THESIS 2013, June 10-12, Chatou France, S. Fukuoka, Keynote

Prediction of three-dimensional movement of gravel particles
in a movable-bed numerical channel
Shoji Fukuoka
Professor, Research and Development Initiative, Chuo University
1-13-27 Kasuga, Bunkyou-ku, Tokyo, Japan 112-8551 :Tel. +81 (03) 3817 1625, Fax: +81 (03) 3817 1625,
e-mail : sfuku@tamacc.chuo-u.ac.jp

We have developed a movable-bed numerical channel which simulates three-dimensional movement ofstreams and various shape gravel particles. Using the numerical channel we performed numericalchannel experiments with graded gravel particles and demonstrated motion of streams and particles
which has been difficult to measure in experiments and fields. Characteristic motion of gravel particles,
sorting mechanism of particles at the bed surface and velocity distributions near the bed are predictedfrom the numerical investigations of the motion of streams and particles.

Key words
Movable-bed numerical channel, three-dimensional motion, gravel bed river, shape of particle, large
particle cluster, vertical velocity distribution, sediment transport rate.

IINTRODUCTIONGravel-bed rivers consisting of large bed materials are generally seen in natural world. Many
researchers have attempted to demonstrate the movement of streams and gravel particles by field
experiments and numerical computation method. However, the prediction method of sediment
discharge and bed variation in rivers with large range of particle sizes have not been established. The
reason is because sediment transport mechanism in gravel-bed rivers is more complicated than that ofsand-gravel bed rivers and measurements in gravel bed rivers are very difficult.
A computational method for solid-liquid multi-phase flow have been developed recently, in which
fluid motion around objects are calculated precisely with fine computational cells smaller thanparticles and solid objects are evaluated by considering as a liquid with the different density.
In this paper, we developed a movable-bed numerical channel with different particle sizes and shapes
which simulated three-dimensional motions of streams and gravel particles by calculating fluid motionin the Eulerian computational method for solid-liquid multiphase flow and calculating gravel particles
motion in the Lagrangian method. We made gravel particles with different sizes and shapes bysuperimposing 8 to 10 small spheres. We conducted movable-bed experiments using the numericalchannel and clarify the motion of streams and gravel particles and hydrodynamic forces acting onparticles, which are difficult to measure in the experiments and fields.

IIOUT LINE OF NUMERICAL METHOD

In the calculation of fluid motion, fluid and solid are calculated as an incompressible
multiphase flow by the equations described below , where the region of gravel particles are considered
as a different density fluid.
∂ui=0,∂x i(1)∂ui∂u1 ∂P∂+ u j i = gi −+{2(ν + ν t )Sij },ρ ∂xi ∂x j∂t
∂x j(2)

THESIS 2013, June 10-12, Chatou France, S. Fukuoka, Keynote
White particles indicate larger particles
Black particles indicate smaller particles

zyx

5m

Before theexperiment

5mt = 200 s
10m

10m

5m

15m

t = 50 s

10m

5m

15m

t = 250 s

10m

15m

15m

12

34Figure 2. Gravel particle model.
ν t = (C S ∆ )2 2 S ij S ij ,Parcentage (%)

Figure 1. Time variation of bed surface.
100806040200101001000

Figure 3. Grain size distributions.
(3)

where ui : i-direction averaged velocity including gravel particle regions in a fluid calculation cell, ρ :density, P : sum of pressure and isotropic component of SGS stress, g i : gravitational acceleration, S ij :strain rate tensor, ν : kinematic viscosity, ν t : SGS turbulent viscosity, C S : Smagorinsky constant (0.173 ), ∆ : calculation grid size. Physical property φ (density ρ , dynamic viscosity µ ) and averaged
velocity ui are calculated as volume-averaged values and mass-averaged values, respectively.
Gravel particle movements are calculated by the Lagrangian method with translational androtational equations of rigid bodies. Fluid forces acting on gravel particles are calculated bymultiplying density to the second term and the third term in the right side of equation (2) andintegrating them within the region of a particle. Contact forces are calculated with each small
superimposed sphere consisting of a gravel particle by the Distinct Element Method and forces acting
on gravel particles are calculated by the summation of the contact forces on the spheres. [1]

