Modeling and Simulation of Rapid Granular Flows. Application to Powder Snow Avalanches

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U

NIVERSITÀ DEGLI

S

TUDI DI

P

ISA

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di laurea in Fisica

Master’s degree thesis

Modeling and Simulation of

Rapid Granular Flows:

Application to

Powder Snow Avalanches

Candidate:

Roberto Lepera

Supervisors: Prof. Fulvio Cornolti

Università di Pisa Dr. Tomaso Esposti Ongaro Istituto nazionale di geofisica e vulcanologia

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S’annuncia col profumo, come una cortigiana, l’Isola Non-Trovata... Ma, se il pilota avanza, rapida si dilegua come parvenza vana, si tinge dell’azzurro color di lontananza...

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Acknowledgements

First, I would like to thank my parents and my sister for their love, encouragement, and financial support throughout my academic career. They represent the real and concrete point of reference during my entire life. This thesis is dedicated to them.

A special thanks goes to supervisor Prof. Fulvio Cornolti, who made me discover the beauty in fluid dynamics, a discipline as complicated as it is profound. At the same time, I am immensely grateful to supervisor Dr. Tomaso Esposti Ongaro who guided me with perseverance and competence in this complex thesis work: at the beginning, I didn’t even know what a granular medium was!

Finally, thanks to all the friends who, in different ways and times, have accompanied me during these (many) years at the University of Pisa. In particular, I would like to mention Dr. Francesco Sardelli, Daniele Daini, and Marco Marino with whom I share the passion for Physics, which has often led us to have very stimulating discussions.

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CHAPTER

1 Introduction

1.1 Overview: granular media

Figure 1.1:Examples and classification of particulate matter. Source:Andreotti et al.(2013).

We observe granular media everywhere in nature, at different scales and dynamic regimes: from sand dunes to cereals, from snow and rock avalanches to interplanetary aggregates like Saturn’s rings and the asteroid belt (Fig.1.1). In spite of this great variety on a granular scale (shapes, sizes, materials), all these particulates share common and fundamental properties at the macroscopic level, manifesting a behavior like a solid or a fluid depending on the situations.

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In this thesis we use the definition of granular medium given byAndreotti et al.(2013), that is a collection of rigid particles of size larger than 100 µm interacting mainly through friction and collisions. Smaller particles, with a diameter between 1 µm and 100 µm, are called powders; other interactions such as van der Waals forces and air drag become important at these scales. Finally, colloids are made up of particles with a diameter below 1 µm, for which the effect of thermal agitation can no longer be neglected.

Granular media is the second most used type of material in industry after water: from the mining to civil engineering (concrete, bitumen, asphalt, etc.), from the chemical industry (e.g., granular fuel and catalysts) to pharmaceutical industry. So, in such areas the study of granular media is motivated by the resolution of problems related to the storage, transportation, flow, and mixing of these substances. Another application domain for granular media is Earth Science since the soil is largely composed of grains. In particular, scientific research dedicates many efforts to describe and predict the risks associated with spectacular and, at the same time, dangerous natural phenomena such as pyroclastic flows, avalanches, landslides, desertification, and soil erosion.

The dynamics of granular fluids is rather complex even under idealized conditions, i.e., spherical and rigid particles, absence of cohesion, elastic collisions, and so on. For example, there is no univer-sally accepted theory that explains even apparently simple phenomena such as the collapse of a pile of dry sand in the laboratory scale. This may seem paradoxical given that the single particle’s motion is described by the well-known laws of classical physics.

We list now the main difficulties encountered in modeling granular flows (Goldhirsch, 2008; Andreotti et al.,2013).

• Large number of particles. Let’s consider a spoonful of sugar (volume V≈1 cm3, diameter of the particles dp ≈ 100 µm), we obtain a number of grains N ≈ V/d3p ≈ 106. On the other hand,

today’s computers cannot simulate approximately more than a million particles simultaneously. Therefore it is practically not feasible to study in this way geophysical phenomena on much larger scales like snow or pyroclastic avalanches. An alternative but also demanding approach is to use space-averaged physical quantities and treat the granular medium as a continuum. • No local thermal equilibrium. It is easy to show that granular media does not reach the local

thermal equilibrium unlike normal fluids, for which the kinetic theory allows to obtain macro-scopic physical quantities starting from the micromacro-scopic ones. Granular media are thus athermic systems, although the temperature of the particles can play a role in the energy balance. For example, let’s consider a granular system with a gravity-driven behavior. At room tempera-ture T=300 K, the thermal energy per particle reads Eth≈kBT≈10−21J; while for a glass bead

with diameter dp=1 mm and a density ρ=2500 kg/m3the variation of gravitational energy on

a granular scale is∆Eg≈mgdp≈10−10J. Therefore the thermal agitation acts on times τthtoo

long compared to those τgto the evolution of macroscopic quantities, being τ≈dp/v∝ E−1/2.

• Lack of scale separation. Another limit imposed on the continuous treatment of granular media is the lack of a clear separation between microscopic and macroscopic length scales. For example, when sand flows down on a pile, the flow thickness is about 10-20 particles. Similarly, the shear bands or the failures of granular soil have typically an extension of a few tens of grain widths. Therefore it becomes difficult to define the control volume on which to average macroscopic quantities.

• Complexity of interactions. The interactions between the particles of the granular medium are made complex by strongly nonlinear phenomena such as friction and inelastic collisions. When grains are immersed in a molecular fluid, hydrodynamic interactions must also be taken into account (e.g., drag forces and buoyancy).

• Energy dissipation. The granular media easily dissipate energy at microscopic level and this makes them very different from the usual systems studied in statistical physics. A rubber ball thrown on the sand does not bounce: its kinetic energy is quickly dissipated by the friction forces between the particles.

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§1.2 − Rapid granular avalanches in geophysics 3

• Different regimes of granular matter. A granular system can exhibit solid, liquid or gaseous behavior depending on the situation. For example, a pile of sand can sustain shear forces under static conditions but the same material can flow down an inclined plane, and it partially "evaporates" if the flow is rapid enough. This can be interpreted as a transition from a frictional regime to a collisional one. Nevertheless, these different flow regimes can also coexist in a single configuration (Fig.1.2).

• Hysteresis. The transition between flow and no flow in a granular medium is a complex phe-nomenon. It may depend on the sample preparation and experimental evidence suggests that both the initial volume fraction and the history of the previous deformation play a role. The physical origin of the hysteresis is still debated.

• Clustering. This term indicates the formation of microstructures in granular gases as has been verified in detailed theories, simulations and experiments. The mechanism proposed for explain this phenomenon is often referred to as "collisional cooling". As in any many-body system also in the granular media density fluctuations are observed, and in the more concentrated regions the collision rate τ−1between particles increases with the density ρ (i.e., τ−1∝ ρ2). Because of the inelastic collisions, in these regions the energy density is dissipated rather quickly, so that there is a local decrease in pressure. This pressure gradient generates a mass flow that feeds the clustering phenomenon, until the process is stopped by other mechanisms.

Figure 1.2:Flow of beads on a pile and different granular regimes in an hourglass. Source:Andreotti et al.(2013).

1.2 Rapid granular avalanches in geophysics

Powder snow avalanches and pyroclastic flows represent two important geophysical applications of models on granular media. In addition to constituting an intriguing theoretical and experimental research field, the study of these potentially destructive avalanches helps us to protect the safety of people in risk areas, and to limit damage to civil structures.

