# Flow dynamics of volumes of large light particles released on submerged steep slopes

Marie Rastello, Jean-Louis Marié, Brivaël Collin, Hervé Bellot, Florence Naaim

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Marie Rastello, Jean-Louis Marié, Brivaël Collin, Hervé Bellot, Florence Naaim. Flow dynamics of volumes of large light particles released on submerged steep slopes. Environmental Fluid Mechanics, 2025, 25 (5), pp.39. 10.1007/s10652-025-10049-x. hal-05235453

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Submitted on 3 Sep 2025

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# Flow dynamics of volumes of large light particles released on submerged steep slopes.

Marie Rastello<fnref n="1" />, Jean-Louis Marié<fnref n="2" />, Brivaël Collin<fnref n="3" />, Hervé Bellot<fnref n="4" />, Florence Naaim<fnref n="4" />

## Abstract

The aim of the study is to carry out two-phase gravity driven flows of large light particles on submerged steep slopes and, from their characteristics, identify the criteria that govern the transition between a purely dense and a mixed (dense-suspended) regime. For that, volumes of large light particles are released without any initial velocity at the top of a 2D flume immersed in a 20 m<fnref n="3" /> tank filled with tap water. The particles are spherical, monodisperse, with two different diameters: 10.6 and 14.4 mm. The volume expansion $\beta$, the height $H$ and the front velocity $U_f$ of the heavy flows are investigated varying the volume released between 0.2 L and 3 L, and the tilt angle of the flume ($\theta$) between 30° and 60°. Two different regimes are observed, one where all the particles move in close contact with each other (dense regime), and as the volume and/or flume angle increases, another where part of the particles are suspended (mixed regime). We find that the overall dynamics of the flow is governed by a buoyancy/drag equilibrium ruled by the densimetric Froude number $Fr = U_f / \sqrt{(\Delta\rho g/\rho_w)H}$, with $g$ the gravity acceleration and $\Delta\rho = \rho - \rho_w$. $\rho$ is the density of the flow and $\rho_w$ that of the ambient fluid: water. The corresponding drag coefficient exerted on the particles volume is found to vary as $C_d \approx 2Fr^{-1.6}$. The key parameter for the onset of the flow of some of the particles in suspension and the transition to the mixed regime proves to be $St_\theta = St \cos\theta / (1 - St \sin\theta)$, with $St = v_s/U_f$ the Stokes number comparing the settling velocity of the particles to the flow front velocity. While pressure remains hydrostatic within the flow in the dense regime ($St_\theta > 0.9$), it increases as suspension occurs in the mixed case and $St_\theta$ decreases. The dynamic pressure at the forehead of the volume then evolves as $P_s \approx \Delta\rho g H \cos\theta (1.9 - St_\theta)^2$.

## 1 Introduction

Flows of granular materials, such as snow avalanches or rock collapse, are common potentially destructive natural hazards. Every year, these extreme events cause severe damage to infrastructures and threaten many lives [Hock et al., 2019, Naaim et al., 2008]. Furthermore, climate

<footnote n="1">¹ Univ. Grenoble Alpes, CNRS, Grenoble INP, LEGI, 38000 Grenoble, France.</footnote>

<footnote n="2">² LMFA, UMR 5509, Univ. Lyon, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, France.</footnote>

<footnote n="3">³ ENS de Lyon, CNRS, LPENSL, UMR5672, 69342, Lyon cedex 07, France</footnote>

<footnote n="4">⁴ Univ. Grenoble Alpes, INRAE, CNRS, IRD, Grenoble INP, IGE, 38000 Grenoble, France.</footnote>
change leads to a destabilization of mountainous areas, making these hazards behavior more
difficult to predict, as they may occur in unexpected places or at unexpected times [Marcer
et al., 2019, Strapazzon et al., 2021]. Therefore, understanding the flow dynamics is of prime
importance as it will lead to a better risk-assessment, a better barriers design, and urban plan-
ning [Hewes and InterAlpine Engineers, 2020]. Natural gravity-driven flows often involve the
motion of a mixture of solid particles and air or water, which can lead to complex behaviors
that are difficult to predict and understand. Indeed, the two-way coupling between the particles
and the surrounding fluid becomes even more intricate with the influence of the inclined slope.
All this leads to non-linear effects that cannot be predicted for now using only analytical or nu-
merical models [Gurjar, 2023]. Field experiments are carried out, with triggering and study of
avalanches in situ, for example at the col du Lautaret (France) or in the Sionne valley (Switzer-
land) [Vallet et al., 2004, Rastello et al., 2011, Thibert et al., 2013a,b]. Due to the size and
velocity of real-life avalanches, these projects are often costly and provide complex data that
can be hard to interpret. In this context, laboratory studies where complexity is reduced and
specific intrinsic mechanisms studied have their interest in bringing key dynamical elements
to light.

