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Granulation of snow: From tumbler experiments to discrete element simulations

2015; Wiley; Volume: 120; Issue: 6 Linguagem: Inglês

10.1002/2014jf003294

2169-9011

Walter Steinkogler, Johan Gaume, Henning Löwe, Betty Sovilla, Michael Lehning,

Fluid Dynamics Simulations and Interactions

Journal of Geophysical Research: Earth SurfaceVolume 120, Issue 6 p. 1107-1126 Research ArticleFree Access Granulation of snow: From tumbler experiments to discrete element simulations Walter Steinkogler, Corresponding Author Walter Steinkogler WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, Switzerland CRYOS, School of Architecture, Civil and Environmental Engineering, EPFL, Lausanne, Switzerland Correspondence to: W. Steinkogler, w.steinkogler@gmail.comSearch for more papers by this authorJohan Gaume, Johan Gaume WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, SwitzerlandSearch for more papers by this authorHenning Löwe, Henning Löwe WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, SwitzerlandSearch for more papers by this authorBetty Sovilla, Betty Sovilla WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, SwitzerlandSearch for more papers by this authorMichael Lehning, Michael Lehning WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, Switzerland CRYOS, School of Architecture, Civil and Environmental Engineering, EPFL, Lausanne, SwitzerlandSearch for more papers by this author Walter Steinkogler, Corresponding Author Walter Steinkogler WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, Switzerland CRYOS, School of Architecture, Civil and Environmental Engineering, EPFL, Lausanne, Switzerland Correspondence to: W. Steinkogler, w.steinkogler@gmail.comSearch for more papers by this authorJohan Gaume, Johan Gaume WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, SwitzerlandSearch for more papers by this authorHenning Löwe, Henning Löwe WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, SwitzerlandSearch for more papers by this authorBetty Sovilla, Betty Sovilla WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, SwitzerlandSearch for more papers by this authorMichael Lehning, Michael Lehning WSL Institute for Snow and Avalanche Research SLF, Davos Dorf, Switzerland CRYOS, School of Architecture, Civil and Environmental Engineering, EPFL, Lausanne, SwitzerlandSearch for more papers by this author First published: 14 May 2015 https://doi.org/10.1002/2014JF003294Citations: 32AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract It is well known that snow avalanches exhibit granulation phenomena, i.e., the formation of large and apparently stable snow granules during the flow. The size distribution of the granules has an influence on flow behavior which, in turn, affects runout distances and avalanche velocities. The underlying mechanisms of granule formation are notoriously difficult to investigate within large-scale field experiments, due to limitations in the scope for measuring temperatures, velocities, and size distributions. To address this issue we present experiments with a concrete tumbler, which provide an appropriate means to investigate granule formation of snow. In a set of experiments at constant rotation velocity with varying temperatures and water content, we demonstrate that temperature has a major impact on the formation of granules. The experiments showed that granules only formed when the snow temperature exceeded −1∘C. No evolution in the granule size was observed at colder temperatures. Depending on the conditions, different granulation regimes are obtained, which are qualitatively classified according to their persistence and size distribution. The potential of granulation of snow in a tumbler is further demonstrated by showing that generic features of the experiments can be reproduced by cohesive discrete element simulations. The proposed discrete element model mimics the competition between cohesive forces, which promote aggregation, and impact forces, which induce fragmentation, and supports the interpretation of the granule regime classification obtained from the tumbler experiments. Generalizations, implications for flow dynamics, and experimental and model limitations as well as suggestions for future work are discussed. Key Points Granulation of snow is temperature dependent Different granule classes can be reproduced in a concrete tumbler The observed granule classes can be modeled with cohesive discrete element simulations by means of aggregation and fragmentation criteria 1 Introduction Snow avalanches exhibit different flow behavior, such as a sheared and fluid-like layer, a plug flow, or a dilute suspended powder cloud [Gauer et al., 2008]. Even different flow regimes can coexist at the same time at different locations inside an avalanche [Sovilla et al., 2008]. The flowing dense core of an avalanche is often