Introduction

Concentrated pyroclastic currents

Small-volume (usually < 107 m3) pyroclastic currents are complex and dangerous volcanic gravitational flows, composed of a hot mixture of gases and particles and generated from lava dome or small-column collapse (Brown and Andrews 2015; Lube et al. 2020). They are capable of flowing over distances typically longer than 3 km and at speeds up to > 100 km h−1 (Brand et al. 2014; Cole et al. 2015; Pollock et al. 2019). Our study focuses on concentrated pyroclastic currents (CPCs), which are two-layer flows composed of a concentrated basal layer (more than ~ 40–60 vol% of particles) and an overriding dilute upper layer (less than ~ 1 vol% of particles) also called the ash-cloud surge (Dufek et al. 2015; Lube et al. 2020). We do not address dilute pyroclastic currents (DPCs) that only exhibit a dilute layer with a bedload region (Dufek et al. 2015; Valentine, 2020; Brosch and Lube 2020).

The basal part of small-volume CPCs tends to be channelized in deep and narrow valleys of volcanoes. They exhibit remarkably long flow runouts compared to other geophysical granular flows of similar volume (Dade and Huppert 1998; Druitt, 1998; Calder et al. 1999; Iverson and Vallance 2001). Previous studies on CPCs have demonstrated that channelization can enhance flow runout by confining the entire mass into a restricted area, preventing rapid lateral spreading and efficient energy dissipation (Woods et al. 1998; Calder et al. 1999; Andrews and Manga 2012; Jessop et al. 2012; Charbonnier et al. 2013; Ogburn et al. 2014; Aravena and Roche 2022). Furthermore, a high interstitial gas pore pressure and related fluidization in the concentrated basal layer has been proposed as an efficient mechanism to reduce the inter-particle friction and also enhance flow runout (Sparks 1978; Wilson 1980; Druitt et al. 2004; Bareschino et al. 2007; Dufek et al. 2015; Lube et al. 2020), as demonstrated by numerous experimental works (Druitt et al. 2007; Girolami et al., 2015; Roche et al. 2010; Rowley et al. 2014; Smith et al. 2018, 2020) and numerical studies (Gueugneau et al. 2017; Breard et al. 2019, 2022; Lube et al. 2019; Aravena et al. 2021). The rapid decrease of pore pressure at coarse flow margins during emplacement and self-channelization of CPCs can explain deposit structures like lateral static levées and steep front lobes (Wilson and Head 1981; Gueugneau et al. 2017; Smith et al. 2018, 2020; Brand et al. 2017). A better understanding of the effect of channelization and friction reduction in small-volume CPCs is crucial to better estimate their inundation area and consequently improve the assessment of hazards they pose along valleys of CPC-generating volcanoes.

The overspill hazard of small-volume CPCs

Even though small-volume CPCs are channelized in valleys of volcanic edifices, most of the fatalities are due to the unpredicted overspill of the flows from the channel confines and a rapid inundation of the interfluves, as the overspills can travel up to a few kilometers from its overflow point (i.e., Volcan de Fuego in 2018; Albino et al. 2020). As detailed in Gueugneau et al. (2021), Kubo Hutchison and Dufek (2021), and Lerner et al. (2022), CPC overspills exhibit various depositional and dynamical characteristics, and here, we distinguish two different types:

  1. (i)

    “CPC overspill,” when the concentrated basal layer of a CPC escapes the valley at a specific location, usually, but not always, accompanied by its upper ash-cloud surge. This overspill continues to flow along valley banks, volcaniclastic terraces, and interfluves, and is usually named overbank flow to distinguish it from its parent valley-confined flow, as at Merapi during the 2006 or 2010 eruptions (Charbonnier and Gertisser 2008; Lube et al. 2011; Gertisser et al. 2011; Charbonnier et al. 2013), at Volcán de Colima (Mexico) in 2015 (Macorps et al. 2018), or at Fuego volcano in 2018 (Albino et al. 2020) (Table 1).

  2. (ii)

    “Ash-cloud surge detachment,” when only the dilute ash-cloud surge decouples from the concentrated basal layer and escapes the confining valley to propagate on its own, as at Montserrat in 1997 (Druitt et al. 2002; Loughlin et al. 2002; Ogburn et al. 2014), Unzen in 1991 (Nakada and Fujii 1993), Merapi in 2010 (Komorowski et al. 2013), or Mount Pelée in 1902 (Gueugneau et al. 2020).

