force moyenne et fluctuations subies par un obstacle
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Force moyenne et fluctuations subies par un obstacleindĂ©formable soumis Ă lâĂ©coulement confinĂ© dâun milieu
granulaireFrançois Kneib
To cite this version:François Kneib. Force moyenne et fluctuations subies par un obstacle indĂ©formable soumis Ă lâĂ©coulement confinĂ© dâun milieu granulaire. MĂ©canique des fluides [physics.class-ph]. UniversitĂ©Grenoble Alpes, 2017. Français. NNT : 2017GREAI024. tel-01688165
THĂSEpour obtenir le grade deDOCTEUR DE LâUNIVERSITĂ GRENOBLE ALPES
SpĂ©cialitĂ© 2MGE : MatĂ©riaux, MĂ©canique, GĂ©nie-civil, ĂlectrochimieArrĂȘtĂ© ministĂ©riel : 25 mai 2016
Présentée par
François KNEIB
ThĂšse dirigĂ©e par Thierry FAUG, chargĂ© de recherches, IRSTEA,et FrĂ©dĂ©ric DUFOUR, professeur, Grenoble INPprĂ©parĂ©e au sein du laboratoire IRSTEA de Grenoble,dans lâĂ©cole doctorale I-MEP2 : IngĂ©nierie - MatĂ©riaux,MĂ©canique, ĂnergĂ©tique, ProcĂ©dĂ©s, Production
Force moyenne et fluctuations subies parun obstacle indĂ©formable soumis Ă lâĂ©coulement confinĂ© dâun milieu granulaire
Mean force and fluctuations experiencedby a non-deformable obstacle subjected tothe flow of a confined granular material
ThÚse soutenue publiquement le 2 juin 2017,devant le jury composé de :Monsieur Farhang RADJAI
Directeur de Recherche, Université de Montpellier, Rapporteur
Monsieur Philippe GONDRET
Professeur, Université Paris-Sud, Rapporteur et Président
Madame Evelyne KOLB
Maßtre de conférences, Université Pierre-et-Marie-Curie, Examinatrice
Monsieur Ămilien AZĂMA
Maßtre de conférences, Université de Montpellier, Examinateur
Monsieur Thierry FAUG
Chargé de Recherche, Irstea, Co-directeur de thÚse
Monsieur Frédéric DUFOUR
Professeur, Grenoble INP, Co-directeur de thĂšse
Monsieur Mohamed NAAIM
Directeur de recherche, Irstea, Co-encadrant de thĂšse
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Remerciements
Je souhaite tout dâabord remercier mes directeurs de thĂšse pour leur accompagnement etleur soutien tout au long de ces trois annĂ©es. Merci Ă Thierry Faug pour sa proximitĂ© etsa disponibilitĂ© malgrĂ© notre distance gĂ©ographique pendant les dix-huit premiers mois.Son ambition et son ouverture dâesprit mâont permis de trouver mon chemin lors deces recherches, sans jamais nuire Ă mon autonomie. Merci Ă FrĂ©dĂ©ric Dufour de mâavoirfait partager sa grande expĂ©rience et de mâavoir aidĂ© Ă prendre du recul Ă de nombreusesoccasions. Merci Ă©galement Ă Mohamed Naaim, encadrant de ma thĂšse, dâavoir cru malgrĂ©moi en mes capacitĂ©s avant mĂȘme le dĂ©but de ces travaux.Je vous suis reconnaissant Ă tous les trois dâavoir tout mis en Ćuvre pour que je puissedĂ©couvrir lâAustralie en seconde annĂ©e de thĂšse. En plus des rencontres enrichissantes Ă la School of Civil Engineering de lâUniversitĂ© de Sydney, le voyage fut inoubliable.
Je tiens Ă©galement Ă exprimer ma gratitude aux membres du jury pour lâintĂ©rĂȘt quâilsont portĂ© Ă mon travail de thĂšse. Merci Ă Philippe Gondret et Farhang Radjai qui ontacceptĂ© de se prĂȘter Ă lâimportante tĂąche de rapporteur, et aussi Ă Ăvelyne Kolb et ĂmilienAzĂ©ma pour avoir participĂ© au jury. Leurs remarques sur mes travaux et les nombreuxĂ©changes que nous avons eus Ă lâissue de ma prĂ©sentation ont Ă©tĂ© trĂšs enrichissants.
Jâai eu lâoccasion de faire de nombreuses rencontres durant ces trois annĂ©es de thĂšse.Alors que certaines se sont rĂ©vĂ©lĂ©es dĂ©terminantes dans lâissue de mes travaux, dâautresont Ă©tĂ© enrichissantes dâun point de vue personnel.Merci en particulier Ă Gilles Nicolet qui mâa accompagnĂ© dans ma dĂ©couverte du domainedes statistiques, nâhĂ©sitant jamais Ă prendre de son temps pour mâaider. Camarades dĂšsles premiers jours, notre collaboration et notre cohabitation mâa Ă©tĂ© fortement utile etagrĂ©able. Avec ton calme lĂ©gendaire et ton sens de lâhumour, tu as su ĂȘtre mon allier auquotidien et dans les moments difficiles. Merci Ă©galement Ă Nicolas Eckert pour son Ćilexpert de statisticien ainsi quâĂ Guillaume Chambon pour son soutien et ses compĂ©tencessur le comportement des fluides complexes. Merci Ă Bruno Chareyre et Emanuele Cata-lano, car les outils quâils ont dĂ©veloppĂ©s avant ma thĂšse on Ă©tĂ© le support Ă bon nombrede mes rĂ©sultats de thĂšse.Jâai Ă©galement une pensĂ©e pour mes camarades de pause, indispensables Ă mon Ă©quilibrepar le sentiment dâĂ©vasion quâils mâont procurĂ©. Merci donc aux « caillouteux » de lâĂ©quipe
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PIER : Franck, Fred (grand gourou), Jean-Baba, David, Sylvain, J2M, JĂ©rĂŽme... Merci Ă MĂ©lanie pour la bonne humeur que tu as dissĂ©minĂ©e dans notre bureau mais aussi pourtes lĂ©gendaires gĂąteaux gourmands.Enfin, merci Ă Fred Ousset dont la place dans ces remerciements ne peut ĂȘtre quâĂ la tran-sition entre mes camarades de boulot et mes amis. Ton esprit de jeune homme tâaidera âjâen suis sĂ»r â dans les grands changements qui tâattendent pour lâannĂ©e 2018, et jâespĂšreque nos expĂ©riences de modĂ©listes amateurs nâen seront que renforcĂ©es.
Merci Ă mes amis qui, parfois mĂȘme sans sâen rendre tout Ă fait compte, mâont soutenupendant ces annĂ©es en sâintĂ©ressant Ă mes travaux, mais surtout par tous les momentsdivertissants quâils mâont procurĂ©. Je pense Ă toute la « bande de KnĂ€kes » et en particulierĂ mes acolytes Luc et Geoffrey. Merci Ă©galement Ă Camille et Tudal nos chers Minimoys,sans oublier Damien le colonel ni William la panthĂšre.Merci Ă ma famille : mes parents qui mâont permis dâarriver jusque lĂ , mes sĆurs Estelleet Marie, ainsi que Patrick et Nicole. Isabelle, Damien, Annabelle, Boris, merci pour vosencouragements sans faille, pour la « pensĂ©e » que vous avez eue pour moi le 2 juin quimâa aidĂ©e â jâen suis convaincu, Ă obtenir de « Beaunes » remarques le jour J.
Jâai choisi de garder cette place de choix en conclusion de mes remerciements Ă cellesqui me sont les plus chĂšres, NoĂ©mie et Anne-Sophie. NoĂ©mie, mĂȘme si certains dĂ©tailsde ce manuscrit tâĂ©chappent encore, sache que tu mâas donnĂ© beaucoup de courage danssa rĂ©daction en mâaidant Ă relativiser tous mes soucis et mâoffrant un nouveau rĂŽle Ă responsabilitĂ©s : celui de papa. Ton arrivĂ©e a Ă©tĂ© un grand bonheur que nous ne sommespas prĂȘts dâoublier. Anne-Sophie, je te remercie pour tes sacrifices sans lesquels lâaventurede la thĂšse nâaurait pas mĂȘme pu commencer. Merci aussi pour ton soutien quotidien,moteur de ma persĂ©vĂ©rance ; pour tes conseils, mâapportant confiance en moi ; et pour tonrĂ©confort, renforçant ma motivation dans les moments de doute. Enfin, merci pour la joieque tu mâa procurĂ©e pendant ces trois annĂ©es, et pour les annĂ©es Ă venir.
Résumé
Les Ă©tudes existantes sâintĂ©ressant au dimensionnement des structures paravalanches sâap-
puient généralement sur des signaux de force lissés dans le temps. Cependant, le caractÚre
hétérogÚne de la neige en écoulement est responsable de fortes fluctuations systématiquement
observées. Cette thÚse a pour objectif de caractériser les fluctuations de force exercées sur un
obstacle surversé par un écoulement granulaire. Des modélisations numériques par la méthode
des Ă©lĂ©ments discrets sont mises en Ćuvre pour simuler lâinteraction entre la neige, reprĂ©sentĂ©e
par un ensemble plan de particules sphériques, et un mur immobile et indéformable. Une par-
ticularitĂ© de ce travail rĂ©side dans lâĂ©tude dâune gamme trĂšs large de rĂ©gimes dâĂ©coulement, de
quasistatique à collisionnel. Deux systÚmes modÚles sont développés dans le but de focaliser la
zone dâĂ©tude en amont de lâobstacle, et de permettre le contrĂŽle des variables macroscopiques de
lâĂ©coulement (vitesse de cisaillement, pression de confinement, tailles des systĂšmes). Le premier
confine les grains entre quatre parois dont lâune impose un cisaillement Ă vitesse constante, alors
que les signaux de force sont mesurés sur la paroi faisant face au cisaillement. Le second systÚme
confine les grains entre deux parois (dont une cisaillante) et une condition aux limites périodique
dans la direction du cisaillement, alors que le mur est immergé dans les grains. Chaque systÚme
est étudié via une analyse moyenne puis les fluctuations sont caractérisées à partir des signaux
de force instantanés.
Le nombre inertiel macroscopique construit sur la vitesse de cisaillement et la pression de
confinement imposĂ©es au systĂšme sâavĂšre ĂȘtre une variable de contrĂŽle Ă la fois de la dynamique
moyenne et des fluctuations dans les deux systÚmes. Une loi empirique a été établie pour chacun
des systĂšmes pour prĂ©dire la transmission de force moyenne sur lâobstacle en fonction du nombre
inertiel macroscopique, et la mesure de grandeurs locales a révélé que la loi rhéologique ”(I)
des milieux granulaires en écoulement est respectée quasiment partout. Les autocorrélations des
signaux de force sur le mur Ă lâĂ©chelle mĂ©soscopique ont rĂ©vĂ©lĂ© lâexistence dâun effet mĂ©moire
des systÚmes aux faibles nombres inertiels, disparaissant avec la transition vers le régime dense
inertiel. Les distributions de forces à trois échelles spatiales différentes sont également contrÎlées
par le nombre inertiel macroscopique : aux régimes lents les distributions sont resserrées et
approchent une forme Gaussienne, aux régimes rapides les distributions sont exponentielles. Des
lois de probabilité log-normales tronquées à trois paramÚtres ont été établies afin de prédire de
façon empirique les distributions de force sur lâobstacle.
Ce travail contribue Ă approfondir les connaissances sur la composante moyenne et les fluc-
tuations de force subies par un obstacle soumis à un écoulement granulaire. Les résultats obtenus
permettent dâenvisager la modĂ©lisation de systĂšmes gravitaires sâapprochant des conditions rĂ©elles
dâĂ©coulements, permettant ainsi la comparaison avec des expĂ©rimentations de laboratoire dans
le but de mieux dimensionner les structures de génie-civil.
Mots clés : écoulement granulaire, obstacle, force, méthodes des éléments discrets, fluctuations,
autocorrélation, distribution.
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Abstract
The existing studies dealing with the design of civil-engineering structures against snow
avalanches are generally based on force times series that are smoothed over time. However the
strong heterogeneity of snow leads to systematic observations of a high level of force fluctuations.
This PhD thesis aims at characterizing the force fluctuations exerted on an obstacle that is
overflowed by a granular flow. Numerical simulations based on the discrete elements method are
implemented to model the interaction between the snow, represented by an assembly of spherical
particles, and a rigid motionless wall-like obstacle. A key feature of this work is the broad range
of flow regimes investigated, from quasistatic to collisionnal. Two model systems are studied in
order to focus on a zone restricted to the upstream of the obstacle, and to allow a full control of
the macroscopic flow variables (shearing velocity, confinement pressure, system sizes). The first
one confines the grains between four walls from which the top one imposes a constant shearing
velocity while the force signals are measured on the wall facing the corresponding displacement.
The second system confines the grains between a static bottom wall, a shearing top wall, and a
periodic boundary condition in the shear direction, while the wall-like obstacle is fully immersed
in the grains. Each system is studied through a time-averaged analysis then the fluctuations are
characterized from the instantaneous force time series.
The macroscopic inertial number built from the shear velocity and the confinement pressure
imposed to the system turns out to be the main control variable of both the mean dynamics and
the fluctuations in the systems. An empirical law has been established to predict the mean force
transmission on the obstacle as a function of the macroscopic inertial number for each of the
two systems, and the measurement of local strain and stress tensors revealed that the granular
flow ”(I)-rheological law is respected nearly everywhere in the samples. The autocorrelations of
force signals on the obstacle at the mesoscopic scale revealed the presence of a memory effect of
both systems at low inertial numbers which vanishes with the transition from the quasistatic to
the dense inertial flow regimes. The force distributions at three different spatial scales are also
controlled by the macroscopic inertial number : for slow regimes the distributions are tightened
and resemble Gaussian shapes, for fast regimes the distributions are rather exponential. Trun-
cated log-normal probability density functions (with three parameters) have been established in
order to predict empirically the force distributions on the obstacle.
This work contributes to advance the knowledge on both the time-averaged and the fluctua-
ting components of the force exerted on a wall subjected to a granular flow. The results enable
to look forward with the modeling of gravity-driven systems approaching real flow conditions,
thus allowing a comparison with laboratory experiments and full-scale measurements, with the
aim of better designing of civil engineering structures impacted by avalanches.
Keywords : granular flow, obstacle, force, discrete elements method, fluctuations, autocorrela-
tion, distribution.
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Table des matiĂšres
1 Introduction 5
1.1 Contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Avalanches de neige et ouvrages de protection . . . . . . . . . . . . 6
1.1.2 Ăcoulements granulaires secs . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Interaction avalanche - obstacle : Ă©tat de lâart . . . . . . . . . . . . 16
1.2 Objectifs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 MĂ©thode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.1 La méthode des éléments discrets (DEM) . . . . . . . . . . . . . . . 24
1.3.2 SystĂšmes modĂšles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 Grille de lecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Ătude de la force moyenne subie par un obstacle soumis Ă un Ă©coulement
granulaire cisaillé 33
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Method and simulated granular system . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Discrete element method and contact laws . . . . . . . . . . . . . . 36
2.2.2 Lid-driven cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Mean force on the sidewall facing to the upper shearing velocity . . . . . . 42
2.3.1 Effect of the cavity length . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.2 Effect of the macroscopic inertial number . . . . . . . . . . . . . . . 44
2.3.3 Toward an empirical model for the mean force . . . . . . . . . . . . 46
2.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Fluctuations de force sur un obstacle en interaction avec un Ă©coulement
granulaire cisaillé 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Force distributions in granular media . . . . . . . . . . . . . . . . . . . . . 56
3.3 The granular lid-driven cavity simulated by DEM . . . . . . . . . . . . . . 58
3.3.1 Macroscopic boundary conditions . . . . . . . . . . . . . . . . . . . 58
3.3.2 Micromechanical parameters . . . . . . . . . . . . . . . . . . . . . . 60
1
2 Table des matiĂšres
3.4 Time-averaged dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.1 Steady vortex and mean force . . . . . . . . . . . . . . . . . . . . . 613.4.2 The local ”(I)ârheology . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Force data autocorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.1 Data recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.2 Force time-series autocorrelation . . . . . . . . . . . . . . . . . . . . 66
3.6 Force distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.6.1 Force distributions at microscale . . . . . . . . . . . . . . . . . . . . 713.6.2 Force distribution at macroscale . . . . . . . . . . . . . . . . . . . . 723.6.3 Force distribution at mesoscale . . . . . . . . . . . . . . . . . . . . 75
3.7 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.7.1 Time-averaged dynamics . . . . . . . . . . . . . . . . . . . . . . . . 773.7.2 Grain-wall force distributions . . . . . . . . . . . . . . . . . . . . . 783.7.3 Quasistatic to dense inertial regime . . . . . . . . . . . . . . . . . . 80
3.8 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Force moyenne et fluctuations sur un mur immergé dans un écoulement
granulaire cisaillé 85
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2 DEM simulation of the wall immersed in a granular flow . . . . . . . . . . 89
4.2.1 Macroscopic boundary conditions . . . . . . . . . . . . . . . . . . . 894.2.2 Micromechanical parameters . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Mean dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.2 Mean force on the obstacle . . . . . . . . . . . . . . . . . . . . . . . 944.3.3 The local ”(I)ârheology . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 Force time-series autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . 984.5 Force distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5.1 Force distributions on the entire obstacle . . . . . . . . . . . . . . . 1004.5.2 Force distributions on portions of the obstacle . . . . . . . . . . . . 102
4.6 Effect of the system length . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.7 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.7.1 Time-averaged dynamics . . . . . . . . . . . . . . . . . . . . . . . . 1074.7.2 Force fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.7.3 Impact for a real-world application . . . . . . . . . . . . . . . . . . 109
5 Conclusion 111
5.1 Contexte et méthode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2 SynthÚse des résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Table des matiĂšres 3
5.2.1 Cavité granulaire entraßnée par cisaillement . . . . . . . . . . . . . 1135.2.2 Mur immergé dans un écoulement granulaire cisaillé . . . . . . . . . 114
5.3 Analyse comparative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.3.2 Two distinct granular systems studied . . . . . . . . . . . . . . . . . 1185.3.3 Grain-wall force distributions . . . . . . . . . . . . . . . . . . . . . 1205.3.4 Force time-series autocorrelation . . . . . . . . . . . . . . . . . . . . 1225.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Discussions et perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Bibliographie 131
4 Table des matiĂšres
Chapitre 1
Introduction
5
6 Chapitre 1. Introduction
1.1 Contexte
1.1.1 Avalanches de neige et ouvrages de protection
Avalanches : déclenchement et écoulement
Une avalanche neigeuse est dĂ©finie comme une masse de neige qui se dĂ©croche puis sâĂ©coulele long dâun versant montagneux. La masse de neige est soumise Ă la pesanteurâqui jouele rĂŽle de force motrice, contre balancĂ©e par le frottement basal. La dĂ©finition de troisphases correspondant Ă trois zones permet de dĂ©crire le dĂ©roulement dâune avalanche (Cf.Fig. 1.1).
Figure 1.1 â Les trois phases de lâĂ©coulement dâune avalanche : initiation, Ă©coulement, arrĂȘt
(© 2014 Canadian Avalanche Center).
Lâinitiation du phĂ©nomĂšne est appelĂ© dĂ©clenchement, et sâexplique par la rupture mĂ©-canique dâun Ă©quilibre dans le manteau neigeux. Cette rupture qui a lieu dans la zonede dĂ©part peut ĂȘtre due Ă une condition climatique particuliĂšre (perte de cohĂ©sion dumanteau par rĂ©chauffement) ainsi quâĂ une surcharge mĂ©canique naturelle (animal, chutede corniche) ou artificielle (dĂ©clenchement prĂ©ventif).
La phase de propagation correspond Ă lâĂ©coulement de la masse de neige le long de lapente. La topographie, les conditions mĂ©tĂ©orologiques, le type de neige formant le man-teau sont autant de facteurs qui influencent grandement le phĂ©nomĂšne de propagation. Ilexiste une classification diffĂ©renciant les types dâavalanches selon les modes dâĂ©coulementsobservables et la composition de la neige (voir Fig. 1.2).
1.1. Contexte 7
Figure 1.2 â Deux types dâĂ©coulements dâavalanches : en aĂ©rosol Ă gauche (©ANENA), dense
Ă droite (photo Alain Duclos).
â Lâavalanche en aĂ©rosol : câest un nuage de neige sĂšche et froide, formĂ© de particules deglace en suspension dans lâair, pouvant atteindre des vitesses de lâordre de 100 m.sâ1.
â Lâavalanche dense : la cohĂ©sion de la neige permet son aggrĂ©gation en grains detaille variable. Ces Ă©coulements ont tendance Ă suivre le relief par la ligne de plusforte pente, et leur forte densitĂ© peut causer des dĂ©gĂąts importants, malgrĂ© desvitesses parfois faibles. Si la teneur en eau liquide est faible, le comportement peutĂȘtre reproduit par un matĂ©riau granulaire sec (câest lâobjet de cette thĂšse, voirpartie 1.1.2), mais si elle est forte le mĂ©lange est hĂ©tĂ©rogĂšne et les modĂšles issus delâhydraulique torrentielle sont plus adaptĂ©s.
â Lâavalanche mixte : câest la combinaison simultanĂ©e dâune avalanche en aĂ©rosol etdâune avalanche granulaire ou dense.
La phase dâarrĂȘt de lâavalanche se produit lorsquâelle rencontre un obstacle ou unepente moyenne suffisamment faible. Elle dĂ©cĂ©lĂšre puis sâarrĂȘte totalement, formant unezone de dĂ©pĂŽt inerte.
Dispositifs de protection et dimensionnement
Face aux dangers pour la population que reprĂ©sentent les avalanches de neige en milieualpin, deux types dâouvrages de protection permanents sont mis en place :
â les ouvrages de dĂ©fense actifs (voir Fig. 1.3), ils interviennent en amont de lâava-lanche, en Ă©vitant le dĂ©clenchement : claies paravalanches, rĂąteliers, filets, reboise-ments ...
â les ouvrages de dĂ©fense passifs (voir Fig. 1.4), ils sont bĂątis en aval de la zone dedĂ©clenchement pour dĂ©vier, freiner ou arrĂȘter les avalanches pendant la propagation :Ă©traves, digues ou murs dâarrĂȘt, dents dĂ©flectrices, tunnels ...
8 Chapitre 1. Introduction
Figure 1.3 â Protections paravalanches actives : claie Ă gauche (photo Böhringer Friedrich), filet
Ă droite (photo Irstea).
Figure 1.4 â Protections passives : mur dâarrĂȘt Ă gauche (Bleie, Ullensvang, NorvĂšge), tunnel
ferroviaire à droite (©SMJ).
1.1. Contexte 9
Figure 1.5 â ReprĂ©sentation schĂ©matique dâune avalanche surversant un obstacle, identifications
des zones (image Chanut [21]).
Lâobjet de cette thĂšse concerne les ouvrages de protection passifs, et plus gĂ©nĂ©ralement lâin-teraction entre une avalanche en Ă©coulementâlent ou rapide, et un obstacle (voir Fig. 1.5).Dans ce type de configuration, une partie de la neige se retrouve bloquĂ©e en amont delâobstacle, formant une zone quasistatique appelĂ©e zone morte. Lâobstacle perçoit alorspar contact direct la force provenant du poids de la zone morte. En fonction des Ă©chellesmises en jeu lors de lâĂ©vĂ©nement (volume de neige, hauteur de lâobstacle), lâensemble zonemorte - obstacle est susceptible dâĂȘtre dĂ©bordĂ© par de la neige en Ă©coulement plus oumoins rapide : on parle de zone inertielle. Lâobstacle subit Ă©galement la force dynamiqueprovenant de la zone inertielle par transmission (et Ă©ventuellement altĂ©ration) via la zonemorte. Le systĂšme ainsi dĂ©crit implique la coexistence entre deux zones dĂ©finies par unmatĂ©riau identique mais des conditions dâĂ©coulement diffĂ©rents. Une transition spatialeaux propriĂ©tĂ©s mĂ©caniques complexes a lieu au sein de la neige entre lâĂ©tat inerte et lâĂ©tatdâĂ©coulement. Cette transition est Ă lâorigine de fluctuations de force sur lâobstacle dâam-plitude et de frĂ©quence Ă©levĂ©es, tel quâexposĂ©s dans la partie 1.1.3.
La caractĂ©risation de la force subie par les structures passives de gĂ©nie civil soumisesaux avalanches est un Ă©lĂ©ment majeur lors de leur dimensionnement. Elle passe par lâĂ©tudede deux types de donnĂ©es complĂ©mentaires. Le premier type est le rĂ©sultat de lissages tem-porels, permettant le dimensionnement en terme de force moyenne, maximale et rĂ©siduelle,facilitant lâĂ©tude des avalanches malgrĂ© leur propriĂ©tĂ© transitoire. Le second type de don-nĂ©es rĂ©sulte des signaux bruts qui comportent des informations temporelles Ă Ă©chelles detemps courtes et permettent dâobtenir des informations sur les fluctuations de force surlâouvrage. Il intervient dans lâanalyse du phĂ©nomĂšne de rĂ©sonance mĂ©canique survenantĂ la frĂ©quence propre des structures, Ă©lĂ©ment critique lors du dimensionnement. Câest surce second type de donnĂ©es instantanĂ©es que porte cette Ă©tude.
10 Chapitre 1. Introduction
1.1.2 Ăcoulements granulaires secs
Les avalanches de neige sont des Ă©vĂ©nements dangereux et souvent rapides qui ont lieudans des environnements hostiles et instables. Pour ces raisons, lâacquisition de donnĂ©es deterrain Ă lâĂ©chelle dâun versant ou mĂȘme dâun couloir est trĂšs dĂ©licate. Afin dâobserver cesphĂ©nomĂšnes dans des conditions reproductibles et contrĂŽlĂ©es et Ă des Ă©chelles acceptables,des expĂ©rimentations en laboratoire dâĂ©coulements sur plans inclinĂ©s sont souvent rĂ©alisĂ©spar analogie aux Ă©coulements in-situ. Dans ce contexte, le choix du matĂ©riau modĂšle estun point important qui nĂ©cessite une Ă©tude de terrain pour obtenir des informations sur laconstitution de la neige en Ă©coulement. Cette section vise Ă montrer que, comme souventutilisĂ©s dans la littĂ©rature (voir [31, 85, 71, 70]), les Ă©coulements granulaires secs sontadaptĂ©s Ă la modĂ©lisation des Ă©coulements de neige dense sĂšche.
De la neige aux grains
Les zones de dĂ©pĂŽt formĂ©es Ă la fin des Ă©vĂ©nements avalancheux sont des terrains propicesĂ lâĂ©tude du matĂ©riau quâest la neige en Ă©coulement. Elles sont en effet situĂ©es dans deszones plus facile dâaccĂšs, et prĂ©sentent lâavantage de pouvoir ĂȘtre analysĂ©es a posteriori,rĂ©duisant ainsi considĂ©rablement la contrainte opĂ©rationnelle. De plus, par opposition aumanteau neigeux avant lâavalanche, le matĂ©riau Ă©tudiĂ© est celui qui sâest rĂ©ellement Ă©coulĂ©,susceptible dâavoir Ă©tĂ© transformĂ© par lâĂ©coulement lui-mĂȘme.
La figure 1.6 montre quatre dĂ©pĂŽts formĂ©s par des avalanches denses, constituĂ©s dâagrĂ©-gats de diffĂ©rentes gĂ©omĂ©tries. En fonction de la composition de la neige et des conditionsdâĂ©coulement, leurs formes peuvent ĂȘtre trĂšs angulaires, se rapprochant de polyĂšdres (voir(a)) ou plus arrondies voire quasi-sphĂ©riques (voir (b,c,d)). Dans ce cas, câest lâĂ©rosionefficace par frottement inter-grains qui a tendance Ă les arrondir lors de lâĂ©coulement.Ă lâorigine, ce sont ces observations dâagrĂ©gats quasi-sphĂ©riques qui ont conduit la com-munautĂ© scientifique Ă considĂ©rer les avalanches comme des Ă©coulements granulaires. Ceconstat a motivĂ© une analyse rĂ©alisĂ©e dans [6], qui vise Ă quantifier la distribution destailles de ces agrĂ©gats pour deux avalanches de neige sĂšche et deux avalanches de neigehumide. Les agrĂ©gats sont le rĂ©sultat de zones fortement cisaillĂ©es au sein de lâavalanche,la dynamique de lâĂ©coulement a donc une forte influence sur leur formation.
Lâassimilation des avalanches Ă des Ă©coulement de grains secs a fait lâobjet de vali-dations rĂ©centes en se basant sur des mesures variĂ©es. En 1999, Pouliquen [77] a Ă©tabliune loi entre Ă©paisseur de dĂ©pĂŽt et pente Ă partir dâĂ©coulements de billes de verre dansun modĂšle rĂ©duit de canal. Cette loi est notamment basĂ©e sur lâexistence de deux anglescritiques Ξmin et Ξmax, qui correspondent respectivement aux pentes en-dessous de laquelleaucun Ă©coulement nâest possible et au-dessus de laquelle aucun dĂ©pĂŽt nâest laissĂ© aprĂšsle passage des grains. En 2008, Rognon et al. [83] ont confirmĂ© lâexistence de ces anglescritiques pour la neige grĂące Ă un canal expĂ©rimental situĂ© au col du Lac Blanc. Les angles
1.1. Contexte 11
(a) (b)
(c) (d)
Figure 1.6 â Exemples de zones de dĂ©pĂŽt dâavalanches denses, formation dâagrĂ©gats ((a) et (c) :
tirées de [3], (b) : ©T. Faug, (d) : tirée de [6]).
12 Chapitre 1. Introduction
obtenus ne sont pas identiques Ă ceux mentionnĂ©s dans [77], mais les diffĂ©rences pourraientĂȘtre expliquĂ©es par les effets de bords qui ne sont pas identiques dans les deux Ă©tudes.Finalement, en 2010, Sovilla et al. [98] ont validĂ© la loi obtenue dans [77] sur des mesuresin-situ de pente et dâĂ©paisseur de dĂ©pĂŽt dâavalanches prises sur le site expĂ©rimental de laSionne grĂące Ă des donnĂ©es acquises par lidar aĂ©rien.
La comparaison des rhĂ©ologies des matĂ©riaux est un autre axe de comparaison entreavalanches et Ă©coulements granulaires secs. Le cisaillement vertical Îł est une donnĂ©e im-portante dans lâĂ©tude des avalanches de neige dense, car couplĂ© Ă la composante de cisaille-ment Ï du tenseur des contraintes, il permet dâavoir accĂšs Ă la rhĂ©ologie de lâĂ©coulement.Les profils verticaux de vitesse dâĂ©coulement de la neige ont Ă©tĂ© mesurĂ©s sur de multiplesdispositifs expĂ©rimentaux Ă Ă©chelle rĂ©elle (voir [47, 32, 56]) ou sur canal dâĂ©coulement(voir [33, 11, 12, 57, 103, 83, 86]). Un point commun Ă ces Ă©tudes est la concentration ducisaillement dans une zone proche du fond, alors que les dĂ©formations au-dessus de cettezone sont bien plus faibles. Certains modĂšles permettent de reconstruire ce comportement(voir [33, 73, 72, 74, 31, 57]), et sont quasiment tous basĂ©s sur un seuil dâĂ©coulement âar-tificielâ Ï0 ou un Ă©coulement biphasique. Seule lâĂ©tude de P.Rognon prend uniquementen compte le caractĂšre granulaire de lâĂ©coulement (voir [83]) pour expliquerâde maniĂšrequalitativeâlâexistence de deux zones de cisaillement Ă partir dâune propriĂ©tĂ© mesurableet intrinsĂšque Ă la neige : le diamĂštre des grains d, non homogĂšne dans la hauteur. Eneffet, cette Ă©tude met en jeu un ensemble bidisperse de grains et Ă©tablit une relation deproportionnalitĂ© entre le cisaillement local et le diamĂštre local des grains : Îł â 1/d.
Le prĂ©sent travail de thĂšse exploite ces similitudes entre les avalanches denses et lesĂ©coulements granulaires, et sâinscrit donc dans le cadre des Ă©coulements de grains rigides,secs et sphĂ©riques tels quâutilisĂ©s dans [77]. Le paragraphe suivant rĂ©sume les rĂ©sultatsexistants concernant ce type dâĂ©coulements.
RĂ©gimes dâĂ©coulement
Un matĂ©riau granulaire est dĂ©fini comme un ensemble de grains de gĂ©omĂ©trie et de taillevariables qui interagissent. Cette simple dĂ©finition englobe des domaines dâĂ©tudes nom-breux, qui nâont pourtant de prime abord pas de points communs. Ainsi, lâindustrie phar-maceutique, le gĂ©nie civil, lâagriculture, la gĂ©ophysique ou encore les chutes de blocs sontautant de domaines concernĂ©s par la comprĂ©hension de ce matĂ©riau complexe. La difficultĂ©principale rĂ©side dans le fait quâaucune loi constitutive universelle ne permet dâĂ©tablir larhĂ©ologie des milieux granulaires. Ă contrario, leur comportement dĂ©pend des sollicita-tions extĂ©rieures qui lui sont appliquĂ©s. Les chercheurs ont donc subdivisĂ© les matĂ©riauxgranulaires en catĂ©gories selon leur mode de sollicitation. Il en rĂ©sulte trois rĂ©gimes quisont dĂ©finis par la maniĂšre dont le milieu granulaire sâĂ©coule (voir Fig. 1.7).