approximated as a granular flow [Roche et al., 2011], sometimes even without taking cohesion into account [e.g., Faug et al., 2009]. A comprehensive understanding of the conditions that define the particle properties and size distribution and the consequent influence on flow dynamics of avalanches are still lacking. Recent studies have shown that the properties of the snow entrained by an avalanche during its downward motion, especially snow temperature, significantly affect flow dynamics [Naaim et al., 2013; Steinkogler et al., 2014a; Sovilla and Bartelt, 2002], mostly by changing the granular structure of the flow. An improved understanding of the driving factors causing the evolution of granule size distributions could help to better understand the hypermobility often observed in landslides and avalanches [Pudasaini and Miller, 2013] as well as the dynamics of powder snow avalanches [Rastello and Hopfinger, 2004]. The relation between particle size distribution and mobility of a granular flow, i.e., velocity and runout, has been emphasized in multiple studies on monodisperse and bidisperse materials [Moro et al., 2010; GDR MiDi, 2004]. Investigations on particle sizes in avalanche deposits (Figure 1) were conducted [Bartelt and McArdell 2009; De Biagi et al., 2012] by collecting particles in the deposition zone of full-scale avalanches. These studies could show that wet avalanches tend to produce larger granules than dry avalanches [Bartelt and McArdell, 2009; Kobayashi et al., 2000]. However, a link between different degrees of particle cohesion and emerging particle size distributions has never been quantified. Figure 1Open in figure viewerPowerPoint Granular structures in the deposition zone of an artificially released avalanche. Quite generally, the dynamics of particle size distributions in cohesive flows must be understood as a competition between aggregation and fragmentation phenomena. Due to the cohesive nature of snow, pieces can aggregate upon collision and stick together to form larger units from smaller ones. Due to the impact energies in a collision, pieces can also break apart and fragment into smaller ones. The competition of both processes is the origin of granulation phenomena in snow, which was so far only investigated by Nohguchi et al. [1997]. In the following, we refer to granulation as comprising both aggregation and fragmentation in contrast to Walker [2007] where the term was reserved for aggregation and enlargement of particle sizes only. Due to the difficulties of investigating these issues within a full-scale field campaign, experiments on snow dynamics are mainly carried out in the laboratory. Most laboratory experiments are however conducted with artificial grains, e.g., glass ballotini [Schaefer and Bugnion, 2013], which do not take interparticle cohesion into account. Studies on wet (i.e., cohesive) granular media [Tegzes et al., 2003; Donahue et al., 2010] have demonstrated the impact of interstitial liquid content on the dynamic flow behavior. These findings are particularly relevant for snow avalanches where the properties and distributions of the granules are fundamental parameters influencing the flow. An interesting experimental approach to granulation of snow is suggested by industrial applications where rotating drums are used for granulation in pharmacy [Vervaet and Remon, 2005; Kristensen and Schaefer, 1987], ceramics processing [Reed, 1995], mineral processing, and fertilizer production. In these disciplines, extensive research on granulation processes [Pietsch, 2003] and the characterization of granule properties and size distributions has been carried out in the last decades [Ennis et al., 1991; Ouchiyama and Tanaka, 1975]. As an appealing side effect, granular flow in rotating drums can be well investigated by the Discrete Element Method (DEM) [Cundall and Strack, 1979]. A DEM approach allows to model the kinematics of a cohesive powder explicitly and predict, e.g., mixing properties in rotating drums [Sarkar and Wassgren, 2009, 2010; Chaudhuri et al., 2006]. Numerical simulations based on DEM were also used recently to model the dense flow of cohesive granular materials by Rognon et al. [2008a], who studied the effect of cohesion on the flow mobility and showed that a plug region could develop if the cohesion is high enough. To our knowledge, the combination of rotating drum experiments and DEM modeling has never been used to address aggregation and fragmentation properties of snow for avalanche applications. It is the aim of the present paper to present a first attempt in this direction to demonstrate the potential of such a combined approach. The paper is organized as follows. We characterize relevant snow cover parameters and parameter thresholds that control the size and properties of granules for different types of snow that formed in a rotating tumbler by ways of laboratory experiments (section 2.1) and numerical simulations (section 2.2). We then compare those results to measurements in the deposition of real-scale avalanches and suggest a diagram of the granulation regimes (section 3). We discuss our results in view of the commonly used terminology of granulation processes [Iveson et al., 2001]. The paper discusses applications of the results and future avenues of research (section 4). 