Table 1 Summary of notable CPCs and morphometry of the corresponding channels they flowed into. For each case, the topographic feature identified as responsible for an overspill is described. The morphometry of PyroCLAST is presented in the last row. H/L refers to the ratio between the initial height (H) of the material over the flow runout length (L)
Full size table

The physics behind CPC overspills remain poorly known despite their danger. Most of our knowledge is based on field studies of CPC deposits (Charbonnier and Gertisser 2008; Lube et al. 2011; Gertisser et al. 2011; Charbonnier et al. 2013; Ogburn et al. 2014; Lerner et al. 2022). These have pointed out that a sudden change in the channel geometry (both from natural causes or the result of human intervention) can potentially reduce the channel capacity, then cause the CPC to overspill. At least four principal topographic features can be distinguished (Table 1):

  1. (a)

    A sharp valley bend (Ogburn et al. 2014; Macorps et al. 2018)

  2. (b)

    A break in slope along the valley (Bourdier and Abdurachmann 2001; Charbonnier and Gertisser 2008)

  3. (c)

    An obstacle obstructing the valley (i.e., sabo dam, lava ridges, or bridges; Charbonnier and Gertisser 2008; Lube et al. 2011)

  4. (d)

    A sudden reduction of the valley width (i.e., constriction; Charbonnier and Gertisser 2008, 2011; Jenkins et al. 2013)

Recently, Kubo Hutchison and Dufek (2021) and Gueugneau et al. (2021) numerically studied the overspill mechanism. Kubo Hutchison and Dufek (2021) have demonstrated that a sinuous valley can cause an important increase of the local flow thickness along the outside of a bend, called superelevation, leading to the overspilling of the channelized flow. Such sudden superelevation was also inferred from field studies at Merapi in 2006 (Charbonnier and Gertisser 2008; Lube et al. 2011). Gueugneau et al. (2021) highlighted the link between the increase of the local CPC volume flux (calculated along the cross-sectional area of the channel) and the occurrence of an overspill. Unsteady flow conditions and increases in flow velocity, thickness, and volume flux can also promote overspilling (Charbonnier et al. 2013; Kubo Hutchison and Dufek 2021). These studies suggest that the CPCs’ properties (volume, flow rate) and the geometry of the valley are key parameters controlling overspills. A deeper understanding of the interaction between the valley-confined CPCs and the topography, as well as the evolution of the internal flow conditions during emplacement, is needed to better characterize the mechanisms of CPC overspills and better assess related hazards.

The interaction between channelized flows and topographic obstacles, such as an oblique plane or a vertical wall, has long been studied experimentally for water-particle flows (Armanini and Scotton 1993; Armanini et al. 1997, Armanini et al. 2011; 2020) or non-fluidized granular flows (Gray et al. 2003; Gray and Cui 2007; Cui et al. 2007; Pudasaini et al. 2007). This issue was discussed for geophysical flows such as snow avalanche, landslides, debris flows, or mud flows to properly design protective structures in mountainous areas (Mancarella and Hungr 2010; Ashwood and Hungr 2016; Iverson et al. 2016; Faug 2021). Although being of primary importance in the process of CPC overspill, the interaction between CPCs and topographic obstacles has never been studied experimentally. The laboratory-scale devices of Roche et al. (2010), Rowley et al. (2014), and Smith et al. (2018) consist of straight horizontal flumes aimed at investigating the internal and depositional processes of CPCs in a confined configuration. On a larger scale, the 20-m-long apparatus PELE (Lube et al. 2015) and the 11-m-long apparatus GRANFLOW-SIM (Bartali et al. 2012; Sulpizio et al. 2016) are straight inclined flumes with a constant slope angle, which are terminated by unconfined flat areas. Hence, none of these existing experimental apparatuses admits a non-straight channel morphology with bends or obstacles, neither flat channel banks to receive an overflow. In this context, a new experimental setup with a non-straight channel morphology is necessary to investigate the interaction between a channelized CPC and a valley bend, and to serve as a benchmark for CPC numerical models (Gueugneau et al. 2020).

Aims of the study

Here, we present the first results of laboratory experiments carried out with a new facility called PyroCLAST (Pyroclastic Current Large-scale Apparatus using Synthetic Topographies) and built to investigate CPC overspills. This study focuses on the relation between the dynamics of channelized parent flows, the occurrence of an overspill, and the dynamics of the overbank flow. In this study, we consider a sharp valley bend (i.e., the most common topographic feature; see Table 1), simplified here by a double 45° bend in opposite directions (see Fig. 1(a)) and positioned halfway along a 5-m-long flume. To explore a broad range of flow conditions at the overspill site, the slope angle is modified from 3 to 15°, while keeping the same initial mass of granular material. We consider initially fluidized granular flows (i.e., with interstitial pore fluid pressure), which are obtained by injecting a vertical air flow at the base of a granular bed in a semi-closed reservoir prior to release into the channel. The properties of the channelized and overbank flows (i.e., front and internal velocity, flow thickness, and discharge rate), as well as the final deposit morphology (i.e., deposit length and thickness), are measured during and after each experimental run, respectively. We first describe the design and conception of the apparatus, and the experimental procedure for each experiment, before presenting the results of the first set of experiments. Results are discussed and put into perspective with previous experimental results of channelized granular flows, but also with natural CPCs.

Fig. 1
figure 1

Illustration of the four main topographic features mentioned to cause a CPC to overspill along volcanic valleys

Full size image

A complementary goal of this project is to build a new experimental database for the benchmarking of numerical models for pyroclastic currents, recently initiated by an international community effort presented in Esposti Ongaro et al. (2020). The first synthetic benchmarks for CPC models conducted by Gueugneau et al. (2021) showed that using a robust reference database to compare models is required to estimate their robustness in simulating channelized CPC processes.

