Spatial and temporal evolution of an experimental debris flow, exhibiting coupled fluid and particulate phases

The internal behaviour of debris flows provides fundamental insight into the mechanics responsible for their motion. We provide velocity data within a small-scale experimental debris flow, consisting of the instantaneous release of a water-granular mixture along a rectangular flume, inclined at 31 degrees. The results show a transition from a collisional, turbulent front to a viscous-type, steady flow body, exhibiting strong fluid-particulate coupling. This is the first time that both the spatial and temporal evolution of the internal mechanics of a small-scale debris flow have been considered. Our results serve as invaluable data for testing two-phase fluid-particulate numerical models.


Introduction
A debris flow is a gravity-driven flow where the interaction of both solid and fluid phases governs the dynamics (Iverson 1997). Debris flows exhibit extremely complex and destructive behaviour (Costa 1984), and a significant amount of research has been dedicated to understanding the governing physical processes (e.g. Takahashi (1981), Iverson (1997), Kaitna et al. (2007), Iverson and George (2014), Pastor et al. (2018)).By nature, the occurrence of debris flows is unpredictable and their behaviour is dependent on a number of physical conditions (such as terrain and material composition). Repeatable, physical models are able to capture the key features of real debris flows, allowing the investigation of flow dynamics in a controlled environment. Furthermore, experimental debris flows are invaluable for the validation of mathematical and numerical models.
Many different experimental set-ups have been employed to investigate one or more aspects of debris flow behaviour, within a wide range of physical scales (Takahashi 1978, Gregoretti 2000, Armanini et al. 2005, Larcher et al. 2007, Iverson et al. 2010, Johnson et al. 2012, de Haas et al. 2015. Large scale ex-periments have the benefit of being directly comparable to real debris flows (Iverson 2015), yet they are expensive, time consuming and complicated to execute. Alternatively, small scale experiments have the advantage of being simple and repeatable, and are capable of reproducing real debris flow features (such as the formation of a distinct granular 'head' and fluid-like 'body') (Paleo Cageao 2014, Turnbull et al. 2015, Lanzoni et al. 2017. Perhaps of most significance, experimental debris flows at a small scale allow for the observation of internal flow dynamics -enabling the calculation of internal velocity profiles, thus revealing the mechanical and rheological behaviour of the fluid-granular mixture. Various small scale experiments have been dedicated to the analysis of the internal velocity profiles within debris flows -typically performed in transparent flumes, with cameras recording the flow (Armanini et al. 2005, Kaitna et al. 2014, Lanzoni et al. 2017, Sanvitale and Bowman 2017. Image processing techniques are applied to the flow images to calculate material displacement and obtain internal velocity profiles. Armanini et al. (2005) examined the velocity profiles within a series of experiments consisting of a recirculating mixture of polyvinyl chloride (PVC) pellets and water in an inclined flume, over an erodible and non-erodible bed. The distinct shapes of the profiles revealed four different granular flow regimes -immature, mature, plug flow and solid bed flow. These definitions have been frequently used to classify the results of experimental debris flows conducted in more recent years (Kaitna et al. 2014, Lanzoni et al. 2017, Sanvitale and Bowman 2017. Vertical velocity profiles can also be compared directly with analytical profiles -based on simplified mathematical models with an assumed rheology (Bagnold 1954) -which may exhibit granular or viscous-type behaviour. In the experiments conducted by Kaitna et al. (2014) and Sanvitale and Bowman (2017), the type of velocity profile within debris flow bodies was found to be dependent on the mixture composition. Alternatively, one single rheological model may be insufficient to describe the velocity profile throughout the whole flow depth. Lanzoni et al. (2017)  Although there are several investigations of the internal behaviour and velocity profiles within experimental debris flows (Kaitna et al. 2014, Paleo Cageao 2014, Kaitna et al. 2016, Lanzoni et al. 2017, Sanvitale and Bowman 2017, we lack information on the internal evolution of such flows. Previous work has only considered the internal behaviour of steady flows, mainly concerning the velocity profiles within the debris flow bodies. This is due to the fact that transient velocity data are subject to error in terms of repeatability, and the vertical profiles cannot be compared with simple mathematical models (which are derived under the assumption of a steady state). However, the temporal evolution of coupled fluid-particulate flows could provide valuable insight into the formation and mechanics of the debris flow head-body architecture. Therefore, we analyse the internal behaviour of a small-scale rapid debris flow, for the entire flow duration. We aim to provide a description of the spatial and temporal internal flow evolution, for fixed experimental conditions. The experiments consist of the dam break release of a water-granular mixture along an inclined flume. We are interested in the internal evolution of a rapid flow that travels along the entire length of the flume (with minimal material deposition), corresponding to the solid bed flow of the regimes defined by Armanini et al. (2005). This represents an extreme case of debris flow propagation in terms of material velocity, at a small scale. We use Particle Image Velocimetry (PIV) to obtain the internal velocity profiles, providing high quality, robust and repeatable data. Data of this quality are rare within the literature, and are ideal for the validation and development of numerical models of debris flows. We utilise the PIV data to infer information on the evolving flow behaviour.
The remainder of this paper is structured as follows. The experimental methodology is detailed in Section 2, including a description of the PIV method.
The velocity data are presented in Section 3, in the form of flow fields and vertical profiles. We consider the deviations of the velocity, in order to identify collisional and non-fluctuating regimes. The implications of the findings are discussed in detail in Section 4, and the results are compared to those of experimental debris flows within the literature. The key findings of this investigation are summarised in Section 5.