Snow avalanches are composed of large amounts of particles that descend a slope due to their relative weight to the surrounding medium, often air. As it propagates and depending on the conditions, the avalanche can or cannot incorporate air into its volume. If it does, it creates huge suspended structures (typically 50-100 m high) [Dufour et al., 2001, Rastello and Hopfinger, 2004]. An example is shown in figure 1a. Sometimes, because the particles are too

<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Avalanche with suspended particles during propagation (photo : C. Vion). (b) Re-
mains of an avalanche after a dense type propagation (photo : M. Rastello).</figcaption></figure>

heavy or the slope too low, the particles only slide or progress on the slope and remain a dense
flow of granular medium, as shown in figure 1b. A mix of both flows is classic in the frame
of large scale avalanches that both have a dense part flowing below and/or after the suspended
part that has been generated by shear with the outer fluid. These avalanches are commonly
called mixed avalanches (dense+suspension).

The broad frame of the behavior of avalanches with suspended particles, so-called powder
snow avalanches, has been tackled and macroscopic scaling laws and predictions have been
given in the simplifying context of self-similar half ellipsoidal volumes of heavy fluid [Beghin
et al., 1981, Beghin and Olagne, 1991]. Extensive work has been done on granular flows (flows dominated by collisions and direct interactions between the particles) [MiDi, 2004, Cassar et al., 2005, Forterre and Pouliquen, 2008]-among others. The results can be used to some extent in the context of dense avalanches. Attempts to cover less dense flows have used brine water [Dai, 2013, Shin et al., 2004, Beghin and Olagne, 1991, Rastello and Hopfinger, 2004] but also suspensions of particles set in suspension before release [Gladstone and Woods, 2000, Gadal et al., 2023]. However, various aspects of the fluid-particle interactions specific to the snow avalanche context still remain to be covered and understood.

It should be noted that particle-driven flows topic far exceeds the avalanche context. An-
other essential connected topic in nowadays world is the underwater transport of sediments
and microplastics by turbidity currents on the edge of the continental shelf and in canyons.
Sediment relocation and microplastic transport is indeed a key topic in the understanding of
fluxes and accumulations [Canals et al., 2006, de Stigter et al., 2007, Pohl et al., 2020, Xie
et al., 2023]-among others.

In this study, we reproduce two-phase volume release gravity-driven flows, where a part
is dense and slides on the slope and in some cases, a part incorporates the surrounding fluid
to form suspended structures. The objective is to investigate the mixed flows that have both a
dense part and a suspended one and to examine the intrinsic connection between both that has
scarcely been addressed in the past. In this context, the influence of slope angle, volume of
particles and particle size is considered. The transition from one state (pure dense flow) to the
other (mix of dense and suspended flow) is one of the keys that is looked into. Here the novelty
is to address this very thin but essential part of the phase space where particles are not set in
suspension by an external operator but by the flow itself through the shear at the interface of
particles flow/external fluid.

The paper is organized as follows. In section 2 the experimental apparatus and processing of the data are described. A map of the experimental parameters investigated (initial volume, flume tilt and particle size) is also presented. Results in terms of force balance and pressure within the flows come in section 3 before a conclusion.

**2 Experimental elements**

The experiments consist in the release of volumes of particles at the upper end of a tilted immersed flume in order to quantify the overall dynamic of the self-organized gravity driven particulate flows that get created. Hence, both flume and particles had to be chosen and characterized very carefully.

**2.1 Experimental device - flume characteristics**

The set-up is an open inclined flume of length *l* = 2.8 m, width *w* = 0.2 m and height *h* = 0.4 m immersed in a 20 *m*³ tank filled with tap water (see figure 2). The absence of lid on the flume prevents any upwards back-current induced by the confinement around the downwards flow from developing. The flume can be titled up to 60°. In the context of the present study the 30° to 60° tilt range has been explored. To prevent the overall sliding of the particles on the bottom of the flume, the floor is covered with a regular roughness using LEGO® plates (regular pattern of cylinders measuring 1.8 mm in height, 4.8 mm in diameter and distant by
<figure><img src="image_3.png" /></figure>

Figure 2: Sketch of the experimental device with the release and the flow phases. The flume represented in black is immersed in a water tank in blue and inclined with an angle $θ$. The length of the flume is $l = 2.8$ m, its height $h = 0.4$ m, and its width $w = 0.2$ m. The release of the volume of particles is achieved by the opening of a box located approximately 0.1 m above the flume floor.