2 Methods 2.1 Experimental Setting To investigate the granulation potential of different snow types, a standard, unmodified concrete tumbler (Figure 2) was used. The tumbler measures 0.6 m at the largest diameter and held 0.4 m3 of snow. The tumbler has two blades acting as mixing elements (Figure 2) and rotated with constant velocity at 0.5 rps. Figure 2Open in figure viewerPowerPoint Granules in motion in rotating tumbler. One of the (two) mixing blades can be seen at the lower left corner. All experiments were conducted in an open-door but roofed laboratory that granted easy and direct access to natural and undisturbed snow, under varying environmental conditions, i.e., different air temperatures and sunny or cloudy skies. A total of 23 experiments with varying initial snow types were conducted and resulted in measurements at 97 time steps. Snow of different types (e.g., new snow particles or melt forms) with different properties (e.g., temperature or density) was used for the experiments. To assess the initial conditions of the investigated snow (temperature, density, grain shape, grain size, moisture content, and hardness) a regular snow profile following the guidelines of Fierz et al. [2009] was conducted before every experiment. Consequently, snow layers with similar properties were collected and a defined volume (0.1 m3) was shoveled into the tumbler. After defining the initial conditions of the added snow, the tumbler was started. Mellmann [2001] identified different flow regimes in rotating drums depending on the Froude number (1) According to that analysis, our experiments should fall into the cascading regime (Fr = 0.275) for cohesionless particles, where the particles are transported upward through solid body rotation, with a downward surface flow of the particles (Figure 3a). However, the presence of the blades and of cohesion forced the regime into a cataracting motion, i.e., individual particles detach from the bed and were thrown off into the free space of the tumbler (Figures 2 and 3b). Figure 3Open in figure viewerPowerPoint (a) Cascading and (b) cataracting regimes for a cohesionless granular material. These regimes mainly depend on the Froude number, particles, wall friction, and the filling degree for a cohesionless granular material (according to Mellmann [2001]). The tumbler was stopped at regular intervals, typically every 5 min which corresponds to 150 rotations, and the properties of the snow or the formed granules were measured. Again, the same procedures and measurement devices as for standard snow profiles [Fierz et al., 2009] were used. Snow temperature was measured with a digital thermometer to an accuracy of 0.1∘C. Snow densities of the granules or fine material in the tumbler were measured with a 100 cm3 snow shovel and a digital scale. If the granules were too small and no sample could be taken with the density shovel, their diameter was measured and they were weighed. This was repeated for multiple granules and an average mass of sample granules was taken. Also, the liquid water content was measured following the guidelines of Fierz et al. [2009] by using an 8X magnification glass and squeezing the snow by hand. Grain shape and size were identified by using a crystal card and a magnifying glass. Additionally, the experiments were filmed with normal and high-speed cameras (videos are provided as Movies S1–S8 in the supporting information). The videos reveal the motion of snow inside the tumbler and the interaction of snow and granules during collisions among themselves and with the tumbler. We defined the initiation of granulation as soon as the snow started to form small snow balls, with a size of approximately 1 cm. If they were not hard enough to sustain collisions among themselves or were destroyed upon touching, they were defined as nonpersistent granules (Figure 4a). On the other hand, hard granules were classified as persistent. Furthermore, we distinguished between persistent-moist granules (Figure 4b), according to snow class dry or moist in Fierz et al. [2009], and persistent-wet granules (Figure 4c), according to wet, very wet, or soaked snow [Fierz et al., 2009]. The experiments were continued and suspended in regular intervals until the formation of new granules could be observed. Experiments were stopped either if no granules formed after an extensive time, i.e., more than 100 min, or if the system reached a stationary state where no further changes in the measurement variables could be observed. Figure 4Open in figure viewerPowerPoint Snow inside tumbler for different experiments where (a) no persistent granules formed, (b) persistent-moist granules formed, and (c) persistent-wet granules formed. Measurements of the size distribution is an elaborate task in the field [Bartelt and McArdell, 2009] and in the tumbler. For our experiments we restrict ourselves to cases where granulation was visually observed and characterized the full size distribution at the end of the experiment by carefully emptying the tumbler and manually evaluating the particle sizes (Figure 5). Figure 5Open in figure viewerPowerPoint Small example set of different granule sizes. 