! 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, SOLIDE ET FLUIDE ENTRE GRANULAIRES MILIEUX LES 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 OLIVIER BRUNOANDREOTTI • YOËLFORTERRE ET 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 = p + ⌧ 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. (2004) matter are nor-; [Lube et al. (2004); Lajeunesse et al. (2004); Balmforthmalized and [Pashias Kerswellet (2004)al. (1996); Zenit; Schowalter (2005); Staron andBalmforth and Christensen Hinch and (2005) Kerswell (1998)], we; (2004)Davidson will; useZenit theet (2005) al. ; Staron and Hinch (2005)], we will use the (2000); Piau (2005); Roussel and Coussot (2005)]. However, the initial height is not the initial radius R0 (rather than the initial height Hinitial0) to normalize radius R0 the(rather slumping than the and initial the height H0) to normalize the slumping and the only length scale affecting the deformation of the column: As demonstrated in Sader and spreading of the column. Hence, we search for scalingspreading laws of relating the column.R Hence,R0 =R0 weand search for scaling laws relating R R0 =R0 and Davidson (2005), its initial radius R plays1=2 a major3=2 role. The initial heightð À setsÞ the value 1=2 3=2 ð À Þ H0 H =R0 to sy sy=qgR0 and g 0g=qg R0 H,0 andH to= theR0 to aspectsy ratiosy=qagR. 0 and g g=qg R0 , and to the aspect ratio a. ð ofÀ theÞ initial compressional¼ stress¼ to be comparedð withÀ Þ the typical¼ length scale related¼ to the yield stress: Ly sy=qg. To single out these two aspects, and prompted by earlier works on slumping¼ of granular matter [Lube et al. (2004); Lajeunesse et al. (2004); Balmforth and Kerswell (2004); Zenit (2005); Staron and Hinch (2005)], we will use the A. A simple prediction based on equilibriumA. shape A simple prediction based on equilibrium shape initial radius R0 (rather than the initial height H0) to normalize the slumping and the Aspreading simple prediction of the column. for the Hence, final shape we search of the for slumping scalingA simple material laws prediction relating can be forR obtained theR final0 =R by0 shapeand of the slumping material can be obtained by 1=2 3=2 ð À Þ assumingH0 H that=R the0 to finals states = resultsqgR0 and fromg theg= equilibriumqg assumingR0 , between and that to the the the aspect final pressure state ratio induced resultsa. from by the equilibrium between the pressure induced by ð À Þ y ¼ y ¼ the variations of the deposit height and the yieldthe stress variations [Pashias ofet the al. deposit(1996); heightRoussel and the yield stress [Pashias et al. (1996); Roussel et al. (2005); Roussel and Coussot (2005)] et al. (2005); Roussel and Coussot (2005)]
@h r @h r A. A simple prediction basedqgh onr equilibriumð Þ s ; shape q(2)gh r ð Þ sy; (2) ð Þ @r ¼ y ð Þ @r ¼ A simple prediction for the final shape of the slumping material can be obtained by andassuming by supposing, that the moreover, final state that results the final from shape the equilibrium canand be by approximated supposing, between moreover,the by pressure a cone that (or induced trian- the final by shape can be approximated by a cone (or trian- glethe in 2D) variations of the deposit height and the yieldgle in stress 2D) [Pashias et al. (1996); Roussel et al. (2005); Roussel and Coussot (2005)] H H h r R r : h(3)r R r : (3) ð Þ ’ R ð@hÀr Þ ð Þ ’ R ð À Þ qgh r ð Þ s ; (2) ð Þ @r ¼ y Integrating Eq. (2) between 0 and R gives immediatelyIntegrating Eq. (2) between 0 and R gives immediately and by supposing, moreover, that the final shape can be approximated by a cone (or trian- gle in 2D)
H h r R r : (3) Downloaded 03 Jul 2013 to 131.111.184.26. Redistribution subjectð toÞ SOR’ R licenseDownloadedð Àor copyright;Þ 03 Jul see2013 http://www.journalofrheology.org/masthead to 131.111.184.26. Redistribution subject to SOR license or copyright; see http://www.journalofrheology.org/masthead Integrating Eq. (2) between 0 and R gives immediately
Downloaded 03 Jul 2013 to 131.111.184.26. Redistribution subject to SOR license or copyright; see http://www.journalofrheology.org/masthead Non Newtonian flows Example of Bingham Collapse
Basilisk granulars are fluids and solids
E. Lajeunesse A. Mangeney-Castelnau and J. P. Vilotte PoF 2005 • Looking for a continuum description! • Lot of recent experiments in simple configurations: shear/ inclined plane, ! with model material (glass beads, sand…)! • Simulations with Contact Dynamics! 