IIINUMERICAL MOVABLE-BED CHANNEL EXPERIMENT

III – 1 Numerical experiment condition
Figure 1 shows the movable-bed numerical channel and definition of coordinate axes used fornumerical analysis. The numerical channel with 15 m long, 1 m wide and 1:20 in bed gradient wasdesigned by considering flow conditions transporting gravel particles, length of bed waves andcomputational load. Four different gravel particles in shape shown in Figure 2 were prepared for bed
materials of movable-bed numerical channel. Five different sizes of particles (40 mm, 50 mm, 70 mm,90 mm, 120 mm) were made by varying the size of small spheres and by superimposing them properly.
The diameters of the various shape particles are given as the diameter of spherical particles with thesame volume. Particles are packed ramdomly to the channel by adjusting supply rate so that theparticle size distribution of the numerical bed channel agrees with the particle size distribution inFigure3. The numerical experiment was lasted for 250s. In the numerical analysis, fluid calculation∆t is set 1.0 × 10-4s, gravel particle movement calculation ∆t is set 1.0 × 10-6s. The grid size for fluid
calculation is 0.01m and other parameters are same values as the authors’ paper. [1]III – 2 The result of numerical experiments
We disscuss at first the flow condition, variation of bed level and sediment transport rate. At
the time t = 250 s, average water depth is 0.23 m, average velocity is 1.92 m/s and Froude number is
1.20. The flow is a supercritical flow and the water surface varies in the same phase as the bed wave.
Figure 1 shows conditions of the bed surfaces in the successive time of the numerical experiment. At t
= 250 s, formations of large gravel cluster and the resulting gravel sorting at the bed are seen at
sections x = 8 m and x = 10.5 m. Figure 4 shows sediment discharge rate of each particle size

D=70mm

D=40mm

Total

3.0
2.0
1.00.0050

100

150200t (s)Figure 4. Sediment discharge rates.
Flow velocityz(m)0.60.50.4

Curve lines indicate streamlines

250

D=120mm

D=70mm

x=8m

Total

151050050100150200250t (s)Figure 5. Number of flowing out particles.
0.6
0.5
0.40.3
ks=70mm0.2
ks=240mm
0.1
01.0
2.0
3.0m
Velocity in x direction ( m/s )D=40mm20z(m)

D=120mm
4.0

Number of flowing outparticles ( 1/s )××10-3( m3/s )Sediment discharge rate
( 10-3 m3/s)

THESIS 2013, June 10-12, Chatou France, S. Fukuoka, Keynote

Flow velocity
ks=240mmm0.30.2
ks=70mm0.101.02.03.0
Velocity in x direction ( m/s )x = 12 m

Log law velocities are shown under both equivalent roughness 70 mm as D60 and 240 mm as twice the maximam particles diameter.
As shear velocity, 0.29m/s averaged from x = 5 m to x = 13 m point is set.

Figure 6. Vertical velocity distributions and flow condition near the bed surfaces.

Curve lines indicate streamlines

Particles of all sizes

Small particles ( D = 40 mm, 50 mm )Figure 7. Trajectories of moving particles.
measured at the downstream end and Figure 5 shows number of particles discharging per unit time at
the downstream end where numbers of particles smaller than 70 mm are dominated. Smaller paricles
are transported actively. However, we can see that sediment discharge rate of large particles is greaterthan that of smaller particles.
Figure 6 shows the vertical distributions of time averaged velocities in the x-direction
measured at the center of the channel at sections x = 8 m and x = 12 m. The velocities are averaged
over 0.15 m lateral range from the channel centerline which is about the largest particle size. Figure 6also shows time averaged velocities of gravel particles moving the lateral range of 0.15m and log-law
velocity distribution shown by assuming ks = 70 mm and 240 mm. The x = 8 m is the section whereclusters of large particles are formed and vertical velocity distributions change gradually from bottom
to upward z-direction and do not follow the log law. On the other hand, at x = 12 m point where large
particles cluster is not developed, vertical gradient of velocity distribution is large and velocity is over
2 m/s at the height of 0.1m from zero velocity level. Velocity distributions of the flow areapproximated to the log law velocity distiribution ( ks = 70 mm ). From these investigation, we foundthat at the location forming large particles clusters, velocity gradually increased from bottom tovertical upward direction in the scale of large particles because there are a number of large pores inlarge particles clusters and large particle clusters give great resistance to flows.
Figure 7 shows trajectories of gravel particles visualized by emitting light on moving gravel
particles at every 1/10s. From this figure, we can find that small particles cannot move over large
projecting particles but come to take detours. Large particles are nearly in rectilinear motion overprojecting particles. Therefore, large particles receive greater flow velocity and are not able to stopeasily on plane bed but can stop at projecting beds as high as large particle size, which means thatlarge particles tend to form clusters. Deep water passages are formed among large particles clusters of
the bed surface and small particles come to move along meandering deeper water passages besidelarge particles. This results in the particle sorting of the gravel bed surface.
Next, we clarify the dynamics of particle motion by investigating how particles behave andreceive forces at the time of their movement and rest. We define the long axis as the axis easiest torotate among orthogonal principal axes and short axis as the axis hardest to rotate. Short axes difinedyContact force