2.2 Discrete Element Modeling 2.2.1 Motivation and Objectives The key ingredient of granulation processes is the competition between aggregation and fragmentation of the constituents upon collisions in the flow. Aggregation and fragmentation rates thereby depend on the kinematics of the flow via collision frequencies, the mechanical properties of the particles, and particularities of the container. Probably, the oldest approach to aggregation-fragmentation phenomena was originally suggested by Von Smoluchowski [1917] in the context of gelation phenomena. The model considers an evolution equation for the size distribution by providing average collision rates and size-dependent probabilities for subsequent aggregation or fragmentation events. These type of models are commonly referred to as population balance models (PBM). They must be regarded as a mean field description since expression for collision rates neither take into account spatial heterogeneities of the flow nor geometrical particularities of the container. On the other hand, flow of granular materials can be conveniently studied within discrete element simulations to correctly capture the kinematics of the particles and spatial characteristics of forces and displacements. Thereby, collisions are taken into account explicitly and, depending on the physical insight of the aggregation and fragmentation processes, particle size distributions can be predicted including particularities of the container, such as blades in a tumbler. It is the connection of DEM and PBM [Barrasso and Ramachandran, 2014] which ultimately allows to upscale the processes for practical applications. DEM can be regarded as a microscopic method to predict the relevant parameters in the PBM models, which has been shown in Reinhold and Briesen[2012]. As a first step in this direction, we start from a cohesive granular system using the DEM by explicitly taking into account the container geometry to address the snow granulation process in a tumbler. In the DEM, a discrete element represents a "snow unit" with a size of a few millimeters. This should not be confused with a "snow grain" in the sense of Fierz et al. [2009], which are commonly an order of magnitude smaller. A DEM approach to tumbler experiments would require to prescribe conditions for aggregation and fragmentation of these snow units under binary collisions subject to experimental conditions of temperature and water content. A generic physical picture of these mechanical processes is almost nonexistent. We therefore start from common mechanical criteria and formulate the model in terms of aggregation and fragmentation parameters. Our choice for the parameters and the hypothesized connection to the experimental conditions will be discussed at the end. 2.2.2 Formulation of the Model The discrete element simulations were performed using the commercial software PFC2D (by Itasca) which implements the original soft-contact algorithm described in Cundall and Strack[1979]. To capture the essential details of snow granulation we carried out two-dimensional simulations of a cohesive granular material, taking into account the tumbler geometry (Figure 2). The simulated system consists of a two-dimensional outer cylinder of radius R = 0.3 m and blades of length lb=0.35R. This configuration corresponds to the experimental setup. The granular samples are composed of about 8000 circular particles of average diameter d = 4.5 mm (thus, R/d ∼ 67), with a grain size distribution polydispersity of ±30 % (diameters ranging from 3 to 6 mm), and a particle density ρp=300 kg/m3. The loading is applied by gravity and rotation of the walls (cylinder and blades). The simulations were performed in the same conditions as the experiments so as to keep the same Froude number which is equal to Fr = 0.3 in our case (ω = 3 rad s−1 and g = 9.81 m s−2). As stated before, this Froude number would correspond to a cascading regime with a cohesionless granular material (Figure 3a) [Mellmann, 2001]. However, the presence of the blades and of interparticle-particle cohesion will considerably modify and force the flow regime from the cascading to the cataracting regime (Figure 3b). The interparticle contact laws used in the simulations