Experimental methodology
A mixture of water and sediment was manually released from behind a lock gate in a rectangular flume of dimension 1.9 × 0.2 × 0.1 m, at an inclination of 31 • (see Figure 1). This angle of inclination corresponds to that of large scale flume experiments at the United States Geological Survey debris flow flume (Iverson et al. 2010), and enables a rapid flow propagation. The mixture consisted of 2.177 kg of sediment and 1.5 l of water, resulting in a total volume of 0.0026775 m 3 , with an initial solid volume fraction of φ s = 0.44. The sediment was composed of multicoloured, crushed glass grit with an angular shape, to represent natural granular material. The particle size distribution is shown in Figure 2.
The mean particle size is d 50 = 0.917 mm, where d x denotes the percentage passing by area. The coefficient of uniformity C U = d 60 /d 10 represents the particle size variety, where d 60 = 1 mm, d 10 = 0.1928 mm and C U = 5 (to the nearest integer). The finer particles are expected to contribute to the viscous effects that are frequently observed in granular flows (Iverson 1997). Sediment of the same grade was permanently fixed onto the flume bed to generate roughness which would produce a no-slip flow. Due to the high friction created by the bed roughness, we found that mixtures with a volume fraction less than φ s = 0.44 did not propagate along the length of the flume (and were therefore not representative of solid bed flows).
A shear box test was conducted to determine the mechanical properties of the granular material. The data obtained for the saturated glass grit are provided in Figure 3, for normal stress values of 30 kPa, 60 kPa, 100 kPa and 130 kPa. The samples were inspected after completion of the tests and no particle crushing was observed. A linear fit is applied to the relationship between the normal stress and the peak shear stress, as shown in Figure 3b. The gradient and the y−intercept of this fit correspond to the internal friction angle and the cohesion of the material respectively. The glass grit was found to be non-cohesive, with an internal friction angle of 39 • . The shear modulus of the material can be approximated as the gradient of the strain-stress curve before the peak values.   This was found to be approximately 2.66 × 10 5 Pa.
At the beginning of each experimental run, 2.177 kg of glass grit was placed behind a lock gate with a cross-sectional area in the shape of a trapezoid, occupying a volume of 0.0017255 m 3 . Subsequently, 1.5 l of water was added slowly to minimise the disturbance to the top of the sediment. Due to its porosity, the sediment was rapidly saturated fully. The initial placement of the sediment and water is depicted in Figure 4, where the bottom layer consists of a mixture of water and glass grit, while the top layer is composed of water only.
To check for repeatability, the debris flow experiments were performed three times. The surface of the sediment phase was marked onto the flume, to ensure that it was placed in the same initial position for each experimental run. To  Figure 5.
Water was poured along the flume bed before and after each experimental run to ensure the removal of any loose sediment that had stuck to the bed.