â Le rĂ©gime quasistatique : les grains ont une inertie quasi nulle qui se traduit par
1.1. Contexte 13
Figure 1.7 â Ăcoulement granulaire, identification des rĂ©gimes quasistatique, dense et collision-
nel (photo issue de [41]).
des contacts frictionnels persistants dans le temps. Les milieux granulaires quasista-tiques sont souvent étudiés par les mécaniciens des sols qui utilisent la théorie de laplasticité pour en prévoir le comportement macroscopique (voir [84]).
â Le rĂ©gime collisionnel : câest lâopposĂ© du rĂ©gime quasistatique, les grains ont uneinertie trĂšs forte qui se traduit par un milieu trĂšs agitĂ© oĂč les grains sont trĂšsĂ©loignĂ©s. Comme les temps de contact sont extrĂȘmement courts, assimilables Ă descontacts binaires, une thĂ©orie dĂ©rivĂ©e de la thĂ©orie cinĂ©tique des gaz a Ă©tĂ© Ă©tabliepour en dĂ©duire les Ă©quations constitutives (voir [51, 46]).
â Le rĂ©gime dense : il se situe entre les deux rĂ©gimes extrĂȘmes, caractĂ©risĂ© par uncomportement âliquideâ. Lâinertie commence Ă avoir de lâinfluence sur la rhĂ©ologie,mais le rĂ©seau de contacts entre les grains peut toutefois percoler Ă travers le milieude maniĂšre transitoire. La majoritĂ© des Ă©coulements gĂ©ophysiques sont concernĂ©s parce rĂ©gime, mais câest aussi celui pour lequel lâĂ©tablissement dâune loi constitutiveest le plus difficile. Câest la raison pour laquelle de nombreuses Ă©tudes ont cherchĂ©Ă comprendre ce type dâĂ©coulements complexes ces derniĂšres annĂ©es.
La recherche dans le monde des milieux granulaires Ă©tant caractĂ©risĂ©e par un large paneldâapplications et donc des systĂšmes physiques diffĂ©rents, le GDR MiDi a unifiĂ© le rĂ©sultatde plusieurs travaux de recherches dans [44] en 2004. Six configurations bidimensionnellesdiffĂ©rentes y sont prĂ©sentĂ©es (voir Fig. 1.8), dont trois confinĂ©es et trois Ă surface libre.Le milieu granulaire (constituĂ© de grains rigides) est cisaillĂ©, soit par gravitĂ©, soit par lecontrĂŽle en dĂ©placement ou en contrainte dâune paroi, et des mesures sont rĂ©alisĂ©es auxparois et au sein des grains afin dâextraire les propriĂ©tĂ©s mĂ©caniques de lâĂ©coulement. Une
14 Chapitre 1. Introduction
Figure 1.8 â Les six configurations dâĂ©coulements granulaires testĂ©es par le GDR MiDi : cisaille-
ment plan (a), cisaillement annulaire (b), chute verticale (c), plan incliné (d), entassement (e),
tambour tournant (f) (illustration tirée de [44]).
grandeur est dĂ©finie sur la base dâune analyse dimensionnelle pour quantifier le rĂ©gimeauquel les grains sâĂ©coulent. Cette grandeur, appelĂ©e nombre inertiel I, doit reflĂ©ter leseffets antagonistes de la vitesse de cisaillement Îł et de la pression de confinement P . Pource faire, deux temps caractĂ©ristiques sont dĂ©finis (voir Fig. 1.9) :
TÎł =1
Îł, (1.1)
le temps typique correspondant à la déformation par cisaillement, et
TP = d
âÏpP, (1.2)
le temps typique de confinement correspondant au rĂ©arrangement (Ă la relaxation) dâungrain sous la pression P (Ïp est la masse volumique des grains). Le nombre inertiel I,grandeur adimensionnelle, est le rapport de ces deux temps :
I =TP
TÎł
=ÎłdâP/Ïp
. (1.3)
Parmi les configurations Ă©tudiĂ©es par le GDR MiDi dans [44], le cisaillement plan sansgravitĂ© (voir Fig. 1.8(a)) met en valeur un comportement intĂ©ressant de lâĂ©coulement vis-Ă -vis du nombre inertiel, Ă condition que les propriĂ©tĂ©s microscopiques des grains soientordinaires (coefficient de restitution e infĂ©rieur Ă 1 et angle de frottement ”p non nul).
1.1. Contexte 15
Figure 1.9 â ReprĂ©sentation schĂ©matique des deux temps caractĂ©ristiques, TÎł et TP (illustration
tirée de [44]). Tγ est le temps nécessaire à une couche de grains pour parcourir une distance d
par rapport à une autre couche par le cisaillement γ. TP est le temps nécessaire à une couche
de grains pour âretomberâ dans une position basse lorsquâelle a atteint une hauteur maximale en
gravissant la couche inférieure.
En particulier, tant que le rĂ©gime collisionnel (gazeux) nâest pas atteint, les propriĂ©tĂ©sintrinsĂšques Ă lâĂ©coulement du milieu granulaire : vitesses, fraction volumique, frottementeffectif ”eff = Ï/P , sont des fonctions univoques de I. Lorsque I est faible, le rĂ©gimeest quasistatique et la rĂ©ponse du systĂšme est indĂ©pendante de la vitesse de cisaillement.Lorsque I augmente, le rĂ©gime dense est atteint et lâinertie entre en jeu : le systĂšme devientdĂ©pendant Ă la vitesse de cisaillement. Lorsque I est Ă©levĂ©, lâĂ©coulement est dans lâĂ©tatcollisionnel. I contrĂŽle donc le rĂ©gime dâĂ©coulement, ce qui signifie quâil y a deux façonsdâeffectuer une transition de rĂ©gime : en modifiant soit la vitesse de cisaillement Îł, soit lapression de confinement P .
Ayant trouvĂ© une relation rhĂ©ologique ”eff = ”eff (I) (dite locale, car en un pointdonnĂ© elle ne dĂ©pend que des grandeurs locales) dans cette configuration simple, il esttentant de vouloir lâĂ©tendre Ă des systĂšmes plus complexes, pour savoir si la rhĂ©ologielocale de tout systĂšme dâĂ©coulement granulaire rĂ©pond Ă cette loi. Mais il sâavĂšre que dansdes systĂšmes plus complexes, la rhĂ©ologie en un point donnĂ© peut Ă©galement dĂ©pendre degrandeurs qui ne sont pas localisĂ©es en ce point. En dâautres termes, il se peut que la rhĂ©o-logie locale soit perturbĂ©e par des grandeurs lointaines, comme la vitesse de lâĂ©coulementen un point gĂ©omĂ©trique diffĂ©rent, ou encore par des longueurs caractĂ©ristiques liĂ©es auxconditions au bord du systĂšme. Dans ce cas on parle de non-localitĂ©, phĂ©nomĂšne Ă©tudiĂ©dans [54, 14].
Enfin, en 2006, Jop et. al. ont proposĂ© une formule thĂ©orique reliant le coefficient defrottement local ” et le nombre inertiel I valable dans le rĂ©gime dense, dans le cas oĂčla non-localitĂ© nâa pas lieu (voir [52]). Ils Ă©tablissant alors une loi rhĂ©ologique empiriquereliant lâinertie de lâĂ©coulement au rapport entre les composantes tangentielle et normaledes contraintes appliquĂ©es aux grains :
Ï/P = ”(I) = ”s +”2 â ”s
I0/I + 1. (1.4)
Câest une fonction Ă double saturation, aux faibles et fortes valeurs de I : ”s est le
16 Chapitre 1. Introduction
Figure 1.10 â La loi rhĂ©ologique ”(I), avec les trois valeurs de paramĂštres du cisaillement plan
donnés dans [52] : ”s = 0.382, ”2 = 0.643, I0 = 0.279.
frottement correspondant au seuil dâĂ©coulement (asymptote lorsque I â 0), ”2 est lavaleur de saturation aux grands I, et I0 est une constante permettant de dĂ©caler la fonctionpar rapport Ă I (voir Fig. 1.10). Sous condition que les propriĂ©tĂ©s microscopiques des grainssoient ordinaires (voir paragraphes ci-dessus) et que la non localitĂ© nâentre pas en jeu, cestrois paramĂštres ne dĂ©pendent que de la configuration gĂ©omĂ©trique du systĂšme (parois,type de sollicitation, dimensionnalitĂ© 2D ou 3D).
1.1.3 Interaction avalanche - obstacle : Ă©tat de lâart
La force exercĂ©e par un Ă©coulement granulaire sur un obstacle immergĂ© constitue un do-maine dâĂ©tude primordial, nĂ©cessaire au dimensionnement des structures de gĂ©nie civil. Enplus des ouvrages de protection paravalanches, les bĂątiments et pylĂŽnes sont Ă©galementconcernĂ©s. La norme europĂ©enne Eurocodes concernant le risque liĂ© Ă lâalĂ©a avalanche deneige recommande la prise en compte de la moyenne temporelle de la pression exercĂ©e surlâouvrage ăP ă. De cette façon lâĂ©coulement de neige est vu comme une masse statique pe-sant sur lâobstacle Ă dimensionner et des coefficients de sĂ©curitĂ© sont utilisĂ©s pour prendreen compte les composantes dynamiques.
Cependant, les signaux temporels de pression sur obstacles soumis Ă des avalanchesont Ă©tĂ© mesurĂ©s et se sont rĂ©vĂ©lĂ©s complexes (voir Fig. 1.11). La pression commencepar augmenter quasi linĂ©airement dans le temps jusquâĂ atteindre un maximum absoluPmax. Pendant cette phase de montĂ©e en charge correspondant Ă lâimpact de lâavalanche,des maximums locaux de pression Ppeak sont observĂ©s. Par la suite, la pression diminuerapidement avant dâatteindre une valeur de pression moyenne quasiment constante ăP ă
1.1. Contexte 17
Figure 1.11 â Signaux de pression P dâavalanches impactant une plaque carrĂ©e de 1m2 sur le
site dâexpĂ©rimental du Lautaret, en fonction du temps normalisĂ© (tmax est le temps auquel le
signal de force est maximal).
correspondant Ă un Ă©tat permanent de lâĂ©coulement, et perturbĂ©e par des fluctuationshaute frĂ©quence ÏP . Enfin, lorsque lâĂ©coulement cesse, le dĂ©pĂŽt en amont de lâobstacle yapplique une pression rĂ©siduelle Pres.
De tels signaux peuventâsâils ne sont pas estimĂ©s a priori dans leur intĂ©gralitĂ©âconduire Ă lâendommagement ou la destruction des structures immergĂ©es. Une meilleurecomprĂ©hension de lâinteraction entre une avalanche et un obstacle est donc nĂ©cessaire pourles dimensionnements. Cela passe en particulier par la mise en Ă©vidence des liens entreles caractĂ©ristiques dâĂ©coulement et les caractĂ©ristiques des signaux de force. Cette thĂšseporte essentiellement sur la comprĂ©hension de la composante fluctuante ÏP des signauxde force.
Force moyenne sur lâobstacle
Le calcul analytique complet de la physique des avalanches reste difficile Ă maĂźtriser, mĂȘmeen faisant des hypothĂšses fortes sur leur composition. En effet, les variables dâĂ©tat au seinde lâĂ©coulement sont Ă lâheure actuelle non mesurables dans des conditions rĂ©elles : lesflux de quantitĂ© de mouvement et le tenseur des contraintes locaux. Aussi, la rhĂ©ologiedu matĂ©riau, câest-Ă -dire la relation locale entre contrainte et dĂ©formation ou vitesse dedĂ©formation, nâest pas Ă©tablie de maniĂšre universelle (voir partie 1.1.2). Dans ce contexte,des mesures de laboratoire ont toutefois permis dâĂ©tablir certaines relations entre les pro-priĂ©tĂ©s macroscopiques des Ă©coulements granulaires secs en rĂ©gime permanent et la forcemoyenne exercĂ©e sur un obstacle immergĂ©. En adĂ©quation avec la rhĂ©ologie des milieux
18 Chapitre 1. Introduction
granulaires dĂ©taillĂ©e en partie 1.1.2, il faut distinguer deux rĂ©gimes : celui oĂč lâinertie estnĂ©gligeable (rĂ©gime quasistatique) et celui oĂč lâinertie joue un rĂŽle important.
Lorsque lâinertie de lâĂ©coulement est faible, la pression moyenne sur lâobstacle estproportionnelle Ă la pression hydrostatique (pression de confinement du milieu granulaire),et la vitesse de lâĂ©coulement nâa pas dâinfluence. Ce comportement a Ă©tĂ© vĂ©rifiĂ© dansplusieurs configurations : Ă©coulement annulaire (voir [2]), Ă©coulement gravitaire vertical(voir [22]), barreau immergĂ© en rotation perpendiculaire Ă son axe (voir [16]). Pour unsystĂšme constituĂ© dâun Ă©coulement gravitaire Ă surface libre rencontrant un obstacle lelong dâune pente, la pression moyenne sur lâobstacle peut alors ĂȘtre estimĂ©e par (voir[40]) :
Phydro = KÏgh cos(Ξ), (1.5)
oĂč Ξ est la pente en amont de lâobstacle, Ï est la densitĂ© moyenne de lâĂ©coulement, g
est lâaccĂ©lĂ©ration de la pesanteur, h est la hauteur de lâobstacle et K est un coefficientadimensionnel qui dĂ©pend de la gĂ©omĂ©trie du problĂšme et des propriĂ©tĂ©s du milieu gra-nulaire.
Lorsque la force dâinertie est grande devant la force de pression hydrostatique, lesgrains sont trĂšs espacĂ©s et interagissent par des chocs rapides. Le milieu granulaire estcollisionnel et peut ĂȘtre vu comme un fluide : la pression moyenne perçue par lâobstacledĂ©pend du carrĂ© de la vitesse de lâĂ©coulement. Cette loi, Ă©galement vĂ©rifiĂ©e dans diversesconfigurations (voir [17, 22, 13, 16]), peut alors sâĂ©crire :
Pdyn = C1
2Ïu2, (1.6)
oĂč u est la vitesse de lâĂ©coulement et C est un coefficient adimensionnel qui dĂ©pend de lagĂ©omĂ©trie du problĂšme et des propriĂ©tĂ©s de lâĂ©coulement.
Finalement, dans le cas dâĂ©tude dâune zone inertielle sâĂ©coulant par-dessus une zonemorte en amont de lâobstacle (voir Fig. 1.5), une approche basique consiste Ă sommer cesdeux contributions pour obtenir la pression totale sur lâobstacle (voir [40]) :
P = KÏgh cos(Ξ) + C1
2Ïu2, (1.7)
Hormis le caractĂšre purement permanent de cette description, la limite Ă cette ap-proche est la forte dĂ©pendance de K et C Ă la taille de lâobstacle, Ă la hauteur de lâĂ©cou-lement et aux propriĂ©tĂ©s mĂ©caniques du milieu granulaire. Ainsi, K = 1 est possible pourun Ă©coulement similaire Ă un Ă©coulement fluide, mais des valeurs supĂ©rieures Ă 10 ont Ă©tĂ©atteintes sous certaines conditions. C varie gĂ©nĂ©ralement entre 1 et 2, et se caractĂ©rise parune dĂ©pendance Ă la fraction volumique locale de lâĂ©coulement granulaire et Ă la formegĂ©omĂ©trique de la surface que lâobstacle prĂ©sente Ă lâĂ©coulement. Une calibration de cescoefficients en fonction du systĂšme Ă©tudiĂ© est donc souvent nĂ©cessaire. De plus, la simple
1.1. Contexte 19
sommation des deux contributions ne permet pas de prendre en compte lâinteraction com-plexe entre les deux zones dâĂ©coulement. Cela revient Ă considĂ©rer que la zone morte âpĂšseâsimplement sur lâobstacle, et transmet sans altĂ©ration la force exercĂ©e par la zone inertielle.
RĂ©cemment, un modĂšle thĂ©orique plus complet a Ă©tĂ© Ă©tabli par Chanut et al. [20]et validĂ© expĂ©rimentalement par Faug et al. [39] pour mieux apprĂ©cier la force exercĂ©epar un Ă©coulement granulaire sec Ă surface libre dans la configuration du mur surversĂ©exposĂ© dans la figure 1.5. Il est basĂ© sur la conservation de la masse et de la quantitĂ© demouvement dans un volume de contrĂŽle correspondant Ă la zone inertielle surmontant lazone morte. Sa particularitĂ© est de prendre en compte la longueur de la zone morte et lecaractĂšre transitoire des signaux de pression prĂ©sentĂ©s dans la figure 1.11. La force surlâobstacle est dĂ©composĂ©e en plusieurs contributions :
F = FNu + Fh0
+ FNw â ”zm
[F Tw + F T
u
]+ Fmv, (1.8)
oĂč FNu est la force due Ă la dĂ©viation de la quantitĂ© de mouvement (en rĂ©gime perma-
nent) entre lâĂ©coulement entrant et lâĂ©coulement sortant, Fh0est la force statique liĂ©e Ă
lâĂ©coulement incident, FNw est la force correspondant au poids de la zone de contrĂŽle, et
”zm
[F Tw + F T
u
]est la force de frottement basal sous la zone morte. Fmv est la force liée
Ă la variation de la quantitĂ© de mouvement Ă travers le temps pour permettre la prise encompte du caractĂšre transitoire des avalanches. Ce modĂšle est particuliĂšrement intĂ©res-sant car ses paramĂštres dâentrĂ©e sont des grandeurs physiques mesurables exprimĂ©es enfonction de :
â Ξ, la pente en amont de lâobstacle,
â Ξmin et Ξmax, les angles limites dĂ©finis dans la partie 1.1.2,
â Ï , un temps caractĂ©ristique de formation de la zone morte,
â h, la hauteur de lâobstacle,
â h0/hâ, le rapport entre lâĂ©paisseur de lâĂ©coulement inertiel entrant et lâĂ©paisseur de
lâĂ©coulement inertiel au niveau de lâobstacle.
Fluctuations de forces sur lâobstacle
Les fluctuations de force dans les milieux granulaires, composantes notĂ©es ÏP dans la fi-gure 1.11, ont fait lâobjet dâun bon nombre dâĂ©tudes ces derniĂšres annĂ©es. Des donnĂ©esin-situ sur le site de la Sionne lors dâĂ©vĂ©nements avalancheux en 1998 et 1999 ont permisdâĂ©tablir un premier lien entre les propriĂ©tĂ©s granulaires de lâĂ©coulement (tailles et vitessesdes grains) et les fluctuations de force sur obstacle (voir [87]). En parallĂšle, des expĂ©ri-mentations de laboratoire ont vu le jour pour Ă©tudier le comportement fondamental des
20 Chapitre 1. Introduction
Figure 1.12 â Spectre de puissance de la force exercĂ©e sur une barre immergĂ©e dans un Ă©coule-
ment granulaire cisaillé (graphique issu de [1]).
matĂ©riaux granulaires. La majoritĂ© dâentre elles mettent en Ćuvre des Ă©coulements lentset des mesures de force aux parois (voir par exemple [67]) ou sur un Ă©lĂ©ment fixe immergĂ©(voir [1, 45, 26]). Les configurations Ă©tudiĂ©es sont souvent des cisaillements dans des gĂ©o-mĂ©tries annulaires de type Couette. Un nombre bien plus restreint dâexpĂ©rimentations ontĂ©tĂ© rĂ©alisĂ©s dans le but dâĂ©tudier les fluctuations dans des Ă©coulements granulaires rapides.Dans ce cas, câest le plus souvent la gravitĂ© qui entraĂźne les grains : dĂ©charges de silosĂ©quipĂ©s de capteurs (voir [62, 42]) ou Ă©coulements Ă surface libre sur plans inclinĂ©s (voir[92]). Dans les Ă©coulements granulaires rapides, lâorigine du phĂ©nomĂšne de fluctuationsest souvent expliquĂ© par des impacts intermittents de particules Ă haute vitesse. Dans lesrĂ©gimes lents par contre, le systĂšme oscille entre des Ă©tats âbloquĂ©sâ (dus aux contraintesstĂ©riques des milieux granulaires), et des Ă©tats de relaxation. Le phĂ©nomĂšne mis en avantcomme responsable des fluctuations est alors le cycle de formation et de rupture du rĂ©seaude forces fortes (Radjai et al. [79]).
La rĂ©alisation des spectres de forces dans les milieux granulaires a Ă©tĂ© faite dans lamajoritĂ© des expĂ©rimentations, sur les parois et sur des capteurs fixes immergĂ©s. MalgrĂ©la variĂ©tĂ© des systĂšmes et conditions dâĂ©coulement Ă©tudiĂ©s, il est saisissant de consta-ter la convergence des spectres vers une loi gĂ©nĂ©rique (voir Fig. 1.12). Les spectres sontconstants aux faibles frĂ©quences, puis une dĂ©croissance en loi puissance dâexposant â2
jusquâaux plus hautes frĂ©quences est observĂ©e. Cette forme de courbe est gĂ©nĂ©ralementcaractĂ©ristique de processus physiques alĂ©atoires, et traduit la complexitĂ© et la richessedes phĂ©nomĂšnes qui ont lieu Ă plusieurs Ă©chelles dans les matĂ©riaux granulaires.
Afin de rĂ©vĂ©ler les propriĂ©tĂ©s mĂ©caniques des milieux granulaires, les distributionsde forces sont souvent utilisĂ©esâen particulier dans les Ă©tudes mentionnĂ©es ci-dessus.Plusieurs choix sont possibles concernant le type de donnĂ©es relevĂ©es et les traitements
1.1. Contexte 21
qui sont effectuĂ©s. La grande majoritĂ© des Ă©tudes exploitent des distributions purementspatiales, câest-Ă -dire que les donnĂ©es sont relevĂ©es sur les forces de contact au sein dâunsystĂšme au repos ou Ă un temps fini dĂ©terminĂ© (âphotoâ du systĂšme). Ă contrario, ce sontparfois les distributions temporelles qui sont utilisĂ©es, ce qui signifie que les Ă©chantillonsde donnĂ©es proviennent de signaux temporels : le signal dâun capteur de force ou le suivide la force exercĂ©e sur une particule. Dans certains cas les distributions sont obtenues Ă partir dâĂ©chantillons combinant des donnĂ©es spatiales et temporelles, via des moyennesdans le temps ou lâespace. Une revue bibliographique sur le sujet sera dĂ©taillĂ©e en partie3.2, mais les principaux rĂ©sultats extraits de la littĂ©rature sur les distributions de forcessont dĂ©crits ci-aprĂšs. On propose ici de distinguer deux modes de mesure :
â les distributions des forces de contact (mesures Ă lâĂ©chelle des grains) dans un premiertemps ;
â les distributions de forces intĂ©grĂ©es dans lâespace (contribution de plusieurs contactssimultanĂ©s) dans un second temps.
Distribution des forces de contact
Les premiĂšres analyses de forces de contact dans les milieux granulaires Ă©taient rĂ©alisĂ©esgrĂące Ă du papier carbone placĂ© aux parois du systĂšme. Il Ă©tait possible de remonteraux forces normales en mesurant la taille et lâintensitĂ© des traces laissĂ©es par les grains(voir [69]). Lâutilisation de matĂ©riaux photoĂ©lastiques (birĂ©fringence induite par contraintemĂ©canique) et de modĂ©lisations numĂ©riques a ouvert lâaccĂšs aux forces de contact entreles grains au sein du milieu granulaire. Plusieurs Ă©tudes croisĂ©es ont dĂ©montrĂ© que dansle cas statique, les distributions des forces de contact grain-grain sont identiques Ă cellesprovenant des forces de contact grain-paroi (voir [95, 96, 80]).
Le comportement des distributions des forces de contact f supĂ©rieures Ă leur valeurmoyenne f est un rĂ©sultat robuste, vĂ©rifiĂ© par de nombreuses Ă©tudes (voir par exemple[61, 81] et Fig. 1.13) : la densitĂ© de probabilitĂ© dĂ©croĂźt exponentiellement jusquâaux forcesles plus Ă©levĂ©es. La probabilitĂ© relativement forte dâavoir des forces bien supĂ©rieures Ă laforce moyenne est gĂ©nĂ©ralement associĂ©e Ă la tendance quâont les milieux granulaires detransmettre une grande partie des forces le long de structures en chaĂźnes appelĂ©es âchaĂźnesde forces fortesâ (voir [79]). Lâintroduction du modĂšle thĂ©orique âq-modelâ par Liu et al.[61] permet de reproduire ce comportement : chaque grain perçoit les forces en provenancedes contacts avec la couche supĂ©rieure, et les transmet via une rĂ©partition alĂ©atoire auxgrains de la couche infĂ©rieure. Plusieurs Ă©tudes ont montrĂ© la sensibilitĂ© de ces distributionsĂ la vitesse de cisaillement (ou au dĂ©viateur du tenseur des contraintes pour les systĂšmesstatiques) : lâaugmentation de la vitesse de cisaillement du milieu granulaire provoque unĂ©largissement des distributions (voir [62, 96, 45] et Fig. 1.13).
22 Chapitre 1. Introduction
Figure 1.13 â Distribution des quantitĂ©s de mouvement P (I) au bord dâĂ©coulements granulaires.
Le capteur, de la taille des grains, est placĂ© sur la surface interne dâun silo se dĂ©chargeant Ă des
vitesses Vf différentes (graphique issu de [62]).
Les distributions des forces de contact faibles (infĂ©rieures Ă f) ont Ă©galement fait lâob-jet dâĂ©tudes, mais aucune loi universelle nâa Ă©tĂ© Ă©tablie. Cette partie des distributions esten effet trĂšs sensible Ă de nombreux paramĂštres : la prĂ©paration de lâĂ©chantillon de grains,la fraction volumique Ίâ au sein du systĂšme (liĂ©e Ă lâĂ©ventuelle vitesse de cisaillement Îł),ou encore lâanisotropie des contacts sont autant de facteurs influents. Ainsi, le comporte-ment exponentiel est parfois continu jusquâaux faibles forces, mais il est souvent observĂ© Ă contrario un plateau aux alentours de f . Dans ce dernier cas, la forme de la courbe pourf < f admet un ensemble de possibilitĂ©s, de la disparition lente au pic de probabilitĂ©lorsque f â 0 (voir [4, 96, 104, 78]). Ă lâheure actuelle, le modĂšle thĂ©orique permettantde prendre en compte cette disparitĂ© de la façon la plus complĂšte est dĂ©crit par Radjai[78] (voir Fig. 1.14). BasĂ© sur la coexistence et lâinterdĂ©pendance des rĂ©seaux de forces fortet faible (voir [79]), il prend en compte les contraintes stĂ©riques des particules et lâaspectalĂ©atoire de la rĂ©partition des contacts sur une sphĂšre.
Distribution des forces intĂ©grĂ©es dans lâespace
Ce travail de thĂšse est focalisĂ© sur lâinteraction milieu granulaireâobstacle, systĂšmedans lequel la force sur lâobstacle est susceptible dâĂȘtre composĂ©e de la contribution si-multanĂ©e de plusieurs contacts grainâobstacle. Les fluctuations temporelles dans cettesituationâpour des Ă©coulements relativement lentsâont Ă©tĂ© Ă©tudiĂ©es par un nombre res-treint dâĂ©tudes expĂ©rimentales, dans des configurations de type Couette (voir [67, 50,45]) et pour une pale tournante dans un milieu granulaire (voir [97]). Les donnĂ©es sontacquises diffĂ©remment selon les Ă©tudes, mais systĂ©matiquement Ă des Ă©chelles spatialessupĂ©rieures Ă la taille des grains : mesure sur la totalitĂ© dâune paroi ou dâun intrus fixe
1.2. Objectifs 23
Figure 1.14 â Distribution des forces de contact f dans un milieu granulaire statique selon le
modÚle théorique Radjai [78] (graphique issu de [78]). Le paramÚtre γ permet de prendre en
compte plusieurs comportements lorsque f < f .
dans [67], [45] et [97] ; mesure des contraintes locales moyennĂ©e sur 260 grains du sys-tĂšme dans [97]. Un comportement commun des distributions de forces se dĂ©gage (voirFig. 1.15) : en comparaison avec les dĂ©croissances exponentielles obtenues Ă lâĂ©chelle desgrains, les distributions se resserrent et tendent vers des lois normales Ă des Ă©chelles demesure plus Ă©levĂ©es. Ce rĂ©sultat est dâautant plus marquĂ© que les Ă©coulements sont lentset sâapprochent de lâĂ©tat quasistatique (forte fraction volumique Ίâ), tandis que lâaug-mentation de la vitesse provoque lâobtention de distributions de formes log-normales [45]Ă exponentielles [50].
1.2 Objectifs
En adĂ©quation avec les rĂ©cents travaux de thĂšse de BenoĂźt Chanut et Paolo Caccamo (voir[21, 18]), cette thĂšse vise Ă Ă©tudier lâinteraction entre un Ă©coulement granulaire sec le longdâune pente et un obstacle plan orthogonal au dĂ©placement. Les travaux prĂ©cĂ©dents sesont concentrĂ©s sur la quantification de la force moyenne subie par lâobstacle, au regarddes zones granulaires formĂ©es Ă lâamont : zone stagnante triangulaire surversĂ©e par unezone en Ă©coulement dynamique. Cependant, les signaux de force sont systĂ©matiquementcomposĂ©s dâune forte composante fluctuante restĂ©e inexplorĂ©e, notĂ©e ÏP dans la figure1.11.
Les fluctuations de force que subissent les structures soumises aux Ă©coulements gra-nulaires sont un ingrĂ©dient essentiel de leur dimensionnement. Ce travail de thĂšse a pourobjectif dâinvestiguer le rĂŽle que joue la transition fluide-solide au sein du milieu granu-laire dans les fluctuations de force sur un obstacle immergĂ©. Il sâagit de les caractĂ©riser
24 Chapitre 1. Introduction
Figure 1.15 â Distribution de force exercĂ©e sur une particule fixe de diamĂštre a (a/d â 2.6)
immergée dans un milieu granulaire en écoulement (expérimentation bidimensionnelle, graphique
issu de [45]). Les différentes courbes correspondent à différentes vitesses de cisaillement.
vis-Ă -vis de paramĂštres physiques macroscopiques choisis afin de couvrir une gamme trĂšslarge de rĂ©gimes dâĂ©coulements permanents : de quasistatique Ă collisionnel. DiffĂ©rentesĂ©chelles spatiales sont considĂ©rĂ©es dans cette Ă©tude : de lâĂ©chelle des grains Ă lâĂ©chelle delâobstacle entier, en passant par une Ă©chelle intermĂ©diaire mĂ©soscopique (quelques grains).
Des systĂšmes modĂšles mettant en Ćuvre lâinteraction entre un Ă©coulement granulaireet un obstacle plan sont imaginĂ©s et modĂ©lisĂ©s informatiquement par la mĂ©thode desĂ©lĂ©ments discrets. Afin de trouver un Ă©quilibre entre rapiditĂ© des calculs et reprĂ©sentativitĂ©physique, les systĂšmes sont restreints Ă des configurations simples en rĂ©gime permanentsans gravitĂ©, correspondant Ă des zones gĂ©omĂ©triques en amont de lâobstacle aussi rĂ©duitesque possible. Lâusage de la mĂ©thode des Ă©lĂ©ments discrets permet de prendre en comptelâaspect granulaire du milieu Ă modĂ©liser, sous des conditions aux limites particuliĂšres.Elle sâavĂšre Ă©galement avantageuse pour acquĂ©rir des donnĂ©es complĂštes, que ce soit auxparois ou au sein du milieu granulaire.
1.3 MĂ©thode
1.3.1 La méthode des éléments discrets (DEM)
La méthode des éléments discrets, formulée dÚs 1979 par Cundall et al. [28], est uneextension de la dynamique moléculaire. ParticuliÚrement adaptée à la modélisation demilieux granulaires, elle prend en compte la complexité géométrique des interactions mul-
1.3. MĂ©thode 25
DĂ©tection des contacts
Calcul des déformations locales aux contacts
Déplacement/rotation des particules par intégration des équations du mouvement
Sommation des forces/moments par particule
Calcul des forces/moments via une loi de contact
Figure 1.16 â Cycle de calcul de la mĂ©thode des Ă©lĂ©ments discrets.
tiples dâun ensemble de particules Ă lâĂ©chelle microscopique. Les particules, supposĂ©eslocalement dĂ©formables, sont dĂ©crites individuellement par leur position, taille, vitessesde translation et de rotation. Le cycle de calcul est dĂ©crit dans la figure 1.16 : Ă chaquepas de temps, les contacts entre particules sont dĂ©tectĂ©s, puis les dĂ©formations normaleet tangentielle dans les repĂšres locaux de chaque contact sont calculĂ©s. Une loi de contactpermet dâen dĂ©duire les forces et moments, puis, aprĂšs sommation par particule, un schĂ©madâintĂ©gration explicite des Ă©quations du mouvement est utilisĂ© pour dĂ©placer et appliquerles rotations aux particules. Le schĂ©ma numĂ©rique est dĂ©crit ci-aprĂšs pour les translations,la formulation est similaire pour les rotations.
Le point de départ est le principe fondamental de la dynamique appliqué à une parti-cule :
x =
âF
m, (1.9)
oĂč x et m sont lâaccĂ©lĂ©ration et la masse de la particule, etâ
F est la somme des forces decontact qui sây applique. La discrĂ©tisation temporelle suivant la mĂ©thode des diffĂ©rencesfinies centrĂ©es Ă lâordre 2 permet dâexprimer explicitement lâaccĂ©lĂ©ration des particules autemps t en fonction de leurs positions x aux temps t+ dt, t et tâ dt :
xt â x
tâdt â 2xt + xt+dt
dt2, (1.10)
dâoĂč :
xt+dt â x
t + dt
(xt â x
tâdt
dt+ x
tdt
). (1.11)
26 Chapitre 1. Introduction
Figure 1.17 â Illustration bidimensionnelle de la loi visco-Ă©lastique.