Powder snow avalanches (PSAs) are gravity-driven flows as high as 200 m descending at frontal velocities of up to 100 m/s. Dry snow PSAs typically develop a dilute frontal region feeding an energetic turbulent suspension, and present a slow sedimenting cloud trailing far behind. Typically they originate from a dense material, which remains hidden and travels behind the front, and for this reason they are also called “mixed” PSAs. (Pudasaini and Hutter,2007;Sovilla et al.,2015)

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Figure 1.3:A powder snow avalanche in Sionne, Switzerland (left) and a pyroclastic flow produced by the Unzen volcano, Japan (right). The similar topography (slope of 35-45 degrees) has produced a comparable phenomenon, with the characteristic turbulent front, the deposit with a concentrated basal layer and front velocity of tens of meters per second. Note the greater diffusion of the dispersed phase in the case of pyroclastic flow due to convection and drag forces.

Pyroclastic flows (or pyroclastic density currents, PDCs) are multiphase, rapid currents of hot gas and volcanic matter (tephra) that moves away from a volcano reaching velocities up to 200 m/s. The gases can reach temperatures of about 1000◦C. They typically touch the ground and hurtle downhill, or spread laterally under gravity. Pyroclastic flows are a common result of certain explosive eruptions, including volcanic column instability and collapse, fountaining produced by caldera collapse, failure of crystallized volcanic domes, and so on. (Gilbert and Sparks,1998;Branney and Kokelaar,2002; Esposti Ongaro et al.,2016)

This two processes present a comparable phenomenology (Fig.1.3-1.4-1.5-1.6), so it is expected that they can be studied in a unitary way by using similar mathematical models. Further research is needed to verify the validity and generality of this insight. However, some differences should be taken into account: e.g., shape, type, and grain-size of the materials involved, the interaction between flow and bottom, and the very different initial temperature of the granular/solid medium.

We now present some data to understand the type and extent of the risks associated with these geophysical events. Millions of snow and ice avalanches falling annually worldwide. In the USA, from 1986 to 2019 dozens of people died each year from snow avalanches; similar data apply to Euro-pean countries such as Austria, France, Switzerland, and Italy. Average annual property damage is approximately $ 400.000, in USA alone. (Pudasaini and Hutter,2007;Colorado Avalanche Information Center,2020)

Volcanic eruptions can ruin vast productive lands, destroy structures and injure or kill the popula-tion of entire cities. Worldwide, around 50 eruppopula-tions occur every year. About a tenth of the world’s population lives within volcanic hazard zones and many lives are regularly lost through volcanic activity. Pyroclastic avalanches are the dominant fatal cause at 5 to 15 km from the volcano. (Pudasaini and Hutter,2007;Brown et al.,2017).

We show the main issues concerning the study of this type of physical phenomena.

• Particle stratification and turbulence. Typically, in rapid flows of dense granular avalanches there is a more concentrated basal region in a solid-liquid regime and an overlying dilute component dominated by collisions, in which air is incorporated into the flow and the effects of the turbu-lence are not negligible. The granular-turbuturbu-lence can be studied by analogous methods to those used for molecular fluids. Some additional complications arise: it is challenging to understand the influence of the particulate phase on fluid turbulence and energy balance between the turbulent, dilute cloud and the dense flow layer.

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§1.2 − Rapid granular avalanches in geophysics 5

Figure 1.4:Schematic view of the conjectured structure of a powder snow avalanche: mainly a concentrated, basal region characterized by a solid-liquid granular regime and an overlying dilute, turbulent component dominated by collisions. The region characterized by the clustering phenomenon is also illustrated. The snow entrainment from the underlying snowpack occurs just behind the front. Source: Adapted fromSovilla et al.(2015).

• Grain-size distribution. The granulometry of these systems varies within many orders of mag-nitude. In geophysical avalanches the diameter of the particles can vary from micrometers (powders and ash clouds) to meters (boulders and rock fragments). Hence, in general a multi-phase treatment of these granular flows is necessary. This approach, in addition to increasing the number of equations to be solved, implies the involvement of new phenomena. Granular materials composed of particles with differing grain sizes, densities, shapes, or surface properties may experience unexpected segregation and mixing during flow.

• Lack of experimental data. We have little information on both initial conditions of granular system (e.g., size, position, energy, composition), and the fully developed flow such as particles velocity and concentration profiles. On the other hand, a more accessible measure is the runout, that is the maximum spatial extension of the material deposit. In some cases, the kinetic energy of the front can be estimated on the basis of the damage caused to buildings, uprooted trees, and so on. • Rheological model, i.e. the expression of the stress tensor as a function of the fluid flow fields. The rheological models proposed in the literature have been developed and tested either in quasi-static conditions or in a slow steady-flow regime (Nedderman,1992). There are no obvious theoretical reasons to believe that these models are also valid for rapid and strongly non-stationary flows. For example, it is challenging the description of shear dilatancy and the shear contraction of a granular medium. In this regard, starting from well-known yield criteria1used in soil mechanics (Mohr-Coulomb, von Mises, Drucker-Prager, etc.), more and more advanced models have been developed as reviewed inSi et al.(2019).

• Scaling issues. Similarity arguments can be used to extrapolate laboratory data on a larger scale. Nevertheless, it’s very difficult to write dimensionless governing equations for rapid, gravity-driven granular flows. A strong obstacle for this transformation is the presence of ad hoc empirical parameters, e.g., in the frictional pressure model (see section1.3).

• Boundary conditions. As in any fluid dynamic problem, even for granular flows there is the thorny question of determining the appropriate boundary conditions. In particular, it is complicated to quantify the interaction between flow and bottom: complex phenomena such as soil erosion, embedding or depositing of material, and impacts with fixed obstacles are involved.

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Figure 1.5:Generalized structure of a dilute PDC with a head intergradational into a body with an overriding mixing zone and trailing wake. Source:Breard and Lube(2017).

Figure 1.6:Scanning Electron Microscopy photo of rounded fragments of crushed snow crystals (left) and volcanic ash from Redoubt volcano, Alaska (right). The snow particles have size dp=0.1−1 mm and

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§1.3 − Structure of the thesis 7

1.3 Structure of the thesis

The objective of this thesis is to review the main modeling aspects of the physics of granular media and to develop a new computational tool useful to simulate rapid granular avalanches over three-dimensional natural terrains. Because such a tool does not exists yet, model validation represents a critical aspect of this work, which is envisaged through the comparison of numerical results with laboratory experiments and large-scale artificially triggered natural avalanches.

The introductory chapter of this work presents the general phenomenology of granular media, from the laboratory scale to the geophysical one and beyond. In particular, the general characteristics and similarities between snow avalanches and pyroclastic flows are outlined, as well as the risks for the population associated with these natural events. The theoretical and applicative motivations for the study of these systems are underlined, including some fundamental problems that still prevent a complete understanding of granular flows. (Branney and Kokelaar,2002;Pudasaini and Hutter,2007; Andreotti et al.,2013)

In the second chapter the general governing equations for granular and molecular fluids are reviewed, by essentially following the formulation ofGidaspow(1994). The balances of mass, mo-mentum, energy, entropy, and enthalpy for a fluid parcel are derived by using the Reynolds transport theorem, for which a proof is provided in the appendix. The fundamental hypothesis of this approach is that, if the granular system have a large number of particles, we can treat it as a continuum and that the volume-averaged quantities are fairly regular fields.