Particle Image Velocimetry
A PIV processing method was applied to the images obtained with the high speed camera. This is an experimental technique used within fluid and soil dynamics, where instantaneous velocity fields are determined by tracking the displacements of individual particles, or groups of particles, within a flow (Adrian 1991, White et al. 2003, Adrian and Westerweel 2011, Pinyol and Alvarado 2017. The method (in two dimensions) involves splitting each image frame into a number of interrogation areas, within which the movement of particles is tracked between subsequent frames. The displacement is obtained by estimating the cross-correlation between the particle positions within each interrogation area, where the true displacement of each particle group must be separated from the noise created by particles overlapping between frames. This is achieved by applying statistical correlation methods to the data, to determine the most likely 'true' particle displacement. An algorithm is then applied to obtain an estimate of the velocity vector field from the displacement values, where certain features of the camera used to obtain the images are taken into consideration. An extensive description of the PIV method can be found in Adrian and Westerweel (2011).
We processed the frames from the camera with the DynamicStudio image processing software to obtain the velocity vectors. The 'Adaptive PIV' option was utilised within DynamicStudio, which automatically adjusts the interrogation area at each frame according to the local particle densities and velocity gradients. This requires the definition of the minimum and maximum values of interrogation areas, which were defined as 32×32 and 64×64 pixels respectively.
The reader is referred to the DynamicStudio user manual for further details on the adaptive PIV method (Dantec Dynamics 2018). The PIV method often requires the addition of seeding particles to track the fluid movement within each interrogation area of the flow images. This is not typically necessary when considering granular flows, as individual particles are easily detected (White et al. 2003). The granular material in the current application is multicoloured -further facilitating the detection of individual grains -and no seeding was required. The PIV analysis was applied to the images of the flow along the side wall, under the assumption of a two-dimensional flow. This is subject to error as the propagating material is unlikely to be uniform across the width of the channel. Furthermore, the flow dynamics are likely to differ somewhat at the flow margin than in the centre, due to the influence of the wall. An alternative option is to use a laser sheet to illuminate a plane in the flow centre, and capture the images in this region for PIV analysis. This method requires the combination of clear particles and a fluid that is refractive index matched, and has been applied recently by Sanvitale and Bowman (2017)  flow, the number of rows of each matrix must be cropped at each frame, to align with the free surface of the flow. This was conducted manually by inspecting the free surface position at each snapshot from the high speed camera. The flow free surface is therefore approximated as a horizontal line.

Overall flow behaviour
Once released from the lock gate, the water-granular mixture rapidly propagated downstream, reaching maximum front velocities in the range of 1 − 1.2 m s −1 .
The main bulk of the flow deposited onto the run-out area, although a thin layer of the granular material was deposited along the bed of the flume. The granular material was fully saturated throughout the flow for all repeats of the same experimental run (Run 1, Run 2 and Run 3). A snapshot from the high speed camera at 0.035 s after the material reached the field of view is shown in

Flow characterisation
The flow behaviour can be characterised by considering the standard deviation e of the velocity from the local average, within that interval: whereū x is the average velocity over N frames (calculated at a single point), and u x is the instantaneous velocity. Low values of standard deviation equate to small variations in instantaneous velocity from the local mean, indicating nonfluctuating behaviour. Conversely, a high standard deviation demonstrates that the averaged velocity profile is not representative of the overall behaviour within the interval, as the flow is rapidly changing. This corresponds to collisional behaviour, which is dominated by fluid turbulence and particle collisions (Bagnold 1954, Johnson andJackson 1987). In the small scale debris flow experiments so that the lower limit is equal toē = 0.15 m s −1 . Accordingly, the dark blue areas in Figure 9 are assumed to correspond to non-fluctuating regions of the flow, while the lighter colours are assumed to represent the collisional regions.
As also shown in Figure  also observed at t = 1.6, with a lower overall velocity magnitude. By t = 2.8 s the velocity has decreased significantly, and the mixture is almost stationary.