Afin de sâaffranchir de lâutilisation de xtâdt, on approxime la vitesse Ă tâ dt
2par :
xtâ dt
2 â xt â x
tâdt
dt, (1.12)
et finalement :
xt+dt â x
t + dt
(xtâ dt
2 +
âF
t
m
). (1.13)
Les positions sont donc connues aux temps i · dt et les vitesses le sont aux temps i · dt2,
i Ă©tant un entier positif. Le modĂšle ainsi dĂ©crit permet le dĂ©roulement de la cinĂ©matiquedes grains en fonction des forces et moments appliquĂ©s. Il reste Ă ajouter la mĂ©caniqueaux contacts, câest-Ă -dire le lien entre dĂ©formations locales et forces/moments de contact,et lâensemble des paramĂštres associĂ©s.
Les expĂ©rimentations dâĂ©coulements granulaires en laboratoire sont souvent rĂ©alisĂ©es Ă lâaide de billes de verre rigides. Afin de faciliter la correspondance future entre les simula-tions numĂ©riques et dâĂ©ventuelles expĂ©rimentations, la loi et les paramĂštres microscopiquessont choisis pour correspondre Ă ce cas de figure. Une loi de contact visco-Ă©lastique estutilisĂ©e et illustrĂ©e par la figure 1.17 : dans les directions normale et tangentielle au plande contact, des raideurs kn et respectivement ks contrĂŽlent la partie Ă©lastique du com-portement. Lâamortissement normal est pilotĂ© par un coefficient de restitution e, et uneloi Ă seuil de frottement ” est utilisĂ©e dans la direction tangentielle. Les choix des quatreparamĂštres, ainsi que la formulation dĂ©taillĂ©e de la loi seront dĂ©crits dans la partie 2.2.1.
Le code de modĂ©lisation par la mĂ©thode des Ă©lĂ©ments discrets YADE (voir [105]) estlargement utilisĂ© pour ce travail de thĂšse. Il a lâavantage dâĂȘtre open-source, permettantainsi le contrĂŽle, la modification et lâimplĂ©mentation de fonctionnalitĂ©s. Il bĂ©nĂ©ficie de cefait dâune large communautĂ© dâentraide et de contributeurs, en grande partie constituĂ©sde scientifiques.
1.3.2 SystĂšmes modĂšles
Le choix de lâutilisation de systĂšmes modĂšles simplifiĂ©s a Ă©tĂ© fait pour deux raisons prin-cipales.
1.3. MĂ©thode 27
La premiĂšre rĂ©pond Ă un souci de limitation du temps de calcul, en se focalisant surune zone restreinte en amont de lâobstacle. De cette façon, il est possible de modĂ©liserfinement cette zone en diminuant la taille des particules. Dans le cas dâun Ă©coulementgravitaire, le temps avant Ă©tablissement de lâĂ©tat permanent au regard de lâinteractionavalanche-obstacle est relativement long. Il faut en effet modĂ©liser le phĂ©nomĂšne dans satotalitĂ©, alors que lâinitiation de lâĂ©coulement le long de la pente ainsi que la force dâimpactpendant lâĂ©tat transitoire ne font pas partie de cette Ă©tude. Lâutilisation de systĂšmesmodĂšles permet dâatteindre bien plus rapidement lâĂ©tat permanent de lâĂ©coulement, et dece fait de limiter le temps de calcul.
La seconde motivation Ă lâutilisation de systĂšmes simplifiĂ©s est la maĂźtrise des para-mĂštres macroscopiques de lâĂ©coulement comme la pression de confinement et la vitesse.Dans le cas de lâinteraction entre un Ă©coulement granulaire et un obstacle, ces paramĂštressont fortement dĂ©pendants Ă la pente et alors difficiles Ă contrĂŽler. En particulier, danslâĂ©tude avancĂ©e de Chanut et al. [20] mentionnĂ©e en partie 1.1.3, la longueur de la zonemorte en amont de lâobstacle est un paramĂštre primordial mais trĂšs sensible Ă la pente.Pour rĂ©pondre Ă cette problĂ©matique, les Ă©coulements dans les systĂšmes choisis ne sontpas induits par la pesanteur, mais par le dĂ©placement de lâune des parois qui contrĂŽlealors le cisaillement et la pression de confinement.
Les deux parties suivantes sont consacrĂ©es Ă la description des deux systĂšmes modĂšlesĂ©tudiĂ©s dans cette thĂšse, Ă savoir : la cavitĂ© granulaire entraĂźnĂ©e par cisaillement, puis lemur immergĂ© dans un Ă©coulement granulaire cisaillĂ©. Ils sont tous deux sans gravitĂ©, etconstituĂ©s de sphĂšres et de murs dĂ©finis dans le mĂȘme plan. Ils peuvent donc ĂȘtre assimilĂ©sĂ des systĂšmes bidimensionnels, mais la formulation DEM et les grandeurs physiques sonttridimensionnelles.
La cavité granulaire entraßnée par cisaillement
La cavitĂ© granulaire entraĂźnĂ©e par cisaillement ou LDC (granular Lid-Driven Cavity, voirFig 1.18) est un systĂšme dont les conditions aux limites sont classiquement utilisĂ©es dansle domaine de la mĂ©canique des fluides. Un assemblage plan de sphĂšres de hauteur initialeH est dĂ©posĂ© par gravitĂ© dans une boite Ă quatre parois. La paroi infĂ©rieure de longueurL est immobile et rugueuse, tandis que les parois latĂ©rales sont lisses, immobiles, et dehauteur infinie. La paroi supĂ©rieure est rugueuse et de longueur infinie. Elle se dĂ©placepar translation constante dans la direction x avec une vitesse U positive, appliquant uncisaillement aux grains. La position de cette paroi suivant lâaxe vertical y est calculĂ©econtinuellement par lâalgorithme de DEM afin dâappliquer la pression de confinement P
au milieu granulaire. Elle repose ainsi sur les grains en leur appliquant une force macro-scopique verticale Fc correspondant Ă la pression macroscopique P = Fc/Ld, oĂč d est lediamĂštre des grains. Il convient de prĂ©ciser quâaucun mouvement de rotation nâest autorisĂ©
28 Chapitre 1. Introduction
Figure 1.18 â Illustration de la cavitĂ© granulaire entraĂźnĂ©e par cisaillement (LDC). La paroi
rouge reprĂ©sente lâobstacle, sur lequel sont faites les mesures.
Ă la paroi supĂ©rieure, et que la pression quâelle exerce est susceptible de ne pas ĂȘtre homo-gĂšne le long de lâaxe x. Le cisaillement ainsi dĂ©fini est simulĂ© pour une pĂ©riode (virtuelle)de dix secondes, dont la premiĂšre est nĂ©cessaire pour atteindre le rĂ©gime permanent alorsque lâenregistrement des donnĂ©es est effectuĂ© pendant les neuf suivantes.
La correspondance avec la description âzone morte - zone inertielle - obstacleâ faitedans les parties prĂ©cĂ©dentes est la suivante : la zone morte est reprĂ©sentĂ©e par lâassemblagede grains, qui sont confinĂ©s et cisaillĂ©s par la paroi supĂ©rieure reprĂ©sentant la zone iner-tielle. Le mur de droite, qui fait face Ă lâĂ©coulement, est lâobstacle sur lequel les signaux deforce F sont enregistrĂ©s. La longueur de la zone dâinfluence de lâobstacle sur lâĂ©coulement,caractĂ©risĂ©e par la zone morte dans le modĂšle de Chanut et al. [20], dĂ©pend de la pentedans les systĂšmes gravitaires et a une forte incidence sur la force subie par lâobstacle. Elleest alors difficile Ă contrĂŽler et Ă mesurer, la technique actuellement utilisĂ©e reste basĂ©e surun critĂšre de vitesse arbitraire. Lâavantage majeur de la cavitĂ© granulaire entraĂźnĂ©e par ci-saillement est la maĂźtrise de la taille de la zone dâinfluence, imposĂ©e par construction aveclâintroduction de la longueur du systĂšme L. Il est Ă noter aussi que les grandeurs macro-scopiques de contrĂŽle : U , P et L, sont ajustables explicitement pour permettre dâĂ©tudierdirectement leur impact sur lâinteraction Ă©coulement granulaire â obstacle. Cependant, cesystĂšme ne permet pas aux grains de surverser lâobstacle, câest-Ă -dire de transiter de lazone morte Ă la zone inertielle. Il est de ce fait susceptible de souffrir dâun phĂ©nomĂšne deblocage des grains dans le coin supĂ©rieur droit de la cavitĂ©, lâĂ©loignant potentiellement delâapplication finale.
1.3. MĂ©thode 29
Figure 1.19 â Illustration du mur immergĂ© dans un Ă©coulement granulaire entraĂźnĂ© par cisaille-
ment (IW). La paroi rouge reprĂ©sente lâobstacle, sur lequel sont faites les mesures.
Le mur immergé dans un écoulement granulaire entraßné par cisaillement
Le mur immergĂ© dans un Ă©coulement granulaire entraĂźnĂ© par cisaillement, IW (ImmersedWall, voir Fig 1.19) est un systĂšme original, imaginĂ© pour correspondre spĂ©cifiquement Ă lâĂ©tude des fluctuations dans lâinteraction avalanche-obstacle. Il peut ĂȘtre vu comme unegĂ©nĂ©ralisation de la LDC Ă des obstacles dont la hauteur h nâest pas infinie, mais infĂ©rieureĂ la hauteur des grains (h < H). Les conditions aux limites supĂ©rieure et infĂ©rieure sontidentiques Ă la LDC : paroi infĂ©rieure rugueuse et immobile, paroi supĂ©rieure rugueuseimposant la vitesse de cisaillement U > 0 et la pression de confinement P par translationssuivant les axes respectifs x et y. La hauteur initiale des grains est H, la longueur dusystĂšme est L, et un mur vertical de hauteur h est immergĂ© dans le milieu granulaire.Le systĂšme dispose dâune condition aux limites pĂ©riodique suivant lâaxe x, ce qui signifieque les grains qui surversent le mur vertical sont de fait rĂ©insĂ©rĂ©s dans la simulation, Ă lâextrĂ©mitĂ© opposĂ©e (voir annotations vertes dans la figure 1.19). Comme pour la LDC,le cisaillement dure dix secondes, dont la premiĂšre est destinĂ©e Ă lâatteinte du rĂ©gimepermanent.
Dans ce systĂšme, la dĂ©finition de la zone morte et de la zone inertielle nâest pas prĂ©-Ă©tablie car les grains sont autorisĂ©s Ă transiter naturellement dâune zone Ă lâautre. Laparoi supĂ©rieure peut donc ĂȘtre vue comme une condition dâentraĂźnement des grains ausein de la zone inertielle, mais la frontiĂšre avec la zone morte se localise librement au seindu milieu granulaire. Contrairement Ă la LDC, la longueur dâinfluence de lâobstacle nâestde fait pas imposĂ©e, mais ce systĂšme a lâavantage de se rapprocher des observations faitessur les Ă©coulements en conditions rĂ©elles en autorisant les grains Ă surverser lâobstacle. Enoutre, lâutilisation de la condition aux limites pĂ©riodique permet de garder la souplessedu choix des grandeurs macroscopiques de contrĂŽle : U , P et L, facilitant la comparaison
30 Chapitre 1. Introduction
de ce systĂšme avec la LDC.
1.4 Grille de lecture
Force moyenne et fluctuations subies par un obstacle indéformable soumis à l'écoulement
confiné d'un milieu granulaire
Chapitre 1 : Introduction générale
CAVITĂ GRANULAIRE MUR IMMERGĂ
Ătudemoyenne
Chapitre 2 : Ătude moyenne de la force subie
par un obstacle soumis à un écoulement granulaire entraßné par
cisaillement
Ătude des fluctuations
Chapitre 4 : Force moyenne et
fluctuations sur un mur immergé dans un écoulement granulaire
cisaillé
Chapitre 3 : Fluctuations de force sur un
obstacle en interaction avec un Ă©coulement
granulaire cisaillé
Chapitre 5 : SynthĂšse et perspectives
ARTICLE
ARTICLE
ARTICLE
ARTICLE
Analyse comparative
Figure 1.20 â Grille de lecture du manuscrit.
AprĂšs cette premiĂšre partie dĂ©diĂ©e Ă lâintroduction gĂ©nĂ©rale de ce travail, la suitedu manuscrit est organisĂ©e en 4 chapitres incluant la conclusion (voir aussi Fig. 1.20).Chaque chapitre est constituĂ© dâun article scientifique, et la description suivante permetde les articuler vis-Ă -vis du contexte gĂ©nĂ©ral de cette thĂšse.
â Le chapitre 2 est dĂ©diĂ© Ă lâĂ©tude exploratoire de la cavitĂ© granulaire entraĂźnĂ©e parcisaillement via des grandeurs moyennĂ©es dans le temps. Le systĂšme ainsi que la
1.4. Grille de lecture 31
loi de contact et les gammes de paramĂštres sont dĂ©crits de maniĂšre dĂ©taillĂ©e. LesrĂ©sultats des mesures et leur rĂ©ponse Ă la variation des paramĂštres macroscopiquesdâĂ©coulement sont prĂ©sentĂ©s et discutĂ©s : lignes de courant, profils spatiaux de forcesaux parois, force moyenne sur lâobstacle. La dĂ©finition dâun nombre inertiel macro-scopique IM , qui reflĂšte les effets antagonistes de la vitesse et de la pression ma-croscopiques, permet de dĂ©finir une expression pour la force moyenne exercĂ©e surlâobstacle. Une Ă©tude de sensibilitĂ© de cette loi aux paramĂštres microscopiques estĂ©galement exposĂ©e.
â Le chapitre 3 sâintĂ©resse principalement aux fluctuations de forces sur lâobstacledans le mĂȘme systĂšme. Une brĂšve description du systĂšme qui reprend les Ă©lĂ©mentsdu chapitre 2 est proposĂ©e, et les rĂ©sultats moyens sont complĂ©tĂ©s par une analyse dela rhĂ©ologie locale. Les fluctuations sont dâabord analysĂ©es via des autocorrĂ©lationsde signaux Ă lâĂ©chelle mĂ©soscopique qui donnent des informations sur la pĂ©riodicitĂ©et la mĂ©moire du systĂšme. Les distributions de signaux de forces Ă trois Ă©chelles sontensuite introduites, et leur comportement vis-Ă -vis du nombre inertiel IM est dĂ©crit.Dans ce chapitre, les limites de la gĂ©omĂ©trie de la cavitĂ© vis-Ă -vis des fluctuationssont mises en avant, annonçant la nĂ©cessitĂ© dâĂ©tudier un systĂšme diffĂ©rent.
â Le chapitre 4 prĂ©sente le mur immergĂ© dans un Ă©coulement granulaire cisaillĂ©, rĂ©-pondant Ă la principale limite du systĂšme prĂ©cĂ©dent. Hormis lâintroduction de lahauteur du mur, les paramĂštres et la loi de contact sont identiques au systĂšme decavitĂ©. Cette partie sâappuie donc sur les outils et rĂ©sultats des chapitres prĂ©cĂ©dentspour Ă©tudier la dynamique moyenne et les fluctuations dans ce nouveau systĂšme. Leslignes de courant ainsi que la rhĂ©ologie locale sont analysĂ©es, et une loi de prĂ©dictionde la force moyenne sur lâobstacle en fonction des paramĂštres macroscopiques estĂ©tablie. Enfin, ce chapitre sâintĂ©resse aux fluctuations via les autocorrĂ©lations et lesdistributions de signaux de forces aux Ă©chelles mĂ©soscopique et macroscopique.
â La conclusion de ce manuscrit fait un rappel des objectifs et de la mĂ©thode employĂ©epour y parvenir. Les rĂ©sultats des chapitres prĂ©cĂ©dents sont ensuite synthĂ©tisĂ©s enmettant lâaccent sur les principaux rĂ©sultats pour chaque systĂšme dans un premiertemps, puis en proposant une analyse comparative des deux systĂšmes dans un se-cond temps. Finalement, certains rĂ©sultats sont discutĂ©s, amenant les perspectivesmajeures de ce travail de thĂšse.
32 Chapitre 1. Introduction
Chapitre 2
Ătude de la force moyenne subie par un
obstacle soumis Ă un Ă©coulement
granulaire cisaillé
33
34Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
Numerical investigations of the force experienced by awall subject to granular lid-driven flows : regimes andscaling of the mean force.Kneib F., Faug T., Dufour F., Naaim M.
Article published in Computational Particles Mechanics.
Abstract Discrete element simulations are used to model a two-dimensional1 gravity-free granular sample, which is trapped between two smooth sidewalls and one bottomrough wall while being subject to a constant shearing velocity at the top under a givenconfinement pressure. This system, inspired by conventional fluid mechanics, is called agranular lid-driven cavity. Attention is firstly paid to the time-averaged dynamics of thegrains once a steady-state is reached. Strong spatial heterogeneities associated with alarge-scale vortex formed within the whole volume of the lid-driven cavity are observed.The mean steady force on the sidewall opposite to the shearing direction is then investi-gated in detail for different cavity lengths, shearing velocities and confinement pressuresat the top. The ratio of the force on the latter wall to the top confinement pressure forceis not constant but depends on both the shearing velocity and the confinement pressure.Above a critical value of the cavity length relative to the wall height and over a wide rangeof both shearing velocity and top confinement pressure, all data merge into a one-to-onerelation between the mean force scaled by the top pressure force and the macroscopicinertial number of the lid-driven cavity. This result reveals the key role played by theinertial rheology of the granular material in the granular force transmission.
2.1 Introduction
The force experienced by obstacles subject to granular flows is of particular interest forseveral problems, e.g. the transport of particles in industrial or agriculture applications[108], the modeling of impact cratering [55], or the design of safe building and protectionstructures able to withstand the avalanche force [38]. For the latter example, it is now well-established that stagnant (quasistatic) zones, namely dead zones, can be formed behindobstacles, while the non trapped grains can continue to flow around the obstacle [39]. Inthose specific configurations, a kind of shear band may appear at the interface betweenthe dead zone and the flowing zone. Up to now, for the particular situation of rigid walls
1The two-dimensional system mentionned here was modeled by a perfectly planar assembly of tree-
dimensional objects : the grains are modeled by spheres and the walls are modeled by cuboid shapes. As
such, their positions coordinates in the z-axis component is fixed to 0, and the rotations of the spheres
are only allowed through the z-axis.
2.1. Introduction 35
normal to incoming free surface flows, a great attention has been paid to the mean (time-averaged) force on the obstacle. Some successful analytical solutions have been developedto predict the time-averaged force on the wall, while accounting for the presence of thedead zone, in either steady [40] or unsteady [20, 39] regimes. One main finding is thatthe apparent weight of the fluid volume disturbed by the wall (influence zone upstream ofthe wall), which includes both the dead zone and the overflowing layer of grains, greatlycontributes to the total force [40, 20, 39, 38]. The apparent weight is the difference betweenthe weight in the slope direction and the effective friction force. The latter force dependson the rheology of the granular material. The main motivation of the present study is tobetter understand the role played by the rheological properties of the granular material(through its friction) in the total force experienced by the wall, under a wide range ofconfinement pressures and shearing velocities imposed by the flow surrounding the deadzone.
A classical geometry in conventional fluid mechanics, namely the lid-driven cavity (seefor instance [91] and references therein) is tackled, with the help of two-dimensional1 (2D)discrete element method (DEM) simulations. Attention is paid to the force experiencedby the vertical wall facing the shearing displacement, as a function of the pressure andthe shearing speed at the top. Instead of directly studying a flow overflowing a wall (asdone in [40, 20]), the lid-driven cavity problem without gravity is considered. This systemallows the investigation of the problem of a quasistatic zone in contact with a wall undervarious flow regimes at reasonable computational times. A wide range of both confinementpressures P and shearing velocities U at the top (mimicking a flowing granular layerabove the dead zone) can be studied. Moreover, the size of the granular sample subjectto P and U is entirely controlled by the size of the lid-driven cavity itself. Analysing adead zone formed upstream of a wall down a slope and subject to a wide range of bothP and U (imposed by a surrounding flow), with the help of periodic DEM simulations,remains challenging. Firstly, the final length of the dead zone reached in the steady state isunknown and greatly depends on both the wall height and the nature of the incoming flow,which makes the size of the simulated system difficult to predict a priori. In particular, ithas been shown in [40] that the length of the dead zone diverges at low slopes. Secondly, itis rather difficult to investigate a wide range of macroscopic inertial number IM â U/
âP
while the present study on the 2D granular lid-driven cavity allows the range of IM tospan four decades (from 6 10â4 to 1.2).
The problem of the force exerted by granular flows impacting obstacles or experiencedby objects in motionless granular materials has been tackled in various granular systems :annular flows [108, 2, 1, 45], unidirectional horizontal [17, 106, 13, 99] or vertical [22]flows, objects penetrating in static granular media [55, 89, 90, 15], flows down inclinesimpacting a wall [40, 20, 39], etc. Rods rotated perpendicular to their axis in a granularbed have been also studied [16, 48]. For rapid (either dense or dilute) flows for which
36Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
inertial forces prevail, a velocity-square dependent force is generally found [17, 106, 13,16]. In contrast, the force is rate-independent when inertia becomes weak [108, 2, 22,16, 48] and the granular force is described as a frictional phenomenon meaning that theforce is proportional to the confinement pressure times the surface of the object. Recentstudies have investigated in detail a broad range of parameters for the specific problemof the object penetration in static granular materials [55, 15]. If inertial effects are notdominant (the velocity-square dependent force does not come into play for sufficiently slowflows), a robust scaling verified for several shapes of objects has been found. The force isproportional to the pressure p [15]: F = α”0pSâ„, where Sâ„ is the object surface normalto the flow direction, ”0 is a constant friction coefficient associated with the granularmaterial at stake and α is a proportionality coefficient which may be much greater thanunity. For the penetration problem, p = ÏgZ is the typical hydrostatic pressure associatedwith the penetration depth Z, where Ï is the bulk density and g = 9.81m.sâ2 is the gravityacceleration. A similar scaling was found earlier for rods rotated perpendicular to theiraxis in a granular bed [16]. The present paper does not deal with the recent results in[49] showing that this classical frictional picture of the granular force may not be valid insome specific systems where the object itself is able to screen the pressure. The presentpaper demonstrates that the force measured at the wall of a lid-driven cavity is consistentwith the traditional frictional scaling, meaning that the force is firstly controlled by thepressure at the top. However, a subtle effect of the rheology of the granular materialthrough imposed U and P will be presented.
The paper is organized as follows. The discrete element method used and the associatedcontact laws are first described in detail, as well as some important results regarding thetime-averaged dynamics of the 2D lid-driven cavity (section 2.2). The time and depthaveraged force exerted on the wall which is normal and opposite to the direction of theshear speed is then investigated (section 2.3). Finally, we discuss the main results andfurther research to be conducted in section 2.4.
2.2 Method and simulated granular system
2.2.1 Discrete element method and contact laws
The Discrete Element Method (DEM) is an extension of the Molecular Dynamics mo-deling method introduced by Cundall & Strack [28]. It is specifically suited to modelgranular media as it allows accounting for the multiple interactions between deformablespheres through contact forces. All numerical simulations in this paper were performedwith the open-source software YADE-DEM, whose implementation details are availableat [105]. At each time step, contact forces are computed from particles relative positionsand rotations, and from specific contact laws accounting for the required contact physics.
2.2. Method and simulated granular system 37
Then an explicit leapfrog integration of the 2nd Newtonâs law allows computing the newposition and velocity of particles.A linear visco-elastic normal force model is used with a Coulomb frictional force alongthe tangential direction. This model is satisfactory for the present study as it can mimickdry contacts regardless to the rate of loading, from a quasistatic regime to a dynamic one(see for instance [27]). The normal component of the interaction force is defined as :
Fn = max(0, knÎŽn + cnÎŽn)n, (2.1)
where ÎŽn is the normal interpenetration between the two particles, ÎŽn the related inter-penetration velocity, kn the contact normal stiffness, cn the viscous coefficient and n thecontact normal unit vector. In accordance with [64], the tangential component of theinteraction force is incrementally computed as :
dFs = (ksÎŽsdt)s
|Fs| †”|Fn|(2.2)
where Ύs is the relative tangential velocity at the contact point, ks the contact tangentialstiffness, ” the contact friction coefficient, dt the time-step and s the tangential unitvector associated with the relative tangential displacement. The normal contact stiffnesskn between two particles 1, 2 is deduced from their Young modulus E1, E2 and theirdiameters d1, d2 :
kn =E1d1E2d2
E1d1 + E2d2. (2.3)
The contact tangential stiffness is determined by the particle Poissonâs ratios Îœ1 andÎœ2 :
ks =Μ1E1d1Μ2E2d2
Μ1E1d1 + Μ2E2d2. (2.4)
For two identical spheres of diameter d, Young modulus E and Poissonâs ratio Îœ,equations (2.3) and (2.4) reduce to :
kn =1
2Ed,
ks = Îœkn.(2.5)
The initial aim is to mimick glass beads behavior, but the real glass Young moduluscannot be used as this would lead to heavy calculation times. It is quite well establishedthat the following dimensionless number should be sufficiently high for the simulations tobe in the asymptotic limit of rigid grains :
N0 =knP0d
=1
2
E
P0
, (2.6)
where the dimensionless number N0 compares the normal contact stiffness to a typicalpressure P0 characterizing the granular system at stake (the value chosen for P0 will be
38Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
discussed hereinafter). Keeping E (or kn) sufficiently large leads to simulations insensi-tive to its value, at least if the mean (macroscopic) behavior of the granular system isconsidered. This generally works well with N0 = 104, as discussed for instance in [40] andreferences therein. However, as local peak pressures are likely to reach very high values,the results sensitivity toward the particle stiffness was investigated by increasing N0 from1.5 104 to 6 104 (see table 2.1).
The contact friction coefficient ” is deduced from the two particle friction coefficients”1 and ”2 :
” = min(”1, ”2) (2.7)
Computing the viscous coefficient cn from the particle parameters â and especiallyfrom the restitution coefficient e â is not trivial. The formulation reported in [88] is used :
e =
exp[âÎČ
Ï
(Ï â arctan 2ÎČÏ
Ï2âÎČ2
)]for ÎČ < Ï0â
2,
exp[âÎČ
Ï
(arctan â2ÎČÏ
Ï2âÎČ2
)]for ÎČ â
[Ï0â2, Ï0
],
exp[â ÎČ
Ωln ÎČ+Ω
ÎČâΩ
]for ÎČ > Ï0,
(2.8)
where ÎČ = cn/meff , Ï20 = 2kn/meff , Ï =
âÏ20 â ÎČ2, Ω =
âÎČ2 â Ï2
0 and 1/meff = 1/m1+
1/m2. But the reciprocal function of equation (2.8) to get a solution for cn(e, kn,meff )
does not allow any analytic solution. Thus, the efficient Newton-Raphson method wasimplemented, which is appropriate as cn(e, kn,meff ) is a monotonic function. The effectof cn is analysed on the results according to the values of e reported in table 2.1. In theDEM model, the time-step is chosen with respect to the minimum natural oscillationperiod amongst all particles (Tmin). The time-step dt is then automatically set up with asafety factor Ndt as :
dt = NdtTmin. (2.9)
The effect of the time-step on the results is studied by modifying Ndt, as reported in table2.1. In accordance with equations (2.3, 2.6, 2.8) and table 2.1, the following ranges ofvalues are used : kn â [6.6 101, 6.6 105]N.mâ1 ; cn â [1.3 10â3, 3.3 10â1]kg.m.sâ1.
To avoid crystallization effect, a slight polydispersity is introduced by choosing thegrain diameters randomly between d(1 â ddisp) and d(1 + ddisp), where ddisp is the po-lydispersity coefficient. Although all results presented in this paper were obtained withddisp = 0.15, it has been verified that they are not influenced by a slight variation of thepolydispersity coefficient by performing some simulations with ddisp = 0.1. The particledensity Ïp is chosen according to the glass density, i.e. 2500 kg.mâ3.
2.2. Method and simulated granular system 39
Table 2.1 â All simulation parameters and corresponding values used. RCP = random close
packing.
Symbol Value(s) Description
d 10â3m mean grain diameter
ddisp 0.1, 0.15 polydispersity coefficient
Ïp 2500 kg.mâ3 grain density
Ί 0.6 estimated RCP (spheres)
e 0.5, 0.75 restitution coefficient
N0 1.5 104, 3 104, 6 104 stiffness safety factor
” 0.27, 0.5 friction coefficient
Ndt 0.15, 0.3 time-step safety factor
Îœ 0.3 Poissonâs ratio
H 30d cavity height
L 0.5H to 10H cavity length
NU 1 to 20 shear velocity coefficient
NP 0.01 to 100 top pressure coefficient
2.2.2 Lid-driven cavity
Geometry and boundaries
The lid-driven cavity is modeled by a planar assembly of spheres trapped in a rectangulargravity-free box (see Fig. 2.1). The box length is L and its height is H. The box heightH was kept constant and equal to 30d in all simulations discussed in the present paper.The two sidewalls are fixed and smooth while the bottom wall is fixed and rough. The topwall is horizontal, rough and rigid. Its infinite length allows the application of a constanthorizontal (shear) velocity U as well as a constant vertical pressure P (through the verticaldisplacement of the top wall satisfying inertial dynamics). Roughnesses are made by gluedgrains. The friction coefficient between walls and particles is equal to the inter-particlefriction. The imposed vertical pressure P directly defines the typical pressure P0 used forthe limit of rigid grains discussed in section 2.2.1 : P0 = P . As a large range of flow regimesis studied, the two dimensionless numbers NU and NP are introduced : U = NU
âgd and
P = NPΊÏpgH. They are convenient to modify the shear velocity and the vertical pressurearound some typical values based on the gravity acceleration g, the grain diameter d, thebox height H, the particle density ÏP and a typical volume fraction Ί. Please note thateven if the system is gravity-free, NP is based on a typical pressure which would act inan identical system under gravity.
40Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
Figure 2.1 â Sketch of the 2D granular lid-driven cavity simulated by DEM.
Figure 2.2 â Example of time series of space-averaged force on the wall, obtained for L/H = 5,
NU = 10, NP = 0.01. The instantaneous force F is highly fluctuating over time. The horizontal
dashed line depicts the value of the time-averaged force, named F , reached in the steady-state
regime (after a short transient of about 1s).
Steady-state regime
Figure 2.2 displays an example of the total normal force (F ) on the wall evolving overtime, obtained for L/H = 5, NU = 1 and NP = 0.01. The tangential component of theforce is much smaller than the normal component for all simulations. The force time seriesshow strong fluctuations. If a moving average is considered (over 0.3 seconds, as shown inFig. 2.2), a transient regime followed by a steady-state regime is observed. F is definedas the mean force experienced by the sidewall facing to the shear velocity averaged overtime and across the wall height, which is reached in the steady-state regime. Thus, F isindependent of the simulation time as soon as the latter is high enough. This mean steadyforce F corresponds to the horizontal dashed line drawn in Fig. 2.2.
The present study is mainly focused on the study of the mean force F over a widerange of velocity U and pressure P . However, three important aspects are first investigated
2.2. Method and simulated granular system 41
in order to highlight the strong spatial heterogeneities within the box, which are causedby the geometry and boundary conditions associated with the lid-driven cavity problem.The main observations associated with the development of a large-scale vortex within thewhole volume of the cavity are reported in the following section.
Large-scale vortex and spatial heterogeneities within the cavity
Firstly, some examples of time-averaged streamlines within the cavity are drawn in Fig. 2.3for L/H = 5 at constant pressure (NP = 1) for two distinct shear velocities (NU =
3, 10). Similarly to classical patterns usually observed with Newtonian fluids flows, alarge-scale vortex appears within the whole cavity. In contrast with recent investigationson the formation of mesoscopic and short-life vortice in granular media (see [82] and [68]),the present study focuses on the macroscopic steady vortex whose size is fully controlledby the cavity size. The vertical dashed line highlights the position of the vortex centerrelative to the box length : the vortex center is almost located at the center of the box atlow shear rates, while it is shifted to the right at high shear rates. The horizontal dashedline depicts the position of the vortex center across the cavity height, which remainsvery close to the upper shearing wall whatever the shear rate. Interestingly, those time-averaged streamlines show that the flow within the cavity ressembles a thick layer of grainsundergoing a quasi simple shear (below the vortex center and moving from right to leftin Fig. 2.3) surmounted by a very thin and rapid inertial layer (above the vortex centerand moving from left to right). The typical thickness of the latter layer is called Ç« (a fewgrains), while the thickness of the former layer is Hâ Ç«, as shown in Fig. 2.3. This pictureis true over around 75% of the cavity length for NU = 3 and around 40% for NU = 10.In addition to the shearing at the top, the roughness at the bottom might be essentialfor the vortex center position (we expect that a smoother but still rough bottom wouldprobably increase Ç«).
Secondly, Fig. 2.4 shows the variation of the time-averaged vertical force on the topwall across the cavity length (along the position x). The data were reconstituted from theposition and the vertical component of the force of each sphere that compose the top wall.It is worth noting that 1
L
â« L
0F topn (x)dx = PLd, where P is the pressure imposed at the
top. The overlapping curves at constant cavity length exhibit a weak effect of the shearvelocity on the top force profile. A strong increase along the cavity length occurs at thetop : the profile is exponential and may become more than exponential when approachingthe position x = L (sidewall facing the shearing velocity). This kind of singular wedgeeffect (caused by a very high confinement in the top right corner of the cavity) prevailswhen the typical length of the wedge reaches the length of the cavity. There might thenexist a critical cavity length above which the magnitude of the force experienced by thesidewall facing to the shearing velocity should not be significantly influenced by this wedge
42Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.0000.0050.0100.0150.0200.0250.030
y
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16velocity [m.sâ1 ]
Ç«
HâÇ«
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.0000.0050.0100.0150.0200.0250.030
y
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45velocity [m.sâ1 ]
Ç«
HâÇ«
Figure 2.3 â Examples of time-averaged streamlines for two shearing velocities : NU = 3 (top)
and NU = 10 (bottom), NP = 1 and L/H = 5 for both plots. The intersection between the
vertical and horizontal dashed lines defines the position of the vortex center.
effect (see discussion in section 2.3.1).Thirdly, the time-averaged force distribution along the wall height is investigated.
