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Granular Flows and Complex Flows With! and Basilisk ! ! granular flows and complex flows with! ! ! ! ! ! ! ! ! Pierre-Yves Lagrée,! and Basilisk ! ! ! Lydie Staron, Stéphane Popinet , ! ! Institut Jean le Rond ∂’Alembert, CNRS, Université Pierre & Marie Curie, 4 place Jussieu, Paris, France ! ! 27/10/14 • What is a granular media?! • size > 100µm! • grains of sand, small rocks, glass PHYSIQUE beads, animal feed pellet, PHYSIQUE S A V O I R S PHYSIQUE ACTUELS LES MILIEUX GRANULAIRES ENTRE FLUIDE ET SOLIDE BRUNO ANDREOTTI, YOËL FORTERRE ET OLIVIER POULIQUEN medicines, cereals, wheat, sugar, ENTRE FLUIDE ET SOLIDE LES MILIEUX GRANULAIRES Sable, riz, sucre, neige, ciment... Bien qu’omniprésents dans notre vie quotidienne, les milieux granulaires continuent de défier l’industriel, de fasciner le chercheur et d’intriguer l’amateur. Pourquoi le sable est-il tantôt assez solide pour former un tas ou soutenir le poids d’un immeuble, et coule-t-il tantôt comme un liquide, lors d’une avalanche ou dans un sablier ? Pourquoi est-il difficile de compacter ou de mélanger des grains ? Comment le vent sculpte-t-il les rides de sable sur la plage et les dunes LES MILIEUX dans le désert ? Longtemps l’apanage des ingénieurs et des géologues, l’étude des milieux granulaires constitue aujourd’hui un sujet de recherche actif à la frontière de nombreuses disciplines – physique, mécanique, sciences de l’’environnement, géophysique et sciences de l’ingénieur. Cet ouvrage s’attache à dresser l’état des connaissances sur les milieux granulaires et à présenter rice...! les avancées récentes du domaine. Issu de cours de Master et d’’école d’’ingénieur, il s’’adresse aux GRANULAIRES étudiants des trois cycles universitaires, aux chercheurs et aux ingénieurs, qui trouveront là une pré- sentation des propriétés fondamentales des milieux granulaires (interactions entre grains, comporte- ment solide, liquide et gazeux, couplage avec un fluide, applications au transport de sédiments et à ENTRE FLUIDE ET SOLIDE la formation de structures géologiques). La description des phénomènes mêle arguments qualitatifs et formels, permettant de pénétrer des domaines aussi variés que l’’élasticité, la plasticité, la physique statistique, la mécanique des fluides ou la géomorphologie. De nombreux encadrés permettent d’appro- fondir certains phénomènes et illustrent les propriétés singulières des milieux granulaires au travers de leurs manifestations les plus spectaculaires (chant des dunes, sables mouvants, avalanches de neige…) Bruno Andreotti, est professeur à l’université Paris VII et effectue ses recherches à l’ESPCI. Ses recherches portent sur l’hydrodynamique, le mouillage et la géomorphogénèse. Yoel Forterre, est chercheur CNRS au laboratoire IUSTI à Marseille. Il travaille sur le comportement des fluides complexes, des milieux granulaires et sur la biomécanique des plantes. Olivier Pouliquen, est chercheur CNRS au laboratoire IUSTI à Marseille. Ses travaux portent 50 % of the traded products sur les matériaux granulaires, les suspensions et fluides complexes. OLIVIER POULIQUEN BRUNO ANDREOTTI ANDREOTTI BRUNO • ET FORTERRE YOËL Série Physique et collection dirigée par Michèle LEDUC SAVOIRS ACTUELS CNRS ÉDITIONS www.cnrseditions.fr www.edpsciences.org BRUNO ANDREOTTI Création graphique : Béatrice Couëdel YOËL FORTERRE ET Ces ouvrages, écrits par des chercheurs, reflètent des OLIVIER POULIQUEN enseignements dispensés dans le cadre de la formation à la recherche. Ils s’adressent donc aux étudiants avancés, aux 00 € chercheurs désireux de perfectionner leurs connaissances ainsi ISBN EDP Sciences 978-2-7598-0097-1 qu’à tout lecteur passionné par la science contemporaine. ISBN CNRS ÉDITIONS 978-2-271-07089-0 CNRS ÉDITIONS Mileu granulaire.indd 1 11/01/11 13:08 PYL 15m 15m PYL PYL spoil tip - Australia PYL spoil tip (boney pile, gob pile, bing or pit heap), «terril» in french PYL PYL PYL pyl PYL Stromboli pyl 1km 500m http://www.pbase.com/image/63044602 2006 Gary Hebert avalanche : le «Frank Slide» 1907 Lofoten Norway 14 photo PYL Nasa 8km 40km http://books.google.fr/books?id=HY6Z5od4-E4C&pg=PA49&dq=granular +flow&hl=fr&ei=lamtTaa_NYyVOoToldcL&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFkQ6AEwCTgK#v=onepage&q&f=true http://www.cieletespace.fr/image-du-jour/5126_la-saison-des-avalanches-sur-mars pyl Non Newtonian flows stress tensor σ = pδ + ⌧ viscous stress tensor ij − ij ij 1 @uj @ui ⌧ =2⌘D strain rate Dij = ( + ) ij ij 2 @xi @xj strain rate second invariant D2 = DijDji p Newtonian fluid ⌘ constant generalized ⌘ function of D2 Newtonian fluid N 1 power law fluid ⌘(D2)=⌘0D2 − Non Newtonian flows strain rate second invariant ⌧ij =2⌘Dij D2 = DijDji p ⌧y yield stress ⌧y N 1 Herschel-Bulkley ⌘(D2)= + ⌘0D2 − 2D2 ⌧y Bingham ⌘(D2)= + ⌘0 2D2 ⌧y Drucker Prager ⌘(D2)= 2D2 Coulomb yield stress ⌧y = µP Non Newtonian flows Example of Bingham Collapse Gerris "fig7_pij.rgb"binary array=748x453 format="%uchar" flipy "fig7_pij.rgb"binary"fig7_pij.rgb"binary array=748x453 array=748x453 