Velocity profiles
The corresponding contour plots for the PIV velocity data are shown in Figure   11. After the initial, fully collisional region of the flow, there is a concentrated area of high velocity at the base at t = 0.3 s. Above this, the upper, collisional layer exhibits some negative velocities. As discussed above, these negative ve- Profiles of horizontal velocity u x against height h are provided in Figure 12 for twelve times ranging between t = 0.035 s and t = 2.8 s, comparing the results from the three different experimental runs. To analyse the error of the timeaveraging process, we fitted an autoregressive (AR) model to the instantaneous data over the 30 frames that were averaged. AR models are used to represent a value in a time series as a weighted sum of previous values in the series, and can be used for forecasting purposes (Brockwell et al. 2002, Box et al. 2015. A time series X n is defined as a linear combination of past observations X n−1 and white noise error terms n : where α i are the model parameters, and p is the number of past observations that are used in the expression of X n . For the current purpose, the parameter p was assumed to be one, so that each term in the AR model is based on the

Shear behaviour
The PIV data can be utilised to obtain profiles of the internal shear strain rate.
Neglecting the horizontal gradients of the vertical velocity u y , the local shear strain rateγ is defined asγ = ∂u x ∂y . (3) The shear rate is approximated at each vertical height h i : where u  Figure 8).
At the final considered time of t = 2.8 s, the shear rate profile decreases linearly until just below the free surface, before increasing towards the surface.
Following Sanvitale and Bowman (2017), the normalised shear rateγ is obtained by dividing by the depth-averaged shear rateγ: where u H and u slip are the values of horizontal velocity at the free surface and the bed respectively. Note that if both u H and u slip are zero, Equation (5)  and Bowman 2017). In addition to velocity profiles, we have also produced velocity flow fields, which clearly present the initiation and evolution of a granular shear layer (see Figure 11).  Table 1, along with the parameters in the current debris flow experiments. Disregarding the spurious values at the free surface (the upper four data points -corresponding to 1.6 mm), the profiles in the present investigation are similar to those of the solid bed flow described by Armanini et al. (2005), from t = 1 s onwards. Solid bed flows occurred for the highest bed inclinations, and were characterised by the shearing flow of the granular phase over a fixed bed.

Velocity profiles within the flow body
Profiles of horizontal velocity exhibited a convex shape, that increased with distance from the bed. The corresponding shear rate profiles decreased with distance from the bed, to a near zero value. The solid bed velocity profiles of Armanini et al. (2005) are compared against those of the current work in Figure   15 -the profiles clearly exhibit the same behaviour. Similar convex velocity profiles have also been recorded in the body of a steady debris flow consisting of gravel and water, in a rotating drum (Kaitna et al. 2014  ter. In the current work, the normalised velocity profiles in the flow body also collapse onto a single curve for each experimental run, from t = 1 s onwards (see Figure 15b), showing the flow similarity over this time. Table 1: A summary of the relevant experimental parameters in the current work, and selected investigations in the literature.
The notation x f refers to the flume run-out length (from the lock gate position), c w is the channel width, d d is the diameter of the rotating drum, θ is the flume inclination, φ s is solid volume fraction, d 50 is the mean particle size and C U is the coefficient of uniformity. The abbreviation n.p. denotes information that was not provided in the literature. We assess the rheological behaviour in the experimental flow body by approximating the dimensionless velocity profile with a granular and viscous scaling. viscous-type velocity profile. Conversely, for C U = 3 a granular profile provided the best fit to the experimental data. A wider grain size distribution promotes particle segregation, which can lead to the finer particles being trapped within the solid matrix. The presence of these fine grains enhances the viscosity of the interstitial fluid, and viscous forces influence flow behaviour (Iverson 1997).
For a more uniform particle distribution, the dominating forces are generally inter-particle granular collisions. In the current experiments, the viscous profile provides the closest fit to the experimental results, despite a relatively small coefficient of uniformity of C U = 5. This is possibly due to the proportion of very small particles with diameters less than 0.5 mm that are present within the mixture (see Figure 2), which add to the fluid viscosity. The results also suggest that the cut-off between granular and viscous-type flow may lie between C U = 3 and C U = 5. A suggested area for future work is to perform further experiments with different values of C U , to test this hypothesis.