Materials and methods

Preliminary analysis

We first present a simple energy balance for an inviscid fluidized granular flow (i.e., with negligible dissipative stresses) in a channel with an obstacle of height \(\Delta z\) by considering the conversion of the flow kinetic energy to the potential energy (Fig. 2(a)). We considered this analysis to design PyroCLAST. For a finite flow mass m, the kinetic energy needed by the flow of velocity vp perpendicular to the obstacle to raise the height \(\Delta z\) satisfies.

$$\frac{1}{2}{m{v}_{p}}^{2}=mg\Delta z,$$
(1)

where g is the gravitational acceleration, so that

Fig. 2
figure 2

Analysis of the minimum flow velocity required to climb an obstacle in a channel bend. (a) Horizontal channel with obstacle of height Δz and perpendicular velocity vp. (b) Horizontal channel with a bend at angle A. (c) Channel inclined at angle alpha. (d, e) Minimum flow velocity vα required to climb the obstacle, from Eq. (6), as a function of the channel slope angle and the angle of the bend

Full size image
$${v}_{p}=\sqrt{2g\Delta z}$$
(2)

This only applies if \(\Delta z\) is significantly larger than the flow thickness. We consider now a bend in a channel with an angle A (Fig. 2(b)). In this configuration, the velocity component \({v}_{p}\) perpendicular to the bend sidewall (green arrow, Fig. 2(b)) as a function of the flow velocity v and the bend angle is

$${v}_{p}=v\mathrm{sin}\left(A\right).$$
(3)

Therefore, combining Eqs. (2) and (3), the minimum velocity needed for an inviscid and incompressible thin flow to overtop the channel bend is:

$$v=\frac{\sqrt{2g\Delta z}}{\mathrm{sin}\left(A\right)}.$$
(4)

If we further consider the slope angle \(\alpha\) of the channel as shown in Fig. 2(c), the projected vertical sidewall height is

$$h=\Delta z\;\cos\;\alpha,$$
(5)

and the minimum flow velocity to climb this height is

$$v_\alpha=\frac{\sqrt{2g\Delta z\;\cos\;\alpha}}{\sin\left(A\right)}.$$
(6)

This theoretical flow velocity is an approximation of the minimum velocity needed to raise the thin flow and potentially generate an overspill since it does not take into account the energy lost by interparticle and particle–wall collisions or friction at impact. The minimum velocity vα decreases slightly from 2.42 to 2.38 m s−1 as the slope angle increases from 0 to 15°, while it decreases significantly when the angle of the bend increases (Fig. 2(d-e)). As a comparison, the maximum velocity of an inviscid gravitational thin flow on a straight slope varying from 0 to 15°, calculated using Eqs. (2) and (5) with \(\Delta\) z the height difference between the starting point and the lowest point of the flow (front), is between 2.2 to 5 m s−1. Because this maximum flow velocity is nearly equivalent or superior to \({v}_{\alpha }\) in a channel with 15-cm-high walls and a 45° bend for the same slopes, this configuration should allow gravitational thin flows to overtop sidewalls at the bend, and therefore was chosen for our PyroCLAST apparatus.

Design of the apparatus

PyroCLAST is designed in a dam-break configuration, for which a granular bed confined in a reservoir is suddenly released in a flume at various inclinations. Many experimental studies in volcanology have considered this configuration (e.g., Roche et al 2010; Sulpizio et al. 2016; Smith et al. 2018), because it allows the generation of rapid dense granular flows, in particular with high Froude numbers and other dimensionless parameters, required to simulate natural CPCs (see Table 2).

Table 2 Principal dimensionless numbers and parameters for CPCs and PyroCLAST
Full size table

To investigate the overspilling of CPCs with our apparatus, the channel morphology was chosen to mimic a valley bend typical of a volcanic edifice (see Table 1). Hence, we designed PyroCLAST with three main elements (Fig. 3): a reservoir that contains the granular material before an experiment, a 1.8-m-long channel with transparent Plexiglas walls that confines the flow after releasing the granular material, and a 3-m-long low-profile channel that comprises the topographic feature. This distal channel section is the key element in the apparatus. To allow flow overspills, and following our preliminary analysis, the second channel has 15-cm-high sidewalls and a 45° bend at a distance of 3 m from the reservoir, followed by another similar bend in the opposite direction 0.55 m downstream. Overspills generated at the first bend spread on a large inclined plane. The inclined plane, made of wood, allows a rapid and easy modification of the channel morphology for future studies.

Fig. 3
figure 3

3D sketches and photographs of the apparatus PyroCLAST: (a) lateral view of the apparatus, (b) view from the end of the channel, (c) 3D sketch in top view of the apparatus showing the three distinct parts (reservoir, transparent channel, wood channel), (d) 3D sketch showing the three parts in side view and the reservoir opening system in the back (sand bag)

Full size image

The channel slope angle can be varied from 3 to 21° by vertically lifting the reservoir within a metal frame, changing the inclination of the two channel sections (Fig. 3). An experiment is triggered by the rapid lifting of a sliding gate that separates the reservoir from the first channel, activated by the vertical fall of a 30-kg sandbag at the rear of the apparatus. To allow the granular material to be initially fluidized, the base of the reservoir consists of a porous plate (2 cm thick, mean diameter of the pores of 20 µm) through which an air flow is injected and maintained throughout the experiment. This permits us to generate pore fluid pressure, which in nature can result from various mechanisms involving particles and gas moving downward and inward respectively. The total thickness of the reservoir base, called the windbox, is 7.6 cm (porous plate included).