Fig. 2.5 shows the normalized pressure variation across the vertical position y, which isexperienced by the right sidewall of the cavity. The curves were obtained by segmentingthe force measurement along the right wall height into 10 parts of same thickness, in asimilar fashion that would stem from data obtained from laboratory tests with 10 forcesensors placed along the wall (the surface of each of them filling a tenth of the wall height).The high increase of the force just below the cavity lid confirms the importance of thewedge effect.
No systematic analysis of the time-averaged streamlines, spatial pressure variationsat the top and against the wall, as functions of L/H, NU and NP is conducted in thispaper. However, the results shown in Figs. 2.3, 2.4 and 2.5 clearly evidence the strongspatial heterogeneities occurring within the cavity, which are associated with the large-scale vortex forming inside the whole volume of the cavity. In spite of those strong spatialheterogeneities, the existence of a scaling of F is investigated thereinafter.
2.3 Mean force on the sidewall facing to the upper shea-
ring velocity
Forces in slow sheared dense granular media are usually described as a frictional pheno-menon, meaning that the forces are proportional to the local confining pressure times the
2.3. Mean force on the sidewall facing to the upper shearing velocity 43
0 1 2 3 4 5 6 7 8 9 10x/H
10-7
10-6
10-5
10-4
10-3
10-2
Ftop
n
NP =1
NU = 1.0
NU = 5.0
NU = 10.0
NU = 15.0
NU = 1.0
NU = 5.0
NU = 10.0
NU = 15.0
NU = 1.0
NU = 5.0
NU = 10.0
NU = 15.0
NU = 1.0
NU = 5.0
NU = 10.0
NU = 15.0
Figure 2.4 â Examples of time-averaged pressure variation along the position x at the top of the
cavity for different cavity lengths and shear velocities. Cavities lengths are displayed by vertical
dashed lines.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Pressure distribution
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
y/H
NU = 1.0 | NP = 25.0
NU = 3.0 | NP = 1.0
NU = 10.0 | NP = 1.0
Figure 2.5 â Examples of normalized spatial pressure variation across the vertical position
y at the right wall of the cavity, for different confinement pressures P and shear velocities U
(L/H = 5). The wall height is segmented into 10 parts for force measurements.
44Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
0 2 4 6 8 10L/H
1.0
1.5
2.0
2.5
3.0
3.5
F/(PLd)
NP =1 NU = 1
NU = 5
NU = 10
NU = 15
Figure 2.6 â Mean force on the right wall relative to the top force, F/(PLd), versus the cavity
length relative to its height, L/H, for different velocities at constant NP = 1 (H/d = 30).
surface of the object (see discussion in the introduction of the paper). The mean forcefrom the top, PLd, should then play a prevailing role in the force transmission from thetop toward the sidewall facing to the upper shear velocity. In the following, the effect ofL, U and P on the mean force F scaled by the top force PLd on the wall is investigated.
2.3.1 Effect of the cavity length
Figure 2.6 shows how the ratio F/(PLd) varies with the cavity relative length L/H.This ratio highly depends on L/H at small L/H values, which confirms some influenceof the strong spatial heterogeneities observed within the cavity and discussed before insection 2.2.2. However, the ratio F/(PLd) becomes only weakly dependent on L/H abovea critical value of L/H around 4 â 5. Data on the force experienced by the left wall ofthe cavity, named Fâ, are not shown here because the magnitude of this force was foundto be much lower than the force F on the right wall, once L/H was sufficiently large.The following values are typically obtained : Fâ/F < 0.05 â 0.10 for L/H above 4 â 5,while Fâ/F could reach 0.2 â 0.25 for L/H = 2. In order to avoid a prevailing effect ofthe cavity length while keeping reasonable computational times (large L leads to highercomputational times), the results presented in the following of the paper were obtainedwith a constant cavity length relative to its height, namely L/H = 5, unless otherwisespecified.
2.3.2 Effect of the macroscopic inertial number
Figure 2.7 shows how the ratio F/(PLd) varies with NU at different values of NP :F/(PLd) undergoes a slow increase with the shear velocity, similarly to the results re-
2.3. Mean force on the sidewall facing to the upper shearing velocity 45
0 5 10 15 20NU
0.5
0.6
0.7
0.8
0.9
1.0
F/(PLd)
NP = 0.01
NP = 0.04
NP = 1.0
NP = 25.0
NP = 100.0
Figure 2.7 â Mean force on the wall relative to the top force, F/(PLd), versus NU at different
NP .
ported by [45] for an object moving in a static granular medium in an annular geometry.Note that decreasing P at constant U leads to a substantial increase of F/(PLd).
Two conclusions are straightforward from Fig. 2.7. Firstly, the force is rate-dependent,in contrast to rate-independent stresses in purely Mohr-Coulomb friction models. Se-condly, it does not correspond to the linear increase of force with the velocity (when thevelocity is not too large) for Newtonian fluids in conventional fluid mechanics. The slowincrease of F with U observed here might be consistent with the ”(I) inertial rheologyproposed by [52] for sheared dense granular materials. It is now well established that flowregimes in sheared dense granular materials are mainly controlled by the inertial num-ber I = tÏ/tÎł (see [44]), which compares the typical time scale of rearrangements underpressure Ï, (tÏ = d/
âÏ/ÏP ), and the macroscopic time scale (tÎł = 1/Îł) linked to the
mean deformation Îł (Îł is the local strain shear rate). The detailed time-averaged map-ping of the local inertial number within the whole volume of the cavity is not quantifiedin this study. This remains a technical challenge regarding data processing because theI(x, z)âmapping is likely to be highly heterogeneous with respect to the shape of thestreamlines and to pressure variation along x at the top, as shown in section 2.2.2. Adirect and much simpler approach is rather adopted by defining the macroscopic inertialnumber of the lid-driven cavity as the macroscopic shear rate U/H times the macroscopictime scale tP = d
âÏ/P associated with the mean top pressure P :
IM =U
HtP =
U
Hd
âÏ
P, (2.10)
where Ï is taken equal to ΊÏp, which corresponds to the mean medium density withinthe whole cavity (see values in table 2.1). It is important to stress that the present studyis focused on how varying U and P affects F/(PLd) (H and d were kept constant).
46Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
Figure 2.8 depicts the ratio F/(PLd) as a function of the macroscopic inertial numberIM . Depending on the set of simulation parameters considered, the top plot in Fig. 2.8shows some scattering, i.e. a slight dispersion of F/(PLd) at identical IM can appear. Inthe bottom plot of Fig. 2.8, the averaged value of F/(PLd) for a given IM is considered,which allows to obtain a very well-defined curve showing that
âšF/(PLd)
â©is entirely
controlled by the inertial number IM , whatever the couple (U, P ). It is worth noting thatsome results obtained here with values of L/H different from 5 (L/H = 7 or 10) are alsoincluded. The latter result means that F is not only controlled by P , but the length of thecavity crucially comes into play. Thus, following the traditional granular force on objectsdiscussed in the introduction, the force F can also be written : F = α”0PHd, where α
would depend on L/H.Fig. 2.8 also allows to verify that the results are insensitive to the value of kn because
this parameter is not constant but varied with P with respect to the limit of rigid grains(see section 2.2.1).A recent study has investigated in detail the influence of the particlesstiffness for steady-state shear granular flows [94]. In particular that study has quantifieda critical stiffness below which the softness of the particles plays a significant role in theconstitutive laws. In the present study, it is verified that kn is kept sufficiently high andweakly influences the one-to-one relation between F/(PLd) and IM shown in Fig. 2.8 bytesting the values reported in table 2.1. The results are presented in Fig. 2.9, confirmingthe weak sensitivity to kn in our simulations. Fig. 2.9 also allows the verification thatboth e and Ndt do not have any influence on the results. However, a strong influence ofthe particle friction ” was found : decreasing ” from 0.5 to 0.27 led to a systematic andremarkable decrease of F/(PLd), whatever the value of IM (see black crosses in Fig. 2.9).This result highlights that even though the mean force is primarily controlled by P andL, the interparticle friction also comes into play. As a consequence, the bulk shear stressanalysis should provide crucial information. This will be done in a future study based onthe work of Singh et al. [93], where the shear stress is deduced from the contact forcesalong the local strain rate eigendirections.
2.3.3 Toward an empirical model for the mean force
The data on Fig. 2.8 can be robustly fitted by the following empirical function (R2 â0.98) :
F
PLd= r1 +
r2 â r1
1 +I0M
IM
, (2.11)
where r1 and r2 would correspond to asymptotic values of the ratio F/(PLd) at lowIM and high IM , respectively. A fit on data displayed in the bottom plot of Fig. 2.8 (seecontinuous line) gives r1 = 0.53 and r2 = 0.86. I0M defines the typical inertial number
2.3. Mean force on the sidewall facing to the upper shearing velocity 47
10-3 10-2 10-1 100
IM
0.5
0.6
0.7
0.8
0.9
1.0
F/(PLd)
NU =1
NU =3
NU =5
NU =7
NU =10
NU =15
NU =20
10-3 10-2 10-1 100
IM
0.5
0.6
0.7
0.8
0.9
1.0
âš F/(PLd)â©
mean over different simulationsr1 +(r2âr1 )/(1+I 0M /IM )
0.00 0.01 0.02 0.03 0.04IM
0.55
0.60
0.65
0.70
0.75
0.80
âš F/(PLd)â©
r1 +bIM
Figure 2.8 â Mean force on the wall relative to the top force, F/(PLd), versus the macroscopic
inertial number IM . Top plot : raw data corresponding to all simulations conducted at different
U and P (H/d = 30) ; for a given IM , some scattering is observed depending on the set of
simulation parameters considered. Bottom plot : same data as in the top plot but by considering
the averaged value of F/(PLd) obtained at identical IM . The continuous line depicts the fit given
by Eq.(2.11) with r1 = 0.53, r2 = 0.86 and I0M = 0.034. Inset : same plot with a linear scale and
the prediction of the linear equation r1 + bIM with b = (r2 â r1)/I0M (valid for the lowest IM ).
48Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
2 3 4 5 6 7kn
1e5
0.5
0.6
0.7
0.8
0.9
1.0
F/(PLd)
IM â0.0012
e=0.5 | ”=0.5 | Ndt=0.3
e=0.75 | ”=0.5 | Ndt=0.3
e=0.5 | ”=0.5 | Ndt=0.15
e=0.5 | ”=0.27 | Ndt=0.3
2 3 4 5 6 7kn
1e5
0.5
0.6
0.7
0.8
0.9
1.0
F/(PLd)
IM â0.0122
e=0.5 | ”=0.5 | Ndt=0.3
e=0.75 | ”=0.5 | Ndt=0.3
e=0.5 | ”=0.5 | Ndt=0.15
e=0.5 | ”=0.27 | Ndt=0.3
0.5 1.0 1.5 2.0 2.5kn
1e2
0.5
0.6
0.7
0.8
0.9
1.0
F/(PLd)
IM â0.6086
e=0.5 | ”=0.5 | Ndt=0.3
e=0.75 | ”=0.5 | Ndt=0.3
e=0.5 | ”=0.5 | Ndt=0.15
e=0.5 | ”=0.27 | Ndt=0.3
Figure 2.9 â Mean force on the wall relative to the top force, F/(PLd), versus the normal
stiffness kn for three values of IM â 0.0012 (top), IM â 0.0122 (center) and IM : IM â 0.6086
(bottom) with different sets of e, ” and Ndt.
2.3. Mean force on the sidewall facing to the upper shearing velocity 49
for which F/(PLd) = (r1 + r2)/2 and is found to be I0M = 0.034. For low IM , Eq. (2.11)reduces to F/(PLd) = r1+bIM , where b = (r2âr1)/I
0M (see dashed line in the bottom plot
of Fig. 2.8). The cross-comparison between the data and this linear law clearly shows thesaturation of F/(PLd) at high IM , which can be captured by Eq.(2.11) with the additionalparameter r2 corresponding to the upper-limiting value of F/(PLd). As a conclusion, theforce on the right wall is dependent on the three different flow regimes, which are classicallycharacterized by the inertial number IM : the quasistatic regime at low IM (IM . 10â3)for which F/(PLd) is constant, the dense inertial regime at intermediate values of IM
(10â3 . IM . 0.3) for which F/(PLd) is a function of IM and the rapid regime at higherIM (IM & 0.3) for which F/(PLd) starts saturating.
It is now well-established that (if non-local effects do not play a role) there exists aone-to-one relation between the inertial number and the bulk friction within the material[52] (enhanced granular rheologies that quantifies the non-local effects introduced in [54]and [14] are not taken into account in this paper). The analogy between the characteristicshape of the data shown in Fig. 2.8, fitted by Eq.(2.11), and the form of the friction lawproposed by Jop et al. (see [52]) for dense sheared granular flows is striking. This mightconfirm the crucial role played by friction, intimately linked to the inertial number, in theforce transmission process. Letâs assume a Iâdependent friction force, known to increasewhen the inertial number increases [52], which is likely to drive the variation of F/(PLd).Assuming the collinearity between the stress and strain rate tensors, the latter frictionforce should (by construction) be oriented from the left to the right in the same directionas the upper shear velocity, so it may produce an increase of F/(PLd) with IM (for thedetailed discussion about the non-collinearity, see [107]). Coming back to the shape of thetime-averaged streamlines within the cavity associated with the large-scale vortex movingclockwise (Fig. 2.3), the thick bottom layer (of thickness H â Ç«) in quasi simple shearmight produce this friction force. In contrast, the friction force stemming from the thinupper layer (of thickness Ç«) is expected to be negligible because the latter layer correspondto a dilute and more inertial flow. The high force experienced by the wall in its upperpart (see Fig. 2.5) is more likely to emerge from a purely inertial force in the form ÏU2Ç«d.The existence of a thick flow in simple shear opposite to the shearing velocity at thetop appears to be consistent with the observation of an increasing F/(PLd) with IM , asobserved in Fig. 2.8.
The inertial forces (proportional to U2) caused by the large-scale vortex moving clo-ckwise do not appear to come into play, meaning that the force is not controlled by thevelocity square (within the range of parameters tested in the present study), in contrastto the velocity-square dependent force generally found for rapid granular flows in someother geometries [17, 106, 13]. Eq.(2.11) demonstrates that the force transmission is ra-ther driven by frictional but rate-dependent processes. The force is proportional to thepressure imposed at the top. However, the proportionality coefficient is not constant but
50Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
depends on the macroscopic inertial number. This result reveals a new granular regimeregarding force transmission process. Further measurements are needed to understandmore in detail the role played by the ”(I) inertial rheology in the force transmission andto possibly develop a model for fully predicting F , instead of Eq.(2.11) which remains anempirical fit of the numerical data.
2.4 Discussion and conclusion
The present paper described the study of a granular lid-driven cavity with a particularattention paid to the force experienced by the sidewall facing to the shear velocity. Becauseof the boundary conditions associated with the lid-driven cavity (two fixed sidewalls, onebottom fixed wall and a moving upper wall at velocity U), a large-scale vortex developedinside the whole volume of the cavity and reached a steady-state regime. Strong spatialheterogeneities stem from this vortex-like dead zone. In particular, the force at the topof the cavity increased exponentially from the left to the right (in the direction of theshearing velocity) with the development of wedge zone of high pressure. As a result, theforce on the right wall was much higher in the upper part of the wall than in the rest ofthe wall height. In spite of this strong spatial heterogeneity, the time-averaged force onthe wall F relative to the mean top force PLd is entirely controlled by the macroscopicinertial number of the cavity IM over a wide range of IM , whatever the couple U andP . From quasistatic to nearly collisional flow regimes, no influence of both the stiffnessand the restitution coefficient used in the contact law between grains was found in oursimulations. For collisional flow regimes, increasing the latter two parameters may slightlyincrease F/(PLd). A significant effect of the interparticle friction was however observed,suggesting the crucial role of friction in the force transmission mechanism. For a givenvalue of the interparticle friction, the data could be described by a robust fit showing thatthe force transmission is fully controlled by a rate-dependent frictional process. Furthermeasurements (mapping of the local inertial number within the whole volume of the cavityin particular) are needed to better understand the one-to-one relation between F/(PLd)
and IM , which might emerge from the ”(I) inertial rheology. This could open a path todevelop a model for predicting the mean force on the wall.
Next step of this research will tackle the force fluctuations on the wall. Since the earlyexperimental work in a granular Couette geometry by Miller et al. (see [67]), force fluc-tuations in sheared granular materials received a great attention in the past recent yearsby analysing force fluctuations on objects moving in a static granular medium [1, 45, 97]or the force at the boundary of a granular flow during silo discharge [62, 42]. Some studiesdirectly tracked the force fluctuations between particles in response to an external per-turbation [26]. Many of those studies support the idea that force fluctuations in sheared
2.4. Discussion and conclusion 51
granular materials are mainly controlled by dynamic force chains which continuously formand collapse, through either repeated cycles of stick-slip events of the bulk granular mate-rial or other (more random) grain-scale dynamics yet to be solved. It is still unclear if thebirth-death evolution remains the prevailing process over a whole range of granular flowregimes from the quasistatic regime to the dense inertial regime. The DEM simulationsof the granular lid-driven cavity offer the possibility to answer the question. Preliminaryanalysis of the temporal force fluctuations have shown that the shape of the force distri-bution is fully controlled by the macroscopic inertial number, further confirming the keyrole played the ”(I)âinertial rheology proposed by [44, 52].
Acknowledgments This work has been partially supported by the LabEx Tec21 (In-
vestissements dâAvenir : grant agreement No. ANR-11-LABX-0030). Thierry Faug andMohamed Naaim are grateful to the financial support by the People Programme (MarieCurie Actions) of the EU 7th FP under REA grant agreement No. 622899 (FP7-PEOPLE-2013-IOF, GRAINPACT).
52Chapitre 2. Ătude de la force moyenne subie par un obstacle soumis Ă un
écoulement granulaire cisaillé
Chapitre 3
Fluctuations de force sur un obstacle en
interaction avec un Ă©coulement
granulaire cisaillé
53
54Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
Force fluctuations on a wall in interaction with a granu-lar lid-driven cavity flowKneib F., Faug T., Nicolet G., Eckert N., Dufour F., Naaim M.
Article submitted in Physical Review E.
Abstract The force fluctuations experienced by a boundary wall subjected to a lid-driven cavity flow are investigated by means of numerical simulations based on the discreteelement method. The time-averaged dynamics inside the cavity volume and the resultingsteady force on the wall are governed by the boundary macroscopic inertial number, thelatter being derived from the shearing velocity and the confinement pressure imposed atthe top. The force fluctuations are quantified through measuring both the autocorrelationof force time-series and the distributions of grain-wall forces, at distinct spatial scalesfrom particle-scale to wall-scale. A key result is that the grain-wall force distributionsare entirely driven by the boundary macroscopic inertial number, whatever the spatialscale considered. In particular, the distributions at wall-scale are found to evolve fromexponential to Gaussian distributions when the macroscopic inertial number is decreased.The transition from quasistatic to dense inertial flow is well identified through remarkablechanges in the shapes of the distributions of grain-wall forces, accompagnied by a loss ofsystem memory in terms of the mesoscale force transmitted toward the wall.
3.1 Introduction
Granular materials are ubiquitous in nature and daily life situations. Depending on thestress level applied, dense packings of grains can exhibit solidlike or fluidlike behaviour.The rheology of dense granular flows is a key question among others in granular physicsthat has attracted increasing attention in the last twenty years [44, 41, 53]. The stillunsolved questions, in particular related to the transition between quasistatic and denseinertial regimes, are relevant to a number of physical problems. Two examples are themodeling of the processes at stake when a full-scale granular flowâsuch as an avalancheor a landslideâinitiates in the release area or ceases in the runout zone on the one hand,and the optimal design of silos to guarantee an efficient transport of particles in foodprocessing or mining industry on the other hand.
Though outstanding progress was recently made regarding the rheology of dense gra-nular flows, most of the existing models were developed to predict the average flow withonly limited attention paid to the fluctuating part (fluctuating trajectories of grains, ve-locity fluctuations, stress fluctuations) of the flow [53]. In a similar manner, many studies
3.1. Introduction 55
about the force experienced by objects immersed in dense granular flows [37] focusedon the average force signal but only a few of them (see for instance [45]) addressed theproblem of the fluctuating part of the force signal.
There exists a large body of statistical studies which are focused on spatial force varia-bility, more particularly on the analysis of the distributions of forces, in static (unsheared)or slowly sheared granular media, thanks to laboratory tests [69, 63, 9, 35, 65, 25], nu-merical simulations [4, 81, 92] and theory [61, 23, 34, 95, 96, 104, 78]. In contrast, thereare only few studies that have tackled the problem of temporal and/or spatio-temporalfluctuations (the distinction between both being not so clear in a number of studies) inslow to fast flows of granular materials [67, 50, 62, 92, 45, 26, 42, 43]. Section 3.2 will givean overview of those studies
The present paper proposes to investigate in detail the fluctuating part of the forceexperienced by a boundary wall subjected to a granular flow. Our study is based on anoriginal system, namely the granular lid-driven cavity, which is simulated by the discreteelement method (DEM). This follows a preliminary analysis of the time-averaged dy-namics of this granular cavity system that was presented in [58]. The force fluctuationsexperienced by the boundary wall are analysed with the help of a systematic characteriza-tion of the probability distributions of grain-wall forces, over a wide range of confinementpressure and shear velocity imparted to the cavity. Moreover, various spatial scales areanalysed including particle microscale, wall macroscale, and mesoscale (intermediate bet-ween the two previous scales).
The present paper is organized as follows. Section 3.2 provides a brief state of the arton past studies dedicated to the distributions of forces in granular media. Following ourinitial numerical study on the time-averaged dynamics of the granular lid-driven cavitysystem [58], Sec. 3.3 gives a summary of the macroscopic boundary conditions and themicroscopic parameters which we are using for this new numerical study. Section 3.4recalls briefly the main results regarding the time-averaged dynamics and enriches themwith analysis of the horizontal position of the vortex formed within the cavity, as well asof the local rheology in the granular bulk. Section 3.5 tackles the temporal fluctuations offorce by analysing in detail the autocorrelation of the force time-series on both the entireheight of the wall and portions of the wall. Section 3.6 is devoted to the analysis of thegrain-wall force distributions at microscale (particle scale), macroscale (entire height ofthe wall) and mesoscale (portion of wall). Finally, the paper ends by discussing the mainresults and some potential implications for basic aspects of granular physics.
56Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
3.2 Force distributions in granular media
The present section provides an overview of the literature about contact force distributionsin granular media for static (unsheared) or slowly sheared systems, as well as for fast flowsystems.
For static granular packings, the qâmodel was early proposed and successful to re-produce the inhomogeneous interparticle contact force distributions observed in the pio-neering experimental and numerical studies [61, 24]. In particular, the qâmodel is able topredict the remarkable exponential decay at large forces generally measured in static gra-nular matter. The high probability (compared to a Gaussian distribution) of having forcesmuch larger than the mean is generally associated with the ability of granular materials todevelop long chainlike structures, the so-called granular force chains, that can support thelarge forces [81]. The probability density function (PDF) of interparticle contact forces fpredicted by the qâmodel in static granular media under gravity takes the following form[61, 24]:
P(f/f) =kk
(k â 1)!
(f
f
)kâ1
exp
(âk
f
f
), (3.1)
where f is the mean and k is the number of downward neighbour particles considered(see much more details in [61, 24, 23]). Considering k = 2 â 3 generally gives goodresults for the forces larger than the mean [101, 4]. However, thanks to technologicalprogress in the field of force sensors, a number of experimental studies identified a rangeof complicated shapes of contact force distributions when approaching forces much belowthe mean (see [4] for a detailed summary of those studies, before 2000, about distributionsat weak forces). In particular, the presence of a plateau, followed by a slight increase atthe smallest forces, was identified under certain circumstances. In order to fit that morecomplicated shape of contact force distributions, the empirical following functional formwas proposed in [69]:
P(f/f) = a
[1â b exp
(âf 2
f2
)]exp
(âÎČf
f
), (3.2)
where a, b and ÎČ correspond to the PDF parameters fitted on the experimental measu-rements made by [69]. A slight modification of Eq. (3.2) was proposed by [9] to interprettheir experimental data on the effect of both packing order (disordered packings versushighly orderedâcrystallineâconfigurations) and interparticle friction on P(f). Finally, itis worthwhile to note that log-normal distributions were reported in some studies [76, 30].
As discussed in a recent review in [78] and a number of references therein, the keyfeatures of the contact force distributions in static or very slowly sheared granular pa-ckings can be summarized as follow : i) the distribution functions fall-off exponentially at
3.2. Force distributions in granular media 57
large forces, ii) a small peak, or plateau, is observed below the mean force, and iii) thevanishingly small forces remain highly probable. In his review about granular force trans-mission in static granular packings, Radjai [78] proposed an elegant model for contactforce distributions and derived an analytical expression for the density function able topredict the three aforementioned features :
P(f) = ÎČ0(1 + Îł0)Îł0 exp(ÎČ0f)
[1 + Îł0 exp(ÎČ0f)]2, (3.3)
where ÎČ0 (homogeneous to a force) and Îł0 (dimensionless coefficient) are the PDFparameters. Under the normalization f = 1, ÎČ0 and Îł0 are linked by the relation ÎČ0 =
(1+ Îł0) ln[(1+ Îł0)/Îł0)]. Though further studies are needed to relate Îł0 to actual physicalproperties of the grains, varying Îł0 allows to cover a wide range of distributions with orwithout the presence of a peak [78].
Among the great number of experimental, numerical and theoretical results concerningthe shape of force distributions in static packings or slowly sheared granular media, a keyresult is that the distribution at small forces (plateau versus peak, maximum value, non-zero value at vanishingly small force, etc.) is found to be very sensitive to the granularsample preparation and shear history the system experiences [4, 78]. While interpretingdistributions of contact forces in static granular packings, Antony [4] concluded that it isrequired to pay a due attention to the shear strain level and any other quantities related toshear history, such as the volume fraction. A theoretical study on static packings proposedby [96] predicted a broadening of the distribution while increasing the shear stress level,moving from distributions with a peak below the mean value to exponential distributions.Two-dimensional granular packings under anisotropic stresses were studied theoreticallyby [104], showing that an increase in the stress anisotropy produced a transition fromdistributions with a peak below the mean to exponential distributions.
A few studies observed the possibility of a crossover in the shape of P(f) from expo-nential to Gaussian at large confining pressure [101, 102, 66]. This change in the shapeof P(f) was attributed to very large deformations of particles [9]. The effect of particlestiffness on distributions of contact forces in static packings was investigated by Eriksonet al. [35]. They found that beyond a high thresholdâaround 30% of deformation, thedistributions became sensitive to the particle stiffness. Decreasing the particle stiffness ledto an increase in the peak and of the slope of the exponential decay at large forces, thusproducing a distribution likely to evolve toward a Gaussian distribution. The increase ofthe peak by decreasing the particle stiffness was also derived from the theory in [96] (notethat this theory predicts an increase of the non-zero value at f = 0). In the limit of rigidgrains (as addressed in the present study), one can conclude that the distributions arevery weaklyânot to say notâaffected by the particle stiffness.
The pioneering laboratory measurements of contact forces distributions in granular
58Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
media (see for instance [61, 69, 63, 66, 9]) were confined to contact forces between a grainpacking and a boundary wall. It is worthwhile to note that some studies paid attention tothe cross-comparison between the grain-grain (inside the bulk) contact force distributionsand the grain-wall contact force distributions. No significant difference was found for staticpackings [80, 95, 96], suggesting that the results for the distributions at the boundary wallsmay generally be extended to the grain-grain contact forces.
For slowly sheared granular packings, the effect of a slight variation of the volumefraction on the shape of distributions was evidenced by the two-dimensional experimentson a granular Couette geometry [50]. The authors observed exponential distributions atthe smallest volume fraction (around 0.78 in their 2D system), while the distributionsexhibited a peak at larger volume fraction (around 0.81). Interestingly, Howell et al. [50]provided a comparison between the distributions at the particle scale to the distributionsaveraged over a great number of particles (⌠260 in their study). For the latter, they foundthat at large volume fraction the distributions became nearly Gaussian. The transitionfrom exponential distributions to narrower distributions with the presence of a peak whileincreasing packing density was also reported in a theoretical study based on the analogywith supercooled liquids and foams [75].
A study on relatively fast granular flows (a silo discharge) identified the effect ofthe velocity (flow rate at the exit of the silo) on the shape of distributions [62]: theauthors observed a broadening of the distribution at large velocities. A similar conclusionwas drawn from the experiments on objects dragged into a static granular medium in anannular cell by Geng et al. [45], who observed broader distributions at large rotation speedsof the annular cell. The theoretical study of OâHern et al. [75] based on the analogy withsupercooled liquids and foams predicted the broadening of distributions with the increaseof the shear stress.
The original system considered in the present paper, namely the granular lid-drivencavity, makes possible the investigation of a wide range of boundary conditions in termsof shearing velocity and confinement pressure, keeping the granular bulk relatively dense.This allows to investigate, for the first time, how the distributions of grain-wall contactforces evolve over a wide range of both shearing velocity and confinement pressure imposedon the granular medium, and considering different spatial scales from a single grain to theentire boundary wall.
3.3 The granular lid-driven cavity simulated by DEM
3.3.1 Macroscopic boundary conditions
A planar assembly of spheres is trapped into a box made of four walls, as sketched inFig. 3.1. The bottom horizontal wall is fixed and rough, while both lateral walls are fixed,
3.3. The granular lid-driven cavity simulated by DEM 59
Figure 3.1 â Geometry and boundary conditions for the lid-driven cavity simulated by DEM.
The granular sample is trapped between four walls, the top one (the lid) applies the shear
displacement U and the confinement pressure P (see text for details).
smooth and spaced by a distance L = 5H. The upper horizontal wall, namely the lid, isrough and has an infinite length that allows for a uniform horizontal displacement, thusproducing a constant shearing of the sample at velocity U . A constant vertical confinementforce Fc is added continuously to the lid whose vertical position is computed through theDEM algorithm. Although the grains reaction to this force is not homogeneous along thelidâbut rather exhibits an exponential shape (see figure 4 of our preliminary study onthis system [58]), an equivalent macroscopic confinement pressure P can be defined fromthe system dimensions : P = Fc/Ld. As the grains are forced to stay inside the cavity andthe force profile is heterogeneous, a perpetual circulation takes place within the wholevolume of the cavity, as further discussed in Sec. 3.4 of the present paper.
The following dimensionless macroscopic parameters are used to run the numericalsimulations over a broad range of confinement pressure P and shear velocity U , namelyNP and NU :
P = NPΊÏpgH, (3.4)
U = NU
âgd. (3.5)
The gravity acceleration g = 9.81 m sâ2 is used for convenience in order to facilitatethe parallel with any potential real laboratory tests in the future, though the numericalsystem considered here is gravity-free. A constant macroscopic volume fraction Ί = 0.6
is considered here, which corresponds roughly to the random close packing of a threedimensional assembly of spheres of width d. The particle density was taken equal toÏP = 2500kg mâ3.
The cavity is initially filled of grains under gravity deposition in order to produce adense granular packing of height H. Once the system reaches the static equilibrium, the
60Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
gravity is set to zero and the shearing starts for a period of 10 seconds. As it will bedefined in Sec. 3.5 (see Fig. 3.6), the data recording starts one second after the shearinghas started. One second typically corresponds to the maximal time needed for the systemto reach a permanent regime in terms of force measured on the sidewall (see also moredetail in [58]). In most of the simulations presented in the present study, H/d was takenequal to 30 with H = 0.3m and d = 1mm. Some results from numerical simulations withH = 0.3m but d = 0.3mm will be discussed in conclusion.
Following the analysis proposed by [44] at the local grain scale, if we consider thetypical time tP = d
âÏ/P associated with the top confinement pressure P and the typical
time tU = H/U equal to the inverse of the macroscopic shear rate, one can define themacroscopic inertial number IM :
IM =tPtU
=dâÏ/P
H/U, (3.6)
where Ï = ΊÏP holds for the density of the granular sample within the cavity.In the present study, NP varied from 0.01 to 100 and NU from 1 to 20, which allows
to investigate a wide range of granular flow regimes defined by IM ranging typically from5 10â4 to 0.6. A very slight variation of Ίââthe volume fraction actually measured inour simulationsâ with the boundary conditions in terms of U and P was observed. Thispoint will be discussed in Sec. 3.7. In addition to IM , another dimensionless parameterthat controls the system is the length L of the cavity relative to its height H. It hasbeen shown in our preliminary study of this granular lid-driven cavity system [58] thatthe constraint L/H â„ 5 should be respected to prevent an effect of the cavity length onthe steady force experienced by the wall facing the shear displacement (namely the rightsidewall shown in Fig. 3.1).
All the results shown in the present paper were obtained for L/H = 5, but theirsensitivity to L/H was investigated by performing simulations with L/H = 7 and L/H =
10. This sensitivity analysis revealed thatâas observed for the mean force, the forcefluctuations remainded not influenced by L/H as soon as L/H â„ 5.
