A study on vertical sorting mechanism in gravel bed rivers with graded sediment using theAPM method
Takatoshi ATSUMI1*, Shoji FUKUOKA2
1

JSPS Research Fellow・Civil, Human and Environmental Engineering, Chuo University, 1-13-27

Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan.
Tel.: +81 3 3817-1615, Email address: a14.c3y6@g.chuo-u.ac.jp
2

Research and Development Initiative, Chuo University. E-mail address: sfuku@tamacc.chuo-u.ac.jp* Corresponding author
Key Words: gravel-bed rivers, sediment transport, vertical sorting, resolved CFD-DEM
1. IntroductionStudies on armoring of grain size materials in gravel bed rivers have been investigated actively. Manymodels have been attempted to evaluate sorting mechanism of bed materials on grain size profiles andshielding effects. There are still issues to be addressed in the modeling of bed variation analysis
method with assumptions for simplifications (e.g., neglecting interactions between moving particles
and assuming no movement of large particles in the riverbed). In bed variation analysis models, thebed load is often evaluated based on the processes of entrainment, transport, and deposition. In
particular, it is important to properly evaluate the entrainment and deposition rate, which are directlyrelated to changes in the riverbed structure.
Physical Experimental observations have limitations in obtaining detailed mechanism of eachparticle motion in graded sediment. On the other hand, numerical approaches have advantages foranalyzing space and time variation motions on each particle motion. Fukuoka et al.1) developed APM
method (a type of resolved-CFD-DEM model) that can directly analyze solid-liquid phase interactions,and obtain trajectories, velocities, and distributions of gravel particles flowing on a fixed bed. They
compared the computational result with experimental results and showed the validation of their APM
model.
In this study, APM method applied for gravel bed with a wide distribution of grain-size of one orderof magnitude in order for better understanding the mechanism of the bed load. And the vertical sorting
mechanism of the bed structure was analyzed and discussed on the basis of the data of particles
movement for each grain size obtained by this numerical computation.
2. Numerical methods
The numerical simulation model of Fukuoka et al. 1) is used for this investigation. Fluid motions weresimulated using the governing equations of a single-fluid model for the solid-liquid multiphase flows
in the Eulerian approach. Particle motions were computed by the Lagrangian approach using DEM
method. The free-surface variation was simulated using the continuity equation of the fluid volume
fraction based on the volume-of-fluid (VOF) method. The particle motion was simulated using themomentum and angular momentum equations as a rigid body. The parameters used in the simulation
re shown in Table 1.

cumulative percentage [%]Table. 1 Simulation parameters.

1.2[m]

Bed slope:1/30

Table. 2 Experimental conditions.

TalbotSimulation(a) d=1.5~5cm(b) d=6~15cm
Particle size [cm]Figure. 1 Computational domain
Figure. 2 Particle size distribution. Figure. 3 Particle shape.
and boundary conditions.
Experimental conditions
Figure 1 shows the Computational domain, coordinate axes and boundary conditions. The
numerical channel is 4.01 m long, 1.01 m wide with a bed slope of 1:30. The periodic boundary
condition was applied to the longitudinal and transverse boundaries. Figure 2 shows the particle size
distribution used in the numerical experiment. The particle size distribution is set based on the Talbot
type and has a spread of one order, d=1.5~15 cm. Figure 3 shows the particle shapes used in thenumerical analysis. In order to speed up the calculation, a single sphere shape was used for particle
sizes d=1.5~5 cm, as shown in Figure-3(a), and for particle sizes d=6~15 cm, several small spheres
without gap were connected to form a spherical shape, as shown in Figure-3(b).
The particles were packed randomly to make a riverbed. The Initial water level was set to 1.2m.
Table 2 shows the experimental conditions. The mean bed elevation was calculated by the bed particles
with velocity less than 0.05 m/s. The water depth and the Froude number were calculated by spatiallyaveraged over the entire simulation area.
4. Simulation Results
Figure 4 shows gravel bed structures on the vertical plane at t=0 and 50s. From the comparison ofFigures. 4 (a) and 4 (b), it can be seen that vertical sorting of particles occurs in the surface layer ofthe riverbed and the particles below z=0.15 m does not shift at all. Figure-5 shows the temporalvariation of the vertical distribution of the solid volume concentration, spatially averaged over theentire analysis area in the x-y plane. Figures 5 (b), (c) and (d) show the solid volume fractions of large,
medium and small particles, respectively. In Figure 5(b), the volume fraction around z = 0.20 m isparticularly reduced. This is due to the vertical upward displacement of large particles, which wereinitially buried in the surface layer. On the other hand, the small particles shown in Figure 5(d) drop
into the pore spaces between the large and medium particles and are displaced vertically downward.
As a result, the volume fraction of the layer above z=0.20m decreases, and the volume fraction below
3.