3.2 mm from each other). The particles that will be flowing are initially grouped into a box. The box is inserted into the flume approximately 10 cm above the flume floor at the top end of the flume. It is then opened instantaneously, thus releasing the particles as a settling suspension with no initial velocity (see figure 2). As they are all sinking vertically, they start organizing and then propagating along the slope thus initiating their own self-organized flow. The volume of particles released $V_o$ is defined as the volume taken by the particles when poured into a large beaker. In this situation, the initial density of the volume of particles relative to water is defined as $\Delta\rho_o = 0.6(\rho_p - \rho_w)$, where $\rho_p$ is the particles density and $\rho_w$ the water density [Sohn and Moreland, 1968].

## 2.2 Experimental device - particles used

The choice of the particles is of great importance so as to reproduce some of the two-phase physical processes observed in real-life avalanches and heavy flows. The particles need to be heavy enough to flow at a reasonable velocity down the flume and light enough to be able to possibly form a suspended structure during propagation. In order to get some first-step canonical results the particles are chosen as spherical as possible and of a single size (e.g. mono-disperse). Good candidates for such properties are hydrogel particles, a sample of which is pictured in figure 3. By definition, hydrogel particles are mostly composed of water and as such should have a low enough density difference with the surrounding water not to sediment too quickly and are of adequate size for our topic (particle diameter $d \approx 1-2$ cm once fully inflated with water). The hydrogel particles<fnref n="1" /> properties have been characterized carefully. Size and circularity distributions have been obtained by using a sample of a few hundreds particles,

<footnote n="1">¹https://www.billeshydrogel.fr/</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 3: Sample of the two types of hydrogel particles used. Left size: $d = 10.6$ mm, right: $d = 14.4$ mm.</figcaption></figure>

taking pictures of it and using a circle-detection tool on Matlab® software. The measured diameters are $d = 10.6 \pm 0.6$ mm for the smaller particles and $d = 14.2 \pm 0.4$ mm for the larger ones. Hence, the particles used can be considered as mono-disperse in size for each set within the precision of our study. Their circularity being $c = 1.002 \pm 0.001$ confirms they can be considered as spherical.

The particles free fall velocity in water has been measured by filming the particles settling into the tank, without any initial velocity, far away enough from the tank sides so that any effects from the presence of a solid wall can be neglected. The recording is for a 2 m-long free fall giving $v_s = 3.3 \pm 0.1$ cm.s$^{-1}$ for the set of smaller particles and $v_s = 4.1 \pm 0.2$ cm.s$^{-1}$ for the bigger ones. The particle Reynolds number $Re_p = \rho_w v_s d/\mu$ is approximately 300 and 600 respectively with $\rho_w = 997$ kg.m$^{-3}$ and a dynamic viscosity $\mu = 10^{-3}$ Pa.s at 25°C. Knowing $Re_p$, the density of the hydrogel particles relative to water $\Delta\rho_p$ can be determined from the balance between buoyancy and drag:

$$ \Delta\rho_p = \frac{3}{4} \frac{\rho_w v_s^2}{gd} C_D \quad (1) $$

where $C_D$ is the drag coefficient for the particle, $g = 9.81$ m.s$^{-2}$ the gravity acceleration. For the range of Reynolds number, $C_D$ is given by Naumann and Schiller [1935] drag correlation:

$$ C_D = \frac{24}{\text{Re}_p} (1 + 0.15 \text{Re}_p^{0.687}) \quad (2) $$

which, substituted in equation 1 gives $\Delta\rho_p = 5.0 \pm 0.4$ kg.m$^{-3}$. The particles density has also been directly measured using a pycnometer and the exact same density difference of 5 kg.m$^{-3}$ found.