are classical [Radjai et al., 2011; Gaume et al., 2011]. The normal force is the sum of a linear elastic and of a viscous contribution (spring-dashpot model), and the shear force is linear elastic with a Coulombian friction threshold (right part in Figure 6). Figure 6Open in figure viewerPowerPoint Schematic of the used contact law. The interpenetration distance between grain i and grain j is denoted δij. The normal and tangential stiffnesses are denoted kn and ks, respectively, gn is the viscous damping coefficient, and μ the friction coefficient. The tensile and shear strengths of the bond are denoted σt and σs. The corresponding mechanical parameters are summarized in Table 1. The value of the normal stiffness kn was chosen in a way that the normal interpenetrations δ at contacts are kept small, δ/d < 10−3, i.e., to work in the quasi-rigid grain limit [da Cruz et al., 2005; Roux and Combe, 2002]. Concerning the normal restitution coefficient e (which is directly linked to the normal viscous damping coefficient gn), we checked that the results presented below, and more generally all the macroscopic mechanical quantities obtained from the simulations, are actually independent of this parameter (in the range 0.1 to 0.9), in agreement with previous studies [da Cruz et al., 2005; Gaume et al., 2011; Gaume et al., 2015b]. Table 1. Mechanical Parameters Used in the Simulationsa kn/P kn/ks μ e ρp σt σt/σs 1.103 2 0.5 0.1 300 kg m−3 0–5 kPa 2 0–20 a akn: normal contact stiffness; P: average pressure; ks: tangential contact stiffness; μ: intergranular friction; e: normal restitution coefficient; ρp: particle density; σt: contact bond tensile strength; σs: contact bond shear strength; Fa: bond formation force, and : bond tensile strength (force). Cohesion was added to the particles by adding a bond to each contact. This bond has specified shear and tensile strengths and , respectively (the subscript f stands for fragmentation). If the magnitude of the tensile normal contact force equals or exceeds the contact bond tensile strength, the bond breaks, and both the normal and shear contact forces are set to zero. If the magnitude of the shear contact force equals or exceeds the contact bond shear strength, the bond breaks, but the contact forces are not altered, provided that the shear force does not exceed the friction limit and provided that the normal force is compressive. At the beginning of the simulation, the cohesion is applied to the whole sample. However, during the experiment, some bonds may break but new bonds may also be created at new contact points. The concept of bond formation between particles and its implementation in DEM is still open [Kroupa et al., 2012]. Hence, many different bonding models currently exist and we have chosen the following simple bond formation criterion that allows us to study the competing effects of fragmentation and aggregation of bonds: (2) where Fa is the bond formation force (the subscript a stands for aggregation), Nij and Sij the normal and shear forces at the contact, respectively. A similar model was already implemented in Brown [2013], Patwa et al. [2014], and Boltachev et al. [2014] with a creation criterion based on the exceedance of a critical interpenetration distance. This criterion is equivalent to our force criterion as the force is a linear function of interpenetration. Furthermore, Siiriä et al. [2011] also used the same force threshold criterion for bond formation to study powder tableting. The contact law which is used in the model is summarized schematically in Figure 6 in which the viscoelastic part is on the right side and the cohesive (plastic) part on the left side, and the ranges of the used parameters are shown in Table 1. Different models of cohesive interaction could also have been used such as the cohesive potential model of Rognon et al. [2008b] (attractive force for short interpenetrations between the grains followed by repulsive force for higher interpenetrations). However, due to the complexity of snow, it would have been difficult, in practice, to link the parameters of this model to snow properties. Our model is simpler and also more straightforward to apply to snow. Its model parameters, namely, the tensile strength [Hagenmüller et al., 2014; Sigrist, 2006] and the bond formation force [Podolskiy et al., 2014; Szabo and Schneebeli, 2007], can be evaluated from laboratory experiments. 