342 (disks,The European Ph polygona,ysical Journal E spheres) in the different configurations? Are there underlying com- mon physical phenomena controlling flow properties in the different geometries? As a result, we expect to identify simple and basic features that could help in developing future model for dense granular flows. g Let us emphasise that this collective work is not a re- view. New results are presented and the paper does not pretend to be exhaustive. First, the paper focus only on steady uniform flows of slightly polydispersed grains, leav- (a) (b) (c) ing aside very important questions such as avalanche trig- gering, intermittent flows or segregation. Second, since the data presented here come from the research group GDR MiDi and collaborators, many important contributions are not included. We refer to them in the references. However, the huge activity in the domain makes the exercise diffi- g g cult. We take refuge behind this excuse for all the contri- butions that have been omitted. GDR MiDi EPJ E 04 (d) (e) (f) 2 Six different configurations Fig. 1. The six configurations of granular flows: (a) plane Dense granular flows are mainly studied in six different shear, (b) annular shear, (c) vertical-chute flows, (d) inclined configurations (Fig. 1), where a simple shear is achieved plane, (e) heap flow, (f) rotating drum. and rheological properties can be measured. These geome- tries are divided in two families: confined and free surface flows. The confined flows are the plane shear geometry (Fig. 1a) where a shear is applied due to the motion of of gravity, this homogeneous state is not achieved in exist- one wall, the annular shear (Fig. 1b) where the material ing experiments [15,16] but is obtained in discrete parti- confined in between two cylinders is sheared by the ro- cle simulations. Most of the results found in the literature tation of the inner cylinder and the vertical-chute flow are obtained imposing the wall velocity and measuring the configuration (Fig. 1c) where material flows due to the shear stress [17–21]. Some are carried out controlling the gravity in between two vertical rough walls. Free surface shear force applied to the moving wall in order to study flows are flow of granular material on a rough inclined the flow thresholds [22]. plane (Fig. 1d), flow at the surface of a pile (Fig. 1e) In the following, we present results of two-dimensional and flow in a rotating drum (Fig. 1f). The driving force discrete particle simulations where Vw is imposed and the is in these last three cases the gravity. In the following, number of grains (size d and mass m) within the cell is we consider successively the six configurations. The data fixed (periodic boundary conditions are used along the comes from different experiments and numerical simula- shear direction). The data are summarised in Table 1. In tions briefly described in a table at the beginning of each one case the volume —the cell width L— and thereby the section. We report for each of them the flowing threshold, density ρ —or the volume fraction Φ— are controlled and the kinematic properties (velocity V (y), volume fraction the pressure P is measured, while in the other case the 2 Φ(y) and velocity fluctuation δV (y) profiles) and the rhe- pressure is controlled and the density is measured. Once ological behaviour, before discussing the influence of the the inter-particle contact laws are fixed, the simulations various experimental or numerical parameters. Both the depend on two parameters: the wall velocity Vw and the notations and the dimensionless quantities naturally used normal stress P or the density ρ. This define a single di- to present the results are given in Appendix A. mensionless number describing the relative importance of inertia and confining stresses, 3 Plane shear flow γ˙ d I = w . (1) 3.1 Set-up P/ρ In the aim of studying flow rheology, the plane shear ! (Fig. 2a) is conceptually the simplest geometry one natu- Both simulations are performed in the limit of rigid grains, rally thinks of. The flow is obtained between two parallel so that the macroscopic timescale L/Vw is much larger rough walls, a distance L apart and moving at the rela- than the microscopic timescales i.e. the elastic and the tive velocity Vw. In the following, we note γ˙ w = Vw/L the dissipative ones. The inter-particle friction coefficient µp mean shear rate. In this configuration, the stress distribu- is null when not specified. The roughness of the walls is tion is uniform inside the sheared layer. However, because made of glued grains similar to the flowing grains. • Looking for a continuum description! • Lot of recent experiments in simple configurations: shear/ inclined plane, ! with model material (glass beads, sand…)! • Simulations with Contact Dynamics! (disks, polygona, spheres)
• Defining a «viscosity»! • Implement it in the Navier Stokes solver Gerris! • Test on exact «Bagnold» avalanche solution! • Test on granular collapse and hourglass The µ(I)-rheology
u(y) N T
T = µN constitutive law? Coulomb dry friction Coulomb friction law
= µP The µ(I)-rheology
u(y) d u falling time I = y P/ displacement time = µ(I)P Coulomb friction law
u = µ(I)P local equilibrium y
d u µ( y )P P/ construction of a viscosity = u y
P. Jop, Y. Forterre, O. Pouliquen, (2006) "A rheology for dense granular flows", Nature 441, pp. 727-730 The µ(I)-rheology Dense granular flows ui,j + uj,i Dij = D2 =0.6 DijDij 0.45 2 u(x, y) v(x, y) (a) 0.55 (b) 0.40 p 0.5 0.35 P falling time τ/PI = dp2D2/ ( p /⇢). u 0.30 | | 0.45 u 0.25 displacement time p h h 0.20 0.4 0.35
=0.00µ(I)0.10P 0.20 Coulomb0.30 friction0 0.05law 0.1 0.15 0.2 0.25 I I Stat J.