As the particles are soft to pressure by hand, it was checked that they behave like hard spheres during the flows. Their deformation rate has been computed as a function of the collision velocity. The collision of hydrogel particles falling alone, freely in the air, on a hard surface has been recorded at 1800 fps using a Phantom® Miro C110 camera. An accurate estimate of the particle velocity and its maximum deformation rate during the collision have thus been obtained. The Poisson's coefficient for the hydrogels has been measured directly on the camera's images using the parallel and perpendicular length of the deformed particles: $v = 0.34 \pm 0.02$. Using Hertzian contact laws [Landau and Lifshitz, 1986], it is possible to link the deformation of the particle and the collision velocity providing a Young's modulus $E = (3 \pm 0.4) \times 10^4$ Pa. This value is coherent with common values for gel material [Lee et al.,
2018]. The flows studied here have velocities of the order of a few centimeters per seconds (see figure 7). For a particle that would come to a direct stop through a direct impact with an obstacle, the deformation that would be observed would be less than 0.5 mm. Hence, the hydrogel particles which are mostly flowing all together and not experiencing violent changes in velocities can be considered as non deformable within the context of our study.

A summary of the hydrogel particle's properties is given in table 1.

<table><caption>Table 1: Geometrical and physical properties of the hydrogel particles used, *d* being the particle diameter, *c* the circularity of the particles, *v*<sub>s</sub> the settling velocity, Δρ<sub>p</sub> the density difference between particles and fluid and *E* the Young's modulus.</caption><thead><tr><th>d (mm)</th><th>c</th><th>v<sub>s</sub> (cm/s)</th><th>Δρ<sub>p</sub> (kg/m<sup>3</sup>)</th><th>E (kPa)</th></tr></thead><tbody><tr><td>10.6 ± 0.6</td><td>1.002 ± 0.001</td><td>3.3 ± 0.1</td><td>5.0 ± 0.4</td><td>30 ± 4</td></tr><tr><td>14.4 ± 0.6</td><td>1.002 ± 0.001</td><td>4.1 ± 0.2</td><td>5.0 ± 0.4</td><td>30 ± 4</td></tr></tbody></table>

## 2.3 Image processing

The volume of particles released is recorded at a frame rate of 60 fps using 2 to 3 cameras (Nikon D5600), depending on the tilt of the flume θ. The recordings are done from the side of the flume as shown in figure 4. Each film sequence is converted into a series of images that is then processed using a standard image processing under Matlab®. The objective of the processing is to measure, as a function of time, three main flow parameters: the front position *X*<sub>f</sub>, the height *H* and the expansion *β* of the volume. For that, the image background is first subtracted to obtain a well-contrasted image of the volume (figure 4-a) after which, it is binarized using a color value threshold (figure 4-b). The threshold value is chosen in order to obtain a satisfactory resolution of the edge of the volume and is kept constant for all the runs. The conversion of the image pixels in physical units is made using a graduated scale fixed all along the lateral side of the flume. The tracking of the volume is initiated on the first image after injection where it is entirely visible and the particles well grouped. The initial position of the front of the volume, its height, its bottom, and its length are extracted manually from this first image. This delimits a region of interest represented by the green rectangle in figure 4-a and 4-b. The determination of *X*<sub>f</sub> and *H* in the next images is then performed automatically. The front position *X*<sub>f</sub> extracted from each image of the film sequence is the location of the more downstream particle detected in the direction of the volume motion. The height determined is the height of the highest particle in the region that covers the length of the volume: the height of the green rectangle. This processing enables to disregard the particles suspended behind the volume in the estimation of *H*.

An example of front position versus time for a volume of 2 L of 10.6 mm diameter particles released in a flume inclined at 37.5° is shown in figure 4-c. The front velocity *U*<sub>f</sub> is deduced from a linear fit of the data and an average of the values thus determined from the different cameras. The height *H* and the expansion *β* of the volume are plotted in figure 4-d and e respectively. We see on this example that *H* varies with time. This is linked to the particles that detach at the top of the volume. The volume is assumed 2D so that *β* is defined as the white surface within the green rectangle times the width *w* of the flume divided by the volume
$V_o$ released. Here, $\beta$ is of the order of 2, meaning that the volume released expanded twice due to the mixing with water. As for the front velocity, the height $H$ and the expansion $\beta$ are calculated for each run by averaging the mean values determined from the different cameras. The processing, as it results, is based on the assumption of a 2D motion of the particles. This assumption is justifiable, provided that the friction forces between the hydrogel particles and the lateral walls are negligible. Referring to the work of Courech du Pont et al. [2003] on granular heap stability, we know that the characteristic length of these lateral wall effects increases with the particles size. According to their results, we can reasonably estimate this characteristic length in our case to be of the order of twice the particle diameter. The flume width $w$ being 200 mm, which is about 14d-19d, it seems reasonable to consider the particles motion as 2D and that the friction forces between the hydrogel particles and the lateral walls of the flume can be neglected.