In particular, the momentum balance takes the same form as the usual Navier-Stokes equation, with the stress tensor and other source terms of governing equations (drag force, rate of energy dissipation, heat flux, and so on) to be determined. In this regard, for the purposes of this thesis, the key aspect is that of the closure relations for a two-phase system consists of a solid-granular component and a continuous-gaseous one. Detailed attention is given to the analysis of the rheological model, i.e. the relationship between stress and strain in the granular medium. In particular, I assume that the shear stress tensor for the granular material has a newtonian form (i.e., proportional to strain-rate tensor) and it is written as a frictional part plus a collisional one (Johnson et al.,1990; Srivastava and Sundaresan,2003;Meruane et al.,2010;Si et al.,2019).

The collisional part is obtained with semi-empirical methods that make use of the kinetic theory, in a similar way to what is done with molecular fluids. This leads to an equation of state analogous to the ideal gas law, in which the thermodynamic pressure (temperature) is replaced by the granular pressure (temperature); the latter quantity provides a measure of the fluctuation of the particles velocity controlled by the collisions, with respect to the average flow motion.

For the frictional part,Johnson and Jackson(1987) have first used empirical expressions for fric-tional pressure (cf.van Wachem,2000). It identifies with the isotropic part of the frictional stress tensor and its action is twofold: on one hand, it prevents that the granular fluid compacts beyond the maximum packing limit, therefore in some way tends to fluidize the granular flow; on the other hand, it is proportional to the coefficient of friction in plastic and pseudoplastic rheological models, then contributing to solidify the system.

The third chapter introduces the Computational Fluid Dynamics tool OpenFOAM (for "Open-source Field Operation And Manipulation"), an open "Open-source C++ based software for the development of numerical solvers, and pre-/post-processing utilities for the simulation of continuum mechanics systems, mainly computational fluid dynamics (CFD) problems. I have developed a numerical model based on the twoPhaseEulerFoam solver (included in the default release), to which a new boundary condition (Johnson and Jackson,1987) and new rheological model have been added followingSi et al. (2019). In addition, a procedure to generate unstructured numerical meshes over a 3D topography have been developed. In more detail, the chapter describes the discretization techniques for partial differential equations through the method of finite volumes, as well as it analyzes the typical file structure of OpenFOAM cases.

The fourth chapter presents the numerical results at a laboratory scale, aimed at model validation. The solver was tested by simulating two benchmark experiments for which direct measurements are available (Mangeney et al.,2010): i.e., the flows arising from the collapse of a rectangular column

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Figure 1.7:Simulation of the collapse of a granular column with dimensions(20×14)cm along a rough horizontal plane. The circles represent the measurements obtained byMangeney et al.(2010), and contour lines indicate the points where the calculated solid volume fraction αs=0.1, at various instants of time. αs=0.62 at t=0.

of granular material, along a horizontal plane (Fig.1.7) and an inclined ramp with an angle of 16◦, respectively. In addition, the kinematics of the front was briefly analyzed in terms of dimensionless quantities, as the size of the initial granular column varies. This procedure allowed us to calibrate the free parameters of the models, as well as to check the validity of the boundary conditions. Particular attention was paid to the sensitivity of the results to the chosen frictional model and the size of the mesh used to discretize the balance equations.

The fifth chapter illustrates the simulation results for two powder snow avalanches (PSAs) on the topography at the Vallée de la Sionne test site (Switzerland). In the first case, a relatively small and channeled flow of ice granules was analyzed, in order to verify the convergence of the simulation, to calibrate the free parameters of the fluid models, as well as to make a preliminary study regarding the sensitivity of the results to the mesh used. Regarding the second avalanche, which was artificially triggered in March 2006, simulation results were compared with the experimental data presented in Sovilla et al.(2008). Simulation results reproduce some key aspects of the PSAs dynamics, such as the development of a bipartite flow (with a concentrated, granular basal flow overlain by a dilute, turbulent particle cloud) and the controlling effect of the topography. However, quantitatively, the model developed still overestimates the avalanche velocity and, thus, its impact.

The last chapter contains conclusions and perspectives that arise from this thesis work. The main results emerged from the simulations are summarized, and possible explanations and remedies to the current limitations are proposed, together with some ideas for further studies on gravity-driven rapid granular avalanches. Finally, potential implications for hazard assessment associated with potential catastrophic phenomena such as powder snow and pyroclastic avalanches are discussed.

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CHAPTER

2 Governing equations

In the first part of this chapter the transport equations are critically reviewed, namely: mass, momentum, energy, enthalpy, and entropy balances for a generic fluid. We essentially follow the approach presented byGidaspow(1994), in which systems with a fairly large number of particles are considered, so as to have physical quantities represented by fields with continuous derivatives. This hypothesis is reasonably verified by a granular medium in the collisional regime, whereby it is possible to define a local statistical equilibrium similarly to what is done in the kinetic theory of molecular fluids. Much more delicate is the continuous treatment of both solid and liquid regimes of the granular system1. These aspects will be explored in the first section.

In the last section, the closure and constitutive relations for a granular fluid are presented. It is about providing an explicit expression for the equations of state, the stress tensors, and the interphase momentum and energy exchange terms. In particular, frictional behavior of the granular medium, which is described by a part of the stress tensor, will be discussed in greater detail. Finally, we will describe the boundary conditions for the velocity and the granular temperature of the solid fraction.

2.1 Granular systems as continuous media

First, we obtain a dynamical parameter that allows us to distinguish the granular solid regime (dominated by friction), from the liquid/gaseous one (characterized by collisions). To do this we follow the approach ofAndreotti et al.(2013). Let’s analyze a rather simple system that can be studied in laboratory scales, which however will allow us to capture some essential aspects of the complex phenomenology of granular flows.

Consider a granular material consisting of spherical particles of diameter d and density ρ under a confining pressure P. The system is confined between two parallel plates by a pressure P exerted on the top plate. Furthermore, the upper plate moves with respect to the lower one with a velocity V; thus in the steady regime the material is subjected to a homogeneous normal stress P, and to a uniform shear rate ˙γ=V/L, where L is the distance between the plates (Fig.2.1).

If the effect of gravity is negligible, for large systems (i.e., L/d1) and rigid particles (i.e., their Young’s modulus E P) there exist only four parameters in the problem: d, ρ, ˙γ, and P. These quantities involve three fundamental units: length, time, and mass. So, in virtue ofΠ theorem, the system is controlled by a single dimensionless number given, for example, by:

I= ˙γd

p P/ρ. (2.1)

This number is called "inertial number". It’s obvious that other expressions can be chosen for the dimensionless parameter, such as the Coulomb number ρd2˙γ2/P≡I2.

1_{As for a qualitative description of the granular regimes, we refer the reader to the section}_1.1_.

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Figure 2.1:Plane share. Source:Andreotti et al.(2013), edited.

Figure 2.2:Two time scales in the rheology of granular media. Source:Andreotti et al.(2013), edited.

In general, the inertial number is an important parameter controlling the rheology of dense granular flows. It can be interpreted as the ratio between two time scales of the system:

I= tmicro

tmacro, (2.2)

where tmicro=d/p P/ρ is a "microscopic" time scale related to the typical time scale of rearrangements

and tmacrois a "macroscopic" time scale related to the mean shear rate (Fig.2.2). To see this, let∆u= ˙γd

the relative velocity of two adjacent layers of grains in a simple shear flow (Fig.2.2, left); so the mean time taken by a particle to cross the grain underneath is d/∆u= ˙γ−1 ≡tmacro. The microscopic time

represents the inertial time taken by a particle to fall into a hole of size∼d under the pressure P (Fig.

