Movement mechanism of boulders composing valley bank and beds by debris flows anddevelopment of debris flow front using the APM method
Hiroki Kato1*, Shoji Fukuoka2
1Civil, Human and Environmental Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo
112-8551, Japan2
Research and Development Initiative, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 1128551, Japan.
* Corresponding author. Tel.: +81 3 3817-1615. E-mail address: a17.jnt5@g.chuo-u.ac.jp.
keyword：debris flow, valley bank and bed erosion, the one-fluid model of solid-liquid multiphase,
APM method

percentage [%]1. Introduction
Clarifying the erosion mechanism of a valley by a debris
flow is important not only for evaluating the amount ofdebris flow that passes through the valley, but also forevaluating the stability of the valley beds and banks after thedebris flow has passed. However, the erosion mechanism ofvalley beds and banks by debris flows has not been fully
clarified.
Iverson et al. [1] measured the total stress and pore waterpressure acting on the channel bed during debris flow
passage using large scale model experiments, and showedthat the momentum of the debris flow is increased byentrainment due to liquefaction of the channel bed. Lyu et al.
[2] focused on bank erosion caused by the passage of debris
diameter [m]flow and compared the flow and erosion between cases
Fig.1 Particle size distribution
where bank erosion is dominant and cases where only bed
erosion occurs. The results show that valley bank erosion
contributes more to the growth of the debris flow front than bed erosion, and that valley bank erosion
occurs after the front of debris flow has passed.
However, it is difficult to measure the forces acting on individual particles and the flow field aroundthe particles in these model experiments, which can only be done by numerical analysis. Euler-Euler
type approach is less computationally demanding than the Euler-Lagrange type approach and analyzes
in the field scale. But it does not elucidate the essential phenomena of the debris flow. In recent years,
with the improvement of computer performance, the Euler-Lagrange type model has been developed.
Fukuda et al. [3] developed the Euler-Lagrange type model (APM method) that can analyze nonspherical particles, and explained the results of the debris flow channel experiment conducted by Itoet al.
In this study, as basic research on the erosion mechanism of valley bank and bed, a straight valley
shape with mobile valley bed and bank is created. The results of the numerical experiment arecompared with the results of the model experiment and the factors causing the movement of individual
particles are discussed.
2. Numerical methods
2.1. Numerical method of fluid motions
In the flow analysis with the stone and gravel particles, a one-fluid model of solid-liquid multiphaseflow is used, where the solid phase is given as a fluid with different densities, and the whole is assumed
to be an incompressible fluid. The basic equation of the flow is shown below.
∂𝑢𝑖=0∂𝑥𝑖(1)

hydrodynamic force

weightnegative contact force
positive contact force

Non-dimensional force [-]resultant force

Fig.2 Initial stateParticle A

detachment(a) Just before the arrivalof the debris flow front

particle A
particle Bflow
Fig.3 State of before thepassage of the debris flow front

Non-dimensional force [-]time[s](a) Particle Adetachment

time[s](b) Particle B(b) Just before particle Astarts to move

Fig.4 Time variation of thelongitudinal non-dimensional force

Fig.5 stone and gravel
around particle A∂𝑢𝑖
∂𝑢𝑖
1 𝜕𝑃
𝜕+ 𝑢𝑗
= 𝑔𝑖 −+{2(𝜈 + 𝜈𝑡 )𝑆𝑖𝑗 }∂𝑡∂𝑥𝑗
𝜌 𝜕𝑥𝑖 𝜕𝑥𝑗
𝜈𝑡 = (𝐶𝑠 ∆)2 √2𝑆𝑖𝑗 𝑆𝑖𝑗

(2)
(3)