3.3.2 Micromechanical parameters
The grain-grain and the grain-wall interactions are handled by a viscoelastic contact lawfor the normal force and an elastic force with a Coulomb threshold for the shear force.The normal contact force Fn and the shear contact force Fs are expressed as :
Fn = max(0, knÎŽn + cnÎŽn)n,
dFs = (ksÎŽsdt)s,
|Fs| †”|Fn|,(3.7)
3.4. Time-averaged dynamics 61
where n is the normal of the contact plane, s is the unity vector along the sheardirection (n · s = 0), kn and ks are the normal and tangential contact stiffnesses, Ύn isthe normal penetration depth, Ύs is the tangential displacement increment, ” is the localfriction coefficient, cn is the normal viscosity coefficient and dt is the timestep.
For a contact between two identical spheres of diameter d, Youngâs modulus E andPoissonâs ratio Îœ, the contact stiffnesses can be computed as :
kn =1
2Ed,
ks = Îœkn.(3.8)
The contact law is governed by four physical parameters : E, Îœ, cn and ”. It hasbeen discussed in a number of studies (see [40, 58] and references therein) that E can bereduced to decrease the total time of calculation without changing the numerical results,as long as we stay in the limit of rigid grains. Such a limit of rigid grains is verified if, forinstance, we take N0 = E/(2P ) = 1.5 104, where P holds for the macroscopic confinementpressure and N0 is a dimensionless number. cn is set in the same manner as in [58], witha restitution coefficient e = 0.5. Most of the simulations were conducted with ” = 0.5 butwe will shortly discuss some results with ” = 0.27 in the conclusion. In order to avoidcrystallization, the grain diameters are homogeneously chosen between d(1 â ddisp) andd(1 + ddisp) where ddisp = 0.15.
3.4 Time-averaged dynamics
The overarching goal of the present paper is to study in detail the force fluctuationsexperienced by the boundary wall that faces the direction of the shearing velocity atthe top. In our previous study of this system [58], we reported an analysis of the resultsregarding the time-averaged dynamics. Before going into the details of force fluctuations,this section proposes to recall some important results reported in [58] and to furtherextend some of the results concerning the time-averaged dynamics.
3.4.1 Steady vortex and mean force
The macroscopic inertial number IM , defined from the boundary condition [see Eq.(3.6)],is the relevant parameter to quantify the flow inertial state and to predict the mean forceexperienced on the sidewall scaled by the force imposed at the top, namely F/(PLd)
(see figure 8 in [58]). More specifically, our previous study showed that the mean forceon the right sidewall is entirely controlled by the changes in granular flow regimes, thelatter regimes being governed by the macroscopic inertial number. In the quasistaticregime at low IM , F/(PLd) is constant. In the dense inertial regime at intermediate IM ,
62Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.00
0.01
0.02
0.03
y
âIM â0.0012|NU =1.0|NP =25.0
0.000.020.040.060.08
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.00
0.01
0.02
0.03
y
âIM â0.0609|NU =10.0|NP =1.0
0.00.20.40.60.8
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.00
0.01
0.02
0.03
y
âIM â0.3043|NU =10.0|NP =0.04
0.00.20.40.60.8
Figure 3.2 â Time-averaged streamlines within the cavity for three different values of IM :
IM = 1.2 10â3 (top panel), IM = 6 10â2 (middle) and IM = 0.3 (bottom).
F/(PLd) is a nearly-linear increasing function of IM . In the rapid regime at higher IM
values, F/(PLd) starts saturating. The measurement of time-averaged local granular flowvelocity-fields revealed the formation of one single vortex occupying the whole volume ofthe cavity. This cavity-scale vortex is caused by the fact this system forces the grains tomove within the cavity volume, without any possibility to escape it.
Figure 3.2 shows examples of the streamlines inside the cavity for low (IM = 1.2 10â3),intermediate (IM = 6 10â2) and high (IM = 0.3) values of IM . The xâposition of thevortex centre along the cavity was extracted from the streamlines (as shown by the verticaldotted lines in Fig. 3.2), thus allowing to deduce the horizontal distance â between thevortex center and the right sidewall. The top graph in Fig. 3.3 shows â scaled by thecavity length L as a function of IM . In the quasistatic regime (IM . 10â2), â is relativelyconstant at an approximative value of L/2, meaning that the vortex is centered. BeyondIM . 10â2, the symmetry of the velocity pattern is lost, as depicted by â/L which startsdecreasing sharply. The vortex centre is more and more shifted toward the right sidewallwhen IM is increased, thus producing an important horizontal asymmetry at the highestIM .
Figure 3.3 depicts how the mean force on the wall F relative to the typical pressureforce PLd imposed at the top evolves with the macroscopic inertial number (see [58] formore details). The variation of both F/(PLd) and â/L with IM reveals the same transition
3.4. Time-averaged dynamics 63
0.2
0.3
0.4
0.5â/L
IM <10â2
IM >10â2
10-3 10-2 10-1 100
IM
0.550.600.650.700.750.800.85
F/PLd
Figure 3.3 â Distance of the vortex center from the right sidewall, â, relative to the cavity length
L, as a function of the macroscopic inertial number IM (top panel). Mean force F on the wall
relative to the pressure force PLd imposed at the top, as a function of IM (bottom panel).
0.2 0.3 0.4 0.5â/L
0.55
0.60
0.65
0.70
0.75
0.80
0.85
F/PLd
IM <10â2
IM >10â2
c1âc2 âL, R2 =0.96
Figure 3.4 â F/(PLd) versus â/L. The grey dashed line is a fit of Eq.(3.9) for IM > 10â2. This
graph reveals that the mean force on the wall is entirely controlled by the distance of the vortex
from the right sidewall.
64Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
from quasistatic to dense inertial regime. While comparing the two curves, a significantcoupling can be detected, which becomes obvious when plotting F/(PLd) as a function ofâ/L in Fig. 3.4. F/(PLd) and â/L are linearly linked, meaning that the mean force on thewall scaled by PLd may be deduced from the vortex position, and vice-versa. It is worthnoting that the scattering at low F/(PLd) in Fig. 3.4 is due to the very low velocity ofthe flow in this region, making difficult the identification of the vortex centre position.
In our previous study, we proposed an empirical scaling for F as a function of IM inthe following form [58]: F = PLd [r1 + (r2 â r1)f(IM)], where f(IM) = (1 + I0M/IM)â1,and r1, r2 and I0M were constant fitting parameters. For the data shown in Fig. 3.3, agood fit is obtained with r1 = 0.53, r2 = 0.86 and I0M = 0.02. The interparticle frictiondoes affect the values of r1, r2 and I0, as it will be discussed in Sec. 3.7.
Considering that the mean force on the wall is controlled by the distance of the vortexcenter from the wall (â being a function of IM as displayed in the top panel of Fig. 3.3),we can specify another scaling that is given by the following relation :
F = PLd
(c1 â c2
â(IM)
L
), (3.9)
where c1 and c2 are two constants that may depend on the micromechanical parametersof the grains. In this specific case c1 = 0.88 and c2 = 0.5, as shown by the dashed line inFig. 3.4.
The above analysis showing the link between the vortex center position and the meanforce on the right sidewall (that faces the shearing direction at the top) further extends theresults reported in [58] concerning the mean dynamics of the granular lid-driven cavity.For given micromechanical grain properties, all macroscopic physical quantities measuredin the cavity can be deduced from the macroscopic inertial number defined on the imposedshearing velocity and confinement pressure.
3.4.2 The local ”(I)ârheology
To further test the rheology, the local effective friction ”loc and the local inertial numberI were measured within the whole volume of the cavity. Detailed steps for the calcula-tion of those local quantities are provided in appendices 3.8 and 3.8. Though the strainfield is rather complicated inside the cavity (see the streamlines in Fig. 3.2), the local”(I)ârheology, given by the relation ”th = ”1 + (”2 â ”1)(1 + I/I0)
â1 [52], was foundto be valid in most of the cavity volume. The values found for the parameters of the”loc(I)âfriction law were ”1 = 0.12, ”2 = 0.40 and I0 = 0.13 (see much more details inappendix 3.8). The difference between the ”loc actually measured and ”th was calculated.Figure 3.5 displays the results in terms of maps of |”loc â ”th|, for two distinct values ofIM ⌠10â2 and IM ⌠6 10â2.
3.5. Force data autocorrelations 65
0.0 0.2 0.4 0.6 0.8 1.0x/L
0.00.20.40.60.81.0
y/H
IM â 0. 0122|NU =10. 0|NP =25. 0
0.000.020.040.060.080.100.120.140.160.18
|”locâ”th|
0.0 0.2 0.4 0.6 0.8 1.0x/L
0.00.20.40.60.81.0
y/H
IM â 0. 0609|NU =10. 0|NP =1. 0
0.000.020.040.060.080.100.120.140.160.18
|”locâ”th|
Figure 3.5 â Maps of |”loc â ”th| for IM = 1.22 10â2 (top panel) and IM = 6.09 10â2 (bottom
panel).
At IM ⌠10â2, a symmetrical bow-like pattern appears across the entire length ofthe cavity (see top panel). Similar symmetrical bow-like patterns (not shown here) weresystematically obtained for IM below 10â2. Above the bow-like pattern and below it (apartfrom the two regions close to the two lateral walls), |”loc â ”th| is nil, which means thatthe sample is well governed by the ”(I)âinertial rheology. For higher values of IM (seebottom panel in Fig. 3.5), the symmetry and the bow-like shape were broken. In otherwords, the region inside the cavity where |”loc â ”th| was nil could extend over the wholeheight of the cavity.
Note that Fig. 3.5 displays the absolute values of ”loc â ”th, thus preventing thedetection of negative values. However, a great number of values of ”loc were found to belower than ”1 for IM below 10â2 (see Fig. 3.18 in appendix), which corresponds to theregion in the center of the cavity in the top panel of Fig. 3.5 (forming the bow-like patternacross the entire length of the cavity). These negative values of ”locâ”th that suggest thepresence of non-locality [54] will deserve further investigation in the future.
In conclusion, the crossover from a bow-like pattern extending across the entire lengthof the cavity to a smaller pattern in the confines of the boundary wall appears to be aclear signature of the quasistatic to dense inertial transition in the system studied here.The rest of the paper is devoted to the influence of the macroscopic inertial number onthe force fluctuations.
3.5 Force data autocorrelations
3.5.1 Data recordings
While analysing temporal fluctuations, data acquisition frequency requires a particularattention : it has to be large enough to be able to capture short-life force events, andsimultaneously small enough to avoid practical memory allocation issues while recording
66Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
Figure 3.6 â Example of time-series of total force on the wall, obtained for NU = 10 and
NP = 25. The instantaneous force F is highly fluctuating over time. The horizontal dashed line
depicts the value of the time-averaged force, named F , reached in the steady-state regime (after
a short transient of about 1s). Inset : same data, zoomed in a time window of 0.05 s.
trends over large durations. The DEM algorithm is not able to model oscillating pheno-mena that occur at frequencies higher than the typical natural contact frequency fc. Inthe present statistical study the record frequency frec was set to 10 kHz and fc typicallyranged from 4.8 kHz to 320 kHz. The frequency corresponding to the DEM timestep wassystematically set to 10fc, avoiding any instability issues. The overlap between frec andthe lowest values of fc may cause the recording of unintended effects caused by particleoscillations. It has been verified that neither the autocorrelations nor the distributions ofthe force time-series presented in the following statistical study are sensitive to frec bycomparing to simulations with a doubled (20 kHz) and a halved (5 kHz) record frequency.
3.5.2 Force time-series autocorrelation
The force time-series on the right sidewall of the cavity show high temporal fluctuations,as displayed in Fig. 3.6. This section aims at studying these fluctuations by analysingthe autocorrelation of force time-series. The first information that will be extracted is theforce periodicity. The latter is for instance a sensitive data while designing civil enginee-ring structures subjected to granular avalanche-flows, if one wants to avoid a resonancephenomenon around the natural frequency of the impacted structure. The second infor-mation arising from force autocorrelation is the critical autocorrelation time âtc, whichcan be interpreted as the time after which the system has forgotten a certain force valueon the wall. In other words, it reflects the typical time during which a future state of thesystem keeps history of its past state regarding the force transmitted to the wall.
3.5. Force data autocorrelations 67
We denote by C(F (t), F (t + ât)) the temporal autocorrelation function, where ât isthe lag between two system states. The critical autocorrelation time âtc is defined withan arbitrary low autocorrelation threshold :
C (F (t), F (t+âtc)) = 0.15. (3.10)
It is worthwhile to note that the above threshold value does not influence the conclu-sions presented here as they are qualitative, focused on main trends. Since the localautocorrelation function C(F (t), F (t+ât)) is likely to be a non monotonic function, it isexpected to cross the threshold many times. In this case, because we focus on the initialautocorrelation decrease corresponding to memory loss and we want to avoid any effectof the signal noise, the lowest value of âtc is kept.
In the following, we present the autocorrelation of force time-series at macroscopicscale, i.e. on the entire height of the wall (Sec. 3.5.2). Then the autocorrelation at meso-scopic scale is studied by splitting the wall height into ten portions of a few grains each(Sec. 3.5.2).
Autocorrelation for the entire height of the wall
Figure 3.7 depicts the autocorrelation of the force signal on the entire height of the rightsidewall, for five values of the macroscopic inertial number IM .
Because of the strong stress localisation taking place at the top right corner of thecavity (see more detail on this wedge effect in [58]), the shearing frequency at the rough-ness (particle) scale is transmitted to the right sidewall, as long as IM is not too high(typically smaller than 10â2). Sinusoid-like oscillations are therefore observed, with a de-creasing period while increasing NU . The inset of Fig. 3.7 shows a collapse of sinusoid-likeoscillations when the autocorrelation function is plotted versus (U/d)ât, the wavelengthassociated with these oscillations being d (the diameter of the grains composing the topwall roughness and shearing the sample).
For the highest IM , this effect is absent, which is in agreement with a more inertialregime for which short-time collisions occur in addition to enduring frictional contactsable to transmit the shear force. The mean free path of grains increases, which makesimpossible any continuous transmission of enduring contacts between grains. Again, wedetect the transition from the quasistatic to the dense inertial granular regime occurringaround IM = 10â2, which is a value similar to the one extracted from the time-averageddynamics in Sec. 3.4.
The general shapes of the autocorrelation functions shown in Fig. 3.7 resemble a lot theones found by Geng et al. [45] in their experimental study of an intruder slowly draggedinto a granular medium (see figure 8 in [45]). We generally observe that C(F (t), F (t+ât))
drops quickly (exponentially) to zero over a time scale âtc and then fluctuate around zero.
68Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
0.000 0.005 0.010 0.015 0.020ât
â0.6
â0.4
â0.2
0.0
0.2
0.4
0.6
0.8
1.0
C(F(t),F(t+ât))
IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
0 1 2 3 4 5 6 7
(U/d)ât
â0.4â0.2
0.00.20.40.60.81.0
C(F(t),F(t+ât))
Figure 3.7 â Evolution of C(F (t), F (t+ât)) over time for five values of Im, where F (t) is the
force time-series on the entire right sidewall. Inset : C(F (t), F (t+ât)) plotted against (U/d)ât
to highlight the oscillations wavelength.
These fluctuations are well explained in our system by the typical frequency U/d asso-ciated with the shearing velocity at the top and the grain size (see discussion above). Wewere not able to find a clear dependency of the critical time âtc on either the macroscopicinertial number IM or any input parameter such as the shear velocity of the lid, or theconfinement pressure at the top. In the following, we focus on portions of the wall.
Autocorrelation for portions of the wall
The force time-series on the right wall is the result of the cumulative contact forces appliedover the entire height of the wall. Consequently, the strong spatial heterogeneity of thegranular lid-driven cavity system may require a more localized analysis by taking intoaccount the position at which individual forces are applied to the wall. We propose hereto investigate the autocorrelation of force time-series on some portions of the wall. Theentire wall height is split into ten slices i â 1...10 of identical size. Each slice has itsassociated force time-series from which the autocorrelation Ci(F (t), F (t + ât)) is thencomputed.
Figure 3.8 shows the autocorrelation of force time-series for each of the ten identicalwall portions, for three values of the macroscopic inertial number. The signal periodicityalready discussed above is still observed at the frequency U/dâassociated with the rough-ness of the top wall moving at velocity U . This is particularly clear in the simulation atIM = 1.2 10â2 for which each temporal autocorrelation curve shows this period, whateverthe vertical slice i considered along the wall. This indicates that the periodic fluctuationsat the top can be transmitted to the lowest position at the bottom of the cavity. Thisresult is consistent with the fact that we could identify a small, yet non-zero, spatial
3.5. Force data autocorrelations 69
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14ât
â0.2
0.0
0.2
0.4
0.6
0.8
1.0
Ci(F(t),F(t+ât))
IM â 0. 0012|NU =1. 0|NP =25. 0
i=1
i=2
i=3
i=4
i=5
i=6
i=7
i=8
i=9
i=10
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14ât
â0.2
0.0
0.2
0.4
0.6
0.8
1.0
Ci(F(t),F(t+ât))
IM â 0. 0122|NU =10. 0|NP =25. 0
i=1
i=2
i=3
i=4
i=5
i=6
i=7
i=8
i=9
i=10
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14ât
â0.2
0.0
0.2
0.4
0.6
0.8
1.0
Ci(F(t),F(t+ât))
IM â 0. 3043|NU =10. 0|NP =0. 04
i=1
i=2
i=3
i=4
i=5
i=6
i=7
i=8
i=9
i=10
Figure 3.8 â Evolution of Ci(F (t), F (t+ât)) over time measured on 10 portions i of wall, for
three values of IM . In each graph, the horizontal dashed line shows the threshold of 0.15 that
was used to derive âtic.
70Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
autocorrelation at IM = 1.2 10â2 (not shown here). At the lowest IM = 1.2 10â3, the per-iodicity caused by the top roughness is present at the highest portion of the wall (i = 10)but quickly fades while going deeper into the sample along the wall, and even disappearsfor the lowest position at the bottom of the cavity. In contrast to the intermediate IM ,the periodic fluctuations from the top wall cannot be transmitted over the entire depthof the wall. This observation is in accordance with the fact that at IM = 1.2 10â3 wedid not find any spatial autocorrelation (not shown here). There might exist some zonesnear the cavity top inside which the grains have enough time to rearrange locally, thusbeing able to relax the high stress caused by jamming close to the wedge. At higher IM ,the periodic fluctuations associated with the roughness of the top wall are lost whateverthe vertical position along the wall, thus further confirming the transition toward a muchmore inertial regime for which the increase of the mean free path between grains preventsthe transmission of those fluctuations (see Sec. 3.5.2).
Figure 3.9 shows the mean value of autocorrelation time, <âtc>, averaged over allportions of the wall. Though there is some data scattering regarding âtic (see grey-coloredcrosses on Fig. 3.9), this plot demonstrates that there exists a IM below which <âtc>
is not zero and beyond which <âtc> vanishes. In other words, it means that beyond aIM (â 10â2) it is not possible to predict a future system state from the past one, thusindicating that the memory of the system is completely lost. This observation was stillvalid by increasing L/H from 5 to 10 (not shown here). The granular lid-driven cavitysystem suggests here a remarkable transition from the quasistatic to the dense inertialgranular regime, which is characterized by a total loss of system memory (not intrinsicto the material) in terms of the force transmission from the top wall toward the rightsidewall.
3.6 Force distributions
The distributions of force time-series on the sidewall give crucial information on fluctua-tions, as they quantify the probability of each force value the sidewall may experience. Inthis section, the analysis of force distributions concerns three spatial scales : the micros-cale focused on individual grain-wall contact forces, the macroscale focused on the totalforce on the wall, and the mesoscale focused on the force experienced by portions of wall.For each scale, the distribution response to the same wide range of IM as tackled in theprevious sections is investigated. All distributions presented in this section were obtainedfrom quantities that were scaled by their time-averaged value.
3.6. Force distributions 71
10-3 10-2 10-1 100
IM
0.0
0.1
0.2
0.3
0.4
ât c
ât ic
<âtc>
Figure 3.9 â Critical time âtic (cross symbols) beyond which the force time-series becomes
weakly correlated (C < 0.15)âfor ten portions i â [1; 10] of the wall height, as a function of IM .
The full circles show <âtc> that is the mean over the ten âtic values. The data were extracted
from the autocorrelation functions plotted on Fig. 3.8
3.6.1 Force distributions at microscale
The distributions of contact forces measured in our DEM simulations, for grains in contactwith the right sidewall, characterize the typical forces repartition applied locally on thewall (grain scale). Recorded contact forces on the wall from all recording times are mergedto form the sample, and then the probability distribution is computed. Let us note f anindividual grain-wall contact force and f the overall mean grain-wall contact force of asimulation. In the following, f = f/f is the scaled grain-wall particle contact force.
Figure 3.10 shows examples of distributions of f obtained with three different valuesof IM but from several distinct dipoles (NU , NP ). The remarkable result is that the dis-tributions collapse very well at each IM , revealing that the repartition of particle contactforces on the wall is entirely controlled by the macroscopic inertial number of the granularlid-driven cavity.
Figure 3.11 shows the distributions of f computed for several IM . We have tested anumber of well-established PDFs to fit these curves, as well as theoretical or empiricalPDFs discussed in the literature for granular contact forces, specifically the ones proposedby [69] [Eq. (3.2)] and [78] [Eq. (3.3)] (see Sec. 3.2). None of those PDFs was able to giveconclusive results over the whole range of IM tested in our DEM simulations. The resultswere conclusive, over the whole range of IM , with a truncated log-normal distributionthat reads as follows :
P(f) =1
S1
fÏâ2Ï(f + f0)
exp
(â [ln(f + f0)â f”]
2
2fÏ2
), (3.11)
72Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
0.0 0.5 1.0 1.5 2.0
f/f
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Den
sity
IM â0.0061|NU =1.0|NP =1.0
IM â0.0061|NU =10.0|NP =100.0
IM â0.0061|NU =5.0|NP =25.0
IM â0.0304|NU =5.0|NP =1.0
IM â0.0304|NU =10.0|NP =4.0
IM â0.0304|NU =1.0|NP =0.04
IM â0.3043|NU =20.0|NP =0.16
IM â0.3043|NU =5.0|NP =0.01
IM â0.3043|NU =10.0|NP =0.04
Figure 3.10 â Probability distributions of contact forces for grains in contact with the wall.
For each IM the curves are collapsing whatever the dipole (U,P ), thus demonstrating that the
boundary macroscopic inertial number fully controls the grain-wall contacts distributions.
where f > 0, f0 > 0, and S is the normalization factor corresponding to the valueat f0 of the survival function of the (untruncated) log-normal PDF. The scale parameterf”, the shape parameter fÏ and the location parameter f0 are monotonic functions of themacroscopic inertial number IM , as shown at the bottom panel of Fig. 3.11.
Apart from the fact that the best fits were obtained with the truncated log-normal dis-tributions over the whole range of IM tested, it remains challenging to provide physically-based arguments to justify the use of such a PDF. A key result of the present study isthat the parameters of Eq. (3.11) are found to be well-defined monotonic functions of IM(see bottom of Fig. 3.11). This opens a path to predict the force distributions empirically,as long as the boundary macroscopic inertial number is known. A question then arises :does that key result too hold for the distributions of the total force on the entire heightof the wall ?
3.6.2 Force distribution at macroscale
Figure 3.12 shows the distribution of F , the total force time-series exerted on the sidewallscaled by its mean F , for seven values of IM ranging from 1.2 10â3 to 0.6. At the lowestIM in the quasistatic regime, the distribution are nearly Gaussian whereas at the highestIM , the PDF has an exponential shape. This result is consistent with the crossover fromexponential to Gaussian PDFs generally observed at high confinement pressures in somepast studies [101, 102, 66] (as reviewed in Sec. 3.2).
At intermediate IM in the dense inertial regime, the distribution has a more complexshape : at a first glance, it may appear as a distribution which would combine some
3.6. Force distributions 73
0.0 0.5 1.0 1.5 2.0
f/f
0.0
0.5
1.0
1.5
2.0
Den
sity
IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0609|NU =10.0|NP =1.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
IM â0.6086|NU =10.0|NP =0.01
10-3 10-2 10-1 100
IM
â1.0
â0.5
0.0
0.5
1.0
1.5
Para
met
er v
alue
f0
fÏ
f”
Figure 3.11 â Top panel : probability distributions of contact forces for grains in contact with the
wall, for different values of IM ; the dashed lines show a fit by a truncated log-normal distribution
(maximizing the likelihood with the Nelder-Mead optimization method). Bottom panel : variation
of the parameters of the log-normal distributions as functions of IM (see Eq.(3.11)).
74Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
0.0 0.5 1.0 1.5 2.0 2.5 3.0
F/F
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Den
sity
IM â 0. 0012|NU =1. 0|NP =25. 0
IM â 0. 0061|NU =1. 0|NP =1. 0
IM â 0. 0122|NU =10. 0|NP =25. 0
IM â 0. 0609|NU =10. 0|NP =1. 0
IM â 0. 3043|NU =10. 0|NP =0. 04
IM â 0. 6086|NU =10. 0|NP =0. 01
10-3 10-2 10-1 100
IM
â2
â1
0
1
2
3
4
5
6
7
Para
met
er v
alue
fw0
fwÏ
fw”
Figure 3.12 â Top panel : PDF of the total force on the right sidewall for eight values of IMranging from 1.2 10â3 to 0.9. A truncated log-normal distribution can be fitted on the data
(dashed lines). Bottom panel : the truncated log-normal parameters variation as a function of
IM .
properties of a Gaussian-like distribution and an exponential-like distribution. However,it was impossible to find a way to model the distributions of total force by an analyticalform that combines both Gaussian and exponential distributions, over the entire range ofIM tested here. The only PDF that could be fitted to the distributions of the total forceon the wall is again the truncated log-normal distribution [see Eq.(3.11)], as shown by thedashed lines drawn in Fig. 3.12.
As for the distributions of f discussed in Sec. 3.6.1, it would be difficult to givea physical interpretation for the truncated log-normal PDF for the distributions of F .However, it is possible to analyse the evolution of the corresponding parameters withthe inertial number. fw
” , fwÏ and fw
0 (the scale, shape and local parameters, respectively)are presented at the bottom panel of Fig. 3.12, as a function of IM . These parameters
3.6. Force distributions 75
follow monotonous paths with IM , allowing to predict the distribution of the force on thesidewall from the inertial number. This result was further confirmed (with the same set ofsimulations as in Fig. 3.10) because several distinct dipoles (NU , NP ) that give the sameinertial number systematically led to the same distribution (curves not presented here).
The bottom panel of Fig. 3.12 displays a jump in the parameters of the truncatedlog-normal distribution between IM = 10â2 and IM = 6 10â2. Further looking at thecurves on the top panel of Fig. 3.12, this jump occurs when the Gaussian shape almostdisappears due to the competition with the exponential shape at low forces, thus displayingan homogeneous repartition (kind of plateau) from F/F = 0 to F/F = 1. This significantchange (for IM ⌠10â2) of the distribution of the total force on the wall, when the lowforces and the mean forces have almost the same probability of occurrence, appears to bean additional marker of the quasistatic to dense inertial transition, concomitant with thetotal loss of system memory discussed in Sec. 3.5.2.
3.6.3 Force distribution at mesoscale
Because of the heterogeneity of the granular sample induced by the cavity boundaryconditions, it is interesting to investigate the force distributions at a mesoscopic scaleâsmaller than the cavity height and greater than the grain scale, as already done for theforce autocorrelation (see Sec. 3.5.2). We note Fi the force time-series on a portion of walllocated at a given i, and Fi its mean. The top panel of Fig. 3.13 displays the distributionsof Fi/Fi obtained on ten wall portions of identical size, for IM â 10â2. The distributionscorresponding to the portions located in the center of the right sidewall collapse into onesingle curve, while the distributions measured close to the top and bottom right cornersof the cavity differ from this master curve. This result reflects the spatial heterogeneityof the cavity over its height, in connection to three zones : a highly sheared zone ofsmall height at the top, a small dead zone at the bottom and (in between) a flowingzone of much larger height in the center. An identical behaviour (curves not shown here)was observed for all values of IM . These three zones, extracted from the distributions ofFi/Fi, are fully consistent with the vertical profile of the time-averaged force measured onthe wall, as depicted in figure 5 of our previous study [58]. The time-averaged force wasrather homogeneous over the same (large) central zone of the wall. It is worth noting thatthe sizes of the bottom and top zones may be sensitive to the grain diameter relative tothe wall height. Further simulations with different grain diameters would be necessary tostudy the potential influence on the size of the boundary (top and bottom) zones identifiedhere.
In the following, the collapse of the Fi/Fi distributions, excluding the extreme (top andbottom) zones of the wall, is exploited to analyse how the distribution at mesoscale evolveswith the macroscopic inertial number. This mesoscale corresponds to a size H/10 = 3d.
76Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Fi/Fi
0.0
0.2
0.4
0.6
0.8
1.0
Den
sity
IM â0.0122|NU =10.0|NP =25.0
iâ€23â€iâ€9i=10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
<Fi/Fi >i
0.0
0.2
0.4
0.6
0.8
1.0
Den
sity
IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0609|NU =10.0|NP =1.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
IM â0.6086|NU =10.0|NP =0.01
Figure 3.13 â Top panel : PDF of force time-series on ten wall portions i for NU = 10 and
NP = 25 (each portion time-series is scaled by its corresponding time-averaged force). The blue,
orange and red curves respectively correspond to i â [1; 2], i â [3; 9] and i = 10. Bottom panel :
mean distribution over i for i â [3; 9] (yellow curves of top panel) for seven values of IM .
3.7. Discussion and conclusion 77
The bottom panel of Fig. 3.13 depicts the mean distribution over seven values of i â[3; 9] (corresponding to the central zone of the cavity excluding the top and bottom highlyinhomogeneous layers), for a wide range of IM . As observed for the force distributionsat micro- and macroscale presented previously (see Sec. 3.6.1 and 3.6.2, respectively),the curves depict shapes that resemble a truncated log-normal PDF. While comparingthe distributions over the wide range of IM , they appear however to be much closerwith each other, meaning they exhibit a weaker dependency toward IM . It has beenverified (curves not shown here) that the distributions at macroscale for the entire walldid remain unchanged if the data near the bottom and top wedges was not considered. Theweaker dependency of the distributions to IM at mesoscale is thus striking, as it cannotbe explained by a wedge effect only (this effect would be detected on the macroscaleotherwise). As a consequence, it can be concluded that similar mesoscale distributionscan result in radically different macroscale distributions through the summation relationthat links the forces on wall portions to the force on the entire height of the wall. Thelatter observation may put emphasis on the crucial role of spatial dependencies betweenmesoscale force signals, that are controlled by the granular flow regime (through the valueof IM). In particular, the Gaussian-like shapes shown on Fig. 3.12 obtained for quasistaticregimes (IM â 10â3) typically reflect the summation of independent mesoscale forcesignals. This is consistent with the fact that no spatial correlation was found at the lowestIM (as detected from the temporal autocorrelation functions at different i shown on thetop graph of Fig. 3.8). Conversely, the evolution toward an exponential-like distributionat intermediate inertial numbers (IM & 10â2) on Fig. 3.12 reveals a spatial dependencybetween the mesoscale force signals. This is consistent with the spatial correlation detectedbetween the temporal autocorrelation functions at different i shown in the mid graph ofFig. 3.8.
At IM â 10â3, the distribution displays a small amount of low forces and a nearlyGaussian repartition centered at about 0.7Fi. Then, with the increase of IM , all distribu-tions quickly converge to an exponential-like decrease with a high number of low forces.Finally, the force distribution on wall portions become independent of IM as soon asIM & 10â2 and i â [3; 9]. This result allows once again the identification of a clear signa-ture of the transition from quasistatic to dense inertial granular flow regime for the cavitysystem studied here.
3.7 Discussion and conclusion
3.7.1 Time-averaged dynamics
The analysis of the time-averaged dynamics in section 3.4 highlighted the crucial roleplayed by the ”(I)ârheology (proposed by [44, 52]) in the granular force transmission
78Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
toward a boundary wall. To further confirm it, we have analysed the slight variation ofthe height H of the cavity system. Up to now, a constant volume fraction Ί = 0.6 closeto random close packing has been considered. The cavity system defined in our study isby construction a volume-free system, meaning that the volume fraction inside the ca-vity may vary. Indeed, though the variation of volume fraction of the granular bulk wassmall, one could measure it. The top graph of Fig. 3.14 displays how the volume fractionΊâ, actually measured in our DEM simulations, evolved with the boundary macroscopicinertial number IM . We observe a plateau (constant Ίâ) at the lowest IM , followed by aslow decrease of Ίâ with log(IM). This slight dilatancy of the granular bulk while increa-sing the inertial number is a robust result for a number of volume-free granular systemsgoverned by the ”(I)ârheology, such as plane shear flows [27], free-surface granular flowsdown inclines [41], annular shear cell flows [36], etc. Finding this dynamic dilatancy lawin the lid-driven cavity system studied here further confirms the key role played by the”(I)ârheology.
Note that the results in the top graph of Fig. 3.14 are shown with two values of both” and d. A decrease of the interparticle friction coefficient produced a slight increase ofΊâ but did not change the qualitative trend regarding the overall evolution of Ίâ withIM . The increase of Ίâ with the decrease of the grain diameter d may be explained bysome boundary effects at the confining walls of the cavity, which are smoothed for smallerd. The bottom graph of Fig. 3.14 shows how the scaling between F/(PLd) and IM , wasinfluenced by the interparticle friction ”. The values of r1, r2 and I0M used for the fittingfunction are slightly changed when ” is divided by nearly two (see detail in bottom panelof Fig. 3.14). Finally, the particle diameter d has a slight influence on the relation in thequasistatic regime, namely when IM is below 10â2. The influence of the grain properties(grain diameter, interparticle friction) will merit more attention in the future.