format="%uchar" format="%uchar" flipy flipy F. Dufour & G. Pijaudier-Cabot, , Int. J. Numer. Anal. Meth. Geomech. (2005)! Non Newtonian flows Example of Bingham Collapse1270 STARON et al. SLUMPING OF A BINGHAM PLASTIC FLUID SLUMPING1271 OF A BINGHAM PLASTIC FLUID 1271 Gerris FIG. 4. Evolution of the normalized position of the front r R0 =R0 (or run-out) and of the top of the column ð À Þ H0 h =R0 (or slump) as a function of the normalized time t t= H0=g for the system displayed in Fig. 2. ð À Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi SLUMPING OF A BINGHAM PLASTIC FLUID 1271 corresponding time evolution of the position of the front in the course of time r t R0 ð ÞÀ and of the slump H0 h t (normalized by R0) is displayed in Fig. 4. During the flow, the upper-right edge of theÀ columnð Þ is preserved (due to locally low compressional stress); in the case of very large a, it can be advected downstream. Closer inspection of the state of the spreading layer reveals the existence of areas of higher viscosities (Fig. 5): A “dead” corner is located at the basis of the initial column, and spreads sideways while the flow decelerates and stops; small patches of higher viscosity also appear at the surface of the spreading layer. The preserved edge forms one of them. The inner deformations are made visible in Fig. 6 by mean of passive tracers; we observe maximum stretching in the vicin- ity of the bottom and in the inner part of the column. As expected, the typical evolution described above is sensitive to both the rheological parameters and the initial aspect ratio. Larger values of the plastic viscosity g induce smaller run-outs but larger flow durations due to the increase of the time scale related to viscous deformation. Through a different mechanism, larger values of the yield stress induce smaller run-outs and smaller slumping times, the material being quickly frozen in a stress state below the yielding value. Finally, a dependence on the initial aspect ratio a, rather than on H or R alone, is observed, as previously stressed in Sader and Davidson 3=2 0 0 3=2 1=2 (2005). In particular, for a small initial aspect ratio, depending1=2 on the value of the yield FIG. 6. Snapshots of the slumping shown in Fig. 2 (a 5,FIG.sy=q 6.gRSnapshots0 0:33, and of theg=q slumpingg R0 shown0:86), in at Fig. 2 (a 5, sy=qgR0 0:33, and g=qg R0 0:86), at ¼ ¼ ¼ stress, no¼ flow may occur at all.¼ All these points are investigated in detail¼ in Sec. IV. t= H0=g 0, 0.85, 1.70, and in the final state, showing inner deformationst= H0=g 0, using 0.85, VOF 1.70, tracers. and in the final state, showing inner deformations using VOF tracers. ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi IV. SCALING LAWS FOR THE SLUMPING ANDIV. SCALING SPREADING LAWS FOR THE SLUMPING AND SPREADING In the analysis of slumping experiments, it is commonIn the practice analysis to of use slumping the initial experiments, height it is common practice to use the initial height of theStaron system (H0 )et as characteristic al J. Rheol length scale against2013of the which system all (H other0) as quantities characteristic are nor- length scale against which all other quantities are nor- malized [Pashias et al. (1996); Schowalter andmalized Christensen [Pashias (1998)et; al.Davidson(1996)1=2 3=2; etSchowalter al. and Christensen (1998); Davidson et al. FIG. 6. Snapshots of the slumping shown in Fig. 2 (a 5, s =qgR0 0:33, and g=qg R 0:86), at FIG. 5. Snapshot showing areas of maximum viscosity close to flow arrest. ¼ y ¼ 0 ¼ (2000)t= ;HPiau0=g (2005)0, 0.85,; 1.70,Roussel and in theand final Coussot state, showing (2005) inner].(2000) However, deformations; Piau the (2005) using initial VOF; Roussel height tracers. is and not Coussot the (2005)]. However, the initial height is not the only length¼ scale affecting the deformation of the column:only length As scale demonstrated affecting in theSader deformation and of the column: As demonstrated in Sader and pffiffiffiffiffiffiffiffiffiffiffi DavidsonIV. SCALING (2005), its LAWS initial FOR radius THER0 plays SLUMPING a majorDavidson AND role. The SPREADING (2005) initial, itsheight initial sets radius the valueR0 playsDownloaded a 03 major Jul 2013 to 131.111.184.26. role. Redistribution The subject initial to SOR license height or copyright; see http://www.journalofrheology.org/masthead sets the value of the initial compressional stress to be comparedof with the the initial typical compressional length scale stress related to tobe compared with the typical length scale related to In the analysis of slumping experiments, it is common practice to use the initial height the yield stress: L s =qg. To single out thesethe two yield aspects, stress: andLy promptedsy=qg. by To earlier single out these two aspects, and prompted by earlier y ¼ y ¼ worksof the on system slumping (H0) of as granular characteristic matter length [Lube scaleetworks al.against(2004) on which slumping; Lajeunesse all other of quantities granularet al.
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