(b) t=50 s(a) t=0 s
Figure. 4 Time change of riverbed structure.
0.7

全粒径(移動粒子)_50s
平均河床高(初期)

z[m]

z[m]

0.6

全粒径_50s

平均河床高(50s)0.50.40.8d=11-15cm_0s
d=11-15cm_50s

0.70.6

d=11-15cm(移動粒子)_50s

0.60.5平均河床高(50s)z[m]全粒径_0s

0.7

0.40.8d=6-10cm_0s

平均河床高(50s)

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.1

000

0.4

0.6

0.8

Solid volume concentration [-]

100.2

0.4

0.6

0.8

1

Solid volume concentration [-]平均河床高(50s)

0.4

0.2

0.2

d=1.5-5cm(移動粒子)_50s

0.5

0.3

0

d=1.5-5cm_50s0.6

d=6-10cm(移動粒子)_50s

0.5

d=1.5-5cm_0s0.7

d=6-10cm_50sz[m]0.80.8000.2

0.40.60.8

1Solid volume concentration [-]

00.2

0.4

0.60.81

Solid volume concentration [-](a) d=0~15cm(b) d=11~15cm(c) d=6~10cm(d) d=1.5~5cm
Figure. 5 Vertical distribution of solid volume concentration.

z=0.20m increases. In Figure 5(c), the volume fraction of medium particles at z = 0.10~0.20m andabove z = 0.30m decreased due to both the rise of buried particles and the drop of exposed particles.
Next, the entrainment and deposition rate, which are related to the armoring of the mixed particles
are calculated. Now consider an arbitrary control volume (C.V.), and the continuity equation for thesolid phase is ,𝜕𝑉𝑠𝑘 (𝑥, 𝑦, 𝑧, 𝑡)
𝑘𝑘
(1)
= 𝑄𝑖𝑛
(𝑥, 𝑦, 𝑧, 𝑡) − 𝑄𝑜𝑢𝑡(𝑥, 𝑦, 𝑧, 𝑡)𝜕𝑡where 𝑉𝑠 :volume of solid phase [m3],𝑄𝑖𝑛 : volume of solid phase flowing into C.V. per unit time
[m3/s] and 𝑄𝑜𝑢𝑡 : volume of solid phase flowing out of C.V. per unit time [m3/s]. k indicates particle
numbers.
If we divide the volume of the solid phase Vs into the volume of moving particles Vsm and the volumeof bed (stationary) particles Vsb, and consider the continuity equation, we can write the following for
Vsm and Vsb respectively.
𝑘𝜕𝑉𝑠𝑚(𝑥, 𝑦, 𝑧, 𝑡)𝑘𝑘(2)= 𝐸 𝑘 (𝑥, 𝑦, 𝑧, 𝑡) − 𝐷𝑘 (𝑥, 𝑦, 𝑧, 𝑡) + 𝑄𝑖𝑛(𝑥, 𝑦, 𝑧, 𝑡) − 𝑄𝑜𝑢𝑡(𝑥, 𝑦, 𝑧, 𝑡)𝜕𝑡𝑘(𝑥, 𝑦, 𝑧, 𝑡)𝜕𝑉𝑠𝑏(3)= 𝐷𝑘 (𝑥, 𝑦, 𝑧, 𝑡) − 𝐸 𝑘 (𝑥, 𝑦, 𝑧, 𝑡)𝜕𝑡𝑘𝑘 (𝑥,(4)(𝑥, 𝑦, 𝑧, 𝑡)
𝑉𝑠𝑘 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑉𝑠𝑚𝑦, 𝑧, 𝑡) + 𝑉𝑠𝑏
where D: deposition rate [m3/s], E: entrainment rate [m3/s]. D and E are the volume of particles
deposited from moving particle to bed particle and the volume of particle detached from bed particle
to moving particle within the C.V. per unit time, respectively.
Figure.6(a)~(c) shows the vertical distributions of the entrainment and deposition rates for respective
particle sizes (d=1.5,6-9,15 cm). The C.V. was set as follows: channel width (4.02 m) in thelongitudinal direction, channel length (1.02 m) in the transverse direction, and 5 cm each in the verticaldirection. And the C.V. to which each particle belongs is determined from the position of the particle
center of gravity. The entrainment and deposition judgement are made for each particle every 10 s.

(a)d=15cm(c)d=1.5cm(b)d=6-9cm
Figure. 6 Vertical distributions of Entrainment and Deposition rate
Entrainment was evaluated when the particle velocity was greater than 5 cm or when the C.V.
belonging to the particle at the time of judgement was different from the C.V. before 10 s. Deposition
was evaluated when the particle velocity was less than 5 cm or when the C.V. belonging to the particle
at the time of judgement was different from the C.V. before 10 s. The amount of entrainment (ordeposition) is given by each particle volume for the C.V. to which it belonged before 10s.The volumeof entrainment and deposition in each C.V. was divided by the observation time of 10 s to obtain E
and D as values per unit time.
In terms of the vertical distribution of D, the larger the grain size is, the higher the vertical positionof deposition occurs, and the temporal change in vertical distribution is relatively small for eachparticle size. In the vertical distribution of E, as in the vertical distribution of D, the larger the particle
size is, the higher the particle detachment occurs. In particular, for all grain sizes, the Entrainment
occurs at higher vertical positions at t=0-10s. Thus, for each particle size, as the water flow began, thesurface particles were detached, and when moving particles were deposited, the large particle sizegroup stayed at a higher position on the surface, and the small particle size group was deposited at alower position, resulting in formation of vertical sorting.
In particular, at t=40-50s, the vertical distributions of E and D tend to be similar for each grain size.
Therefore, from equation (3), the time variation of the vertical distribution of the volume of bed
particles for each grain size becomes small, and the bed structure reaches an equilibrium state.
5. Conclusion
A numerical movable-bed experiment of the mixed particle size was conducted to investigate thevertical distribution of the entrainment and deposition rate, and to better understand the formation
mechanism of vertical sorting.
6. Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP20J22413.
References[1] Fukuoka, S., Fukuda, T., Uchida, T., 2014. Effects of sizes and shapes of gravel particles on
sediment transports and bed variations in a numerical movable-bed channel, Adv. in Water Res.,72, 84-96, https://doi.org/10.1016/j.advwatres.2014.05.013.