3.7.2 Grain-wall force distributions
As already mentioned in Sec. 3.2, Howell et al. [50] found exponential distributions atthe smallest volume fraction while the distributions exhibited a peak at larger volumefraction. Considering the one-to-one relation between IM and Ίâ shown in Fig. 3.14, theevolution of the distributions with IM depicted in Figs. 3.11 and 3.12 are in qualitativeagreement with the evolution of the distributions with the volume fraction reported byHowell et al. [50] in their 2D granular Couette experiments.
While varying IM and the spatial scale of interest, the probability distributions ofgrain-wall forces which we found in the present study (summarized in Fig. 3.15) cover thewide range of shapes discussed in the literature overview proposed in Sec. 3.2. In their two-dimensional granular Couette experiments, Howell et al. [50] compared the distributionsat the particle scale to the ones averaged over a great number of particles, and found that
3.7. Discussion and conclusion 79
10-3 10-2 10-1 100
IM
0.48
0.49
0.50
0.51
0.52
0.53
0.54
0.55
Ίâ
H/d=30 | ”=0. 5
H/d=90 | ”=0. 5
H/d=30 | ”=0. 27
10-3 10-2 10-1 100
IM
0.5
0.6
0.7
0.8
0.9
F/PLd
”0. 5 : r1 =0. 53|r2 =0. 86|I 0M =0. 02
”0. 27 : r1 =0. 4785|r2 =0. 7962|I 0M =0. 0223
H/d=30 | ”=0. 5
H/d=90 | ”=0. 5
H/d=30 | ”=0. 27
Figure 3.14 â Volume fraction Ίâ (top panel) and scaled mean force on the sidewall F/(PLd)
(bottom panel) actually measured in the numerical simulations, as a function of IM : d = 0.001
and ” = 0.5 (blue curve), d = 0.0003 and ” = 0.5 (green curve), d = 0.001 and ” = 0.27 (red
curve). In the bottom panel, the fits proposed in [58] are drawn to show how ” affects the fitting
parameters r1, r2 and I0M .
80Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
0.0 0.5 1.0 1.5 2.0 2.5 3.0force/mean force
0.0
0.5
1.0
1.5
2.0
Den
sity
IM â 0. 0012 | macroscaleIM â 0. 0012 | mesoscaleIM â 0. 0012 | microscaleIM â 0. 6086 | macroscaleIM â 0. 6086 | mesoscaleIM â 0. 6086 | microscale
Figure 3.15 â PDFs of grain-wall forces at three different spatial scales for two extreme values
of IM .
at large volume fraction the distributions became nearly Gaussian. The latter observationappears to be consistent with the evolution of the distributions for IM = 1.2 10â3 (largestvolume fraction in this numerical study) when the spatial scale is increased, as shown inFig. 3.15.
A key result of the present study is that the value of the boundary macroscopic inertialnumber and, to a lesser extent, the spatial scale (macro, micro, or meso) considered, arekey inputs that predetermine the grain-wall force distributions. Note that doubling L/H
(not shown here) did not change the distributions. All measured distributions could bemodeled by truncated log-normal PDFs whose parameters were fully controlled by IM .This opens a path to being able to predict the grain-wall force distributions over a widerange of IM .
3.7.3 Quasistatic to dense inertial regime
The analysis of the granular lid-driven cavity problem showed that the macroscopic iner-tial number IM fully controlled all the time-averaged quantities (the mean scaled forceon the wall, the position of the vortex, and the bulk volume fraction) and the force fluc-tuations (through the analysis of probability distributions). Moreover, a clear transitionwas observed at IM around 10â2 in the time-averaged dynamics of the cavity system.This transition was identified by an increase of the scaled steady force, a displacementof the vortex position toward the wall, a loss of symmetry in the |”loc â ”th| maps, anda macroscopic dilation of the granular bulk. The autocorrelation of force time-series atmesoscale allowed to reveal a total loss of system memory (not intrinsic to the material)beyond a value of IM ⌠10â2. Finally, we identified drastic changes in the evolution of the
3.8. Appendices 81
Figure 3.16 â Illustration of the kernel smoothing used for regular mesh fields calculations.
shape of probability distributions beyond a IM once again close to 10â2. All these resultsare interpreted as a salient signature of the transition from quasistatic to dense inertialflow regime in granular media.
Acknowledgments This work has been partially supported by the LabEx Tec21 (Inves-tissements dâAvenir : grant agreement No. ANR11-LABX-0030). Thierry Faug and Moha-med Naaim are grateful to the financial support by the People Programme (Marie CurieActions) of the EU 7th FP under REA Grant Agreement No. 622899 (FP7-PEOPLE-2013-IOF, GRAINPACT). François Kneib and Thierry Faug would like to thank PierreJop for instructive discussions.
3.8 Appendices
Kernel smoothing As the granular sample is made of discrete particles involving dis-continuities, the construction of classical fields often used in continuous mechanics requirea particular attention. In this paper, a spacial kernel smoothing is used, as described inFig. 3.16.
A 2D set of points mj spaced by d constitutes a regular mesh over the whole cavity.The physical quantities are computed at each mesh point from the particles quantities,according to the particle-point distance. Thus, any (scalar or tensor) quantity Qi which isassociated with each individual sphere Si can also be associated with all individual meshpoints mj, forming a continuous field. This quantity at the point mj, noted Qj, is the
82Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
Figure 3.17 â Illustration of the tesselation for the per-particle volume calculation.
result of a spacial kernel smoothing with a Gaussian kernel :
Qj =1â
i
N (Dij)
â
i
QiN (Dij), (3.12)
where Dij are the distances between the spheres centers and the mesh points, and N isthe Gaussian function of mean 0 and standard deviation d/2. It is worth noting that theuse of a Gaussian kernel is arbitrary and any classical kernel such as a simple rectangularfunction should not modify the results. The choice of the standard deviation follows theresult presented in [107]: the coarse-graining width should be of the order of magnitude ofthe grains diameter to avoid any oscillation effect and at the same time to limit sidewalleffects.
Local stress and strain rate tensors The per-particle stress tensors and strain ratetensors used in the current paper are computed thanks to the algorithms that are partof the YADE-DEM open-source code (see [105] and [19]). The space inside the cavityis discretized with the help of a VoronoĂŻ tesselation algorithm, as shown by the dottedlines in Fig. 3.17. In this way, it is possible to compute quantities based on an equivalentcontinuum of the cavity space. Each sphere is then associated with a list of neighboursforming a bounding polyhedral contour CÇ« with their centroid positions, and a boundingvolume VÏ which does not overlap with the neighbours.
Making the assumption that the spheres are at static equilibrium, the following Love-Weber expression can be applied to each sphere in order to compute their associated local
3.8. Appendices 83
stress tensors based on the contact forces :
Ï =1
VÏ
â
cn
~fc â~lc, (3.13)
where cn is the ensemble of contacts on the sphere, ~fc are the contact forces and ~lc arethe vectors linking the sphere center to the contact points.
The local strain rate tensor D is defined as D = Ç«
ât, where Ç« is the strain tensor
computed from two system states shifted by a short lag ât. For a particular sphere, thedisplacements of all neighbours during ât are computed. Then the average displacementgradient < âd~r > is obtained from the integration of the displacement d~r along CÇ« (see[19] for more details) :
< âd~r >=1
VD
â«
CÇ«
d~xâ ~nds, (3.14)
where VD is the volume associated with the contour CÇ«, and d~r is linearly interpolated onthe segment between two successive neighbours. Finally, the strain tensor is given by thesymmetric part of < âd~r >.
Inertial number and effective friction coefficient The stress tensors Ï and strainrate tensors D computed according to Sec. 3.8 enable the computation of the spatial fieldsof inertial numbers I and effective friction coefficients ”loc. The following formulations arethe same as described in [10] and are applied at each field point m.
The norm ||A|| and the deviatoric component AâČ of a matrix A are defined as :
||A|| =â
Tr(AAT )
2, (3.15)
AâČ = Aâ Tr(A)
3I3, (3.16)
where I3 is the identity matrix of size 3. The inertial number I and the effective frictioncoefficient ”loc are :
I = dâÏp
||DâČ||âp
, (3.17)
”loc =Tr(ÏâČ
DâČ)
3
1
p||DâČ|| . (3.18)
In the above relations, p denotes the pressure that is defined from the decomposition ofthe Cauchy stress into the isotropic pressure and the deviatoric stress : Ï = pI3 + Ï
âČ.Figure 3.18 displays ”loc as a function of I for each point m that belongs to a selectedregion of the cavity not too close to the boundary walls (see its definition in the caption ofFig. 3.18) and for all simulations of this studyâover the whole range of IM tested. Finally,
84Chapitre 3. Fluctuations de force sur un obstacle en interaction avec un
écoulement granulaire cisaillé
10-4 10-3 10-2 10-1 100
I
0.1
0.2
0.3
0.4
0.5
”
”1 =0. 12|”2 =0. 4|I0 =0. 13”th(I) =”1 + (”2 â”1)/(1+ I0/I)
1.22e-036.09e-031.22e-026.09e-029.13e-023.04e-016.09e-01
IM
Figure 3.18 â Test of the local ”(I)ârheology for all simulations of this study, including all
IM . The points come from a region of the cavity selected to approach the conditions of a simple
shear flow. This region corresponds to the upper half region along y that is centered in a band
of length L/2 along x.
the empirical following form proposed by [52] is used to fit all the points, as drawn by thesolid line on Fig. 3.18 :
”th(I) = ”1 + (”2 â ”1)1
1 + I0/I, (3.19)
where ”1 = 0.12, ”2 = 0.40 and I0 = 0.13. Note that a great number of points falls below”1 at low IM , suggesting that non-locality [54] is present. This point will need furtherinvestigation in the future.
Chapitre 4
Force moyenne et fluctuations sur un
mur immergé dans un écoulement
granulaire cisaillé
85
86Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
écoulement granulaire cisaillé
Mean force and fluctuations on a wall immersed in asheared granular flowKneib F., Faug T., Dufour F., Naaim M.
In progress.
Abstract The time-averaged dynamics and the force fluctuations in a system compo-sed of a wall-like obstacle subjected to a granular flow under different shearing velocityand confinement pressure are investigated by means of numerical simulations based onthe discrete elements method. The mean force on the wall is controlled by the boundarymacroscopic inertial number which is derived from the confinement pressure, the shearingvelocity and the height of grains sheared above the wall. An empirical law is thus definedto predict the mean force as a function of the boundary macroscopic inertial number,whatever the flow regime and the wall height. The force fluctuations on the wall are cha-racterized by the analysis of the force autocorrelations at mesoscale on the one hand, andby the distributions of grains-wall forces at both macroscale and mesoscale on the otherhand. A modified macroscopic inertial number that takes into account a shearing acrossthe whole height of grains is found to fully control the force fluctuations on the wall. Thedistributions evolve from Gaussian-like shapes to broader asymmetric log-normal shapeswith the increase of this macroscopic inertial number. The collapse of all distributionsabove a certain value of the macroscopic inertial number, and the concomitant loss ofsystem memory revealed by the force autocorrelations, are two indicators of a remarkabletransition from quasistatic to dense inertial flow regime. The latter result is further em-phasized by a detailed study of the local rheology, showing that the ”(I) rheological lawgoverns the system in a large part of the granular bulk.
4.1 Introduction
Many industrial processes and geophysical problems involve granular matter at rest orin movement. More specifically, the force exerted on solid objects subjected to flowinggranular matter is an important topic that has attracted increasing attention over therecent years. The time-averaged force on objects of different sizes and shapes was stu-died in various configurations (see for instance [108, 2, 22, 16, 90]). The drag force onimmersed solid objects in dense free-surface granular flows was firstly investigated by theexperimental work of Wieghardt [108]. In his pioneering experiments, Wieghardt found onthe one hand that the drag force was only weakly influenced by the flow velocity, and onthe other hand that a quasistatic pile of grains formed upstream the solid object. These
4.1. Introduction 87
findings were confirmed later in different configurations from low to intermediate shearrates, including annular shear flows [2], free-fall chute flows [22], immersed rods rotatingalong an axis perpendicular to its main axis [16], etc. All these studies showed that, whe-never the obstacle is in contact with a quasistatic bulk of grains, the drag force linearlydepends on the confinement pressure and the object size rather than on the flow velocity.This picture is however no longer valid once the granular flows become fast, more diluteand collisional, as shown for instance in [8, 106]. In this situation, it was found that theforce acting on the obstacle evolved with the square of the relative grains-object velocity.
A free-surface flow of granular materials down a slope and coming across a perpendi-cular wall-like obstacle is a common situation in the field of natural hazards related tofull-scale gravity-driven granular flows. In particular, protection structures against snowavalanches or landslides are built to limit their propagation through the decrease of theirkinetic energy. In this situation, a roughly triangular stagnant zone of grainsâa so-calleddead zoneâis formed upstream the obstacle, and a more inertial zone made of flowinggrains can overcome the dead zone (see [38] and references therein). In-situ force measu-rements of snow avalanches against a square plate were conducted by Thibert et al. [100],Berthet-Rambaud et al. [7], and Baroudi et al. [5]. The time series signals systematicallydisplay complex shapes. After a fast rise, the force reaches a peak, and then decreases toa plateau. The latter plateau reflects a permanent regime, disturbed by high amplitudeand high frequency fluctuations. A full prediction of the entire force time series remainschallenging due to measurement issues and the complex rheology of granular materials.Faug et al. [40] proposed a rich semi-analytic model able to predict the average force on awall-like obstacle in steady regime. It is based on the mass and momentum conservationequations applied to a control-volume corresponding to the dead zone plus the overflowinginertial zone. This model has the particularity to take into account the length of the deadzone. Such a model was extended with some success to transient flows that mimic naturalavalanche-flows [20, 39]. However, while this model (based on conservation equations intheir depth-averaged form) predicts well the force signals smoothed over time, it does nothandle the fluctuating component which remains poorly documented in this flow-obstacleconfiguration. The overarching aim of the present paper is to analyse force fluctuationsexperienced by a wall facing a sheared granular flow.
Most studies dedicated to force fluctuations in granular materials concerned quasi-static or slowly sheared systems, in Couette-like geometries (see [1, 45, 26]). Only a fewnumber of papers addressed the problem of force fluctuations for rapid granular flows.The experiments were based on the discharge of silos equipped with a force sensor, at aboundary wall, whose size was the typical size of the grains [62, 42]. The measurementsof contact force distributions in quasistatic or slowly sheared systems were initially com-puted with the help of carbon paper placed at a system boundary [69], but the use ofphoto-elastic disks in laboratory experiments, as well as numerical modelling, opened the
88Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
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path to measuring the contact forces inside the granular bulk. Some cross-comparisonswere performed in [80, 95, 96, 76], showing thatâat least in the static caseâthe grain-grain contact force distributions from the bulk were identical to the grain-wall contactforce distributions. A robust result concerns the shape of the contact forces f distributionsfor values greater than the mean f : they decrease exponentially up to the highest valuesof f (see for instance [61, 62, 81]). This behaviour is sensitive to the shear velocity (or thedeviatoric stress tensor in static cases), as the inertia tends to broaden the distributions(see [62, 96, 45]). For the lowest forces (f < f), different behaviours depending mostlyon the sample history are observed from vanishing to very high probabilities near f = 0.Some empirical or theoretical models were proposed to fit the entire range of observeddistribution shapes for quasistatic systems [69, 67, 78].
A few studies tackled the distributions of forces that were integrated through a spatialdimension, for slow to intermediate flow rates. It means that the data sets were syste-matically recorded on surfaces that were greater than the grain size. The configurationstested were the Couette geometry [67, 50, 45], as well as a rotating immersed rod [97].A remarkable trend emerged from all these studies : while increasing the spatial scaleconsidered, the exponential force distributions narrow and finally move toward Gaussian-like distributions. The latter result is especially pronounced for very slow flows, whereasintermediate shear rates result in exponential to log-normal shapes.
The present paper is dedicated to the numerical modelling of the force exerted ona wall-like obstacle immersed in a sheared granular flow. The originality of this work isthe wide range of flow regimes investigated, from nearly quasistatic to collisional flows,through dense inertial flows. In order to control the flowing conditions and to focus on thegrain-wall interactions, an innovative planar and gravity-free system is designed. Whilethe spherical grains are trapped across the vertical direction between two rough walls,a periodic boundary condition is used in the horizontal direction. The obstacle is a wallwhich is fully immersed in the granular bulk and orthogonal to the shearing direction.First, the modelled system (boundary conditions, relevant macroscopic parameters), aswell as the numerical simulations based on the discrete element method, are described inSec. 4.2. Then, the mean (time-averaged) dynamics in terms of velocity fields, scaling ofthe mean force experienced by the wall, and spatial maps of the local ”(I)ârheology, isanalysed in Sec. 4.3. The fluctuations are quantified through both the autocorrelation ofmesoscopic force signals associated with the system memory (Sec. 4.4), and the distribu-tions of force time series at macroscopic and mesoscopic spatial scales (Sec. 4.5.1). Theeffect of the system length relative to its height is discussed in Sec. 4.6, before concludingthe paper with an extended discussion on the main results and perspectives (Sec. 4.7).
4.2. DEM simulation of the wall immersed in a granular flow 89
4.2 DEM simulation of the wall immersed in a granular
flow
4.2.1 Macroscopic boundary conditions
A planar sample of spheres is confined between two walls at the top and the bottom, andby a periodic boundary condition linking the right and the left borders of the system,as depicted by the vertical dashed lines in Fig.4.1. The bottom wall is fixed, rough,and its length is equal to the system length L. A constant macroscopic pressure P isapplied to the system. It is controlled by the vertical position of the rough top wall,continuously computed by the DEM algorithm and applying a vertical force on the sample.The periodicity of the system also concerns the top wall which has a constant horizontalvelocity U shearing the sample. The top wall remains perfectly horizontal by preventingits rotation in the (x, y) plane (see Fig.4.1). The roughness of these walls is made of fixedaligned spheres that are identical to the ones forming the granular medium. A thin andsmooth vertical wall of finite height h, namely the obstacle, is immersed into the spheresassembly. The spheres are initially deposited under gravity to form a rectangular densepacking of height H greater than h. Then, the gravity is set off and the shearing startsfor a period of ten seconds over which the data are collected. Only the data arising fromthe latest nine seconds are exploited because the first second is the observed time neededby the system to reach a permanent regime. In the present system, the grains are allow tooverflow the obstacle (by contrast to the lid-driven cavity problem studied in [58]), thusmimicking a granular flow passing over a rigid wall.
The following dimensionless macroscopic parameters are used to run the numericalsimulations over a broad range of confinement pressure P and shear velocity U :
NP =P
ΊÏpgH, (4.1)
NU =Uâgd
. (4.2)
The mean grain diameter d was taken equal to 1 mm and H/d = 30, and a slightpolydispersity was introduced by picking randomly the grain diameters between 0.85d
and 1.15d. A constant macroscopic volume fraction Ί = 0.6 is considered in these twodefinitions, as it roughly corresponds to the random close packing of a three-dimensionalassembly of spheres. We will discuss in more detail the mean volume fraction of thegranular sample in the concluding section of the paper (Sec. 4.3). The particle densitywas taken equal to ÏP = 2500 kg mâ3, and although the system is gravity-free, the gravityacceleration is considered here (g = 9.81 m sâ2) for convenience to compare the resultswith laboratory experiments in a future work.
90Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
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Figure 4.1 â Geometry and boundary conditions of the periodic sheared system simulated by
DEM. The granular sample is trapped between two horizontal walls and a periodic boundary
condition in the horizontal xâdirection (dashed lines). The top wall applies the shear displace-
ment U and the confinement pressure P (see text for details), and force measurements are done
onto the sidewall of height h, immersed into the granular sample.
The boundary macroscopic inertial number IM can be defined from the typical timeassociated with the top confinement pressure tP = d
âÏ/P and the typical time equal to
the inverse of the macroscopic shear rate tU = H/U :
IM =tPtU
=dâÏ/P
H/U, (4.3)
where Ï = ΊÏP holds for the density of the granular sample.
In the present study, NP ranged from 0.01 to 100 and NU from 1 to 20, which allowsto investigate a wide range of granular flow regimes defined by IM ranging typically from1.2 10â3 to 6 10â1.
The length of the periodic cell relative to the initial height of the sample, L/H, is aparameter that may also influence the system dynamics. It has been shown by similarDEM simulations on another system, namely the lid-driven cavity (see details in [58, 59]),that the mean dynamics and the force fluctuations are no longer dependent on L/H assoon as L/H > 5. In this paper L/H is taken equal to 5 unless otherwise specified, andthe sensitivity of the results to L/H will be discussed in detail in Sec. 4.6.
Simulations with various obstacle heights (h â 5d, 10d, 15d, 20d) were performed toanalyze the influence of this parameter on the time-averaged results.
4.2.2 Micromechanical parameters
The discrete elements method, the contact laws and the associated parameters used in thispaper are the same as described in [58]. The general DEM algorithm is described in [28].A visco-elastic contact law is used for the normal component of the inter-particle forces(see [88]), while a classical coulomb friction threshold drives the tangential component inthe same fashion as in [64]. The following set of equations describes the micromechanical
4.3. Mean dynamics 91
model used in the present study, for the grain-grain and the grain-wall interactions :
Fn = max(0, knÎŽn + cnÎŽn)n,
dFs = (ksÎŽsdt)s,
|Fs| †”|Fn|,(4.4)
where n and s are the unity vectors along the contact normal and shear directions res-pectively, kn and ks are the normal and tangential contact stiffnesses, Ύn is the normalpenetration depth, Ύs is the tangential displacement increment, ” is the local friction co-efficient, cn is the normal viscosity coefficient and dt is the time-step. The four physicalparameters kn, ks, cn and ”, are chosen to fit the behaviour of glass beads. As discussedin [40] the contact stiffnesses are reduced to decrease the total time of calculation, but thelimit of rigid grains was respected with the help of an adimensional number N0 = 1.5 104.For two grains of same size, kn was thus set regarding the macroscopic pressure and N0 :kn = N0Pd, and ks was set with a poisson ratio Μ = 0.3 : ks = Μkn. cn is set in the samemanner as in [58] with a restitution coefficient e = 0.5, and ” was taken equal to 0.5.
In accordance with the motivation of this studyâi.e. the force fluctuations experiencedby an obstacle immersed in a granular flow, the current paper is mainly focused on F (t),the normal force time-series exerted on the vertical sidewall over a broad range of IM .Before going into the details of the force fluctuations (see Secs. 4.4 and 4.5), the time-averaged dynamics of the system (mean force F on the wall, streamlines, vertical velocity-profiles and local ”(I)ârheology within the granular sample) is studied in the next section.
4.3 Mean dynamics
4.3.1 Velocity field
The time-averaged granular flow velocity streamlines are displayed on Fig.4.2 for threedifferent inertial numbers IM and one obstacle height h = 20d. The norm of the localgrain velocity vg scaled by the top wall shearing velocity U is displayed on the colourscale. For all simulations the flow velocity fits the top and bottom boundary conditions,ranging from zero near the bottom to U in the vicinity of the top wall. At low inertialnumbers (IM â 1.2 10â3) the flow velocity shows a symmetric pattern along the systemlength. Near both corners at the bottom of the system, very low flow rates take placein front of the obstacleâlikely at the origin of the streamlines instabilities. They arethe markers of nearly triangular dead zones of height h forming near the obstacle andgradually disappearing while moving farther from the obstacle. In a central region (50d .
x/d . 100d) the streamlines are horizontal and parallel with each other, revealing thatthe flow experiences a quasi-horizontal shear without any significant vertical displacement.Back to past studies on free-surface gravity-driven granular flows passing over a rigid wall
92Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.0000.0050.0100.0150.0200.025
y
IM â0.0012|NU =1.0|NP =25.0
0
1
vg /U
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.0000.0050.0100.0150.0200.0250.030
y
IM â0.0609|NU =10.0|NP =1.0
0
1
vg /U
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14x
0.0000.0050.0100.0150.0200.0250.0300.035
y
IM â0.3043|NU =10.0|NP =0.04
0
1
vg /U
Figure 4.2 â Time-averaged streamlines for an obstacle height of h = 20d (thin vertical bars
on each border of the system), for three different values of IM : IM = 1.2 10â3 (top panel),
IM = 6.09 10â2 (middle) and IM = 0.3043 (bottom). The colour scale represents the norm of
the local grain velocity scaled by the top wall shearing velocity, vg/U .
[40, 20, 39, 38], the latter homogeneous region mimics the incoming flow, co-existing witha similar (nearly triangular) dead zone formed upstream the obstacle (the wall on theright side).
However, the response of the system velocity streamlines to the increase of IM is tricky.While the top panel at the lowest IM exhibits a symmetric pattern, the middle and thebottom panels of Fig.4.2 display different behaviours. As the obstacle is also duplicatedthrough the periodic boundary condition along x, a local vortex of height h starts formingon the left side of the system. In the present context, this phenomenon is considered asan artifact caused by the grains flowing through the periodic boundary condition andincoming with high velocity from the top of the vertical left sidewall. The shape of thisvortex varies with the macroscopic boundary conditions. In particular, the vortex lengthincreases with IM , disturbing progressively the above picture of the central region whereshear is quasi-horizontal. The latter region becomes narrower when increasing IM , andmay totally disappear at the largest IM . This peculiar feature of the periodic flow-obstaclesystem studied here is further investigated in the following paragraph, focusing on thevertical velocities of grains.
Figure 4.3 depicts the vertical component Vy of the grain velocity along the systemlength, averaged over time and height, and scaled by the top wall shearing velocity U .This measurement is presented for seven values of inertial numbers, from quasistatic to
4.3. Mean dynamics 93
0.0 0.2 0.4 0.6 0.8 1.0x/L
â0.03
â0.02
â0.01
0.00
0.01
0.02
Vy/U
IM â0.0012|I âM â0.0037|NU =1.0|NP =25.0
IM â0.0061|I âM â0.0183|NU =1.0|NP =1.0
IM â0.0122|I âM â0.0365|NU =10.0|NP =25.0
IM â0.0609|I âM â0.1826|NU =10.0|NP =1.0
IM â0.0913|I âM â0.2739|NU =15.0|NP =1.0
IM â0.3043|I âM â0.9129|NU =10.0|NP =0.04
IM â0.6086|I âM â1.8257|NU =10.0|NP =0.01
Figure 4.3 â Profile of the vertical component of the grains velocity Vy, averaged over time and
grains height H, scaled by the top wall shearing velocity U . The profile is plotted for an obstacle
height h of 20d and seven values of IM from 1.2 10â3 to 0.6.
collisional flow regimes.
For IM †1.2 10â2 all profiles collapse, revealing that the global (scaled) grain velocityis not influenced by the macroscopic conditions in the quasistatic regime. Near the (leftside) incoming flow region (x/L < 0.3), Vy is negative with a high slope gradient, inducedby the grains that continuously fill the bottom of the system. Between x/L = 0.3 andx/L = 0.85, Vy/U is nearly close to zero with only a very small positive slope gradient. Thisregion, which we call a plateau, characterizes the homogeneous sheared zone describedjust above. Finally, from x/L = 0.85, the increase of the slope gradient marks the presenceof the dead zone formed upstream of the obstacle (on the right side) overflowed by thegrains. The presence of a plateau in the center of Fig. 4.3 is a sign that the system lengthshould be large enough to guarantee that the flowing direction of the granular assemblyis orthogonal to the obstacle. Thusâif non-local effects do not play a prevailing role, theforce exerted on the obstacle should not be influenced by its periodic duplicate (on theleft side). The effect of increasing L/H (fixed to 5 here) will be discussed in Sec. 4.6.
As shown at the middle and bottom panels of Fig.4.2, the vortex formed in the left sideof the system produces vertical movements of the granular flow that are inhomogeneousthrough the system height for simulations done with IM â„ 6.09 10â2. As a consequence,the averaging over height done in Fig.4.3 to obtain the Vy/U profile for the correspondinginertial numbers cannot be quantitatively interpreted. Qualitatively, the plateau is stillobserved up to IM = 9.13 10â2 but the two curves obtained for the largest IM (IM â„ 0.3)are highly disturbed by the vortex, thus preventing the occurrence of the plateau inthe collisional regime. We will show in Sec. 4.6 that increasing L/H does not change thisobservation : it is not possible to produce a plateau for L/H = 10 at high inertial numbers.
94Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
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This flow inhomogeneity along the main flow direction (xâaxis direction) observed for highIM makes difficult, in the collisional regime, the direct comparison of this system to free-surface gravity-driven granular flows passing over a wall. The influence zone downstreamof the duplicate obstacle (on the left side of the periodic system) is too long so that itdisturbed the granular kinematics in the whole volume of the system up to the obstacle(on the right side) of interest here. However, it will be shown in the following of the paperthat an analysis of the force (time averaged value and fluctuations) experienced by thelatter obstacle can be done continuously, over the whole range of IM covered by this study.In particular, a robust scaling of the mean total force exerted on the obstacle is proposedin the next section.
4.3.2 Mean force on the obstacle
The main feature of the present system is that the height H of the granular sample ishigher than the height h of the obstacle, by contrast to the lid-driven cavity problem in[58]. As a consequence the following flow dynamics can be observed : grains are trappedupstream the obstacle (the right sidewall), thus forming a dead zone, and other grainsare able to flow over the obstacle, as discussed in Sec. 4.3.1. Based on the height H â h
of the sheared layer passing over the obstacle, a new macroscopic inertial number IâM canbe defined :
IâM =dâ
Ï/P
(H â h)/U. (4.5)
In the above definition, the dead zone is assumed to cover the entire height of the wall.The grains passing over the wall are sheared between the top wall of vertical positionH on the one hand, and the top of the dead zone of null velocity and height h on theother hand. Introducing this macroscopic inertial number allows us to obtain a remarkablescaling of the mean force F on the obstacle, scaled by the pressure force Phd associatedwith the wall height, as a function of IâM , as shown in Fig.4.4.
A collapse of all the data is observed, including four obstacle heights h â 5d, 10d, 15d, 20d.This result reveals that the shearing velocity and the top pressure can be wrapped intoa macroscopic inertial number which predicts the scaled mean force on the sidewall. It isenhanced by the construction of IâM , showing that the inertial number corresponding tothe grains overflowing the sidewall plays a crucial role in the force applied on the obstacle.
The scaled mean force, F/(Phd), shows a quasi-linear evolution with IâM at low andintermediate IâM . Though a scattering of the data appears at high IâM , a saturation thenoccurs from around IâM = 0.3 (see the linear plot in the inset of Fig. 4.4). The followingrelation fits well the data points, as shown by the solid line drawn on Fig. 4.4 :
F
Phd= râ1 +
râ2 â râ1
1 +Iâ0M
IâM
, (4.6)
4.3. Mean dynamics 95
10-3 10-2 10-1 100
I âM
1.5
2.0
2.5
3.0
F/Phd
h=5d
h=10d
h=15d
h=20d
0.0 0.5 1.0 1.5 2.0
1.5
2.0
2.5
3.0
Figure 4.4 â Time-averaged force on the obstacle F scaled by Phd as a function of IâM , the
inertial number corresponding to the height of grains overflowing the obstacle. Four obstacle
heights are presented : h â 5d, 10d, 15d, 20d. Inset : same data with linear x-axis scale.
where râ1 = 1.4 and râ2 = 3.3 are the two asymptotic values of F/(Phd) at low and highIâM , respectively. The constant Iâ0M = 0.3 is the inertial number for which F/(Phd) =
(râ1 + râ2)/2. It is worth noting that the values of râ1, râ2 and Iâ0M were obtained for the
configuration presented in this paper, and may be influenced by the micromechanicalgrain parameters (d, ”, etc.), as well as by the system configuration. In particular, theeffect of L/H on the mean force scaling shown in Fig. 4.4 will be discussed in Sec. 4.6.
It is worthwhile to note that the data saturation (concomitant with some scattering)and the plateau which vanishes in the mean Vy profiles (see Sec. 4.3.1) both occur atIâM â 0.3. This observation is a signature of the transition from the dense inertial regimeto a more collisional regime, where enduring frictional grain-grain contacts are replacedby short-life contacts due to the strong increase of the inertia and the increase of the meanfree path of particles. The role played by the local ”(I)ârheology, proposed by [52, 44],on the force transmision toward the wall-like obstacle is further analysed below.
4.3.3 The local ”(I)ârheology
The form of the empirical scaling showing a one-to-one relation between the scaled meanforce F/(Phd) on the wall and the boundary macroscopic inertial number IâM , given byEq. (4.6), suggests that the local ”(I)ârheology comes into play. To further investigatethe rheology, the time-averaged values of the local inertial number I and the local effectivefriction ”loc were measured inside the whole volume of the granular system.