2.2, right). By using the Newton’s law in the vertical direction, we can write: m ¨z ∼md/t2_micro = Fz,

with m∼ρd3and Fz∼Pd2which given tmicro ∼d/p P/ρ.

The inertial number enables a quantitative classification of the flow regimes (Fig.2.3). In fact, small values of I (I→0) correspond to the quasi-static regime, in the sense that macroscopic deformation is very slow compared with the microscopic rearrangements. Large values of I (I & 1) indicate a rapid and dilute flow regime. The dense-flow regime lies in between. Note that the transition from the quasi-static regime to the inertial one can occur either by increasing the shear rate or reducing the normal stress.

So we have now a new tool to justify a continuous treatment of granular systems, at least in the quasi-stationary conditions. For I 1 it is expected that normal stress will no longer be able to pack the granules (equivalently, the shear rate is sufficiently large), resulting in a purely collisional regime, whereby the macroscopic quantities averaged over the control volume can be defined through a kinetic formalism analogous to the case of molecular fluids.

On the other hand, for I 1 the granules interact almost exclusively by means of contact forces, with a typical time of rearrangement much less than that associated with macroscopic evolution. In this situation, it is reasonable to think that the volume averaged stress is sufficiently regular (Fig.2.3, below) on scales of length Ld and that therefore the dynamic evolution in this regime can be well described by a continuous model.

Finally, the description of the "liquid" regime (I.1) is more delicate and there is not sufficient consensus in the literature on which formalism is more suitable. In depth averaged models (not

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§2.2 − Transport equations 11

Figure 2.3:Above: flow regimes as a function of the inertial number I. Below: the evolution of the force network with I in two-dimensional simple-shear flows; the lines represent the normal forces between grains. A force chain consists of a set of particles within a compressed granular material that are held together and jammed into place by a network of mutual compressive forces. Between these chains are regions of low stress whose grains are shielded for the effects of the grains above by vaulting and arching. A set of interconnected force chains is known as a force network. Source:Andreotti et al.(2013), edited.

covered in this thesis) the rheological model can explicitly depend on I (Andreotti et al.,2013). Another solution is to represent this regime as a sort of "overlap" of the two aforementioned ones, and more precisely to write the stress tensor as the sum of a frictional part and a collisional part (van Wachem, 2000;Si et al.,2019). Therefore the continuous treatment of the liquid regime is inherited, in a sense, from that used for the solid and gaseous ones.

2.2 Transport equations

In light of the above analysis, Reynolds’ transport theorem can be used to study the balance of a generic physical quantity per unit of volume ψ:

d dt Z V(t)ψ dV= Z V(t) ∂ψ ∂t + ∇·(ψv) dV, (2.3)

where v is the velocity of control volume V(t), also known as flow or macroscopic velocity, and d/dt=∂/∂t+v·∇denotes the material derivative. In appendixAa proof of Eq. (2.3) is presented.

The use of the volume fraction (i.e., the volume of a constituent divided by the volume of all con-stituents) is common in the literature to describe the behavior of a multiphase flow. This is a natural choice if the volume occupied by a phase cannot be occupied by the remaining phases, at the same spatial position at the same time. Accordingly, if Viis the volume occupied by phase i, we define its

volume fraction αias:

Vi = Z

V(t)αidV, with

∑

_i αi

=1. (2.4)

2.2.1 Mass balance

The mass miof the fluid i, contained in a volume V(t), is linked to its density ρiby the following

relation:

mi= Z

V(t)αiρidV. (2.5)

In absence of chemical reactions and mass inflow/outflow in the system, the conservation of mass implies that ˙mi=0. Then, the Reynolds theorem (2.3) with ψ=αiρiand the contradiction argument

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applied to an arbitrary volume V(t)leads to the continuity equation for the phase i: ∂(αiρi) ∂t + ∇·(αiρivi) =0 (2.6) or, equivalently: d(αiρi) dt +αiρi∇·vi =0. (2.7)

2.2.2 Momentum balance

The momentum balance derives from the Newton’s second law of motion applied to the fluid element of volume V(t): dPi dt = d dt Z V(t)αiρividV=Fi. (2.8)

This equation states that the rate of change of momentum of phase i equals the resultant force Fiacting

on it: Fi= Z S(t)Ti·dΣ + Z V(t)αiρifidV + Z V(t)bijdV, (2.9)

where S(t)is the boundary of V(t). The first term in the right side of Eq. (2.9) accounts for the surface forces, the second one the external bulk forces and the last one the bulk forces of interaction between phases. Note that the sum of the interaction forces is clearly zero by virtue of Newton’s third law, namely:

∑

ij

bij=0. (2.10)

In a orthogonal cartesian coordinate system(x, y, z) the stress tensor Ti is represented by the

matrix:

Ti=





Tixx Tixy Tixz

Tiyx Tiyy Tiyz

Tizx Tizy Tizz



. (2.11)

Let δAαbe a small portion of S(t), oriented in the α direction, and let δFiβbe the force acting on it

along the β direction. Hence, the generic element Tiαβin (2.32) is given by:

Tiαβ= δFiβ δAα ≈ ∂Fiβ ∂Aα . (2.12)

Moreover, it’s possible to show that conservation of angular momentum implies that the stress tensor is symmetric: Tiαβ =Tiβα. By using the divergence theorem, we can rewrite the first integral of the Eq.

(2.9) as: Z S(t)Ti ·dΣ= Z V(t) ∇·TidV. (2.13)

An application of the Reynolds transport theorem (2.3) with ψ=αiρivi, the use of Eq. (2.13) in (2.9)

and the usual contradiction argument leads to three momentum balance scalar equations for each phase i: ∂(αiρivi) ∂t + ∇·(αiρivi⊗vi) = ∇·Ti+αiρifi+bij, (2.14) . αiρidvi dt = ∇·Ti+αiρifi+bij. (2.15) The explicit form of Ti, fi, and bijis obtained by means semi-empirical closure relations and/or the

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§2.2 − Transport equations 13

2.2.3 Energy balance

Consider a closed system (i.e., ˙mi =0) in local thermal equilibrium. The internal energy balance

moving with the phase i is given by the first law of thermodynamics: dEi dt = dQi dt −P dVi dt +Di+Qij where Ei= Z V(t)αiρieidV. (2.16)

P is the thermodynamic pressure of the continuous phase, Diis the energy dissipation rate, Qijis the

rate of interphase heat exchange, and eiis the specific internal energy per unit of mass. The rate of

heat transfer ˙Qiis related to the heat flux qiby the following relation:

dQi dt = − Z S(t)αiqi·dΣ= − Z V(t)∇·(αiqi)dV. (2.17)

Moreover, we define the specific energy rates diand qij:

Di = Z

V(t)didV; Qij= Z

V(t)qijdV. (2.18)

Now, let’s apply the Reynolds theorem (2.3) and the usual contradiction argument on an arbitrary volume V(t), thus the Eq. (2.16) becomes:

∂(αiρiei) ∂t + ∇·(αiρiviei) =αiρi dei dt = −∇·(αiqi) −P ∂αi ∂t + ∇·(αivi) +di+qij. (2.19)

Recall that for a molecular fluid the dissipation rate d=T:∇v. However, for now we do not make this term explicit since the treatment of dissipation is more delicate for granular media. This problem will be addressed in more detail in the section2.3.