µ(Ι) (c) kinetic regime
µ2
µ2 µ1 .Mec µ(I)=µ1 + I0/I +1 µµs µ1 0.32 (µ2 µ1) 0.23 I0 0.3 1
quasi-static h. P. Jop, Y. Forterre, O. Pouliquen, (2006) "A rheology for dense granular flows", Nature 441, Ipp. 727-730
Figure 1. (a) Ratio of shear to normal stress τ/P as a function of the P07020 (2006) dimensionless shear rate I = ud/h P/ρ in simulations of plane shear. (b) τ/P as a function of I = ud/h√gh estimated at the base for flow down inclined planes. ! (c) Sketch of the dependence of the friction coefficient µ with dimensionless shear rate I =˙γd/ P/ρ. ! stays trapped is t t ,wecancompute the time averaged volume fraction φ: mean − micro t φ +(t t )φ φ = micro min mean − micro max . (4) tmean It follows that the volume fraction varies linearly with the dimensionless shear rate I = tmicro/tmean: φ = φ (φ φ )I. (5) max − max − min Typical values are φmax =0.6andφmin =0.5. Equations (3)and(5)representconstitutive equations that can be applied to predict different flow configurations. In the next section we discuss the predictions made with this approach and compare them with experimental observations.
3. Different flow configurations
3.1. Plane shear The first important test of the rheology concerns simple plane shear without gravity. The plane shear configuration is shown in figure 2(a). A granular layer of thickness h
doi:10.1088/1742-5468/2006/07/P07020 4 The µ(I)-rheology
ui,j + uj,i D = D2 = DijDij ij 2 u(x, y) v(x, y) p falling time I = dp2D2/ ( p /⇢). | | displacement time p = µ(I)P Coulomb friction law
construction of a viscosity based on the D2 invariant and redefinition of I
µ(I) ⌘ = p construction of a viscosity p2D ✓ 2 ◆
P. Jop, Y. Forterre, O. Pouliquen, (2006) "A rheology for dense granular flows", Nature 441, pp. 727-730
1.1. LE MODÈLE y [m] les coordonnées spatiales et t [s] le temps, voir la figure 1.1). Ce système s’écrit sous la forme ∂ h + ∂ (hu)+∂ (hv)=P I t x y − ⎧ 2 2 ⎪ ∂t (hu)+∂x hu + gh /2 + ∂y (huv)=gh(S0x Sf x) , (1.1) ⎪ − ⎨⎪ % &2 2 ∂t (hv)+∂x (huv)+∂y hv + gh /2 = gh(S0y Sf y) ⎪ − ⎪ ⎩⎪ avec S = ∂ z(%x, y) et S =& ∂ z(x, y), 0x − x 0y − y où 2 – g =9.81 m/s est la constante de gravité, – P (t, x, y) [m/s] l’intensité de la pluie, – I(t, x, y) [m/s] le taux d’infiltration de l’eau dans le sol, – S⃗ = S ,S R2 le terme de frottement qui dépend de la loi de frottement f f x f y ∈ choisie' (voir le chapitre( 3), – z(x, y) [m] la topographie. Enfin, l’opposé de S0x (respectivement de S0y) est un nombre sans dimension qui représente la variation de la topographie selon x (respectivement selon y). Il est plus communément appelé pente selon x (respectivement selon y). Saint-Venant Savage Hutter Gerris @h @(Q) @Q @ 5Q2 g Q tel-00531377, version 3 - 1 Feb 2011 + =0 + ( + (h2)) = ghµ(I) @t @x @t @x 4h 2 Q | | Fig. 1.2: Adhémar Jean-Claude Barré de Saint-Venant (extrait de Debauve [1893, p.432]).