For that purpose, the strain (D) and stress (Ï) tensors at local (grain) scale werecomputed, using a spatial kernel smoothing method and tesselation techniques (see much
96Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
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Figure 4.5 â The local ”(I)ârheology for this study, extracted from all simulations. The points
come from a region of the granular flow-wall system selected to approach the conditions of a
simple sheared granular sample.
more details in [59]). The pressure p within the granular medium was defined by :
p =1
3Tr(Ï). (4.7)
The local effective friction was calculated as :
”loc =1
3
Tr(ÏâČD
âČ)
pâDâČâ , (4.8)
where DâČ = D â 1
3Tr(D)I3 holds for the deviatoric strain tensor and Ï
âČ = Ï â pI3
is the deviatoric stress tensor. We define I3 as the unit tensor and âAâ =â
Tr(AAT )/2
for any A. Finally, the local inertial number was computed using the relation :
I = d
âÏPpâDâČâ. (4.9)
Then, for each position (x, y), the measured ”loc was compared to a ”th(I) derivedfrom the relation proposed by Jop et al. [52], which reads as follows :
”th = ”1 + (”2 â ”1)1
1 + I0/I, (4.10)
where ”1, ”2 and I0 are parameters dependent on the mechanical grain properties. Inour study, we found that the best fitting parameters for ”th were ”1 = 0.17, ”2 = 0.48
and I0 = 0.18, as shown in Fig. 4.5.Figure 4.6 displays the spatial maps of |”loc â ”th| for three values of IM . First, it is
generally observed that the granular sample obeys the local ”(I)ârheology in most ofthe volume of the granular system, except in the regions close to the sidewalls in the two
4.3. Mean dynamics 97
0.0 0.2 0.4 0.6 0.8 1.0x/L
0.00.20.40.60.81.0
y/H
IM â 0. 0012|NU =1. 0|NP =25. 0|h/d=20
0.000.020.040.060.080.100.120.140.160.18
|”locâ”th|
0.0 0.2 0.4 0.6 0.8 1.0x/L
0.00.20.40.60.81.0
y/H
IM â 0. 0609|NU =10. 0|NP =1. 0|h/d=20
0.000.020.040.060.080.100.120.140.160.18
|”locâ”th|
0.0 0.2 0.4 0.6 0.8 1.0x/L
0.00.20.40.60.81.0
y/H
IM â 0. 3043|NU =10. 0|NP =0. 04|h/d=20
0.000.020.040.060.080.100.120.140.160.18
|”locâ”th|
Figure 4.6 â Spatial maps of |”loc â ”th| for three values of IM : IM = 1.2 10â3 (top panel),
IM = 6.09 10â2 (middle) and IM = 0.3 (bottom). See text for the definition of ”loc and ”th.
corners at the bottom of the system. Second, at low IM the patterns resulting from this|”loc â ”th| mapping is symmetric, while it becomes asymmetric at the highest IM . Atthe highest IM , the region in the left corner at the bottom, in which the measured ”loc
discards from the ”th derived from the ”(I)ârheology, grows in size and tends to extendin the middle of the granular system. Those observations are fully consistent with thestreamlines shown in Fig. 4.2. The regions with a dead zone (whatever IM) correspond toregions where the granular material does not obey the ”(I)ârheology.
Back to the analogy with gravity-driven free-surface flows passing over a wall (asstudied in [40, 20]), our study reveals that the incoming flowing grains able to overtopthe wall would obey the ”(I)ârheology, while the grains trapped inside the dead zone(upstream of the wall) would not. The deviation from the ”(I)ârheology can thus helpto demarcate the dead zone from the coexisting (more inertial) flowing zone above.
The region (in the left corner at the bottom), where a vortex-like structure formsat high IM , grows in size with IM and also corresponds to a region where the granularmedium does not obey a ”(I)ârheology.
From now on, the paper tackles the force fluctuations on the obstacle, through analy-sing the autocorrelation of force time-series (Sec. 4.4) and the distributions of grain-wallcontact forces (Sec. 4.5). All data obtained for the next analysis was extracted from nu-merical simulations with an obstacle height h = 20d. Keeping the obstacle high enoughto catch collective effects of the grains (in order to avoid discretization issues), while theheight available for the grains to overflow is one third of the total height H of the sample,made the results clearer. The results extracted from the simulation with h/d = 5, 10 or
98Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
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15 (not shown here) were very similar overall.
4.4 Force time-series autocorrelation
In this section, the force fluctuations are investigated by analysing the autocorrelation offorce time-series on portions i of the wall. Information about the memory of the system
(we will define below what we call memory of the system) is then extracted through acritical autocorrelation time âtic.
The force signal record frequency is set to 10 kHz, this value being high enough tocapture short-time events so that the results do not depend on this parameter (see moredetail in [59]). To study the potential heterogeneity of the force signals along the obstacleheight, the sidewall is split into five portions i â 1, 2, 3, 4, 5 from bottom to top, ofidentical size h/5. Each slice has its associated force time-series from which the autocor-relation function Ci(F (t), F (t+ât)) is then computed, where ât is the lag between twosystem states. The critical correlation time âtic is extracted from each slice. It can beinterpreted as the delay after which the system has forgotten a certain value of force onthe wall. In other words, it reflects the typical time during which a future state of thesystem keeps history of its past state regarding the force transmitted to the wall. Thecritical correlation time âtic is defined with an arbitrary low correlation threshold :
Ci(F (t), F (t+âtic)
)= 0.15. (4.11)
As the local autocorrelation function Ci may be a non-monotonic function, it is possiblethat the threshold value is reached many times. In this case the lowest value of âtc iskept, avoiding the noise to have an influence on the results.
The top panel of Fig. 4.7 shows the autocorrelation of force time-series for each ofthe five identical portions of the wall, for a low value of the macroscopic inertial num-ber (1.2 10â3). For this granular regime, the autocorrelations decrease slowly to reacha vanishing value between approximatively 0.2 s to 1.5 s, depending on the wall portionconsidered. Although an observable trend suggests that bottom portions are correlatedfor longer times (lower slopes) than top portions (higher slopes), no general rule could bededuced, because this behaviour is not strictly monotonic.
The values of autocorrelation times âtic for each portions of the wall, as well as theirmean over iâcalled <âtc>, are displayed on Fig.4.7. A scattering of the âticâvaluesaround <âtc> is observed, showing that autocorrelation times are inhomogeneous alongthe height of the sidewall height. However, the average value <âtc> shows a remarkabledecreasing trend with IM . This behaviour may be interpreted as a decrease of the systemmemory while increasing the inertial number, which is in accordance with a collisionalregime where short-life contacts are dominant compared to long-lasting enduring frictional
4.4. Force time-series autocorrelation 99
0.0 0.5 1.0 1.5 2.0ât
â0.2
0.0
0.2
0.4
0.6
0.8
1.0
Ci(F(t),F(t+ât))
IM â0.0012|NU =1.0|NP =25.0|h=20d
i=1
i=2
i=3
i=4
i=5
10-3 10-2 10-1 100
IM
0.0
0.2
0.4
0.6
0.8
1.0
1.2
ât c
ât ic
<âtc>
Figure 4.7 â Top panel : evolution of Ci(F (t), F (t+ât)) over time for five identical portions of
the wall height i â 1, 2, 3, 4, 5 from bottom to top, and IM = 1.2 10â3. The grey dashed line is
the threshold used to obtain âtic. Bottom panel : critical time âtc beyond which the force time
series become weakly correlated (âtc < 0.15) for the five portions of the wall, as a function of
IM . <âtc> is the mean over the five âtic values of a simulation.
100Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
écoulement granulaire cisaillé
contacts. Near IM = 1.2 10â2 the mean autocorrelation time nearly vanishes, thus showingthat the system is unable to keep memory of a past state. The granular medium, whichis at the origin of the force transmission from the boundary conditions to the sidewall,handles here the signature of a remarkable cross- over from quasistatic to dense inertialregime. It is important to note that what we call the memory of the system is not anintrinsic memory of the granular material, but is rather system-dependent. This becomesobvious when those âtic (and <âtc>) are shown to be influenced by the length of thesystem, as it will be discussed in Sec. 4.6.
4.5 Force distributions
In the following, we analyze how the distributions of force time-series evolve with theboundary macroscopic inertial number, at macroscale (the entire wall ; Sec 4.5.1) as wellas at mesoscale (portions of wall ; Sec 4.5.2) .
4.5.1 Force distributions on the entire obstacle
This section investigates the distribution of the total force experienced by the sidewall, onthe whole surface facing the granular flow. This measurement gives information about thecollective effect of the granular bulk on the obstacle. This analysis is crucial for real-worldapplications, while for instance designing civil engineering structuresâin the case of mainanchors that hold the whole obstacle, or for technical reasons when the installation of localforce sensors is not possible (see a discussion in Sec. 4.7.3). The total force time-seriesF is obtained by summing all grain-wall contributions at each recording time. This forcescaled by the mean force F is noted F = F/F , and its probability distribution density iscomputed.
The top panel of Fig.4.8 shows the distributions of F , for seven values of IM rangingfrom 1.2 10â3 to 0.6. The distribution shapes follow a nearly Gaussian law at low IM
(= 1.2 10â3). A slight asymmetry appears while increasing IM : the decreasing tail at highforces becomes more and more important. At the same time, from IM â„ 1.2 10â2, theprobability to have weak or vanishing force signals on the sidewall increases. A systematicfit of the distributionsâactually measured in DEMâusing a shifted (and truncated) log-normal function, called P(F ), provides very good results over the whole range of IM
tested, as depicted by the dashed lines on top of Fig. 4.8 :
P(F ) =1
S1
FÏ
â2Ï(F + 1)
exp
(â [ln(F + 1)â F”]
2
2FÏ
2
), (4.12)
where F” is the scale parameter, FÏ is the shape parameter, and the shift (truncation) is 1.S is the normalization factor corresponding to the value at 1 of the survival function of the
4.5. Force distributions 101
0.0 0.5 1.0 1.5 2.0 2.5 3.0
F/F
0.0
0.5
1.0
1.5
2.0
Den
sity
IM â 0. 0012|NU =1. 0|NP =25. 0
IM â 0. 0061|NU =10. 0|NP =100. 0
IM â 0. 0061|NU =5. 0|NP =25. 0
IM â 0. 0061|NU =1. 0|NP =1. 0
IM â 0. 0122|NU =10. 0|NP =25. 0
IM â 0. 0609|NU =10. 0|NP =1. 0
IM â 0. 0913|NU =15. 0|NP =1. 0
IM â 0. 3043|NU =10. 0|NP =0. 04
IM â 0. 6086|NU =10. 0|NP =0. 01
10-3 10-2 10-1 100
IM
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Para
met
er v
alue
FÏ
F”
Figure 4.8 â Top panel : distribution of the total force on the right sidewall for seven values of
IM ranging from 1.2 10â3 to 0.6. The dashed lines show fits using a (truncated) log-normal PDF
given by Eq. (4.12). Bottom panel : variation of f” and fÏ, the two parameters of Eq (4.12), as
function of IM .
102Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
écoulement granulaire cisaillé
(untruncated) log-normal PDF. The evolution of these parameters with IM is displayedon the bottom panel of Fig.4.8.
Three distinct simulations, done for different macroscopic shear rates and confinementpressures U, P but keeping the inertial number constant (IM = 6.1 10â3), give identicaldistributions, as shown in the top panel of Fig.4.8 (orange-colored curves). This revealsthat the boundary macroscopic inertial number is the only parameter which controls thedistribution of the total force on the wall. Such a key result is further confirmed by the twofitting parameters F” and FÏ, displayed at the bottom of Fig.4.8, that evolve monotonicallywith IMâwhatever U, P. It can be concluded that Eq. (4.12), fed with the parameterscontrolled by IM (bottom of Fig.4.8), provides an empirical model capable of predictingthe total force distribution for the granular flow-wall interaction system studied here.
Moreover, the DEM simulations show that all distributions of F collapse when IM â„6.1 10â2, as shown in the top of Fig.4.8 (blue curves). This observation becomes clearerin the bottom of Fig.4.8, as both F” and FÏ reach a plateau from IM = 6.1 10â2 up tothe highest value of the inertial number achieved in this study. This saturation of thetotal force distribution is likely to reflect a fast flow state largely influenced by inertialeffects. It occurs above a certain value of IM between 1.2 10â2 and 6.1 10â2, which closelymatches the value of IM at which the loss of system memory was identified in Sec. 4.4.
In order to analyse the distributions of the force at mesoscopic spatial scale, thustaking into account the potential heterogeneity over the system height, the next sectiontackles the distributions of force time-series recorded on portions of the obstacle height.
4.5.2 Force distributions on portions of the obstacle
In the same manner as in Sec. 4.4, the sidewall is split into five portions i â 1, 2, 3, 4, 5of same height. We note Fi the force time-series associated with the portion i and Fi itsmean. The upper panel of Fig.4.9 shows the distribution of Fi/Fi, the force arising fromeach portion scaled by the corresponding mean, for IM = 6.1 10â2. The distributionsof the force time-series from the three portions located in the center of the sidewall areequivalent (orange lines), unlike the two located at the top and bottom (red and bluelines, respectively). This difference between the portions i â 2, 3, 4 on the one side,and the portions i = 1 and i = 5 on the other side, is observed for any IM (not shownhere). This result reveals that the mesoscopic force distributions are homogeneous alonga large area at the centerâalong the yâaxis (vertical) directionâof the obstacle, whereassome modifications near the weak forces appear at the top and the bottom. In the bottomcorner, at the interface with the bottom rough wall, few grains can be trapped and exertenduring forces on the obstacle, thus decreasing the probability of weak forces on thetime-series. Near the top of the obstacle, the increasing probability of small forces can beexplained by the shortening of the thickness of the dead zone, thus causing the obstacle
4.6. Effect of the system length 103
to experience directly the force of the flowing grains.
Because of the collapse of the distributions for portions i â 2, 3, 4 is observed forall values of IM , the mean (called <Fi/Fi>i) of the distributions over these three centralportions of the sidewall was computed, and its variation with the macroscopic inertialnumber was computed. The results are shown on the bottom panel of Fig.4.9. Similarlyto the distributions on the entire height of the obstacle (see Sec. 4.5.1), the resulting curvesdepict log-normal-like shapes. Near IM â 10â3, the distribution displays a small amountof low forces and a repartition which approaches a Gaussian-like shape. The number ofweak forces increases with IM , shifting the maximum probability density toward the lowforces. This behaviour can be explained by a finite-size effect. At low inertial numbers theforce signal may arise from the summation of several enduring contact forces, which isknown to produce a Gaussian-like shape. By contrast, at higher inertial numbers the forcesignal arises from a lower number of contacts, thus increasing the probability of weak (ornull) forces. Finally, all the force distributions on wall portions collapse for IM â„ 6.1 10â2.The latter observation reveals thatâexcluding the effects related to the top and bottomendsâthe force distributions at mesoscale become independent of IM as soon as a certainIM is reached. This further confirms the result presented in Sec. 4.5.1 for the entire heightof the obstacle, where the collapse of distributions was interpreted as a signature of thetransition from quasistatic to dense inertial regime.
4.6 Effect of the system length
The results that have been presented up to now were extracted from simulations with aconstant length of the system (L/H = 5 and H = 30d, where d = 1 mm). This sectioncompares those results to additional ones from simulations where the length of the systemwas doubled (L/H = 10, while keeping H = 30d and d = 1 mm).
The simulations at L/H = 10 did not allow to produce a plateau at high IM aroundzero in the curves of Vy/U versus x/L (not shown here), the results being similar to theones shown in Fig. 4.3. Doubling the length of the system did not allow to reduce theeffect of the vortex-like structure (generated by the duplicate obstacle on the left side),which starts disturbing a lot the entire volume of the system for IM larger than 0.3.This proves to be a signature of the transition toward the collisional regime, thus makingquestionableâbeyond that IM ⌠0.3âthe analogy between the periodic system studiedhere and a free-surface gravity-driven granular flow passing over a rigid wall.
Figure 4.10 shows the scaled mean force, F/(Phd), as a function of IâM for L/H = 5
and L/H = 10. The empirical scaling given by Eq. (4.6) is slightly influenced by the ratioL/H. Doubling the length of the system yields a slight increase of the scaled mean force,but the empirical scaling that relates F/(Phd) to IâM is still robust, provided that râ1 and
104Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
écoulement granulaire cisaillé
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Fi/Fi
0.0
0.2
0.4
0.6
0.8
1.0
Den
sity
IM â0.0609|NU =10.0|NP =1.0
i=1
i=2
i=3
i=4
i=5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
<Fi/Fi >i
0.0
0.2
0.4
0.6
0.8
1.0
Den
sity
IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0609|NU =10.0|NP =1.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
IM â0.6086|NU =10.0|NP =0.01
Figure 4.9 â Top panel : distribution of force time-series experienced by five equal portions i of
the sidewall for NU = 10 and NP = 1 (each portion time-series is scaled by its corresponding time-
averaged force). The blue, orange and red curves correspond respectively to i = 1, i â 2, 3, 4and i = 5. Bottom panel : mean distribution over i for i â 2, 3, 4 (orange curves of top panel)
for seven values of IM .
4.7. Discussion and conclusion 105
10-3 10-2 10-1 100
I âM
1.5
2.0
2.5
3.0
3.5
F/Phd
L/H=5
L/H=10
0.0 0.5 1.0 1.5 2.0
1.5
2.0
2.5
3.0
3.5
Figure 4.10 â Scaled mean force F/(Phd) versus IâM for L/H = 5 and L/H = 10. Although
they are not discriminable in this plot, data from each L/H result from four values of h/d : 5,
10, 15 and 20. Grey solid line : same fit as in Sec. 4.3.2 ; grey dashed line : fit for L/H = 10,
with râ1 = 1.5, râ2 = 4.1 and Iâ0M = 0.3.
râ2, in Eq. (4.6), become slightly dependent on L/H.The top and bottom panels of Fig. 4.11 compare two extremes distributions of grain-
wall forces at macroscale (entire height of the obstacle) and mesoscale (portions of obs-tacle) respectively, for L/H = 5 and L/H = 10. The distributions, besides at macroscalefor very low IM , are weakly influenced by the system length, and even totally independenton this parameter when the saturation occurs at intermediate to high IM .
Overall, doubling the length of the system does small changes to the main resultsregarding the time-averaged dynamics and the distributions of grain-wall forces. However,moving from L/H = 5 to L/H = 10 has a significant effect on the variation of <âtc> withIM . As shown in Fig. 4.12, while <âtc> is largely positive at small IM for L/H = 5, ittends to approach zero for L/H = 10 at the same IM . This put emphasis on the fact thatthe positive value of <âtc> at small IM can be interpreted as a signature of a memoryactivated by the system (memory of the system) in terms of the force transmission towardthe wall, but not of a memory that would be intrinsic to the granular material itself.
4.7 Discussion and conclusion
The present paper described periodic numerical simulations based on DEM of a granularflow passing over a wall-like obstacle, over a wide range of confinement pressure andshearing velocity imposed at the top of the granular flow. The main results concerning thetime-averaged dynamics and the force fluctuations are summarized and further discussedin this concluding section. We end the paper by exploring the potential implications of this
106Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
écoulement granulaire cisaillé
0.0 0.5 1.0 1.5 2.0 2.5 3.0
F/F
0.0
0.5
1.0
1.5
2.0
Den
sity
IM â0.0012|NU =1.0|NP =25.0|L/H=5
IM â0.0012|NU =1.0|NP =25.0|L/H=10
IM â0.0609|NU =10.0|NP =1.0|L/H=5
IM â0.0609|NU =10.0|NP =1.0|L/H=10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
<Fi/Fi >i
0.0
0.2
0.4
0.6
0.8
1.0
Den
sity
IM â0.0012|NU =1.0|NP =25.0|L/H=5
IM â0.0012|NU =1.0|NP =25.0|L/H=10
IM â0.0609|NU =10.0|NP =1.0|L/H=5
IM â0.0609|NU =10.0|NP =1.0|L/H=10
Figure 4.11 â Distributions of the total force on the entire height of the obstacle (top panel)
and mean, over i for i â 2, 3, 4, of the distributions on portions of obstacle (bottom panel),
for L/H = 5 (thick lines) and L/H = 10 (thin lines). In each panel, we present the results for
IM = 1.2 10â3 (the lowest IM tested) and IM = 6.09 10â2 (IM beyond which the distributions
collapse).
4.7. Discussion and conclusion 107
10-3 10-2 10-1 100
IM
0.0
0.2
0.4
0.6
0.8
1.0
1.2
ât c
ât ic
<âtc>L/H=10
<âtc>L/H=5
Figure 4.12 â Critical times âtic (and their mean <âtc>) beyond which the force time series
become weakly correlated (âtc < 0.15), for L/H = 5 and L/H = 10.
basic knowledge for a real-world problem that is the interaction of full-scale granular flowsin nature, such as avalanches, in interaction with civil engineering structures (Sec. 4.7.3).
4.7.1 Time-averaged dynamics
The mean force experienced by a wall (of height h) in contact with a dead zone subjectedto a granular flow (of incoming height H) passing over the wall obeys the followingscaling : F = f(IâM)Phd, where IâM is a macroscopic inertial number built on the boundaryconfinement pressure P and the shearing velocity U over the height difference Hâh, andf is a function which saturated at both low and high IâM . It was found that the bulk ofthe granular flow was controlled by the local ”(I)ârheology, while the dead zone formedupstream of the wall was not. At very high IâM , the ”(I)ârheology became, however,much less relevant. A vortex-like structure, inside which the local ”(I)ârheology wasnot verified, formed downstream of the duplicate obstacle (on the left side of the periodicsystem simulated) and grew in size, thus producing a disturbance that propagated over theentire volume of the system. This was interpreted as the transition towards the collisionalregime.
Figure 4.13 shows how the mean volume fraction Ίâ of the system, actually measuredin the DEM simulations, varies with the macroscopic inertial number IâM . The curve hasthe typical shape of the Ί(I)âconstitutive law generally observed in a number of othersystems, such as plane shear flows [29], inclined flows down a slope [41, 40], or annularshear cell flows [36]. Note that those curves result from simulations at different obstacle
108Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
écoulement granulaire cisaillé
10-3 10-2 10-1 100
I âM
0.40
0.42
0.44
0.46
0.48
0.50
0.52
0.54
Ίâ
h=5d
h=10d
h=15d
h=20d
h=25d
10-3 10-2 10-1 100
IM
0.40
0.42
0.44
0.46
0.48
0.50
0.52
0.54
Ίâ
Figure 4.13 â Volume fraction Ίâ actually measured in the DEM simulations, as a function of
IâM . Inset : Ίâ versus IM .
heights. The inset of Fig. 4.13 that shows Ίâ versus IM with a noticeable scattering at highIM , further confirms that the inertial number IâM defined on the shearing height H â h
above the obstacle is the relevant parameter for this system (by contrast to the lid-drivencavity problem, as investigated in [58]). Note that the relation Ίâ(I
âM) was found to be
slightly dependent on L/H for very high IâM (not shown here).
4.7.2 Force fluctuations
In the present study, the analysis of the force fluctuations was performed through twomain measurements. Firstly, the autocorrelations of force time series were computed at aspatial mesoscopic scale, intermediate between the grain size and the entire wall height(considering five portions of the wall, identical in size). Those temporal autocorrelationswere found to be inhomogeneous through the wall height. A criterion based on the mean ofan autocorrelation threshold through the wall height (<âtc>) was however able to reflectthe system memory. Secondly, the distributions of force signals were computed at bothmacroscale (wall height) and mesoscale (considering the same portions of the wall as usedfor the autocorrelations). The boundary macroscopic inertial number IM imparted to thegranular system was found to be the control parameter of both types of measurements,thus highlighting remarkable trends that are summarized below.
For the lowest values of IM tested, the system exhibited a significant memory, and nar-rowed distributions from normal-like to log-normal-like shapesâdepending on the spatialscale consideredâwere observed. This result typically reflects the quasistatic flow regime :the wall experiences a constant and large number of enduring contacts with the grains.This increases the time needed for a force value to be forgotten by the system on the one
4.7. Discussion and conclusion 109
hand, and restricts the range of actual force values on the wall on the other hand.
While increasing IM , the overall memory of the system decreased and the distributionswidened. The inertial effect started to come into play, with faster motions of particles cau-sing shorter contact times with the wall. In comparison with the quasistatic regime, thebroadening of the statistical range of force values is thus interpreted as the consequenceof (i) shorter and scarcer contacts with the wall near the weak force values on the onehand and (ii) more intense interactions near the highest forces on the other hand. Themeasurements of force fluctuations showed a saturation near 10â2 †IM †6 10â2, ma-terialized by the collapse of force distributions and the vanishing of the system memory.In accordance with the well-established time-averaged dynamics of dense granular flows,as reviewed for instance in [41, 53], this saturation was interpreted as a clear signature ofthe transition from quasistatic to dense inertial regime.
4.7.3 Impact for a real-world application
The periodic immersed wall system studied in the present paper was motivated by theunderstanding of force fluctuations experienced by walls impacted by full-scale granularflows in nature, such as snow avalanches. Though the system is still a simplified model(planar and gravity-free in particular) of that geophysical problem, there are importantoutcomes regarding potential applications and perspectives.
First, the analysis of the mean dynamics of the system suggests that there exists arelatively wide range of flow regimes under which the mean force of a full-scale granularflow on a rigid wall of surface Sw should be controlled to first order by the confinementpressure (F/Sw ⌠P ). However, the ratio of the impact pressure to the confinementpressure is not constant. It is controlled by the inertial number IâM , increasing from avalue higher than 1 at low IâM up to a value around 3â 4 when IâM approaches unity. It isworthwhile to note that, in their simulations of a wall impacted by a free-surface granularflow down an incline (see [40]), Faug et al. found a ratio of the mean force on the wallto the pressure-dependent force of the incoming flow (associated to flow-depth) around4 at similar inertial numbers close to unity, namely below a value for which velocity-square inertial force would become dominant. This suggests a strong analogy betweenthe periodic immersed wall system studied here and the more complicated problem of agravity-driven free-surface flow overflowing a wall, and shows that the value of F/Sw toP (namely F/(Phd) in the present study) appears to be robust.
Second, it is interesting to stress that the force distributions on the entire wall heightwere found to be much more narrow at low IMâ, thus indicating that slow flows can exerta relatively steady thrust on the wall (mainly centered over time on the mean value), whilethe wall immersed in a faster flow will experience highly time-varying levels of force. Thisrobust result at the scale of the entire height of the wall put emphasis on two types
110Chapitre 4. Force moyenne et fluctuations sur un mur immergé dans un
écoulement granulaire cisaillé
of force fluctuations in the pressure-dependent force regime. Any structure impacted byavalanches should be designed against both types of force fluctuations as a given avalanchetrack can produce either fast or slow dense flows, depending on the quality of the snowinvolved.
The present study paves the path for short-term research that will analyse, with thehelp of a similar approach, the force fluctuations for more complicated systems consistingof a rigid wall impacted by a gravity-driven free-surface avalanche-flow down an incline(as studied in [20] for the mean force). Finally, long-term research will tackle the problemof deformable walls impacted by granular flows.
Acknowledgments This work has been partially supported by the LabEx Tec21 (In-vestissements dâAvenir : grant agreement No. ANR11-LABX-0030). Thierry Faug andMohamed Naaim are grateful to the financial support by the People Programme (Ma-rie Curie Actions) of the EU 7th FP under REA Grant Agreement No. 622899 (FP7-PEOPLE-2013-IOF, GRAINPACT).
Chapitre 5
Conclusion
111
112 Chapitre 5. Conclusion
5.1 Contexte et méthode
Le risque naturel que reprĂ©sentent les avalanches de neige fait lâobjet dâimportantes re-cherches depuis quelques dizaines dâannĂ©es. Afin de mieux dimensionner les structuresde gĂ©nie civil soumises Ă cet alĂ©a, la comprĂ©hension de lâinteraction complexe entre unmilieu granulaire et un obstacle est nĂ©cessaire. Ă lâheure actuelle, des modĂšles avancĂ©s per-mettent de prĂ©voir la force moyenne subie par un mur soumis Ă une avalanche. Cependant,les fluctuations de force de frĂ©quence et dâamplitude Ă©levĂ©es, systĂ©matiquement relevĂ©essur lâobstacle, restent peu documentĂ©es. Ce travail de thĂšse sâintĂ©resse au problĂšme desfluctuations de force : il Ă©tablit un cadre dâexpĂ©rimentations adaptĂ© Ă leur Ă©tude, et fournitdes Ă©lĂ©ments qui permettent de les caractĂ©riser en fonction des grandeurs macroscopiquesqui gouvernent lâĂ©coulement granulaire.
Les Ă©coulements granulaires secs ont Ă©tĂ© mis en Ćuvre numĂ©riquement par la mĂ©thodedes Ă©lĂ©ments discrets. Deux systĂšmes modĂšles ont Ă©tĂ© dĂ©veloppĂ©s dans le but de focaliserla zone dâĂ©tude en amont de lâobstacle, et de permettre le contrĂŽle des variables ma-croscopiques de lâĂ©coulement (vitesse de cisaillement, pression de confinement, taille dusystĂšme). Le premier a Ă©tĂ© lâobjet des chapitres 2 et 3 : les grains sont confinĂ©s par quatreparois, dont lâune impose un cisaillement Ă vitesse constante. Les signaux de force ont Ă©tĂ©mesurĂ©s sur la paroi faisant face au cisaillement. Le chapitre 4 sâest focalise sur le secondsystĂšme : le cisaillement est toujours imposĂ© par la diffĂ©rence de vitesse entre deux paroisopposĂ©es, mais la condition aux limites pĂ©riodique dans lâaxe du cisaillement permet auxgrains de surverser un unique mur immergĂ© sur lequel sont rĂ©alisĂ©es les mesures. Chacundes systĂšmes a Ă©tĂ© dans un premier temps analysĂ© par une approche moyennĂ©e dans letemps, puis les fluctuations ont Ă©tĂ© caractĂ©risĂ©es Ă partir des signaux de force instantanĂ©s.
5.2 SynthÚse des résultats
Un nombre consĂ©quent de mesures a Ă©tĂ© effectuĂ© lors de lâĂ©tude des systĂšmes modĂšles ;elles visent Ă caractĂ©riser la dynamique moyenne ou les fluctuations lors de lâinteractiondu milieu granulaire avec lâobstacle.
â Dynamique moyenne : lignes de courant, profils de force aux parois, force moyennesur lâobstacle, fraction volumique, tenseur des contraintes et taux de dĂ©formationlocaux, nombre inertiel.
â Fluctuations : autocorrĂ©lation et distribution des signaux de force sur lâobstacle,Ă trois Ă©chelles spatiales diffĂ©rentes.
Lâobjet de cette partie est de synthĂ©tiser les rĂ©sultats mentionnĂ©s prĂ©cĂ©demment dans lesarticles de cette thĂšse, en distinguant dâabord les deux systĂšmes. Une analyse comparativedes deux systĂšmes est prĂ©sentĂ©e par la suite, sous forme dâun article de confĂ©rence.
5.2. SynthÚse des résultats 113
5.2.1 Cavité granulaire entraßnée par cisaillement
Nombre inertiel IM
Le nombre inertiel macroscopique IM â U/HâP unit les conditions aux bords rĂ©gissant
lâĂ©coulement : vitesse de cisaillement U , pression de confinement P et hauteur du systĂšmeH. Il sâest avĂ©rĂ© fondamental dans la comprĂ©hension du systĂšme de cavitĂ©, en tant quevariable de contrĂŽle de la plupart des mesures. Il gouverne en particulier lâinfluence delâinertie des grains sur la force moyenne ressentie par lâobstacle, mais Ă©galement toutesles mesures caractĂ©risant les fluctuations. De plus, la variation du nombre inertiel macro-scopique a souvent rĂ©vĂ©lĂ© que les transitions de rĂ©gimes dâĂ©coulement ont des incidencesmarquĂ©es sur lâinteraction avalancheâobstacle.
Dynamique moyenne
Une Ă©tude poussĂ©e de la dynamique moyenne du systĂšme a Ă©tĂ© menĂ©e dans un pre-mier temps afin dâen comprendre les comportements vis-Ă -vis des conditions aux limites(pression de confinement, vitesse, taille).
LâĂ©tude des lignes de courant et des profils de forces aux parois pour la cavitĂ© ontrĂ©vĂ©lĂ© la complexitĂ© et lâhĂ©tĂ©rogĂ©nĂ©itĂ© spatiale de la cavitĂ©. Un vortex unique se formedans tout lâespace de la cavitĂ©, mettant en mouvement la grande majoritĂ© des grains. Laposition du centre de ce vortex sur lâaxe horizontal dĂ©pend du rĂ©gime dâĂ©coulement : ilest centrĂ© aux rĂ©gimes quasistatiques et se rapproche de lâobstacle avec lâaugmentationde IM . Sur le mur faisant face au cisaillement, la force moyenne est rĂ©partie de façonnon rĂ©guliĂšre : elle est quasiment nulle proche de la paroi infĂ©rieure, trĂšs Ă©levĂ©e dans lescouches supĂ©rieures, et Ă peu prĂšs homogĂšne dans les couches intermĂ©diaires.
La pression de confinement P et la longueur du systĂšme L gouvernent au premierordre la force moyenne sur lâobstacle F , mais une composante dĂ©pendante de la vitessede cisaillement doit Ă©galement ĂȘtre prise en compte pour une description totale. Une loide saturation permettant de prĂ©dire F en fonction de IM et de L a Ă©tĂ© Ă©tablie, suggĂ©rantque F porte la signature de la loi rhĂ©ologique locale ”(I) des Ă©coulements granulaires.Ce rĂ©sultat a Ă©tĂ© confirmĂ© par des mesures rhĂ©ologiques locales dans les Ă©coulements,rĂ©vĂ©lant que la loi rhĂ©ologique ”(I) est respectĂ©e quasiment partout dans le systĂšme. LesrĂ©gions oĂč cette loi nâest pas respectĂ©e correspondent principalement aux zones prochesdes murs verticaux et Ă©galement Ă une fine bande horizontale centrĂ©e verticalement auxrĂ©gimes lents. LâĂ©volution de la fraction volumique macroscopique moyenne Ίâ avec IM
suit la mĂȘme forme classique que les Ă©coulements Ă volume libre, confirmant le rĂŽle de larhĂ©ologie ”(I) dans la force subie par lâobstacle.