By scalar multiplication of both sides of Eq. (2.14) by viand by using again the Eq. (2.6), the balance

of the specific kinetic energy per unit of mass ki=v2_i/2 is also obtained:

∂(αiρiki)

∂t + ∇·(αiρiviki) =αiρi

dki

dt =vi·∇·Ti+αiρivi·fi+vi·bij. (2.20) Finally, the sum of Eq. (2.19) and Eq. (2.20) gives the total energy balance of the phase i:

∂[αiρi(ei+ki)] ∂t + ∇·[αiρivi(ei+ki)] =αiρi d(ei+ki) dt = = −∇·(αiqi) −P ∂αi ∂t + ∇·(αivi) +di+qij+vi·∇·Ti+αiρivi·fi+vi·bij. (2.21)

2.2.4 Enthalpy balance

Let us rewrite now the internal energy equation in terms of enthalpy hi =ei+P/ρi. Inserting this

expression in Eq. (2.19) we obtain that:

∂(αiρihi) ∂t + ∇·(αiρivihi) =αiρi dhi dt =αi dP dt − ∇·(αiqi) +di+qij. (2.22) Similarly, using the Eq. (2.21), we easily get the following equation for hi+ki:

∂[αiρi(hi+ki)] ∂t + ∇·[αiρivi(hi+ki)] =αiρi d(hi+ki) dt = =αi ∂P ∂t +vi· (αi∇P+ ∇·Ti) − ∇·(αiqi) +di+qij+αiρivi·fi+vi·bij. (2.23)

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2.2.5 Entropy balance

The specific internal energy eidepends upon the specific entropy siand upon the specific volume

ρ−1_i , that is:

ei=ei(si, ρ−1_i ). (2.24)

Using the definitions of temperature and pressure, total derivative of Eq. (2.24) gives: dei dt =Ti dsi dt −P dρ_i−1 dt = P ρi 1 αi dαi dt + ∇·vi , (2.25)

where the mass balance equation (2.6) was used. By inserting the last term of (2.25) in the internal energy balance (2.19) we obtain that:

∂(αiρisi) ∂t + ∇·(αiρivisi) =αiρi dsi dt = − ∇·(αiqi) Ti + di Ti +qij Ti . (2.26)

We consider again a closed fluid parcel with volume V(t), boundary surface A(t), and moving with velocity vi. If σiis the rate of production of entropy (per unit of volume), the integral form of the

entropy balance becomes: d dt Z V(t)αiρisidV + Z A(t) αiqi·dA Ti = Z V(t)σidV. (2.27)

The Reynolds transport theorem, the divergence theorem, and the usual contradiction argument applied to Eq. (2.27) lead to following entropy balance:

∂(αiρisi) ∂t + ∇·(αiρivisi) =αiρi dsi dt = −∇· αiqi Ti +σi. (2.28)

In virtue of the second law of thermodynamics σi ≥0, and σi =0 only for reversible processes. A

comparison of Eq. (2.26) and Eq. (2.28) yields:

σi = − αiqi·∇Ti T_i2 + di Ti +qij Ti ≥0. (2.29)

Since by definition the dissipation rate di ≤0, the inequality (2.29) implies that:

αiqi·

∇T

T ≤qij. (2.30)

For example, in absence of interphase heat exchange and if we use the Fourier’s law for heat conduc-tion q= −k∇T, (2.30) restricts the sign of the thermal conductivity to be positive.

2.3 Closure and constitutive relations

Above we presented the transport equations for a continuous medium, leaving some source terms in a general form. Now, we mainly discuss closure and constitutive relations for the granular phase. In particular, we provide some semi-empirical expressions for the frictional part of the stress tensor, one of which partially derived from a micromechanical description of contact forces between particles.

As regards the collisional behavior of the more dispersed component, it is possible to use an approach based on kinetic theory similarly to what is done for molecular fluids; however this theory also requires empirical input.

The central result of the kinetic approach is the derivation of a balance equation for the granular temperature (also called granular energy), which describes the transport of the particles’ velocity fluctua-tions within the flow. Another remarkable result that we will show is an equation of state that relates collisional pressure and granular temperature; this consititutive equation looks formally analogous to the ideal gas law.

From now on, unless otherwise advised, we will limit ourselves to studying the behavior of a two-phase system. The quantities relating to the solid fraction will be indicated without subscript, while those associated with the gaseous phase with the subscript g.

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§2.3 − Closure and constitutive relations 15

2.3.1 Momentum equation closure

Let’s consider the momentum equation2.14. First, we provide an expression for the source term fi

which takes into account the external bulk forces acting on phase i (solid or gaseous):

fi=g−αi∇P (2.31)

where g is the gravitational acceleration and P represents the thermodynamic pressure. The first term in RHS is easy to understand, since we will deal with gravity-driven currents in the present study. As for the second term, it is far from intuitive and we refer the reader to the derivation obtained by Enwald et al.(1996).

As already mentioned above, for example in the section2.1, the stress tensor for the granular phase must take into account the different regimes (solid, liquid or gaseous) in which the flow can occur. One way to do this is to write the stress tensor T as the sum of a collisional component Tcand a frictional component Tf (Si et al.,2019, and authors cited therein), i.e.:

T=Tc+Tf (2.32)

It should be remembered that, as pointed bySrivastava and Sundaresan(2003), Eq. (2.32) is not straightforwardly supported by a physical principle but it captures merely the two extreme limits of granular flows.

Collisional law

When the flow is dominated by collisions between the particles, the kinetic theory may be em-ployed to formulate the stresses acting on the averaged flow (Gidaspow,1994;van Wachem,2000), from which the following expression for the collisional stress tensor is obtained:

Tc= (−pc+λc∇·v)I+2µcS (2.33)

where pcis the collisional pressure, µcis the shear viscosity representing the effect of excessive mo-mentum flux due to collisions, λcis the bulk viscosity representing the effect of excessive momentum flux due to dilatation or contraction of the granular material, I is the identity tensor, and S is the rate of deviatoric strain tensor of the granular material, defined as follows:

S= 1

2 h

∇v+ (∇v)Ti−1

3(∇·v)I, (2.34)

where the superscript T indicates the transposition operator. The parameters pc, µc, and λcpresent

in Eq. (2.33) all depend on the granular temperature Θ, which was introduced to represent the kinetic energy of the granular particles controlled by velocity fluctuations (Gidaspow,1994). More preciselyΘ≡ h(δv)2i/3, where the angle brackets indicate the statistical average and δv represents

the fluctuation of the particles’ velocity.

The kinetic theory provides the following balance equation for granular temperature: 3 2 ∂(αρΘ) ∂t + ∇·(αρvΘ) = ∇·(κ∇Θ) +Tc:∇v−J (2.35)

where κ is the heat conductivity of the granular material (also called granular conductivity) and J is the dissipation rate due to inelastic particle collisions. The first term on the right side of Eq. (2.35) quantifies the diffusion of particle fluctuation energy, while the second term represents the production of particle fluctuation energy due to shear in the granular material.

An important underlying assumption of Eq. (2.35) is that work done by the frictional component of particle stresses is transformed directly into conventional thermal energy and it does not contribute to the fluctuation energy of the particles (Johnson and Jackson,1987).