Experimental procedure

We chose to focus our experiments on the influence of the initial pore pressure and the slope angle on the dynamics of the channelized flow and overspill generation. We selected five slope angles, 3, 6, 9, 12, and 15°, at which three to five experiments with initially fluidized granular material were run. In addition, three non-fluidized experiments at 9 and 12° were performed for comparison with those involving initial pore pressure. Experiments were recorded using four GoPro cameras and one high-speed camera to investigate the flow front position and velocity of both the parent channelized and overbank flows. The flow front position was measured by correlating each GoPro video frame at different locations across the channel length. Images recorded with the high-speed camera at 1000 frame/s (dashed red square in Fig. 4(A-C)) were analyzed by particle image velocimetry (PIV) using PIVlab on Matlab (Thielicke and Sonntag 2021). The flow thickness (Fig. 4(B-D)) and flow velocity profiles at the sidewall (Fig. 4(D)) were measured for each frame at 1.5 m from the reservoir (vertical black line, Fig. 4(A, B)). By integrating the velocity profile over the flow thickness for each frame (Fig. 4(C, D)), we deduced the temporal evolution of the volume flux at that particular location. Note that this discharge rate is an approximation as the flow velocity varies across the channel due to wall friction (Fig. 4(C)).

Fig. 4
figure 4

Configuration for data acquisition during the experiments. (A) Schematic cross section showing where the PIV measurements are performed (dashed red rectangle). (B) Example of an image processed by PIV to extract the horizontal displacements (green arrows). (C) 3D diagram of the flow configuration in the channel, showing the 3D flow velocity field and the section recorded for PIV. (D) Horizontal flow velocity measured at the channel wall as a function of position from the base, calculated from the PIV frame presented in (B)

Full size image

We used synthetic, subspherical borosilicate glass beads of 45–90 μm diameter (Table 1 in Supplementary material). We chose this material because it has several advantages compared to natural PC particles: (i) it is well sorted and contains a negligible amount of very fine (< 20 µm) particles and is therefore simple to fluidize at laboratory scale (i.e., negligible cohesion effect), it is easy to use and recycle, and it does not pose a health hazard, and (ii) their internal properties are well known (i.e., density, shape, and grain size distribution), which helps in characterizing physical processes involved during flow emplacement. The glass beads have a density of ~ 2500 kg m−3, while granular beds have a bulk density of ~ 1500 kg m−3 with a solid volume fraction of ~ 0.6. We chose a constant bulk volume of particles of 0.09 m3, corresponding to a mass of ~ 135 kg of beads with a bed height of 40 cm in the reservoir. The beads belong to group A of Geldart’s classification (Geldart 1973), meaning that the granular bed is fully fluidized (i.e., pore fluid pressure equal to the lithostatic pressure) at the minimum air fluidization velocity Umf with negligible expansion. In our experiments, Umf is 0.01 m s−1, corresponding to an air flow rate of about 0.5 m3 s−1 (measured at onset of bed expansion) supplied by a compressor equipped with a dryer unit. To avoid particle cohesion effects caused by the ambient air humidity, particles were dried in an oven at 80 °C for 40 to 90 h prior to each experiment, depending on the degree of humidity of the room, before being immediately loaded into the reservoir a few minutes before the experiment. For all experiments of this study, the initial temperature of the granular medium was 50–60 °C, while the ambient temperature and the relative humidity were 19–21 °C and 55–60%, respectively.

Results

Dynamics of channelized flows

We first analyze the channelized flow kinematics at varying channel slope angles and with or without initial pore pressure. The position of the front of the initially fluidized flows (measured along the channel central axis) and the corresponding flow front velocities are shown in Fig. 5(a), b. For comparison, the front positions and velocities of non-fluidized flows with 9 and 12° channel slope angles are also presented. To allow a comparison with future experiments and numerical model benchmarking, the front position is normalized by the initial height of the granular bed in the reservoir h0 = 0.4 m. Results show that at channel slope angles of 9–12° (i.e., only comparable angles between the two types of flows), the normalized flow runout increases from x/h0 = 2.5–3 for non-fluidized flows to x/h0 = 4–5 for initially fluidized flows.

Fig. 5
figure 5

Kinematic data of non-fluidized and initially fluidized flows. Temporal evolution of (a) the flow front position x normalized by the initial bed height h0, (b) the front velocity vf, and (c) the front Froude number Frfront.(d) Maximum flow runout as function of the slope angle. Temporal evolution at 1.5 m from the reservoir (calculated using PIV) of (e) the normalized flow thickness, (f) the mean velocity at sidewall v_s, (g) the internal Froude number Frint, and (h) the flow discharge rate