where 𝜌 and 𝑢𝑖 are the volume average density and the mass average velocity, respectively, taking
into account the volume fraction of the particles in a computation grid. 𝜈 is the kinematic viscosity,
𝑔𝑖 is the gravitational acceleration, 𝑃 is the sum of the isotropic components of pressure and SGSstress, 𝑆𝑖𝑗 is the strain rate tensor, 𝜈𝑡 is the turbulent viscosity, and 𝐶𝑠 is the Smagolinsky constant.
∆ is the computation grid size. The free water surface was simulated using the volume of fluid (VOF)
method.
2.2. Numerical method of gravel motions
The motion of the particles is solved using the equation of motion of a rigid body, and the contact
judgment and contact force of the gravels are evaluated using the distinct element method for eachgravel particle. The hydrodynamic forces on the particles are calculated directly by volume integrationof the pressure and diffusion terms in the equation of motion of the flow field. The basic equations areshown below.
𝑓𝑀𝑟̈𝑖 = 𝑀𝑔𝑖 + 𝐹𝑖 + 𝐹𝑖𝑐
𝑓𝑐
𝜔̇ 𝑖 ′ = 𝐼𝑖−1
′ 𝑗 ′ {𝑅𝑗 ′ 𝑖 (𝑇𝑖 + 𝑇𝑖 ) − 𝜀𝑗 ′ 𝑘 ′ 𝑙 ′ 𝜔𝑘 ′ 𝐼𝑙 ′ 𝑚′ 𝜔𝑚′ }

(4)
(5)

where, where the index 𝑖 denots the components of the global coordinate system fixed on the space,
the indices 𝑖 ′ –𝑚′ denote𝑔 the components of the local coordinate system fixed on a rigid body, M is
the mass of a particle, 𝑟𝑖 is the position of the center of gravity of the rigid body, 𝜔𝑖 is the angularvelocity, 𝐹𝑖 is the force acting on the surface of the particle, and Ti is the torque on the center of thegravity generated by the force. The superscripts 𝑓 and 𝑐 are the components of the−1fluid forces andcontact forces between particles, respectively, 𝑔𝑖 is the gravitational acceleration, 𝐼𝑖 ′ 𝑗′ is the inverseof the matrix consisting of the components of the moment of inertia tensor in the local coordinate
system, 𝑅𝑗′ 𝑖 is equal to 𝒆𝑗′ ∙ 𝒆𝑖 (𝒆𝑗′ and 𝒆𝑖 are unit basis vectors), and 𝜀𝑗′ 𝑘 ′ 𝑙′ is the Levi–Civita
symbol.

4flow

z [m]

flow

3.5
32.5

𝜃 =20°

2
16x[m](a) Particle with a diameterof more than 0.5m

20x[m]

𝜃 =20°
242832(b) Particle with a diameter ofless than 0.5m

Fig.6 Position of the center of gravity for each of
the particles

Non-dimensional force [-]z [m]total contact force
resultant force
weighthydrodynamic force
positive contact forcenegative contact force

time[s]Fig.7 Time variation of the nondimensional vertical force
acting on particle B

Experimental conditions of the numerical movable-bed channel
The numerical movable-bed channel used in this study is a straight channel of 60 m in length and 8
m in width. The slope of the channel was set to 20°. The coordinate axes are the x-axis in thedownstream direction, the y-axis in the transverse direction, and the z-axis in the upward direction(the z-axis direction is referred to as the vertical direction). We used non-spherical particles of thevalley bed and bank consisting of 10 different particle sizes, with the largest particle size being 1.0 m
and the smallest being 0.1 m, as shown in Figure 1. All the particles are equal in shape and density.
The diameters of non-spherical particles were defined as the diameters of spheres with the samevolume.The particles were randomly dropped into the channel, and the initial channel form was set
up with the slope of the valley bank at about 30 degrees and is shown in Figure 2. In the experiment,a constant flow rate of 10.0 [m3 ⁄s] was continuously applied at the upstream end of the channel. No
sediment was supplied to the flowing water. At the downstream end, the pressure was set to zero andthe sediment was discharged out freely. The debris flow front was formed by taking particles from thevalley bed and bank.
3.

4. Movement mechanism of valley bank and bed particles
We focused on particles that had moved more than 5 meters in the longitudinal direction from thesection where the debris flow front had passed. In Figure 3, the blue particles are the valley bankparticles and the yellow particles are the valley bed particles. The number of valley bank particles andbed particles we focused on was 38 and 14, respectively.
4.1. Movement mechanism, of valley bank particles
The main force causing the movement of 38 sample particles of the valley bank was as follows: 1particle was due to hydrodynamic forces, 5 due to collision forces, 23 due to a decrease in resistanceforce, and 9 due to other factors. The decrease in the resistance force implies the decrease in the contact
force and brought the start of the longitudinal movement of the valley bank particles. Figure 4
demonstrates time variations of the longitudinal force acting on particles A and B shown in Figure 3.
The positive contact force and fluid force on particle A in Figure 4(a) increase with the arrival of thedebris flow front. However, the negative contact force also increases, and the resultant force becomeszero and the particle does not start to move. At about 18 seconds after the arrival of the debris flow
front, particle A starts to move, although the force is smaller than when the debris flow front arrived.
Figure 5 shows the state of the gravel around particle A just before the arrival of the debris flow front
and just before the start of movement. The red particle is particle A. Figure 5 shows that just before
the arrival of the debris flow front, the particles on the downstream side of the debris flow are firmly
interlocked, taking placement where they can produce a large resistance force. On the other hand, justbefore particle A starts to move, the surrounding stones and gravels break down their alignment.
Particularly, the particles on the downstream side have been shifted, and particle interlocking has taken
worse alignment. It is thought that the resistance force of the site is smaller than that before the arrivalof the debris flow front, and this causes the valley bank particles to start moving and the scale of debris
flow front to grow.