## 2.4 Map of the runs

The range of volumes $V_o$ and tilt angles $\theta$ that have been investigated in the present study are summarized in table 2 and plotted under a phase diagram in figure 5. Experiments for the

<table><caption>Table 2: Run characteristics and associated symbols.</caption><thead><tr><th>Given Name</th><th>Particle size</th><th>Volume span</th><th>Angle span</th><th>Symbol</th></tr></thead><tbody><tr><td>AR</td><td>d = 10.6 mm</td><td>2 L</td><td>32° - 56°</td><td>◇</td></tr><tr><td>VR</td><td>d = 10.6 mm</td><td>0.2 L-3 L</td><td>45°</td><td>□</td></tr><tr><td>AVR<sub>s</sub></td><td>d = 10.6 mm</td><td>0.5 L-3 L</td><td>32° - 52°</td><td>○</td></tr><tr><td>AVR<sub>t</sub></td><td>d = 14.4 mm</td><td>0.5 L-3 L</td><td>32° - 52°</td><td>▽</td></tr></tbody></table>

10.6 mm particles include three types of runs: one at a constant volume $V_o = 2$ L and angles ranging from 32° to 56° by one or two degree steps (red diamonds); one at a constant tilt angle $\theta = 45°$ and volumes ranging from 0.2 to 3 L by 0.1 L steps (green squares); one for five tilt angles $\theta = 32.3°, 37.5°, 42.4°, 47.8°, 52.5°$ and six volumes each $V_o = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0$ L (blue circles). Only this latter type of runs was performed with the 14.4 mm particles as well (magenta triangles). In what follows, the red diamonds will refer to AR (angle runs), the green squares to VR (volume runs), the blue circles to AVR_s (angle-volume runs for the smaller particles) and the magenta triangles to AVR_t (angle-volume runs for the larger particles).

## 3 Results

Once released, the flow of particles $V_o$ organizes more or less quickly, taking a shape characterized by an aspect ratio $\gamma = L/H$ (figure 6) and a volume expansion $\beta = V/V_o$. The time needed to take this shape can vary from one run to another, since the release of the particles is performed manually. Although the shape may change slightly as the volume flows down on the inclined plane, it has been observed that $\gamma$ remains approximately constant and of the same order of magnitude as $L_o/H_o$ the initial aspect ratio out of the basket. From this observation, it follows that the average height $H$ can be estimated as:
<figure><img src="image_5.png" /><figcaption>Figure 4: Image processing steps: a) Image of the volume of particles after subtracting the image background; (b) same image after applying a color threshold; the green square designates the region of interest including the volume, the red line the position of the floor; c) position of the front of the volume with time; d) volume height with time; e) volume expansion with time.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5: Phase diagram: experiments with the smaller particles and variation of the slope angle: ◇, variation of the initial volume: □, mixed variation of both angle and volume: ○, ▽: experiments with larger particles and mixed variation of both angle and volume. See table 2 for details. The dotted lines materialize the emergence of the particles detachment regime observed visually on the flows: blue for 10.6 mm particles, magenta for 14.4 mm particles.</figcaption></figure>

$$H \approx \sqrt{V/\gamma w} \qquad (3)$$

where w is the flume width and V is equal to βV₀. The expansion β, the height H and the front velocity Uf of the volume for AVR_s and AVR_l are plotted in figure 7 as a function of the tilt angle θ (a, c, e) and the volume released V₀ (b, d, f). We see that for small particles, β is approximately constant, of the order of 2 up to 40° and increases afterwards, reaching approximately 3 for 55°. This behavior is explained by the fact that above 40°, particles detach from the volume under the effect of the shear between the flow and the ambient fluid. It forms a suspension at the rear part of the volume and increases the mixing with the ambient water. This clearly appears on the chronophotography shown in figure 9: the shape which is approximately semi-elliptical up to θ = 40° (schematic in figure 6), exhibits a tail in its upper part when the suspension forms at the rear part of the volume. As expected, the increase of β with θ is weaker in the case of the larger particles. Indeed, these particles being heavier, they start to detach at higher tilt angles (higher front velocities). This will be quantified later on through a parameter depending on the Stokes number St = vs/Uf comparing the settling velocity of the particles to the flow front velocity.