Full size image

The front velocity vf of initially fluidized flows reveals distinct phases (Fig. 5(b)): (i) a first phase of acceleration during ~ 0.15 s after release, then a short period of nearly constant velocity during ~ 0.1 s, and a second stage of acceleration until ~ 0.5 s on average (notice that all experimental flows show about the same temporal evolution); (ii) a transitional phase during which the flows have different kinematics depending on the channel slope angle: another stage of acceleration at 12 and 15°, a constant velocity at 9° and 6°, or a deceleration at 3°; (iii) an impact phase during which the flows interact with the oblique side of the channel bend, causing a short but rapid drop in the flow velocity; (iv) a constant deceleration phase until the flows stop, except for experiments at 12 and 15° channel slope angles for which another acceleration of the flow front is observed after the bend at 1.5–2 s, followed by a final constant deceleration. Note that the second stage of acceleration during phase (i) for all experiments, and the terminal velocity peak observed after 2.5 s for experiments at 15° slope, are caused by surface waves, which are clearly visible on videos, traveling faster and eventually overtaking the flow front, as observed in debris flow experiments (Zanuttigh and Lamberti 2007). Surface waves push the flow front further downstream, increasing the maximum flow runout that draws a linear relationship with the channel slope angle (Fig. 5(d)).

To better characterize the regime of our experimental flows, we calculated the Froude number of the flow front \({Fr}_{\mathrm{front}}= \frac{{v}_{\mathrm{f}}}{\sqrt{H g \mathrm{cos}\alpha }}\), with α, the slope angle; g, the gravity; vf, the flow front velocity; and H, the vertical flow thickness (Fig. 5(c)). A mean flow thickness of 0.05 m is set as the characteristic thickness H, obtained by averaging the thicknesses at the flow heads in Fig. 5(e), as described hereafter. The flow front velocity and the front Froude number show similar temporal evolutions, in particular with Fr >  > 1 during most of the flow emplacement. These values of Froude number indicate that the flows are supercritical and mostly driven by their inertia, except at very late depositional stages, and therefore prone to surface wave formation as confirmed by our observations.

To further characterize the flow dynamics, we present in Fig. 5(e-h) PIV measurements, at 1.5 m from the reservoir, of the temporal flow thickness, the mean velocity at sidewall vs, and the flow discharge rate. In addition, the internal Froude number \({Fr}_{\mathrm{int}}= \frac{{v}_{\mathrm{s}}}{\sqrt{H g \mathrm{cos}\alpha }}\) calculated from \({v}_{\mathrm{s}}\), the mean velocity at sidewall (Fig. 5(f)), is also presented. Overall, the higher the channel slope angle, the higher the flow velocity and flow discharge rate. The flows exhibit important variations in their thickness and velocity, enabling us to differentiate a head, a body, and a tail. For each channel slope angle, these flow properties show similar patterns as shown in Fig. 5(e, f): (i) when the flow front reaches the probing area, the thickness rapidly increases and reaches a first plateau (increasing with the slope angle), while the mean velocity \({v}_{\mathrm{s}}\) also increases rapidly and reaches a maximum value that increases as the channel slope angle increases (up to 1.2 m s−1 on average at 15°); (ii) following this plateau, the thickness increases again, which marks the transition to the flow body with maximum flow thicknesses; (iii) after the passage of the body, the flow thickness remains constant for the low to medium slope angles (< 9°), but shows a decrease of up to 30% for the steepest slopes (12 and 15°), while the flow velocity decreases in each case. We identify this last phase as the passage of the tail of the flow. By integrating the flow velocity along the flow height with the channel width, the evolution of the flow discharge rate (DR) is obtained (Fig. 5(h)). Two well-developed peaks in discharge rate reflect the passage of the flow head and the flow body. The higher the channel slope angle, the higher the DR, causing stronger second peaks of discharge rates at high slope angles. We will discuss below the correlation between this second peak in the DR and the generation of overbank flows.

Generation of flow overspill

We have observed flow overspills at the passage of the channel bend only for initially fluidized flows and at varying slope angles from 9 to 15°. Two different types of flow overspill can be distinguished: (i) a front splash after the passage of the flow front, forming a thin immobile deposit < 3 mm, and (ii) a second overspill, which we call overbank flow, occurring after the front splash in experiments with 12 and 15° channel slope angles.

The front splash is caused by the complex behavior of the flow front interacting with the bend section: it runs up vertically along the bend sidewall, in a jet-like behavior as previously described for wet granular flows impacting a vertical wall (Armanini and Scotton 1993; Armanini et al. 2011), and is simultaneously deflected laterally. Once the jet reaches its maximum height, its falls back on the channel bank and transitions into a reflected wave inside the channel, also observed in wet granular flow experiments (Armanini and Scotton 1993; Armanini et al. 2011; Iverson et al. 2016). The wave rolls laterally and falls on the flow’s free surface, parallel to the bend side wall (Fig. 6(a)). When the front has passed the obstacle at slope angles 12–15°, the overbank flow starts forming at the first bend and propagates on the bank in the same direction as the main channel axis (see arrows in Fig. 6(b-d)). It reaches its maximum runout approximately at the middle of the bend, and its outer limit is subparallel to the bend sidewall. The thickness of the overbank flow deposit varies from ~ 0.2 cm at the front to ~ 3 cm along the edge of the channel at 15°.