Fluid-particulate coupling
The current experiments were performed with a significantly higher content of fluid than for the majority of similar, debris flow experiments (see Table 1). Despite this, the behaviour observed within the flow body is comparable to results presented in the literature, as discussed above. In terms of volume fraction, As opposed to monosized spheres, the current experiment consists of crushed glass of varying diameter. The angular shape allows the interlocking of grains and adds extra frictional resistance that is not present for spherical grains. For angular, crushed material, inter-particle shearing is significant, in addition to the shearing between the material and the bed. Therefore a flow consisting of realistic granular material exhibits lower velocities than that of glass spheres.
Furthermore, the dilation and contraction of the crushed glass particles regulates the motion of the water, enhancing the coupling between the two phases.
Experiments involving spherical particles are beneficial in terms of simplicity, and allow the investigation of a wide range of factors affecting flow behaviour.
However, the flow dynamics can be significantly different from that of a realistic granular material, as shown by the qualitative difference between the current experimental results and those of Paleo Cageao (2014). The relatively high water content in these experiments has particular relevance to subaqueous debris flows, which have wet heads. Based on the present experiments, subaqueous debris flows may exhibit markedly different flow behaviour at the head than current models (derived from subearial debris flows) predict.

Velocity profiles within the head-body transitional region
In the transitional region between the head and body of the experimental debris flow, it is comprised of a concentrated lower layer and a more dilute upper layer (see t = 0.3 s and t = 0.6 s in Figure 7). The corresponding velocity profiles show that in the lower layer, the velocity increases with distance from the bed to a velocity maximum towards the top of the layer. Above the maximum, the velocity decreases rapidly and exhibits negative values due to shearing between the layers, and the inability of the PIV software to produce accurate velocity values. These profiles share similarities with those observed in the steady state profiles of some submarine gravity currents, where differences in density drive a dense fluid through a less dense, ambient fluid (Simpson and Britter 1979, Kneller et al. 1999, Lowe et al. 2002. For some sediment-laden flows, notably high concentration turbidity currents, the settling of sediment can result in a layer of high concentration at the bed, while the upward mixing of turbulence produces a dilute upper layer that entrains sediment (Postma et al. 1988, Stevenson et al. 2018). In the internal profiles of these flows, the velocity maximum is located at the top of the lower layer due to the balance of the shear at the bed and at the interface between the dense fluid and the ambient fluid (Kneller et al. 1999). These profiles are observed in steady state flows, and above the interface between the two layers of material the velocity steadily approaches a zero value. This overall shape is similar to the internal velocity profiles in the current experimental flow at t = 0.3 s and t = 0.6 s (see Figure   12). Although the flow is transient at these times, and shows large fluctuations in the upper layer, the analogy to high concentration turbidity currents provides a deeper understanding of the mechanism responsible for the observed velocity profiles. Furthermore, it has been postulated that the transport of sediment in high concentration turbidity currents is a result of the interaction between a high concentration lower layer, and a turbulent upper layer (Postma et al. 1988).
This has potential relevance to the formation of the observed architecture in the current flow. However, it should be noted that two-layer turbidity currents are only a subset of natural systems (e.g. Paull et al. (2018)), and many flows are likely to exhibit a more gradual stratification (Peakall and Sumner 2015).

Experimental limitations
As discussed in Section 2.1, the PIV method requires the detection of individual particles over multiple frames in order to produce accurate velocity vectors. This wasn't possible at the head of the flow due to the low particle concentration and their turbulent behaviour. The velocity values recorded at the flow free surface were also subject to error, due to an overlying layer of water where particles were not present. The fact that particle tracking was not accurate at the front of the flow suggests that some particles were transported away from the flume walls in the cross stream direction, as a result of the high fluid turbulence. This implies that the two-dimensional flow approximation is subject to error, particularly at the front of the flow. Furthermore, the presence of the side wall may influence the flow dynamics. Despite these limitations, the experiments showed a high degree of repeatability, as shown in Figure 12. The small differences between the different runs at certain times may be a result of a delay in the opening of the lock gate, or variability in material composition.

Conclusion
The influenced strongly by the coupling of the granular and liquid constituents. Indeed, in reality granular-fluid coupling plays a vital role in debris flow dynamics (Iverson 2003, Iverson et al. 2011. The experimental flow in the current work is therefore more analogous to realistic debris flows than experiments involving idealised spherical particles, and the data could provide valuable validation for the development of two-phase numerical models. Regarding the non-fluctuating flow body, a viscous-type profile is able to capture the velocity throughout the majority of its depth. The granular material in the current experiments has a coefficient of uniformity C U of 5. Integration with the work of Sanvitale and Bowman (2017) suggests that the transition from a granular to a viscous-type flow profile takes place between a coefficient of uniformity of 3 and 5.