2.2.3 Dimensional Analysis Snow is a material generally characterized in terms of stresses rather than forces. Hence, the tensile and shear strengths and (forces) are scaled by the bond surface to obtain the tensile and shear strengths σt and σs (stresses): (3) Realistic values of the tensile and shear strengths for cohesive snow can be found in the literature [Mellor, 1974; Jamieson and Johnston, 1990, 2001; Gaume et al., 2012, 2015aa] and belong to the range 0–5 kPa. Furthermore, following Rognon et al. [2008b], different dimensionless numbers are used to quantify the intensity of the interparticle strength. These numbers compare the contact bond tensile strength and the bond formation force Fa to two typical forces of the system. The first number ηf (fragmentation number) characterizes the fragmentation potential of a bond after a collision due to the cataracting regime of the tumbler: (4) where mp is the particle mass and 〈v〉=〈Rp〉ω the average particle velocity (〈Rp〉 is the average particle radial position). The fragmentation number thus compares to the impact force of an elastic body. This number is nevertheless different from that used in Rognon et al. [2008b] for a gravitational shear flow which compares to the gravitational force mg. In our case, the main source of granule breakage are granule collisions, so our fragmentation number was modified accordingly. The second number ηa (aggregation number) characterizes the potential of formation of a new bond inside a granular assembly submitted to a pressure P (5) This number compares Fa to the average normal force Pd2 due to pressure. According to this definition and to Rognon et al. [2008b], a transition between different cohesive flow regimes should depend on these two numbers ηf and ηa and should occur for values close to unity. In our study, was varied between 0 and 0.05 N and Fa between 0 and 1 N. The collisional force is equal to N (〈Rp〉≈0.2) and the average pressure force is equal to Pd2≈0.018 N by taking P = ρgR, the hydrostatic pressure at the bottom of the tumbler. The fragmentation number ηf typically varies in the range 0.5–4.5 and the aggregation number ηa in the range 0.01–10. 2.2.4 Simulation Procedure and Granule Definition Simulations were performed for 6 different values of and 15 of Fa (and thus different values of ηf and ηa) for a total of 90 simulations. At the beginning of each simulation, cohesion is applied to the whole sample so it forms a single large block similar to a cohesive slab in the release zone of an avalanche. The simulations were stopped after 30 revolutions, corresponding to an apparent steady state (almost constant size distribution). At each quarter revolution, the granules are identified as a cluster of particles linked by cohesive bonds and the size of these different clusters was computed. The procedure of cluster identification is provided as supporting information. As the granules may not be perfectly circular, the maximum transverse length was retained, similar to what was measured experimentally. Hence, for each simulation, the complete granules size distribution was available as a function of time. In addition to the granules size, the average number of cohesive bonds per granule was computed. Figure 7Open in figure viewerPowerPoint Snow density measurements (bulk density from 100 cm3 shovel) for all individual experiments (gray lines) and time of each measurement (gray and colored markers) for persistent-moist (colored squares) and persistent-wet (colored triangles) granules. 3 Results In this section we discuss the measured snow cover parameters and their temporal evolution during the experiments and identify the most relevant parameters for the granulation process (section 3.1). Depending on the persistence of the formed granules, all conducted experiments are assigned to three granulation classes. This classification also provides the basis for the DEM simulations which supplement the measurements (section 3.2). We summarize the main results of experiments and modeling in Figure 15 and finally present a combined view of experimental, modeled, and real-scale avalanche size distributions (section 3.3). 3.1 Evolution of Snow Parameters The following graphs (Figures 7 to 12) display all individual experiments (gray lines), the time of each measurement (gray and colored markers), and whether nonpersistent (gray circles), persistent-moist (colored squares), or persistent-wet (colored triangles) granules were recorded. Most experiments were initialized with a snow density between 200 and 350 kg m−3 (Figure 7). Densities of persistent granules reached similar values as observed in the deposition of real-scale avalanches (400 kg m−3). Densities larger than 550 kg m−3 were only observed for persistent-wet granules. For the slush experiment (snow density of 951 kg m−3) water was added after persistent-wet granules had formed. In many experiments with no persistent granulation, the density of the remaining fine, ungranulated snow also reached values up to 450 kg m−3. The error for the snow density measurements was around ±10 kg m−3. A strong dependency of granulation on snow temperature could be observed (Figure 8). Once a temperature of −1∘C was reached, granulation occurred very fast (colored markers in Figure 8). Multiple experiments below the threshold of −1∘C were run for an extensive duration (more than 100 min), yet no persistent gr