Fig. 6
figure 6

Sequence of images from the top of the bend section of the apparatus for an experiment with a 12° slope angle, showing two types of overspills: (a) the front splash and (b–e) the overbank flow that partially covers the front splash. Time t = 0 s is defined when the overbank flow starts to spill over in (a)

Full size image

The formation of these flow overspills is related to varying channelized flow conditions and occurs at very specific periods during the emplacement of the parent flows. By carefully observing the videos of each experiment, we can define a flow overspill sequence (Fig. 7):

  1. A)

    The very thin and fast flow head reaches the bend.

  2. B)

    The flow front impacts the oblique channel wall and propagates both vertically and laterally, causing the formation of a roll back wave. This also causes an important variation of the flow thickness across the channel, with a flow height along the impacted side higher than along the opposite side of the bend. In the following, we name this local and rapid thickness increase a superelevation.

  3. C)

    For experiments at 3 and 6° slope, the jet height is lower than the channel bank height and no overspill occurs. However, at 9° or higher, the jet height exceeds the channel wall height and the granular material overflows on the channel banks, toward the distal bend section (see Fig. 6), to create a front splash that immediately freezes.

  4. D)

    For experiments at 12° slope and higher, the front roll back wave inside the channel slowly transitions to a granular jump (Gray et al. 2003; Gray and Cui 2007) at the beginning of the bend. Simultaneously, the flow thickness along the impacted side of the bend rapidly increases and finally exceeds the channel height. This causes the channelized flow to overspill on the channel bank to form a secondary flow. The latter initiates first at the proximal point of the bend, and propagates downstream over the channel bank as an overbank flow, while its flow direction is that of the straight channel axis (see also Fig. 6.). Note that the granular material does not reach the opposite side of the bend from the onset of the overbank flow until almost the end of the flow propagation.

Fig. 7
figure 7

Diagrams and photographs of an experiment at 12° slope angle illustrating the morphological evolution of the flow front impacting the bend. (A) Fast and thin flow head approaching the bend; (B) flow front impacting the bend and ramping up the channel margin vertically; (C) the front rolls back partially toward the flow inside the channel, while another part overspills on the channel bank; (D) the flow body reaches the bend and causes the flow to overspill and generate an overbank flow

Full size image

Overbank flow dynamics

Overbank (OB) flows occurred at 12 and 15° channel slope angles (Fig. 8(a, b)). As for their parent channelized flows, OB flow runout distance increased significantly when the slope increased, from x/h0 = 0.5–1 at 12° to x/h0 = 1.8–2.1 at 15° (Fig. 6(a)). However, the temporal evolution of the velocity of OB flows differs significantly from that of the parent channelized flows since OBs do not interact with any obstacle but spread on an unconfined inclined surface. In fact, these OB flows share some similarities with the non-fluidized channelized flows (Fig. 5(a, b)): an initial acceleration phase followed by a short constant velocity phase and then a constant deceleration. However, the Froude number at the front \({Fr}_{\mathrm{f}\_\mathrm{OB}}= \frac{{v}_{\mathrm{fOB}}}{\sqrt{H g \mathrm{cos}\alpha }}\) =  ~ 2–5, with a typical thickness H = 0.01 m (approximately the average thickness of OB flows), is closer to that of the parent channelized flows (Fig. 8(c)). This shows that OB flows are also supercritical and therefore prone to surface wave instabilities (Fig. 6(c, d)).

Fig. 8
figure 8

Kinematic data of the front of the overbank flows. (a–c) Temporal evolution of the normalized flow front position, the front velocity, and the front Froude number. (d) Maximum runout of front splashes and overbank flows as function of the slope angle. (e) Runout of the overbank flows as function of runout of their parent channelized flows

Full size image

There appears to be a correlation between the slope angle and the runout of the OB flows (Fig. 8(d)). In contrast, the maximum extent of front splashes shows significant variability with the slope angle. OB flow properties are dependent on those of their parent channelized flows. There is a good correlation (R2 = 0.89) between the channelized flow runout Rcf and the OB flow runout Rob (Fig. 8(e)):

$${R}_{\mathrm{ob}}=1.5{R}_{\mathrm{cf}}-7.4$$
(7)

This shows that for Rob = 0, the minimum channelized flow runout required to generate an overspill is Rcf = C/1.5 = 4.9 m. Equation (7) shows that the occurrence and runout of an OB flow can be expected from the parent channelized flow properties.