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It has been clearly shown in the studies mentioned above that the expressions for the collisional pressure, the shear viscosity, the bulk viscosity, the granular heat conductivity, and the dissipation rate determined by a kinetic theory can generally be written as:

pc=Π(α, e)ρΘ (2.36)

µc=M(α, e)ρdΘ1/2 (2.37)

λc=Λ(α, e)ρdΘ1/2 (2.38)

κ=K(α, e)ρdΘ1/2 (2.39)

J=Ω(α, e)ρd−1Θ3/2−Ξ(α, e)ρΘ∇·v (2.40)

where e∈ [0, 1]is the coefficient of restitution and quantifies the degree of elasticity of the collisions; d is the diameter of the granular particles; andΠ, M, Λ, K, Ω, and Ξ are all dimensionless and positive functions of e and α. The value of these latter coefficients varies slightly depending on the model used: unless otherwise advised, the choices ofSi et al.(2019) were followed in this thesis work.

Eq.2.36is very interesting because it is formally analogous to the ideal gas law, in which the granular temperature (pressure) takes the place of the thermodynamic temperature (pressure). Many other arguments can be used to justify the definition of a granular temperature (Goldhirsch,2008). At the same time, although it is not directly measurable, the granular temperature appears as a fundamental quantity in the determination of all the collisional parameters just studied.

Finally, remember thatSi et al.(2019) introduced a correction to the restitution coefficient, to also take into consideration the torque due to tangential contacts between the particles, which is usually neglected in the conventional kinetic theory. Moreover theΛ and M coefficients were modified by the factor 1+ξd|∇α|to take also into account anisotropic collisions.

Frictional law

When a granular flow is dominated by continuous contacts among particles, it cannot be treated as an extreme case of the kinetic theory. However, we still adopt the general form of a rheological model in continuum mechanics, decomposing as usual the frictional stress tensor Tf into an isotropic and a deviatoric part (Johnson and Jackson,1987;van Wachem,2000), namely:

Tf = −pfI+τf (2.41)

where pf is the frictional pressure and τf is the particle shear stress. Frictional pressure will be described in more detail below (see Eq.2.48). For cohesionless granular materials, the frictional law is often written as:

τf =2µfS (2.42)

where µf ≡τ/S is the frictional viscosity, τ≡

q

(τf: τf)/2, S≡

√

2S : S, and the symbol ":" denotes the double inner product (S : S≡SijSij). Moreover, we assume that the generalized shear stress τ has

the following functional form: τ=τ(pf, α).

An usual approach to establish a frictional law for dense granular flows in the quasi-static regime is to apply the well-known yield criterion of granular materials in soil mechanics, including Mohr-Coulomb, von Mises, Drucker-Prager, and Cam-Clay criteria. The simplest expression for τ can be given according to the Mohr-Coulomb criterion (Schaeffer,1987;Johnson et al.,1990;Nedderman, 1992):

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where φ is the internal friction angle of the granular material: it is the angle, measured between the normal force and resultant force, that is attained when failure just occurs in response to a shearing stress; its value is determined experimentally. In the following we will refer to Eq. (2.43) as the "Johnson-Jackson-Schaeffer model".

To include the effect of shear dilatation and contraction of the granular material, a more elaborate frictional model based on the smoothed Cam-Clay yield criterion was developed bySrivastava and Sundaresan(2003), also known as "Princeton model":

τ= pf 1+ n−1 nS sin φ∇·v sin φ (2.44) where n=      √ 3 2 sin φ if ∇·v≥0 1.03 if ∇·v<0 (2.45)

It is important to note that Eq.2.44represents the dilatant effect of sheared granular materials rather well, but its accuracy is quite questionable if the shear contraction should also be correctly represented (Si et al.,2019). This consideration will be justified below.

Let us now analyze the salient features of the rheological model developed byChristoffersen et al. (1981), which is based on a micromechanical description of granular medium behavior. Most of the simulations presented in this thesis have been carried out with this frictional model. Considered is a sample of cohesionless granular material, consisting of rigid particles, and which is subjected to overall macroscopic average stresses. On the basis of the principle of virtual work, and by an examination of the manner by which adjacent granules transmit forces through their contacts, a general representation is established for the macroscopic stresses in terms of the volume average of the tensorial product of the contact forces and the vectors which connect the centroids of adjacent contacting granules. Similarly, the corresponding kinematics is examined and the overall macroscopic deformation rate and spin tensors are developed in terms of the volume average of relevant microscopic kinematic variables.

In summary,Christoffersen et al. (1981) obtained the following general relation between the macroscopic stress Tf _{and the bulk rate of strain S:}

" τ pf 2 −sin2φ # (Tf+pfI): S τ = τ pf cos 2 φ ∇·v (2.46)

Substituting Eq. (2.42) into Eq. (2.46), and remembering Eq. (2.41), we obtain:

τ= pf   s sin2φ+ cos 2_φ 2S ∇·v 2 +cos 2 φ 2S ∇·v   (2.47)

Christoffersen et al.’s model is very close to Srivastava and Sundaresan’s model for description of shear dilatancy (∇·v>0), but it has an improved accuracy for description of shear contraction (∇·v<0). We also note that the three frictional models just described in the Eqs. (2.43)-(2.44)-(2.47), respectively, are equivalent if the granular flow is incompressible, i.e. when∇·v=0.

Fig. (2.4) shows a plot of the functional relation between the dimensionless quantities τ/pf and

(∇·v)/S. The experimental data ofShibuya et al.(1997) are also included for comparison. It emerges that Eq. (2.47) best represents the measures, in all the regimes considered. More generally, in the literature it is reported that granular materials often have overestimated fluidity as compared to the experimental observations, and the reason is often related to a drastic drop of the frictional stress. Owing to these reasons, the model ofChristoffersen et al.(1981) was employed in this study.

We must now determine the expression of the frictional pressure pf. In other words, it is a matter of evaluating the effective particle pressure due to enduring contacts according to the flow conditions.

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Figure 2.4:Comparison of the normalized frictional coefficients from different frictional laws with experimental data. Source:Si et al.(2019).

It is evident that the effective particle pressure depends closely on the volume fraction of the granular material, but a directly validated expression based on prime principles does not available in literature.

Experimentally, granular materials become static at close-packed volume fraction, so in this regime the frictional pressure is expected to diverge. This, on the one hand, prevents the particles (considered rigid) from compacting further and on the other hand, indirectly, increases the frictional viscosity in order to stop the flow. At the other extreme, the pressure vanishes at loose-packed volume fraction.

In light of these observations, the following expression introduced byJohnson et al.(1990) is often suggested: pf =    Fr(α−αmin) γ1

(αmax−α)γ2 if αmin<α<αmax

0 if α≤αmin

(2.48)

where Fr ≡ ηρgd, αmax is the close-packed volume fraction, and αminis the loose-packed volume

fraction. The values of αmax and αmindepend on the arrangement pattern and size distribution of the

granular particles in a particular problem. η, γ1, and γ2are empirical constants.

It may be necessary to point out that Eq.2.48is essentially a conceptual model based on a limited understanding of the physics. The generality of the expression is thus quite uncertain but, without a better choice, we choose to follow previous studies (Johnson et al.,1990;van Wachem,2000;Si et al., 2019) which are based on observations at laboratory scales, and expect Eq.2.48to be reasonably good for gravity-driven granular flows.

