D075). As the particle moves between 7.5 < x < 45particle size increases, 𝐹̅𝑓𝑥 acting on non-spherical m.
particles such as A135, B135 and C135 are about 2.0 times
larger than the 𝐹̅𝑓𝑥 acting on the spherical particle (D135).
The larger the particle size and the more the shape deviatesfrom the sphere, the greater fluid force and a particle
acceleration.
Fig. 8 shows the longitudinal component of the collision
force |𝐹𝑐𝑥 | that reduces the longitudinal component of theinstantaneous particle velocity. The effect of particle shapeon the mean value of |𝐹𝑐𝑥 | acting on a small particle (d =75 mm) is very small, however, that on a large particle (d =135 mm) differs greatly among shapes. These are two Figure. 8 The longitudinal component
reasons. First, except for A135, the mean value of |𝐹𝑐𝑥 | also of the collision force acting on eachparticle.
increases as the shape deviates from the sphere. Second, thelarge sphere (D135) rolls over the bed roughness moreeasily, and the collision force of D135 is lower than that of D105.
4. Conclusions
Saltation motions of single particles with various volumes and sizes in a turbulent flow over aroughened bed were analyzed to clarify relationships among particle postures, instantaneous velocity,
fluid forces and collision forces acting on a particle, and particle volumes and shapes. As the particle
diameter increases, effects of the particle shape on its motion become significant. Non-sphericalparticles are transported with a large projected area in the flow direction and are accelerated by thelarge fluid force. However, when they hit the channel bed, they generate a large collision force anddecelerates. Therefore, the instantaneous velocity of non-spherical particles varies widely. In addition,
non-spherical particles have more three-dimensional motion than spherical particles.
These results would contribute to the understanding of fundamental mechanisms in gravel-bed
rivers such as the momentum exchange between saltating particles and deposited particles.
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