The observed trends are independent of the volume released, as shown in figure 7-b, but the higher the θ, the higher the β. The emergence of the particles detachment regime has been visually identified on the film sequences. It is materialized in the phase diagram (figure 5) by the blue dotted line for the smaller particles and the magenta dotted line for the larger ones. These limiting lines are consistent with the trends in figure 12-a-b. As seen in figure 7-c-d-e-f,
<figure><img src="image_7.png" /><figcaption>Figure 6: Schematic of the volume of particles in its moving frame of reference with θ: the slope angle, $U_f$: flow front velocity, $H$, $L$: height and length of the flow and $v_s$ the settling velocity of the particles. The volume of particles width is $w$.</figcaption></figure>

the behavior of the height $H$ and the front velocity $U_f$ as a function of $\theta$ and $V_o$ is similar. Both increase with $\theta$ and $V_o$. This increase is more significant for $\theta$ greater than 40° and $V_o$ greater than 2 L, whatever the particles. However, the heights are somewhat lower in the case of heavier particles, as could be expected. In addition, we measured the aspect ratio $\gamma = L/H$ for all runs and used it to estimate the average heights $H$ from equation 3 and the experimental values of $\beta$. The heights thus calculated for the $AVR_s$ runs are plotted in figure 8-a-b with blue filled circles. We note that they provide an estimation of the measurements in agreement with the flow observations, within 4% uncertainty.

The non-dimensional numbers that are relevant to characterize the motion of the volumes of particles are the Stokes number $St$, the Reynolds number $Re = \rho_w U_f H/\mu$ and the densimetric Froude number $Fr = U_f / \sqrt{(\Delta\rho g/\rho_w)H}$. The height $H$ of the volume varying typically between 5 and 15 cm, $Re$ is of the order of $10^3$ to $10^4$. This means that inertial effects are dominant over viscous effects. Under these conditions, it is reasonable to assume that the motion of the volume of particles, when established, is driven by a balance between its submerged weight and the drag force that exerts on it, a balance that reads:

$$ \Delta\rho V g \sin\theta = \frac{1}{2} \rho_w C_d U_f^2 H w \quad (4) $$

from which we deduce that:

$$ C_d = 2 \frac{\Delta\rho V g \sin\theta}{\rho_w H w U_f^2} \quad (5) $$

This balance assumes that the friction effects between the system of particles and the walls can be neglected. The question of lateral walls has been briefly discussed in the end of subsection 2.3 and it has been concluded that their effects can be reasonably disregarded. The question of the friction with the bottom remains. The floor is covered with a regular roughness whose characteristics have been given in section 2. The height of the cylinders constituting the roughness is 1.8 mm, but given the spacing between them: 3.2 mm and the diameter of the particles,
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Figure 7: a): $\beta$ as a function of flume tilt $\theta$. b): $\beta$ as a function of volume released $V_o$. c): $H$ as a function of flume tilt $\theta$. d): $H$ as a function of volume released $V_o$. e): $U_f$ as a function of flume tilt $\theta$. f): $U_f$ as a function of volume released $V_o$. ○: experiments with the small particles, ▽: experiments with the large particles. See table 2 for details.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 8: Comparison of the height of the volume as measured for small particles: ○ and as calculated from equation 3: ●; uncertainty between estimations and measurements ≈ 4%.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 9: Chronophotography of 4 flows for different tilt angles. From top to bottom, $\theta = 33.4^\circ$, $\theta = 39.3^\circ$, $\theta = 46.5^\circ$ and $\theta = 53.4^\circ$. For all, $V_o = 2.0$ L. The line at the bottom of the figure is for time. Graduations are every 2 s.</figcaption></figure>
the "effective" roughness seen by the particles is at maximum 0.5 mm for the smallest ones ($d$ = 10.6 mm) and 0.4 mm for the largest ones ($d$ = 14.4 mm). Therefore, particles are never trapped in the space between two cylinders and their slow down near the wall remains weak. As a proof, we observe that some particles are left behind the volume of particles, but the number of the particles forming this "tail" in contact with the wall is very small compared to the number inside the volume (see for example figure 4-a-b). Moreover some particles visible in the "tail" are particles that detached from the volume in the suspension phase and redeposited behind on the floor. Hence, as a first approximation, we also neglect the friction with the floor.