Discussion

Influence of the slope angle on channelized flow dynamics

We discuss the dynamics of our experimental flows in light of earlier studies on confined granular flows of spherical particles and with smooth boundaries. Theoretical (Brodu et al. 2015; Zhu et al. 2020) and experimental (Louge and Keast 2001; Holyake and McElwaine 2012; Heyman et al. 2017; Roche et al. 2021) works have revealed a richness of flow dynamics, which depends essentially on the channel slope angle and the balance between driving gravitational forces and resisting forces at the smooth rigid boundaries (i.e., the channel base and lateral walls). After release, the granular material accelerates, decelerates, and stops if the slope angle is less than ~ 15°. At steeper slope angles, the flow accelerates and propagates in a so-called immature regime until it acquires a steady fully developed (SFD) regime once resisting forces balance driving forces. The transition from the immature to the SFD flow regime occurs at longer distance as the slope angle increases. SFD flows have a dense core caused by inelastic collapse of the expanded granular material, but lower particle concentrations at their boundaries, which is a signature of a supported flow regime, and they often exhibit longitudinal rolls (Brodu et al 2015). As shown by both theoretical (Zhu et al. 2020) and experimental (Roche et al. 2021) investigations, the effective friction coefficient (µ) of these flows with smooth boundaries increases with the slope angle and it depends essentially on a dimensionless number analogous to a Froude number (Fr). Interestingly, the µ(Fr) curve for these plug-like flows with high shear velocities along the smooth boundaries shares strong resemblance with the µ(I) rheological curve for granular flows over rough substrates and with negligible slip velocities (cf. Breard et al. 2022). We rely on these findings to discuss our experimental results. We acknowledge, however, that care must be taken regarding the interpretation of the dynamics of confined, initially fluidized flows with smooth boundaries whose physics have not yet been investigated in detail, in contrast to non-fluidized flows. Our experiments were carried out at gentle slope angles ≤ 15°. Therefore, the non-fluidized flows remained in the immature regime and stopped in the straight upper channel upstream of the bend. In contrast, initially fluidized flows, with their reduced internal stresses (cf. Breard et al. 2022), could reach the bend, either in the immature (12 and 15°) or the SFD (3, 6, and 9°) regimes. Then, their dynamics changed drastically through interaction with the bend and they eventually stopped downstream. In a straight channel, these flows might have reached the SFD regime (12 and 15°) or remained steady (3, 6, and 9°) for a while before they would have behaved as non-fluidized flows once defluidized.

Results obtained with initially fluidized flows show a proportional relationship between channel slope angle and flow runout (Fig. 5(d)). This proportionality has already been demonstrated in previous studies for confined non-fluidized (e.g., Pouliquen 1999; Gray et al. 2003; Pudasaini et al. 2007) or fluidized flows with similar channel slope angles (< 20°; e.g., Chédeville and Roche 2015; Aravena et al. 2021). The runout of initially fluidized flows increases by a factor of ~ 2 from 3 to 15° slope angles, in agreement with the simulations of Aravena et al. (2021) and with the experiments of Chédeville and Roche (2015). In the latter publication, the authors show that the runout increases exponentially beyond channel slope angles of 15–20°, which have not been explored in our study. However, simulations of Aravena et al. (2021) show that the increase in runout distance may be self-limited at an increasing channel slope angle because the high flow velocity due to fluidization causes lateral spreading of the flow and rapidly reduces its thickness, which induces faster pore pressure diffusion and transition to the non-fluidized flow regime.

Flow interaction with the channel bend

The front splash overspill occurs at slope angles ≥ 9° when the flow front impacts the bend and rises vertically before falling back both on top of the channelized flow and on the channel bank. The presence of a reflected wave is in agreement with the experiments of Armanini and Scotton (1993) and Armanini et al. (2011)who have demonstrated that flows with Fr > 1 tend to form thin and fast fronts that run up the obstacle in a jet-like behavior, which transition to a reflected wave eventually. These results suggest that our initially fluidized flows had a fluid-like behavior at least until the impact with the channel bend.

As shown by our simple analysis with the thin flow theory in “Design of the apparatus,” the minimum front velocity of an inviscid flow needed to generate a vertical rise of 15 cm along an oblique 45° channel bend decreases from 2.42 m s−1 at a 3° slope angle to 2.39 m s−1 at 15°. Since fluidization drastically reduces effective friction, and that the experimental flow front thickness (~ 1 cm) is much smaller than the channel depth, we can postulate that our initially fluidized experimental flows behave as inviscid thin flows, and that pore pressure has not decreased significantly at the time of impact with the bend. Measured flow front velocities (Fig. 9(a)) are ~ 0.5–1 m s−1 higher than the theoretical minimum velocity \({v}_{\mathrm{p}}\) obtained from Eq. 6 (blue dashed line, Fig. 9(a)), suggesting that experimental flows have enough energy before the impact to overtop the channel wall, even at a 3° slope angle. The theoretical height H the experimental flows could reach calculated from Eq. 6, with a channel angle \(\vartheta\),

Fig. 9
figure 9

(a) Theoretical minimum flow front velocity needed to overtop the sidewall (dashed blue line) and measured velocities in experiments (square = measured, red asterisk = average value at given slope angle) as a function of the slope angle. (b) Theoretical height that a flow can reach according to the measured front velocity (squares = calculated value, red asterisk = average value at given slope angle)

Full size image
$$H=\frac{\left(\sin\left(\vartheta\right)v\right)^2}{2g\;\cos\;\alpha}$$
(8)

is ~ 10–16 cm higher than the 15-cm height of the channel sidewall (black dashed line, Fig. 9(b)). However, as seen in “Results,” flow overspills are only observed at channel slope angles ≥ 9°. Therefore, our data suggest that at impact with the oblique channel wall at the lowest slope angles of 3 and 6°, (i) pore fluid pressure has decreased significantly (because of smaller flow velocities and thicknesses compared to higher slope angles) and/or (ii) energy dissipation through collisions between the particles and with the sidewall was non-negligible. In fact, the flow energy was high enough to cause overspill only in experiments at slope angles ≥ 9°, and we can define the minimum velocity to generate an overspill with our configuration between 2.9 and 3.1 m s−1 (Fig. 9(a)).