070

Nondimensionalizedhydrodynamic force
and contact force
in x - direction

Velocity

0.0ox20

Hydrodynamic force

0.4Angle (z0.80.6)

Long axis (black)
Short axis (white)

x-direction velocity( m/s )THESIS 2013, June 10-12, Chatou France, S. Fukuoka, Keynote-20

Angle between short axis and z - axis

50

Angle between long axis and x - axis
30-0.40time ( s )Settlement

0.4 -0.4

0time ( s )Movement

0.4 -0.4

Small particles ( D = 40 mm , 50 mm , 70 mm )0time ( s )Settlement

0.4 -0.4

00.4
time ( s )Movement

Large particles ( D = 90 mm , 120 mm )Figure 8. Conditions of long axis
Figure 9. Time variation of the particles velocities,
and short axis of particles
hydrodynamic forces, contact forces andon the bed surface.
particles direction concerning movment and settlement.
in such manner indicate almost the same direction as the normal vector to the most flat plane ofparticles. Long axes and short axes of the particles are shown in Figure 8. Figure 9 shows time
variations of averaged gravel particles velocities in x – direction and contact forces and hydrodynamic
forces acting among gravel particles nondimensionalized by x - component gravity force and angles
between a long axis and x - axis and angles between short axis and z – axis. The time zero in Figure 9
is selected at the time when a particle bigins to move and comes to a stop. These values are evaluatedby averaging individual particles motion. When large particles stop, the angle between long axis and x- axis becomes smaller. Apparently, large particles incline long axes toward lateral direction easy toroll downstream in their moving stage, and in their stopping stage, they incline the long axes to thedownstream direction. Moreover, large particles tend to direct a flat plane upward at the instant of thestopping in order to decrease the angle between short axes and z - axis. These dynamic features arealso seen in small particle motion, but not so clear compared to large particles. This is because that
smaller particles at the time of movement and rest are affected by the irregular shape of gaps and water
passages formed by surrounding larger particles at the bed surface. Time variations of long axes andshort axes at the movement stage of larger particles tend to be opposite to that at the time of thestopping stage. Hydrodynamic forces acting on larger particles increase a little after the stopping in
the process of the particle stop, on the other hand, hydrodynamic forces acting on smaller particles
decrease after stopping. The reason of the differences is considered that small particles can stop at
small velocity regions behind stationary large particles, while large particles stand high positions andrecieve large hydrodynamic forces and therefore cannot stop easily but do by the collision and supportof surrounding large particles.

IVCONCLUSIONS

We developed the movable-bed numerical channel with various shape particles. The numericalchannel was used to investigate the movement of streams and gravel particles near the bed surface.
Following main conclusions were obtaind:
1. The movable-bed numerical channel experiment drew a new methodology to understand three
dimensional motions of particles, vertical velocity distribution near the bed surface, effects of shapeand size of particles on the movement and the rest of gravel particles, the sorting mechanism of thebed surface and sediment transport rate.
2. Large particles move high position in rolling and saltating manner along the bed surface and receive
large flow velocity, therefore cannot stop easily. Large particles stop by the collision force withsurrounding large particles and tend to form large particle clusters. Small particles move deeper
water passages beside large particle clusters. This mechanism is important for particle sorting on thebed surface.
3. Vertical velocity distributions at the location forming large particles clusters has a convex form
upward in a larger particle diameter range near the surface and on bed surface where large particle
clusters are not formed, velocity distribution follows approximately the log law velocity distribution.

Ⅳ[1]

REFERENCES
Fukuda, T., Fukuoka, S. and Uchida, T.; (2012). – Three-dimensional numerical modeling forthe motion of stones with different sizes and shapes in streams. Proceedings of tenthinternational conference on hydroscience and engineering.