Fluctuations
Les fluctuations de force sur le mur faisant face au cisaillement ont été étudiées via
114 Chapitre 5. Conclusion
lâutilisation dâoutils statistiques Ă des Ă©chelles spatiales variables. LâautocorrĂ©lation dessignaux de forces Ă lâĂ©chelle mĂ©soscopique a mis en avant une propriĂ©tĂ© importante de lacavitĂ© : la frĂ©quence liĂ©e Ă la rugositĂ© de la paroi supĂ©rieure est directement observable,sauf dans les rĂ©gimes trĂšs inertiels. Ce phĂ©nomĂšne liĂ© au blocage des grains dans le coinsupĂ©rieur droit se transmet dans toute lâĂ©paisseur du systĂšme jusquâĂ la paroi infĂ©rieureaux rĂ©gimes intermĂ©diaires, alors quâaux rĂ©gimes lents elle nâest observable quâen surface.La mesure de lâautocorrĂ©lation a Ă©galement permis dâobserver le phĂ©nomĂšne de mĂ©moiretemporelle du systĂšme, existant pour les cisaillements Ă faible inertie mais disparaissantlors de la transition vers le rĂ©gime dense inertiel.
Lâinvestigation des distributions de forces sur lâobstacle est un Ă©lĂ©ment majeur de lacaractĂ©risation des fluctuations. Quelle que soit lâĂ©chelle spatiale considĂ©rĂ©e, le nombreinertiel IM sâest rĂ©vĂ©lĂ© ĂȘtre le paramĂštre de contrĂŽle de ces mesures. Les distributionsde forces Ă©voluent entre des formes Gaussiennes aux rĂ©gimes les plus lents et des formesexponentielles aux rĂ©gimes les plus rapides. La transition entre ces deux types de densitĂ©sde probabilitĂ©s nâest pas triviale, mais les distributions ont pu ĂȘtre approchĂ©es numĂ©ri-quement par des fonctions log-normales tronquĂ©es dont les paramĂštres sont des fonctionsmonotones de IM . En concordance avec la littĂ©rature, lâĂ©chelle spatiale de la mesure a Ă©ga-lement une influence sur les distributions. Il a donc Ă©tĂ© relevĂ© que les Ă©chelles plus larges(macroscopique, Ă lâĂ©chelle de lâobstacle entier) ont tendance Ă resserrer les distributions,tandis quâen diminuant lâĂ©chelle des mesures (jusquâĂ lâĂ©chelle microscopique des grains),les distributions ont tendance Ă sâĂ©largir.
5.2.2 Mur immergé dans un écoulement granulaire cisaillé
Nombre inertiel IM
Ce systĂšme dispose dâune grandeur gĂ©omĂ©trique supplĂ©mentaire par rapport Ă la cavitĂ© :la hauteur du mur h. De ce fait, le nombre inertiel IM a Ă©tĂ© complĂ©tĂ© par une dĂ©finitioncorrespondant au cisaillement des grains au droit de lâobstacle : IâM â U/(H â h)
âP .
La dĂ©finition de IâM considĂšre alors que tout le cisaillement au niveau de lâobstacle estconcentrĂ© dans la rĂ©gion qui le surplombe (H â h). Tandis que IM sâest rĂ©vĂ©lĂ© ĂȘtre lagrandeur de contrĂŽle des fluctuations, lâintroduction de IâM a Ă©tĂ© fondamentale dans lacomprĂ©hension de la dynamique moyenne du systĂšme.
Dynamique moyenne
LâĂ©tude de la dynamique moyenne de ce systĂšme a permis de rĂ©vĂ©ler sa rĂ©ponse Ă la variation des conditions aux limites dâĂ©coulement (pression de confinement, vitesse,longueur, hauteur de la paroi).
Les lignes de courant et les profils de vitesse verticale le long de lâaxe horizontalfournissent un premier aperçu de la phĂ©nomĂ©nologie moyenne du systĂšme Ă mur immergĂ©.
5.2. SynthÚse des résultats 115
Dans les rĂ©gimes lents Ă intermĂ©diaires, des conditions proches du cisaillement simple sontobservĂ©es dans la majeure partie du systĂšme. Lorsque lâinertie augmente, on constatecependant une recirculation locale dans la zone infĂ©rieure au point de rĂ©injection desgrains (partie gauche du systĂšme). Dans ce cas, une forte composante verticale de lavitesse perturbe la situation de cisaillement simple.
La hauteur de la paroi h et la pression de confinement P sont les variables qui entrenten jeu au premier ordre dans la force moyenne subie par lâobstacle F . Cependant, ladescription prĂ©cise de F nĂ©cessite dâintĂ©grer une composante dâinertie de lâĂ©coulementvia IâM . Une loi de saturation permet de prĂ©dire F/Phd en fonction de IâM , suggĂ©rant laprĂ©sence de la loi rhĂ©ologique locale ”(I) dans le systĂšme. Ce rĂ©sultat est confirmĂ© parles mesures locales de dĂ©formation et de contrainte ainsi que par le comportement de lafraction volumique Ίâ avec IâM . La loi ”(I) est respectĂ©e localement dans le systĂšme saufau droit de lâobstacle pour les cisaillements lents et le comportement classique de Ίâ desĂ©coulements Ă volume libre est vĂ©rifiĂ©.
Fluctuations
Les mĂȘmes mesures statistiques que pour la cavitĂ© ont Ă©tĂ© utilisĂ©es pour le mur immergĂ©.Le phĂ©nomĂšne de mĂ©moire temporelle du systĂšme a Ă©tĂ© observĂ© via lâautocorrĂ©lation designaux de forces Ă lâĂ©chelle mĂ©soscopique. Les cisaillements Ă rĂ©gimes faibles provoquentun effet de mĂ©moire, signifiant quâune valeur de force met un temps non nĂ©gligeable Ă ĂȘtre âoubliĂ©eâ par le systĂšme. En revanche, ce comportement disparaĂźt totalement aveclâaugmentation de lâinertie et le passage dans le rĂ©gime dâĂ©coulement dense inertiel.
Ă lâĂ©chelle macroscopique (intĂ©gration sur tout lâobstacle), le comportement des dis-tributions de force Ă©volue de façon univoque avec le nombre inertiel macroscopique IM .Les formes sont Gaussiennes aux rĂ©gimes lents et sâĂ©largissent de maniĂšre asymĂ©triquelorsque IM augmente. Une saturation de cette Ă©volution (superposition des distributions)est observĂ©e Ă partir dâune valeur de IM proche de celle Ă laquelle la mĂ©moire du systĂšmedisparaĂźt, traduisant la transition du rĂ©gime quasistatique vers le rĂ©gime dense inertiel. Lesdistributions ont pu ĂȘtre approchĂ©es par une loi empirique de forme log-normale tronquĂ©edont les paramĂštres sont fonction de IM . La rĂ©duction de la dimension spatiale dâintĂ©gra-tion des mesures produit un effet sur les distributions qui est en accord avec la littĂ©rature.Ainsi, les distributions Ă lâĂ©chelle mĂ©soscopique prĂ©sentent des formes plus larges et moinssymĂ©triques quâĂ lâĂ©chelle macroscopique, sâĂ©loignant de la forme Gaussienne quel que soitle rĂ©gime.
116 Chapitre 5. Conclusion
5.3 Analyse comparative
Lâarticle de confĂ©rence figurant ci-aprĂšs reprend les rĂ©sultats clĂ©s de cette thĂšse, en pro-posant une analyse comparative des deux systĂšmes vis-Ă -vis du large panel de mesureseffectuĂ©es.
5.3. Analyse comparative 117
Force fluctuations experienced by a boundary wall sub-jected to a granular flow in two distinct systemsKneib F., Faug T., Dufour F., Naaim M.
Conference paper accepted for Powder and Grains 2017.
Abstract The present paper describes a numerical study of force fluctuations experien-ced by a boundary wall subjected to a granular flow on two distinct systems, namely alid-driven cavity and an immersed wall system. Though the two systems exhibit differenttime-averaged dynamics, the force fluctuations experienced by the boundary wall showrobust features in terms of the shapes of the probability density distributions and auto-correlation functions, under a wide range of boundary confinement pressure and shearingvelocity imparted to the granular flow at the top of the system. This study identifies thekey link between the grain-wall force fluctuations and the ”(I)ârheology while movingfrom quasistatic to inertial regime.
5.3.1 Introduction
The problem of an assembly of grains in interaction with a boundary wall is relevant fora number of real-world applications involving granular materials, such as the transport ofparticle in food processing or mining industry, the impact of large-scale avalanche-flowswith civil engineering structures in geophysics, the design of technological earthmovingequipments used to work with the soils in agriculture, etc. Understanding the force ex-perienced by a boundary wall in interaction with a granular flow is crucial but little isknown about the fluctuating part of the granular force under slow to fast flow regimes. Thedistributions of grain-grain or grain-wall contact forces (noted f) in static or quasistaticsystems received a great attention, as reported in [4, 78] and a number of references the-rein. A salient feature in static granular media is a relatively robust exponential decreaseat large forces, while the distributions at small forces may show very different behaviours(absence or presence of a plateauâeven a peak, non-zero probability at f = 0), dependingon many parameters associated with the shear history of the system. By contrast, only afew studies were devoted to force fluctuations at stake in flows of granular materials (seefor instance [42]).
The present study investigates the distributions of grain-wall forces, as well as forcetime-series autocorrelation, when a granular flow interacts with a rigid boundary wall,under a wide range of macroscopic boundary shearing velocity U and confinement pressureP imparted to the granular bulk. The approach is based on the cross-comparison betweentwo distinct granular flow systems simulated by discrete element method (DEM). The two
118 Chapitre 5. Conclusion
systems and their time-averaged dynamics are briefly described in section 5.3.2. Section5.3.3 is devoted to the grain-wall force distributions on the entire height of the wall and onportions of wall. The analysis of force time-series autocorrelation is addressed in section5.3.4. Finally, a conclusion is given in section 5.3.5.
5.3.2 Two distinct granular systems studied
Flow-wall geometries and methods
This section gives a brief summary of the two systems studied, both involving a boundarywall that is subjected to a planar and gravity-free granular flow. The first system, namelythe granular lid-driven cavity, is a closed system for which the grains are forced to stayinside the cavity, thus producing a perpetual recirculation of the grains in the wholevolume. The second system, namely the immersed wall granular system, is an openedsystem which enables the grains to escape by passing over an immersed wall, while othergrains are trapped behind the wall-like obstacle. Both systems are simulated thanks toDEM, using the open-source YADE software [105].
The contact forces between the spherical particles of diameter d were modelled usinga damped linear spring law for the normal direction and a linear spring with a staticCoulomb threshold law for the tangential direction. Four microscopic parameters wererequired for the simulations presented here : the normal stiffness kn, the tangential stiffnesskt, the interparticle friction coefficient ” and the normal restitution coefficient e. Allsimulations were conducted in the limit of rigid grains, varying kn and kt accordingly,with respect to the macroscopic pressure imparted by the top wall to the system. Weconsidered ” = 0.5 and e = 0.5. Full information on how the values of those parameterswere implemented is given in our initial study of the lid-driven cavity system [58].
For both systems, we defined the two dimensionless following parameters : NU =
U/âgd and NP = P/(ΊÏP gH) where H is the height of the system (equal to 30d with
d = 1 mm in the simulations presented here), and ÏP was the particle density takenequal to 2500 kg mâ3 (corresponding to the density of glass beads generally used insmall-scale laboratory experiments). The systems lengths L were set to five times theheight of grains : L/H = 5. A polydispersity of ±15% was given to the grains in orderto avoid crystallization. Note that we considered typical pressures ΊÏP gH (where Ί wasa typical volume fraction taken equal to 0.6 for a dense three-dimensional packing) andtypical velocities
âgd in order to define NP and NU , though the systems simulated by
DEM were gravity-free in the present study. This choice was made for convenience iflaboratory experiments under gravity are developed in the future and compared to theDEM simulations presented in the paper. For both systems, NU and NP were typicallyvaried in the ranges [1; 20] and [0.01; 100], respectively.
Based on the boundary conditions imposed at the top in terms of shearing velocity U
5.3. Analyse comparative 119
Figure 5.1 â Geometry and boundary conditions for the granular lid-driven cavity simulated by
DEM
and confinement pressure P , a macroscopic inertial number can be defined as the ratiobetween the typical time tP = d
âÏ/P , associated with the top confinement pressure P ,
and the typical time tU = H/U , equal to the inverse of the macroscopic shear rate overthe height of the sample : IM = U
Hdâ
Ï/P .
The granular lid-driven cavity
The two-dimensional granular lid-driven cavity simulated by DEM consists of a planarassembly of spheres trapped into a box made of four walls, as sketched in Fig. 5.1. Theperpetual recirculation of the grains within the cavity (closed system) produced stream-lines showing the formation of a vortex in the whole volume of the cavity. The mean forceon the right sidewall (see Fig. 5.1) was found to fit the empirical following scaling :
F =
(r +
âr
1 + I0M/IM
)PLd, (5.1)
where r = 0.53, âr = 0.33 and I0M = 0.2 in this study (see detail in [58, 59]). Thevalues of r and âr depend on the mechanical properties of the grains, for instance theinterparticle friction, as reported in [59] where results from DEM simulations with ” = 0.27
are presented too.
The immersed wall granular system
The immersed wall granular system consists of a horizontal flow of spheres passing overa wall of height h = 20d, as sketched in Fig. 5.2. This system was simulated by DEMusing periodic conditions. The immersed wall-like obstacle led to the formation of a deadzone behind the wall, this dead zone co-existing with a sheared granular flow over it. Asa result, the mean force on the wall was found to obey the empirical following relation :
F =
(râ +
ârâ
1 + Iâ0M /IâM
)Phd, (5.2)
where râ = 1.4, ârâ = 1.9 and Iâ0M = 0.3 in the present study (see detail in [60]). Thevalues of râ and ârâ depend on the mechanical properties of grains. IâM is the macroscopic
120 Chapitre 5. Conclusion
Figure 5.2 â Geometry and boundary conditions for the immersed wall granular system simu-
lated by DEM. Periodic conditions were used, thus explaining the presence of a wall on the left
side (lighter-colored area), which is a duplicate of the wall (of interest in our study) on the right
side
inertial number defined with the difference between the height of the system (the heightof grains) and the immersed wall height : IâM = U
HâhdâÏ/P .
Distinct time-averaged dynamics
Because the two systems presented above are by construction different, they show twodifferent time-averaged dynamics. First, a standing vortex-like structure is formed insidethe cavity volume [58, 59], while longitudinal streamlines with a curvature in the vicinityof the wall take place in the immersed wall system [60]. Second, the mean force scalingsallowed us to identify different relevant spatial scales specific to each system. The relevantpressure force is PLd for the cavity (see Eq. (5.1)), while it is Phd for the immersed wallsystem (see Eq. (5.2)). The relevant spatial scale for the macroscopic inertial number isthe system height H for the cavity (see Eq. (5.1)), while it is the height difference H â h,namely the height of the granular layer overflowing the wall, for the second system (seeEq. (5.2)). Much more details about the specific features of the lid-driven cavity and theimmersed wall system can be found in [58, 59] and [60], respectively.
However, we show in the following how these two distinct systems exhibit very similarânot to say, for some aspects, identicalâkey features in terms of the force distributionsand the force time-series autocorrelations on the boundary wall, and we will identify arobust signature of the transition from quasistatic to dense inertial granular flow regime.
5.3.3 Grain-wall force distributions
At the scale of the entire wall
Figure 5.3 shows the distributions of total force on the boundary wall for the lid-drivencavity (top panel) and the immersed wall system (bottom panel) for different values ofIM . First, it is worthwhile to note that the distributions are entirely controlled by thevalue of IM , whatever the pair (U, P ) (see the curves for IM = 6.1 10â3 obtained with
5.3. Analyse comparative 121
0.0 0.5 1.0 1.5 2.0 2.5 3.0
F/F
0.0
0.5
1.0
1.5
2.0
Den
sity
LID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITY IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0061|NU =10.0|NP =100.0
IM â0.0061|NU =5.0|NP =25.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0609|NU =10.0|NP =1.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
IM â0.6086|NU =10.0|NP =0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0
F/F
0.0
0.5
1.0
1.5
2.0
Den
sity
IMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALL IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =10.0|NP =100.0
IM â0.0061|NU =5.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0609|NU =10.0|NP =1.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
IM â0.6086|NU =10.0|NP =0.01
Figure 5.3 â Distributions of total force experienced by the boundary wall for the lid-driven
cavity (top panel) and the immersed wall system (bottom panel), for different values of IM
three different pairs (U, P ) for the two systems). Second, it is found that they generallyevolves from an exponential-like form at high forces (much larger than the mean) withthe presence of a plateau/peak at small forces for high IM , toward a Gaussian-like format the lowest IM . We observe a difference between the cavity and the immersed wall atthe highest IM : the cavity exhibit purely exponential shapes without any peak, while thepeak at small forces is still present in the immersed wall system.
At the scale of a portion of wall
For both systems and a given IM , our simulations showed that the distribution of theforce Fi on each portion i of the boundary wall was independent of the vertical locationalong the boundary wall, excluding the extreme locations at the top and the foot of thewall (not shown here : details can be found in [59] for the cavity and [60] for the immersed
122 Chapitre 5. Conclusion
wall). As a result, we could average the distributions and extract robust mean values ofdistributions as functions of IM , as depicted in Fig. 5.4. The top panel shows the resultsfor the lid-driven cavity, while the bottom panel holds for the immersed wall system.
The force distributions shown in Fig. 5.4 collapse nicely into one single master curve,independent of IM , once IM becomes greater than 10â2, whatever the granular systemconsidered. This remarkable result, showing that there exists a threshold value of IM
which is independent of the system geometry, suggests that the distributions on portionsof wall are robust to identify the transition between quasistatic and dense inertial granularflow regimes.
5.3.4 Force time-series autocorrelation
The autocorrelation function C(t, t + ât) of force time-series on the wall was analysedfor the cavity and the immersed wall system. It was generally observed that C(t, t +ât) dropped quickly (exponentially) to zero over a time scale âtc and could fluctuatearound zero (see details in [59, 60]). Note that very similar C(t, t + ât) functions wereextracted from an experimental study on the force experienced by an intruder draggedinto a granular medium in an annular geometry [45].
Well-defined fluctuations associated with the transmission of the frequency U/d as-sociated with the roughness of the wall at the top shearing the granular sample couldsometimes be identified in the case of the cavity, as detailed in [59]. By contrast, thiseffect was never detected for the immersed wall. This observation illustrates the fact thatthe cavity can produce high confinement pressure on the right top corner (see [58] formuch more discussion).
We could not find any clear relation between the critical time âtc beyond which theforce time-series became weakly correlated, and the inertial number IM , considering theentire height of the boundary wall subjected to the granular flow. However, by consideringthe autocorrelation function for portions of wall (as done for the distributions in section5.3.3), we found that each âtc
i (the critical time corresponding to a vertical location i
along the boundary wall) had systematically a non-zero value for IM below 10â2, whileit was nil beyond IM = 10â2. This result being robust whatever the system considered,we defined the critical time ăâtcă averaged over all the locations i considered along theboundary wall. The latter is plotted in Fig. 5.5 as a function of IM for the two granularsystems. It is remarkable that both curves display very similar behaviours.
The autocorrelation function C(t, t +ât) can be seen as a measure of the memory ofthe system through time ât. High values of C indicate that a future state of the system(in terms of force applied to the wall) depends on the past state of this system, while itdoes not when C tends toward zero. As such, we can conclude from Fig. 5.5 that a futurestate of the system may strongly depend on the past state below IM â 10â2 while it does
5.3. Analyse comparative 123
0.0 0.5 1.0 1.5 2.0 2.5 3.0
<Fi/Fi >i
0.0
0.2
0.4
0.6
0.8
1.0
Den
sity
LID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITYLID-DRIVEN CAVITY IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0609|NU =10.0|NP =1.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
IM â0.6086|NU =10.0|NP =0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0
<Fi/Fi >i
0.0
0.2
0.4
0.6
0.8
1.0
Den
sity
IMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALLIMMERSED WALL IM â0.0012|NU =1.0|NP =25.0
IM â0.0061|NU =1.0|NP =1.0
IM â0.0122|NU =10.0|NP =25.0
IM â0.0609|NU =10.0|NP =1.0
IM â0.0913|NU =15.0|NP =1.0
IM â0.3043|NU =10.0|NP =0.04
IM â0.6086|NU =10.0|NP =0.01
Figure 5.4 â Distributions of the force experienced by a portion of the boundary wall for the
lid-driven cavity (top panel, H split into 10 portions) and the immersed wall system (bottom
panel, h split into 5 portions). The curves drawn in both graphs correspond to the mean value
obtained from all the vertical locations i along the wall, excluding the extreme locations at the
top and the foot of the boundary wall
124 Chapitre 5. Conclusion
10-3 10-2 10-1 100
IM
0.0
0.1
0.2
0.3
0.4
<ât c>
LID-DRIVEN CAVITYIMMERSED WALL
Figure 5.5 â Critical time ăâtcă beyond which the force time-series become weakly correlated,
as a function of IM . We report the mean over the âtci values for i â [1; 10] in the case of the
lid-driven cavity (cross symbols) and for i â [1; 5] in the case of the immersed wall system (circle
symbols)
not beyond this critical IM . The latter result being robust whatever the granular systemconsidered, this reveals another clear signature of the transition from the quasistaticregime (system with a memory) to the dense inertial regime (full loss of memory).
Note that the non-zero value of ăâtcă below IM â 10â2 is higher for the immersed wallthan for the cavity. This difference may be explained by the fact that the grains against theboundary wall are moving along the cavity wall (perpetual recirculation), while they arenot in the immersed wall system because they are trapped in the stagnant zone formedagainst the immersed wall. Therefore, memory effects may be more pronounced in theimmersed wall system.
5.3.5 Conclusion
The present paper paid a particular attention to the force fluctuations experienced by theboundary wall, considering a wide range of macroscopic boundary conditions in terms ofshearing velocity and confinement pressure imposed to each system. By comparing theresults between the two distinct systems, we could extract some key features which ap-peared to be independent of the system at stake. In particular, our study showed that thedistributions measured on portions of wall (of size around 3d) moved from an exponentialdecrease at large forces with a shoulder at small forces, which were independent of IMfor IM greater than 10â2, toward IM -dependent Gaussian-like distributions while IM wasdecreased. Moreover, we could detect a system memory for the lowest IM , which disap-peared while increasing IM beyond 10â2. Those features were obtained from two distinctgranular flow systems, characterized by a different time-averaged dynamics in terms of
5.3. Analyse comparative 125
streamlines and mean force scalings on the boundary wall. Those results then appear asa robust signature of the transition between quasistatic and dense inertial granular flowregimes, revealing and quantifying the link between the ”(I)ârheology proposed by [52]and the force fluctuations experienced by a boundary wall.
Acknowledgments This work has been partially supported by the LabEx Tec21 (In-vestissements dâAvenir : grant agreement No. ANR11-LABX-0030). Thierry Faug and Mo-hamed Naaim acknowledge the financial support by the People Programme (Marie CurieActions) of the EU 7th FP under REA Grant Agreement No. 622899 (FP7-PEOPLE-2013-IOF, GRAINPACT).
126 Chapitre 5. Conclusion
5.4 Discussions et perspectives
La comparaison des fluctuations de force au sein des deux configurations Ă©tudiĂ©es danscette thĂšse a mis en valeur certains rĂ©sultats communs, qui ne semblent donc pas dĂ©pendredu systĂšme Ă©tudiĂ©. Les comportements des distributions Ă lâĂ©chelle mĂ©soscopique (â 3
grains) sont similaires : elles sont de forme log-normales et dĂ©pendent de IM aux rĂ©gimeslents, alors quâelles sont de forme quasi-exponentielles et indĂ©pendantes de IM aux rĂ©gimesrapides. Du rĂ©gime quasistatique au rĂ©gime dense inertiel, les distributions Ă lâĂ©chellemacroscopique (tout lâobstacle) sont trĂšs proches entre les deux systĂšmes. Enfin, lâeffetmĂ©moire des systĂšmes est prĂ©sent au rĂ©gime quasistatique, sâeffaçant totalement lors de latransition vers le rĂ©gime dense inertiel. Lâensemble de ces similitudes dans des systĂšmesoĂč la dynamique moyenne est pourtant totalement diffĂ©rente, confirme lâimportance dela loi rhĂ©ologique locale ”(I) dans les fluctuations de force sur un obstacle soumis Ă unĂ©coulement granulaire (voir partie 1.1.2).
Un certain nombre de perspectives Ă©mergent alors, avec comme objectif dâappliquerles rĂ©sultats de cette thĂšse Ă des systĂšmes se rapprochant progressivement de la situationrĂ©elle dâun obstacle surversĂ© par un Ă©coulement granulaire. Une description non exhaustivede ces perspectives est prĂ©sentĂ©e ci-aprĂšs.
Longueur des systĂšmes
Parmi les paramĂštres Ă©tudiĂ©s dans ce travail, la rĂ©ponse des systĂšmes Ă la variationde la longueur L est particuliĂšrement intĂ©ressante. La motivation premiĂšre de lâutilisa-tion de la cavitĂ© (LDC) par rapport Ă un systĂšme gravitaire a Ă©tĂ© la maĂźtrise de la zonedâinfluence de lâobstacle sur lâĂ©coulement. Les rĂ©sultats montrent bien dans ce cas queL, bien quâintervenant par construction dans la force moyenne subie par lâobstacle, nâapas dâinfluence sur les fluctuations de forces. Lâintroduction du systĂšme pĂ©riodique Ă murimmergĂ© (IW) modifie ce comportement car L a un effet sur la force moyenne comme surles fluctuations : modification lĂ©gĂšre du comportement F/Phd = f(IâM) et des distribu-tions ; modification radicale de lâeffet mĂ©moire. DĂšs lors, L apparaĂźt comme une limitede ce systĂšme de laquelle il sera difficile de sâaffranchir, tel que le montrent les lignes decourant dans la partie 4.2.
Il serait alors intĂ©ressant dâĂ©tudier la rĂ©ponse dâun systĂšme qui ferait un pas de plusvers lâapplication finale, câest-Ă -dire oĂč lâĂ©coulement des grains est induit par la gravitĂ©. Onpourrait sâintĂ©resser dans un premier temps aux fluctuations dans le rĂ©gime permanent, enimaginant un dĂ©bit de grains constant (condition pĂ©riodique ou rĂ©servoir alimentĂ© conti-nuellement). Dans un second temps, le caractĂšre transitoire des Ă©coulements granulairesnaturels (pic de force suivi dâun plateau puis dâune rĂ©siduelle, voir partie 1.1.3) pourraitĂȘtre pris en compte en rĂ©alisant des lĂąchers de masses de grains en amont. Les rĂ©sultatsapportĂ©s par le prĂ©sent travail seraient alors un bon point de comparaison. Le problĂšme
5.4. Discussions et perspectives 127
10-3 10-2 10-1 100
IM
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
ÏF/ F
h=15d
h=20d
cavité
Figure 5.6 â Ratio de lâĂ©cart type des signaux de force ÏF aux forces moyennes F en fonction du
nombre inertiel macroscopique IM . Les résultats sont représentés pour la cavité ainsi que deux
hauteurs de murs immergés.
de la non-maĂźtrise des grandeurs macroscopiques se poserait (et notamment la consti-tution dâun nombre inertiel macroscopique IM), mais les outils amenĂ©s par cette thĂšsepermettent de relever des grandeurs locales (dont le nombre inertiel local), et pourraientdonc rĂ©pondre Ă cette problĂ©matique.
Rapport entre fluctuations et force moyenne
Dans la dĂ©marche de prise en compte des fluctuations pour de meilleurs dimensionne-ments des structures de gĂ©nie-civil, la mesure du rapport entre lâamplitude typique desfluctuations (lâĂ©cart type des signaux) et la valeur de la composante moyenne de la forceest un Ă©lĂ©ment essentiel. Ce coefficient de variation est tracĂ© dans la figure 5.6 pour toutela gamme de rĂ©gimes dâĂ©coulements Ă©tudiĂ©e dans ce travail.
Aux rĂ©gimes lents, lâamplitude des fluctuations reprĂ©sente 25% de la valeur moyenne dela force, et ce rapport augmente logarithmiquement avec le nombre inertiel. Aux rĂ©gimesrapides, lâamplitude des fluctuations prend alors une forte importance, jusquâĂ des rap-ports de lâordre de 80% pour les murs immergĂ©s et 150% pour la cavitĂ©. Dans les rĂ©gimesdâĂ©coulements dense inertiel et collisionnel, les fluctuations jouent donc un rĂŽle majeurdans la force sur lâobstacle, et sont par consĂ©quent un ingrĂ©dient essentiel au dimensionne-ment des structures. En particulier, alors quâune saturation du coefficient de variation estobservĂ©e pour les murs immergĂ©s aux forts IM , lâamplitude typique des fluctuations peutdĂ©passer la valeur moyenne du signal pour les situations trĂšs confinĂ©es localement (cas dela cavitĂ© avec recirculation). Les fluctuations reprĂ©sentent alors la majeure partie de la
128 Chapitre 5. Conclusion
composante du signal et leur prise en compte dans les dimensionnements est primordiale.Les modélisations réalisées dans cette thÚse et les coefficient de variation présentés
dans la figure 5.6 portent toutefois sur des écoulements plans. La généralisation à troisdimensions pour se rapprocher de la plupart des écoulements réels est discutée ci-aprÚs.
Dimensionnalité
Les Ă©coulements in-situ sont de toute Ă©vidence en trois dimensions spatiales, et la rĂ©a-lisation dâexpĂ©rimentations de laboratoire dans des conditions planes nâest pas aisĂ©e. Lepassage des systĂšmes numĂ©riques plans dĂ©crits dans ce travail Ă des systĂšmes pleinementtridimensionnels est donc une Ă©volution majeure Ă envisager pour permettre les pontsvers lâingĂ©nierie. Dans ce cas, et si les effets de bords ne sont pas dominants, on peutsâattendre Ă des rĂ©sultats moyens qualitativement semblables car ils sont lissĂ©s par na-ture. Les rĂ©sultats concernant les fluctuations sont toutefois susceptibles dâĂȘtre modifiĂ©squantitativement voire qualitativement pour deux raisons :
â le nombre de grains en contact avec lâobstacle changerait drastiquement avec uneinfluence sur les distributions ;
â les contraintes stĂ©riques inter-particules sont plus fortes en trois dimensions, aug-mentant le frottement effectif par phĂ©nomĂšne de blocage et modifiant donc la rhĂ©o-logie du milieu granulaire.
Il faudrait alors analyser lâinfluence du passage de 2 Ă 3 dimensions sur les rĂ©sultats, et enparticulier comment sont modifiĂ©es les relations posĂ©es par cette thĂšse entre les grandeursmacroscopiques moyennes et les fluctuations.
GĂ©omĂ©trie et dynamique de lâobstacle
Lâobjet de cette thĂšse concerne lâĂ©tude de systĂšmes oĂč lâobstacle est un mur dont lasurface efficace fait face Ă lâĂ©coulement. Dans les configurations Ă©tudiĂ©es, lâobstacle est unobjet adjacent au fond fixe et rugueux exerçant un frottement basal Ă lâĂ©coulement. Unezone morte de taille variable se forme alors naturellement Ă proximitĂ© du coin formĂ© parlâintersection du fond et de lâobstacle. Les risques naturels mettant en jeu des Ă©coulementsgranulaires concernent cependant des obstacles de toutes formes qui, en plus dâĂȘtre surver-sĂ©s, sont potentiellement contournĂ©s par lâĂ©coulement. LâĂ©tude dâobstacles de gĂ©omĂ©triesdiffĂ©rentes serait une Ă©volution intĂ©ressante de cette thĂšse, sâappliquant par exemple auxcas de pylĂŽnes ancrĂ©s dans le sol ou Ă des ouvrages de dĂ©flexion des Ă©coulements gra-nulaires. Le systĂšme de mur prĂ©sentĂ© ici pourrait alors ĂȘtre modifiĂ© en immergeant desobstacles de formes diverses qui ne seraient pas en contact avec le fond, laissant passerles grains de part et dâautre. La comparaison des rĂ©sultats avec ceux obtenus dans cettethĂšse pourrait mettre en relief le rĂŽle jouĂ© par le blocage strict des grains dans le coininfĂ©rieur par rapport Ă des Ă©coulements dĂ©viĂ©s par lâobstacle.
5.4. Discussions et perspectives 129
Dans ce travail, lâobstacle est considĂ©rĂ© comme un Ă©lĂ©ment fixe et parfaitement indĂ©for-mable avec lequel les grains interagissent. Cependant, le passage Ă des problĂšmes in-situimplique lâutilisation de structures aux propriĂ©tĂ©s mĂ©caniques qui leur sont propres. Ilserait alors utile par la suite de modĂ©liser lâobstacle en prenant en compte des com-portements rĂ©alistes (modules dâĂ©lasticitĂ©, endommagement, modes de vibration ...), etdâanalyser lâinfluence rĂ©ciproque de lâobstacle avec le milieu granulaire. Dans cette optique,lâanalyse de la rupture des structures en fonction des conditions dâĂ©coulement est un pointsensible particuliĂšrement intĂ©ressant Ă Ă©tudier. Dans un premier temps, des obstacles plusrĂ©alistes pourraient intĂ©grer les systĂšmes en rĂ©gime permanent couverts par cette thĂšsepour Ă©tudier lâeffet des fluctuations sur les vibrations mĂ©caniques naturelles de lâobstacle.Des systĂšmes gravitaires pourraient ĂȘtre envisagĂ©s dans un second temps afin de prendreen compte le caractĂšre transitoire des Ă©coulements et lâeffet successif des diffĂ©rentes phasesdes signaux de force (pic, plateau, rĂ©siduelle) sur la rĂ©sistance de lâobstacle.
130 Chapitre 5. Conclusion
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