In this thesis we will try to extend the validity of Eq.2.48in geophysics for the study of snow avalanches. In the absence of useful experimental data to calibrate the parameters, a maximum threshold for pf has been established in order to guarantee that α . αmax at least in one cell of

the computational domain. We will explore these aspects in the section4.2dedicated to multiscale simulations.

Interphase momentum transfer

Interphase exchange coefficients are derived from semi-empirical correlations. Their validity can depend on the flow regime, thus including also correction for turbulence and different particle concen-trations. In this thesis the gas-solid drag model adopted byGidaspow(1994) has been implemented, neglecting other exchange terms:

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where Kdis the interphase momentum transfer coefficient. For αg<0.8, we use the Ergun equation:

Kd=150

α2µg

αgd2

+1.75ρgα|v−vg|

d (2.50)

where µgis the dynamic viscosity of the gaseous phase. If αg≥0.8, we use the Wen-Yu model:

Kd= 3 4Cd ααgρg|v−vg| d α −2.65 g (2.51) Cd=    24 Res h 1+0.15(Res)0.687 i if Res <1000 0.44 if Res ≥1000 (2.52) Res = αgρgd|v−vg| µg (2.53)

2.3.2 Enthalpy equation closure

We present here the closure relations for the enthalpy equation2.22. A general formula widely used in the literature for the interphase heat exchange is:

qsg= −Kht(T−Tg) (2.54)

where T and Tgare the thermodynamic temperatures of the granular phase and the gaseous one,

respectively. The gas-particle heat transfer coefficient Khtwas obtained from the Ranz-Marshall model

(Ranz and Marshall,1952):

Kht=Nus 6αkg d2 (2.55) Nus =2+0.6Re1/2Pr1/3 (2.56) Re= ρgd|v−vg| µg , Pr = cpgµg kg (2.57)

where k is the thermal conductivity and cpis the specific heat at constant pressure.

The heat flows qihave been modeled with a Fourier-like law, i.e.:

qi = −αiki∇Ti (2.58)

As mentioned in section2.3.1, we follow the approach ofJohnson and Jackson(1987) and affirm that the dissipation rate in the granular phase is due to the friction forces only, so:

ds=Tf:∇v (2.59)

Finally, the thermal and caloric equations of state used for the phase i are the following:

ρ=constant, ρg= P e RTg (2.60) Ti= hi cpi (2.61) where eR≡R/M indicates the specific gas constant, R is the gas constant, and M is the average molar mass of air.

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Figure 2.5:Control volume (dashed line) for derivation of the energy-flux boundary condition. Source:Johnson and Jackson(1987), edited.

2.3.3 Boundary conditions

The boundary conditions at the walls for the granular phase are now presented. For the laboratory experiments, following the choices ofSi et al.(2019), the no-slip condition and the zero gradient condition for the granular temperature was used, i.e.:

ˆn·∇Θ=0 (2.62)

where ˆn is the normal unit vector to the wall. This means that the boundaries do not supply or absorb granular fluctuation energy. It must be remembered that the no-slip condition is usually not the best choice for studying the behavior of a granular flow. However, in the present case, such a choice seems reasonable because a layer of the same particles was glued to the bottom in the experiment.

In the snow avalanche simulations was used the Johnson and Jackson wall boundary conditions (Johnson and Jackson,1987;Reuge et al.,2008). The condition on the slip velocity can be obtained by equating the tangential force per unit area exerted on the boundary by the particles to the correspond-ing stress within the particle assembly close to the boundary. So we obtain that:

µ ˆn·∇vt= − πφραg0

2√3αmax

Θ1/2_v

t−pftan δ (2.63)

where vtis the component of the slip velocity (i.e., v−vwall) parallel to the wall, µ=µc+µf is the

total dynamic viscosity of granular medium, φ represents the specularity coefficient, g0is the radial

distribution function, and δ indicates the angle of friction between the surface and the particulate material.

The value of φ depends on the large-scale roughness of the surface and varies between zero for perfect specular collisions and unity for perfectly diffuse collisions. g0, also called pair correlation

function, describes how the number of particles varies as a function of distance from a reference particle. In this thesis the granular material is assumed to consist of cohesionless rigid spheres, so it seemed appropriate to use the radial function proposed by Carnahan-Starling (Carnahan and Starling, 1969), i.e.: g0(α) = 1 1−α+ 3α 2√1−α + √ α 2(1−α)3 (2.64)

The second boundary condition is obtained from energy balances over the control volume shown in Fig. (2.5) whose upper and lower faces have unit area and whose depth will be allowed to tend to zero, then: κ ˆn·∇Θ= πφραg0 2√3αmax v2tΘ1/2− √ 3πραg0(1−e2w) 4αmax Θ 3/2 _(2.65)

where ewparticle-wall restitution coefficient which quantifies the degree of elasticity of the collisions

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CHAPTER

3 Introduction to OpenFOAM

The balance equations presented in the two previous chapters are nonlinear and, even for simple systems, an analytical solution for them is not available. In fact, for example, the momentum equation for the solid phase has an even greater complexity than the well-known Navier-Stokes equation for molecular fluids, of which a general analytical solution is not known.

An alternative approach is to look for approximate solutions of the discretized equations by using numerical simulations. For this purpose we have used OpenFOAM® 7, an open source software C++ based for the development of numerical solvers, and pre-/post-processing utilities for the sim-ulation of continuum mechanics systems, mainly computational fluid dynamics (CFD) problems. With the help of sufficient abstraction in OpenFOAM, users can focus on the mathematical model of the problem without considering too implementation details. This allowed us to modify the default solver twoPhaseEulerFoam with relative ease, adding a new rheological model (Eq.2.47) and a new boundary condition (Eq.2.63).

In this chapter we will introduce the discretization procedures based on the Finite Volume Method (FVM) and their implementation in OpenFOAM. Furthermore, we will describe the minimum struc-ture of an OpenFOAM case, the procedures for creating a mesh with the utilities blockMesh and snappyHexMesh, as well as how to set up the discretization schemes by analyzing those used in the simulations.

3.1 FV discretization procedures

The partial differential equations (PDEs) that we want to solve contain derivatives of scalar, vector, and tensor fields with respect to time and space. In this section we will show how a tensor field is constructed in OpenFOAM and how the derivatives of these fields are discretized into a set of algebraic equations to be solved numerically.

In general, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. The numerical approach adopted by OpenFOAM and chosen for this thesis work is based on the Finite Volume Method (FVM). This can be described by defining three main procedures (Versteeg and Malalasekera,2007;Moukalled et al.,2016):

• spatial discretization, dividing the spatial domain into finite volumes, or cells;

• temporal discretization, dividing the time domain into a finite number of time intervals, or steps; • solving algebraic systems generated by the discretization procedure.

Time discretization is required for transient problems: the time interval is divided into a finite set of time steps of duration∆t, which can vary according on some condition calculated during the

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Figure 3.1:Parameters in finite volume discretization. Source:Greenshields(2015).

simulation. Spatial discretisation requires the subdivision of the domain into a number of cells, also called control volumes, which form the computational mesh. OpenFOAM uses only contiguous cells, i.e. cells that do not overlap each other, and that completely fill the spatial domain.

Fig.3.1shows a typical example of two contiguous cells. The values of the calculated quantities are usually stored at the cell centres (or centroids) P and N, although they can also be stored on faces or vertices. Each cell is delimited by a set of flat faces, indicated from now on with the generic label f . OpenFOAM uses unstructured meshes, i.e. there are no constraints on the orientation and number of cell faces. This allows for greater freedom in mesh generation and manipulation, especially on complex or time-varying domain geometries.