Since $V = \beta V_o$ and $\Delta\rho = \Delta\rho_o/\beta$, $C_d$ in equation 5 simplifies in:

$$C_d = 2 \frac{\Delta\rho_o V_o g \sin\theta}{\rho_w H w U_f^2} \qquad (6)$$

The drag coefficient calculated from this expression and our data is plotted in figure 10 as a function of the densimetric Froude number $Fr$. We see that under that form all the data collapse well onto a single curve whose equation is:

$$C_d \approx 2Fr^{-1.6} \qquad (7)$$

This good data grouping attests that an approach neglecting the friction forces with the walls and considering the balance between submerged weight and drag forces to describe the motion of the particles volume seems grounded. Using equations 4 and 7 the evolution of the front velocity $U_f$ as a function of $V_o$, $\sin\theta$, $\beta$ and $H$ is expressed in the form:

$$U_f = \sqrt{\frac{\Delta\rho_o g \beta^4 V_o^5}{\rho_w H^9 w^5} (\sin\theta)^5} \qquad (8)$$

The values calculated from equation 8 for the two runs AR and VR are shown in figure 11. The average uncertainty between the measured and estimated values lies between 10% (figure 11-a) and 15% (figure 11-b). This shows that the estimations are satisfactory and confirm the relevancy of our assumptions. The governing dynamics led by a $Fr$-number can recall other phenomena including groups of interacting particles which are also connected with $Fr$ numbers. Among others, the drag coefficient arising from the interaction between dense snow avalanches and obstacles is exposed to be connected with a $Fr$ number by Kyburz et al. [2022]. The types of sand bed forms in flumes or rivers has also been shown to be ruled by a Froude number [Engelund and Fredsoe, 1982, Charru et al., 2013].

The question is now to determine which parameter may trigger the detachment of the particles at the upper part of the volume, for example, when the tilt of the flume is increased. We may reasonably think that it occurs when the settling velocity $V_s \cos\theta$ perpendicular to the flume direction is no longer large enough to overcome the velocity $U_f - V_s \sin\theta$ of the particles in the direction parallel to the flume and thus maintain the particles close to the boundary (see figure 6). The parameter that quantifies this transition is therefore:

$$St_{\theta} = \frac{V_s \cos\theta}{U_f - V_s \sin\theta} = \frac{S_t \cos\theta}{1 - S_t \sin\theta} \qquad (9)$$

Figure 12-a shows the evolution of the densimetric Froude number as a function of $St_{\theta}$. Two regimes are clearly visible. When $St_{\theta} > 0.9$ the particles submerged weight is strong
<figure><img src="image_16.png" /><figcaption>Figure 10: Drag coefficient $C_d$ as a function of the densimetric Froude number $Fr$. Experiments with the smaller particles and variation of the slope angle: ◦, variation of the initial volume: □, mixed variation of both angle and volume: ○. ▽: experiments with larger particles and mixed variation of both angle and volume. See table 2 for details. Solid line: $C_d \approx 2Fr^{-1.6}$ (equation 7).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 11: Comparison of the front velocity as measured (open symbols) and as calculated from equation 8 (x): a) front velocity versus sinus of the tilt angle $θ$, uncertainty between estimations and measurements is ≈ 10%; (b) front velocity versus the volume released $V_o$, uncertainty between estimations and measurements is ≈ 15%.</figcaption></figure>
enough to maintain the particles grouped on the bottom. The particles remain grouped inside the volume and in contact with the boundary. In this case the densimetric Froude number $Fr$ is approximately constant, with an order of magnitude of 1.2. This trend is a little less visible for the large particles because the $Fr$ values are more scattered. This scattering is mainly due to the lower contrast of the blue color of the large particles compared to the red for the smaller ones ending in a higher uncertainty on $H$. However the trend is clearly visible under the adimensional form in 12-b. In contrast, when $St_θ < 0.9$, the inertial effects overcome the submerged weight effects. Particles start to detach, leading to a particle suspension at the rear part of the volume. $Fr$ then increases up to 2 as the inertial effects become more important compared to the submerged weight effects. The transition between the two regimes is represented by the dotted lines in the phase diagram (figure 5). Evidently, the transition comes later for heavier particles (magenta triangles) as their settling velocity is larger.

<figure><img src="image_18.png" /><figcaption>Figure 12: a) $Fr$ number as a function of $St_θ$. Solid lines are: $2.4 - 1.3St_θ$ for $St_θ < 0.9$ and $1.2$ for $St_θ > 0.9$. b) $Fr$ non-dimensionalized by its value for $St_θ > 0.9$ as a function of $St_θ$; solid line for $St_θ < 0.9 : Fr/\sqrt{2cos(θ)} = 1.9 - St_θ$ and 1 for $St_θ > 0.9$. Experiments with the smaller particles and variation of the slope angle: ◇, variation of the initial volume: □, mixed variation of both angle and volume: ○, ▽: experiments with larger particles and mixed variation of both angle and volume. See table 2 for details.</figcaption></figure>