4.2. Movement mechanism, of valley bed particles
The movement of particles that make up thevalley bed were analyzed in the same way as thatof the particles that form the valley bank. The 14
velocity[m⁄s]specimens were examined. The main factor that z [m]caused the particles to start moving in thelongitudinal direction was hydrodynamic force(2 specimens), collision (1 specimen), reduced
debris flow front
resistance (11 specimens), and other factors (0specimens). The main factor in the movement of
the particles that make up the valley bed is adecrease in the resistance force. Figure 4(b)x [m]shows that, when the debris flow front reaches
particle B, both the hydrodynamic force and thepositive contact force increase, and the negative
contact force also increases, like particle A, so theFig.8 Contour of flow velocityresultant force is zero and the particle does notstart moving. After that, the particle started to move with less force than when the debris flow frontarrived.
Figure 6 shows change in the position of the center of gravity for each of the particles of interest in
the valley bed. Figure 6 shows that the particles on the surface of the valley bed first move verticallyupward, and then begin to move longitudinally. It is thought that the particles of the valley bed begin
to move because they decrease contact force in the negative direction by moving vertically upward.
In addition, for particles in the valley bed, the smaller particles moved to a lower position in relation
to the initial valley bed. It is thought that the smaller particles drop among the larger particles.
The particles on the valley bed surface began to move at about 18.6 seconds. This is the time when
the rearmost part of the debris flow front is passing through the location of the particles of interest. In
other words, the particles on the surface of the valley bed begin to move not when the debris flow
front arrives and exerts a large force, but when the debris flow front passes by.
Figure 7 shows the time variation of the non-dimensional vertical force acting on particle B. The
absolute value of the contact force in the negative direction increases with the arrival of the debris
flow front. It is thought that the force is exerted to the vertical downward direction from the particles
constituting the debris flow front, which acts as a resistance force and prevents the particles fromstarting to move.
Figure 8 shows the velocity contour map during the passage of the debris flow front. It is seen thatthe velocity of the debris flow front is small, while the velocity of the following flow is large. This issimilar to the experimental results of Lyu et al. [2]. Unstable gravel and stone particles on the bed of
the channel after the debris flow front passed through are supplied into the front of debris flow by thefast flow of the following stream.
5. Conclusion
In this study, the erosion mechanisms of individual particles in a valley were investigated using theAPM method. The results of numerical experiments were similar to those of previous experimental
studies. The particles on the bank and bed of the valley are eroded due to a decrease in resistance force.
Valley bank particles had reduced resistance due to the discharge of downstream particles, and valley
bed particles had reduced resistance due to moving to a relatively higher position than the surrounding
particles. It is clear that particles initiating migration by such a mechanism is transported to the debris
flow front by the following flow and that a debris flow develops.
References[1] Richard M. Iverson, Mark E. Reid, Matthew Logan, Richard G. LaHusen, Jonathan W. Godt andJulia P. Griswold: Positive feedback and momentum growth during debris-flow entrainment ofwet bed sediment, Nature Geoscience, Volume 4, pp.116-121, 2011[2] Richard M. Iverson, Mark E. Reid, Matthew Logan, Richard G. LaHusen, Jonathan W. Godt andJulia P. Griswold: Positive feedback and momentum growth during debris-flow entrainment ofwet bed sediment, Nature Geoscience, Volume 4, pp.116-121, 2011
[3] Fukuoka, S., Fukuda, T., Uchida, T., 2014. Effects of sizes and shapes of gravel particles onsediment 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.