An overbank flow follows the front splash in experiments with 12° and 15° channel slopes, because the thickness of the flow body increases to the point that the flow overspills on the channel bank. This sudden thickness increase is due to several factors. The thickness variation of a granular flow impacting an obstacle has been studied in earlier experimental works. A granular jump was observed for granular flows at Fr > 1 impacting an obstacle perpendicular (Gray et al. 2003; Pudasaini et al. 2007) or oblique (Cui et al. 2007; Gray and Cui 2007) to the direction of flow propagation. Gray et al. (2003) and Pudasaini et al. (2007) showed that the granular jump increases rapidly in thickness after impact and propagates upslope, similarly to pure fluid flows (Savage and Hutter 1989). Gray et al. (2003), and later Tiberghien et al. (2007), Faug et al. (2012), and Caccamo et al. (2011, 2012), showed that a granular jump can evolve into a basal stagnation zone (i.e., where the granular material is static) overlain by the flowing material, which overtops the obstacle. Our experimental flows with Froude numbers of 3–5 were prone to granular jump. However, we could not investigate this phenomenon because the channel side walls at the bend were not transparent. Furthermore, the frontal roll back wave fell back onto the flow (see Fig. 7) and the flow discharge rate was not constant. These two processes did not permit us to investigate in detail the variation of flow thicknesses at the impact with the bend and, therefore, prevented us from documenting the dynamics of the granular jumps.

Implications for natural PCs and perspectives

Our experiments involving a simple configuration suggest that a channel bend can be very efficient to cause overspill of CPCs. We have demonstrated that an initially fluidized granular flow impacting an oblique channel wall can generate an overspill, even when propagating on gentle slopes ≤ 15°. Overspill can occur even if the channelized flow does not fill the entire channel width (Gueugneau et al. 2021; Kubo Hutchison and Dufek 2021). Future studies on CPCs overspill and evaluation of channel capacity should focus on determining the conditions that promote the formation of a superelevation wedge in the vicinity of a sudden topographic change. Using up-to-date, high-resolution DEMs, an automatic recognition of hazardous overspill zones along valleys of PC-generating volcanoes from numerical simulations could lead to a new generation of enhanced and dynamic hazard and risk maps. The fact that a granular jump is one of the factors contributing to the formation of overbank flows confirms the ability of shallow-water numerical models to simulate CPC features, since such models are able to reproduce hydraulic jumps (as seen for example in Gueugneau et al. 2021). The possible presence of a basal stagnation zone, however, would require the use of other modeling approaches, such as two-phase continuum (Aravena et al. 2021), multiphase discrete element (Lube et al., 2019; Breard et al., 2022), or smooth-particle hydrodynamics (Zhu et al., 2021) simulations.

Further experimental studies should address some key parameters in more detail. First, the channel geometry: as shown by earlier studies, the angle and the sinuosity of the bend (Peruzzetto et al. 2021; Kubo Hutchinson and Dufek 2021) as well as the shape of the topographic feature (obstacle, break in slope, constriction; Gueugneau et al. 2021) control the generation and type of overspills. Comparison with the results of other existing experimental facilities such as that of Sulpizio et al. (2016), for which flows are only confined along half of their path, could better constrain the effect of flow confinement on the flow runout. It would be also important to run experiments with polydisperse granular materials, in order to investigate the dynamics of two-layer flows, and in particular the detachment of the upper dilute part at impact with the different topographic features (Fig. 1).

Finally, our study provides an experimental dataset for current and future benchmarking initiatives of mass flow models. The PyroCLAST apparatus can be used in future studies to (i) improve our capability of assessing PC inundation zones over complex topography by investigating the interplay between the PC dynamics and their responses to topographic changes, and (ii) quantify how changes in topography can modify the local flow rheology. A cross comparison with other large-scale facilities specifically designed to study the effect of channelized flows at break-in-slope (Sulpizio et al. 2016) is also another potential application with PyroCLAST.

Conclusions

We observed two types of flow overspills from experimental, initially fluidized channelized granular flows over low to medium channel slope angles (9 to 15°) and interacting with a 45° oblique bend: (i) a front splash at the impact of the flow front with the bend and (ii) an overbank flow at the passage of the flow body, provided the slope angle was high enough (12 and 15°). Results of our experiments have demonstrated that the slope angle enhances the channelized flow runout and favors the formation of overspills at the bend, due to faster flow propagation and related mass flow rate. The flow velocity is the key parameter, and in the configuration chosen, a minimum flow front speed of ~ 2.9–3.1 ms−1 is required to generate overspills, which corresponds to a minimum flow runout of ~ 4.9 m in the channel. The formation and dynamics of overbank flows are linked to the dynamics of their parent confined flows, and a correlation exists between their respective flow runouts. Overbank flow is caused by both the formation of a granular jump and a peak in discharge rate due to flow unsteadiness, which contribute to an increase of the granular material thickness along the impacted side of the bend and cause an overspill. The formation of a superelevation wedge should be carefully studied in natural valleys to help define zones of potential hazardous CPC overspills. Finally, the results of this study constitute a dataset for current and future benchmarking of numerical models of CPCs.