As long as particles are grouped within the volume, we can expect the dynamic pressure: $P_s = 1/2ρ_wU_f^2$ at the forehead of the volume (point $S$ in figure 6) to be of the order of the hydrostatic pressure at the bottom of the volume: $ΔρgHcos(θ)$. It follows that $Fr ≈ \sqrt{2cos(θ)}$, which is verified in figure 12-b. This is no longer the case when particles detach and form a suspension behind the core volume ($St_θ < 0.9$). $Fr$ is then greater than $\sqrt{2cos(θ)}$, which means that $P_s$ also increases. From the trend in figure 12-b: $Fr/\sqrt{2cos(θ)} = 1.9 - St_θ$, we can deduce that $P_s$ increases as:

$$P_s \approx \Delta\rho g H \cos(\theta) (1.9 - St_\theta)^2 \quad (10)$$

Therefore, things behave as if the particles moving in suspension at the rear part of the volume would contribute to create an extra pressure that would add to the initial hydrostatic
pressure. This result resembles the excess pressure (interstitial or pore pressure) reported in Cassar et al. [2005]. Here, the excess pressure is maximum when $St_θ = 0.46$ and is equal to about 1 times the hydrostatic pressure $ΔρgHcos(θ)$. In Cassar et al. [2005] the excess pressure (pore pressure) never exceeded 25% of the hydrostatic pressure. The two flow situations are therefore not comparable. In a horizontal configuration ($θ = 0$), $St_θ$ reduces to $St$. This recalls the suspension thresholds described by Niño et al. [2003] in connection with Bagnold [1966] criteria.

# 4 Conclusion

In this study, we reproduce in the laboratory, two-phase gravity driven flows of large light particles on submerged steep slopes. The objective is to identify the criteria that govern the transition between the dense and the mixed (dense-suspended) regimes. The size of the particles is 10.6 mm and 14.4 mm, with a density: $Δρ_p = 5kg/m^3$ slightly larger than the one of ambient water. The height $H$, the volume expansion $β$ and the front velocity $U_f$ of the flows of particles are measured using video cameras, varying both the volume $V_o$ of particles released and the slope angle $θ$.

As $Re$ number is large ($10^3$ to $10^4$), the inertia effects dominate over the viscous effects and $Re$ number is not observed to be a key parameter for the flow. The volume motion, when established, proves to be governed by an equilibrium between buoyancy and inertia: the submerged weight of the overall volume of particles is balanced by the drag force exerted on it. The friction forces between the particles and the walls are considered to be small enough to be neglected. The corresponding drag coefficient is found to depend on the $Fr$ number as $C_d \approx 2Fr^{-1.6}$, which yields a front velocity $U_f \approx \sqrt{Δρ_o g β^4 V_o^5 (sinθ)^5 / ρ_w H^9 w^5}$. The good agreement between this front velocity and the measured ones, both in the dense and in the mixed regime, attests the relevancy of this simple force balance.

The parameter that governs the transition between the purely dense and the mixed (dense-suspended regime) proves to be $St_θ = St \cosθ/(1 - St \sinθ)$. This parameter quantifies the ratio between the settling velocity of the particles perpendicular to the flume direction (here: 3.3 cm/s and 4.1 cm/s) and the incoming shear velocity of water in the direction parallel to the flume. $St_θ > 0.9$ is the situation where the settling velocity overcomes the incoming shear velocity, so that all particles can remain in close contact sliding on the slope. Below this limit, the settling of the particles is not sufficient to maintain them all close to the boundary. As a result, a portion of the particles get set in suspension by the shear at the flow interface: the transition from the dense to the mixed flow behavior occurs.

In the dense regime ($St_θ > 0.9$) the densimetric Froude number is constant of order of magnitude 1.2, meaning that the dynamic pressure $P_s$ at the forehead of the volume of particles is of the same order as the hydrostatic pressure. This is no longer the case in the mixed regime ($St_θ < 0.9$), where the densimetric Froude number increases as the inertial effects get larger than the submerged weight effects. As a result, the pressure $P_s$ increases significantly above the hydrostatic pressure. This increase corresponds to an excess pressure that seems linked to the motion of the particles in suspension.
Acknowledgements

Alexis Buffet is thanked for technical support. We acknowledge financial support from the French National Research Agency Grants ANR-19-CE30-0041/PALAGRAM and ANR-19-IDEX-UGA/LOGSO. For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

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