The first step of the FV discretization process is to integrate the terms of PDEs over a cell volume V. Most terms that contain spatial derivatives are then converted to integrals over the cell surface S bounding V by Gauss’s theorem:

Z

V∇ ?φ dV= Z

SdS?φ (3.1)

where φ indicates a generic tensor, S is the surface area vector, and the symbol?represents in the RHS any tensor product (inner, outer, and cross) and in the LHS the respective derivatives (divergence, gradient, and curl). Thereafter, the surface integrals are linearized using schemes that can be specified within the code or can be set through an input file and stored within a fvSchemes class object. The main discretization methods implemented in OpenFOAM for the various PDEs terms will be described in more detail in the paragraph3.1.1.

Finally, the discretization procedure creates a set of algebraic equations which are commonly expressed in the following matrix form:

[A][xn] = [bo] (3.2) where[A]is a square matrix,[xn_]_{is the column vector of dependent variable at the new time step, and}

[bo]is the source vector at the old time step. In principle, the system (3.2) could be solved by using standard linear algebra techniques, however this would be too expensive in terms of computation time due to the large size of the matrix[A]. The alternative approach of most of the OpenFOAM solvers is to adopt segregated solution algorithms. The idea is to solve the equations one-by-one, resolving the coupling by means of an iterative procedure.

Most fluid dynamics solvers use either the pressure-implicit split-operator (PISO), the semi-implicit method for pressure-linked equations (SIMPLE) algorithms, or a combined PIMPLE algorithm. These algorithms are iterative procedures for coupling equations for momentum and mass conservation. In the simulations presented in this thesis, we have used the solver twoPhaseEulerFoam which is

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§3.1 − FV discretization procedures 23

Figure 3.2:Discretisation of PDE terms in OpenFOAM. Source:Greenshields(2015), edited.

based on the PISO and PIMPLE algorithms, suitable for transient problems.

The OpenFOAM code is written by keeping a high level of abstraction, allowing the developers to write new models and algorithms with relative facility. For this purpose, OpenFOAM provides by default an array template class List<Type>, which allows you to create a list of objects of class Type that inherits the functions of Type. For example, a list List of vectors vector is List<vector>. The lists of tensor classes are defined by the template class Field<Type>. For better code readability, all instances of Field<Type> are renamed using typedef declarations such as scalarField, vectorField, and so on.

The algebraic operations performed between Fields are obviously subject to restrictions linked to the number of their elements. Furthermore, OpenFOAM encourages the user to attach dimensional units to each tensor in order to perform a dimensional check, thus avoiding the implementation of operations without physical meaning. Units are defined using the dimensionSet class and a tensor with units is defined using the dimensioned<Type> template class.

The vectors[xn]and[bo]of Eq. (3.2) contain a list of values defined at locations in the geometry, i.e. a geometricField<Type>, or specifically volField<Type> when using FV discretisation. The ma-trix[A]contains the list of coefficients of a system of algebraic equations, and it is described by a class of its own, called fvMatrix. fvMatrix<Type> is created by discretizing a geometricField<Type> and then inherits the <Type>. The newly defined class supports many of the algebraic operations between matrices such as addition, subtraction, and multiplication.

Each term in a PDE is represented using the static function classes finiteVolumeMethod (or fvm) and finiteVolumeCalculus (or fvc). They contain functions that represent differential operators such as divergence, Laplacian, time derivative, and so on, that discretize geometricField<Type>s. The functions of the class fvm compute the implicit derivatives and return an fvMatrix<Type>; fvc functions compute explicit derivatives and other explicit operations, returning a geometricField<Type>.

Table3.2lists the main functions that are available in the fvm and fvc classes to discretise the most common terms of PDEs.

3.1.1 Discretization schemes for PDEs terms

In this section we will describe some basic discretization methods for PDEs terms. The subscripts N and P refer to the fields calculated in the cell centres, whereas the subscript f refers to those calculated

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on the faces (see Fig.3.1).

Laplacian term

The Laplacian term is first integrated over a control volume and then linearized as follows: Z

V∇ · (Γ∇φ)dV= Z

SdS· (Γ∇φ) =

∑

_f ΓfSf · (∇φ)f (3.3)

The last term in the RHS of Eq.3.3is implicit if the length vector d between the center of a reference cell P and the center of a neighboring cell N is orthogonal to the face plane f (see Fig.3.1):

Sf· (∇φ)f = |Sf|

φN−φP

|d| (3.4)

In case of non-orthogonal meshes, an explicit corrective term is added, which is obtained by interpo-lating the calculated gradients at the cell centers.

Convection term

The convection term is linearized in a similar way to the Laplacian one: Z V∇ · (ρvφ )dV = Z SdS· (ρvφ ) =

∑

f Sf · (ρv)fφf ≡

∑

f Ffφf (3.5)

The φf surface field can be evaluated through a variety of schemes implemented in OpenFOAM.

• Central differencing (CD), is second-order accurate but unbounded, φf is defined as follows:

φf = fxφP+ (1− fx)φN (3.6)

where fx≡ f N/PN, f N is the distance between f and the cell centre N, and PN is the distance

between cell centres P and N.

• Upwind differencing (UD), is bounded but first-order accurate; this scheme determines φf

from the direction of flow:

φf =

(

φP for F≥0

φN for F<0

(3.7) • Blended differencing (BD), are schemes that combine UD and CD in an attempt to preserve

boundedness with reasonable accuracy:

φf =γ(φf)CD+ (1−γ)(φf)UD (3.8)

where γ is called blending coefficient and it is evaluated by OpenFOAM through the Gamma differencing scheme. OpenFOAM offers also other well-known schemes such as van Leer, SUPERBEE, MINMOD, etc.

Divergence

The divergence term described below is explicit and distinct from the convection term, in that it does not involve the product of a velocity and a dependent variable. It is discretized in the usual way:

Z

V∇ ·φ dV= Z

SdS·φ=

∑

_f Sf ·φf (3.9)

The fvc::div function can take as its argument either a surface<Type>Field or a vol<Type>Field; in the latter case, φf is interpolated to the face by central differencing (Eq.3.6).

Modeling and Simulation of Rapid Granular Flows. Application to Powder Snow Avalanches

U

NIVERSITÀ DEGLI

S

TUDI DI

P

ISA

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di laurea in Fisica

Master’s degree thesis

Modeling and Simulation of

Rapid Granular Flows:

Application to

Powder Snow Avalanches

Contents

Acknowledgements

CHAPTER

1

Introduction

1.1

Overview: granular media

1.2

Rapid granular avalanches in geophysics

1.3

Structure of the thesis

CHAPTER

2

Governing equations

2.1

Granular systems as continuous media

2.2

Transport equations

∑

2.2.1

Mass balance

2.2.2

Momentum balance

∑

2.2.3

Energy balance

2.2.4

Enthalpy balance

2.2.5

Entropy balance

2.3

Closure and constitutive relations

2.3.1

Momentum equation closure

2.3.2

Enthalpy equation closure

2.3.3

Boundary conditions

CHAPTER

3

Introduction to OpenFOAM

3.1

FV discretization procedures

3.1.1

Discretization schemes for PDEs terms

∑

∑

∑

∑