E3S Web of Conferences 415, 02023 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502023

Experiments on and numerical calculations of debris flows
passing through narrowed and widened channel regions
Jun’ichiro Takahama1*, and Kouhei Shintani2
1Division of Environmental Sciences, Graduate School of Life and Environmental Sciences, Kyoto Prefectural University, Kyoto

606-8522, Japan
2River Division, Hokuriku Regional Development Bureau, Ministry of Land, Infrastructure, Transport and Tourism, 1-1-1 Misaki-cho, Chuo-ku, Niigata 950-8801, JapanAbstract. We clarified the dynamics of debris flows in areas wherein the topography changes abruptly.
First, the dynamics of debris flow through a constriction were investigated by experimentally generating
non-uniform flow conditions. Then, numerical calculations were performed using an erosion/deposition rateequation that considers the pressure gradient in a constriction, and compared with the experimental data.

1 Introduction
When the topography changes abruptly, such as innarrow regions of canyons, debris flow would beexpected to vary significantly. However, few detailed
studies have appeared on this phenomenon. The
erosional and depositional characteristics of debris
flows are often evaluated based on the magnitude of thecorrelation between the gravity term in the equation ofmotion and the shear stress (yield stress) on the river bed
surface; channel transitions have not been considered.
Here, we describe experiments in which flow became
non-uniform as debris passed through a narrowed
channel region [1], and present a method for evaluating
erosion and deposition based on effects of channeltransitions. We also describe relevant applications [1].

2 Overview of channel experiments
We created narrowed and widened areas in the middle
of a straight channel (length, 6 m; width, 10 cm). toobtain a saturated movable bed. We then supplied
sediment and water on a constant basis. The channel
slope was 15º, the water supply rate was 531 cm3/s, thesediment supply rate was 235 cm3/s; and the sediment
proportion was 0.31.

3 Equilibrium slope with a pressureterm and the erosion and deposition
velocity equations

where subscripts s and w denote the amount ofsolid/liquid motion and water flow, respectively. B is thechannel width, h is the thickness of the fluidized bed ofeach layer, v is the mean flow rate of each layer, θ is theriver bed gradient, ρ is the mean density of each layer, gis gravitational acceleration, P is the pressure acting on
the side of each layer, and τb is the shear stress acting on
the river bed surface[3]. In the equations, βw and βs are
momentum correction coefficients for each layer, and γd
is a shape factor reflecting the correlation between
concentration and flow rate. All coefficients wereassumed to be 1. Terms associated with the curvature of
the river bed surface were omitted. The erosion rate sT
was obtained by extending [2] the erosion rate equationof Egashira et al. [4] to a two-layered flow, as follows:

where vt is the mean flow rate of the entire two-layered
flow and θe is the equilibrium slope corresponding to
the mean concentration of all layers (ct). As the total
fluidized bed thickness is ht, we obtain:

Motion equations for two layers [2] were combined toobtain the following motion equation for all layers:
where cs is the mean volume concentration of gravel
moving in the bed and ϕ is the internal friction angle. If* Corresponding author: jtaka@kpu.ac.jp© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 415, 02023 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502023

we add a pressure term to the motion equation (1), andif the equilibrium slope at this point is θep, we obtain thefollowing equation:

this slope reflects the balance of the parameters.
Therefore, the equilibrium slope in a narrowed region
can be larger or smaller than in other regions, explaining
the downward convex shape of the experimental curve.

4 Conclusions
We experimentally examined debris flows during
channel transitions, and modeled abrupt channeltransitions by introducing the flow pressure gradient to
the equilibrium slope. We used the model to evaluate theresults of experiments. We also conducted experiments
on fixed beds; our intriguing results will be summarized
during the presentation.

where ht = hs +hw , Pw0 = Pw/cosθep , and Ps0 = Ps cosθep.
By introducing θep into the erosion and deposition
velocity equation, i.e., Equation (2), the effects ofnarrowed and widened channels can be considered whenevaluating erosion and deposition. In such a case, theequilibrium slope θep that considers the pressure gradientis expressed using θe, as follows

Equation (7) shows that θep is smaller than θe in areas
with sloping water surfaces (such as debris flow fronts),
and that these areas are more susceptible to erosion. In
contrast, erosion is suppressed and sedimentation occurs
under certain conditions. Figure 1 shows theexperimental data derived when debris flowed throughnarrowed regions, and the numerical calculations for
erosion and deposition rates, when the equilibrium slope
was that of Equation (7). The x-axis is parallel to theinitial river bed. The top row shows the channel width
in longitudinal section. The river bed is represented byblack circles (●), indicating that the bed is degraded in
narrowed regions. This is also evident in the calculation(thick solid line). However, the calculated free surface
is linear; this does not mirror the experimental values.
Regarding the effect of introducing a pressure term intoa movable bed, debris entering a narrowed region isaffected by the sidewalls, as narrowing graduallyincreases and flow height thus increases. Thus, flow
depth becomes larger than that of flow just entering thenarrowed regions (and is thus just upstream of thatregion). This causes a pressure difference between thedebris flow upstream (where the flow depth is smaller)
and that in the narrowed section (where the flow depthis greater); pressure is directed upstream. In other words,
gravity (a shear stress on the river bed surface) acts inthe downstream direction but the yield stress andpressure act in the upstream direction. Thus, theequilibrium slope becomes larger than that in a nonnarrowed region, given that the slope reflects thebalance between the different effects. After the debris
flows through the narrowed regions, it is released fromthe effects of side walls as the new region graduallywidens, and thus has more room to escape; this reduces
the depth. In turn, this causes a pressure difference
between the flow in the narrowed region (larger flowdepth) and that in the widened region (smaller flowdepth); this directs pressure downstream. In otherwords, gravity and pressure both act downstream but theyield stress acts upstream. The equilibrium slope issmaller than that in the non-narrowed region because

Fig.1 Comparison of experimental results of debris flow in aconstriction with calculation results in which a pressure
gradient is introduced into the erosion and deposition rateequation (on a movable bed)

2

E3S Web of Conferences 415, 02023 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502023

References
1.

2.

3.
4.

K. Shintani, Experimental study on the generationand flow phenomena of debris flows, Master'sThesis, Gifu University (2006)
J. Takahama, Y. Fujita, K. Hachiya, K. Yoshino,Application of two layer simulation model forunifying debris flow and sediment sheet flow andits improvement, in Debris-Flow Hazards
Mitigation: Mechanics, Prediction, andAssessment, Rickenmann & Chen (eds.),
Rotterdam: Millpress (2003)
S. Egashira, K. Miyamoto, T. Itoh, Annual Journalof Hydraulic Engineering 41 (1997)
S. Egashira, N. Honda, Y. Ando, Annual Journalof Hydraulic Engineering 40 (1996)

3