Le système de Saint-Venant a été introduit dans un cadre unidimensionnel par l’in- génieur des Ponts et Chaussées Adhémar Jean-Claude Barré de Saint-Venant (figure 1.2) dans un Compte Rendu à l’Académie des Sciences (voir de Saint Venant [1871]). Pour décrire l’écoulement de l’eau dans un canal dont la section est rectangulaire et le fond plat et horizontal, il propose le système suivant
∂h ∂hu + =0 ∂t ∂x ∂hu ∂ gh2 (1.2) + hu2 + =0 ∂t ∂x 2 ) * 20 Gerris is a free finite volume code by Stéphane Popinet! one part of the code is a Shallow Water solver
n 2 n+1 n+1 Q⇤ Q @ Q g 2 Q Q⇤ Q + ( + (h )) = 0 = gh⇤µ(I⇤) t @x h 2 t Q | ⇤| Audusse et al.
finite volume Rieman Solver + well balanced 1.1. LE MODÈLE y [m] les coordonnées spatiales et t [s] le temps, voir la figure 1.1). Ce système s’écrit sous la forme ∂ h + ∂ (hu)+∂ (hv)=P I t x y − ⎧ 2 2 ⎪ ∂t (hu)+∂x hu + gh /2 + ∂y (huv)=gh(S0x Sf x) , (1.1) ⎪ − ⎨⎪ % &2 2 ∂t (hv)+∂x (huv)+∂y hv + gh /2 = gh(S0y Sf y) ⎪ − ⎪ ⎩⎪ avec S = ∂ z(%x, y) et S =& ∂ z(x, y), 0x − x 0y − y où 2 – g =9.81 m/s est la constante de gravité, – P (t, x, y) [m/s] l’intensité de la pluie, – I(t, x, y) [m/s] le taux d’infiltration de l’eau dans le sol, – S⃗ = S ,S R2 le terme de frottement qui dépend de la loi de frottement f f x f y ∈ choisie' (voir le chapitre( 3), – z(x, y) [m] la topographie. Enfin, l’opposé de S0x (respectivement de S0y) est un nombre sans dimension qui représente la variation de la topographie selon x (respectivement selon y). Il est plus communément appelé pente selon x (respectivement selon y).
Saint-Venant Savage Hutter @h @(Q) @Q @ 5Q2 g Q tel-00531377, version 3 - 1 Feb 2011 + =0 + ( + (h2)) = ghµ(I) @t @x @t @x 4h 2 Q | | Fig. 1.2: Adhémar Jean-Claude Barré de Saint-Venant (extrait de Debauve [1893, p.432]). Q Le système de Saint-Venant a été introduit dans un cadre unidimensionnel par l’in- I = d génieur des Ponts et Chaussées Adhémar Jean-Claude Barré de Saint-Venant (figure 1.2) 2 dans un Compte Rendu à l’Académie des Sciences (voir de Saint Venant [1871]). Pour h P/ décrire l’écoulement de l’eau dans un canal dont la section est rectangulaire et le fond plat et horizontal, il propose le système suivant
∂h ∂hu + =0 ∂t ∂x ∂hu ∂ gh2 (1.2) + hu2 + =0 ∂t ∂x 2 ) * 20
µ(I)=µs µ mu = 0.45 ! µ(I)=µs + I0 mu = (0.4 + 0.26/(0.4/In +1)) ! I +1 mu = (0.4 + 0.26*exp(-0.136/In))! /I µ(I)=µs + µe 1.1. LE MODÈLE y [m] les coordonnées spatiales et t [s] le temps, voir la figure 1.1). Ce système s’écrit sous la forme ∂ h + ∂ (hu)+∂ (hv)=P I t x y − ⎧ 2 2 ⎪ ∂t (hu)+∂x hu + gh /2 + ∂y (huv)=gh(S0x Sf x) , (1.1) ⎪ − ⎨⎪ % &2 2 ∂t (hv)+∂x (huv)+∂y hv + gh /2 = gh(S0y Sf y) ⎪ − ⎪ ⎩⎪ avec S = ∂ z(%x, y) et S =& ∂ z(x, y), 0x − x 0y − y où 2 – g =9.81 m/s est la constante de gravité, – P (t, x, y) [m/s] l’intensité de la pluie, – I(t, x, y) [m/s] le taux d’infiltration de l’eau dans le sol, – S⃗ = S ,S R2 le terme de frottement qui dépend de la loi de frottement f f x f y ∈ choisie' (voir le chapitre( 3), – z(x, y) [m] la topographie. Enfin, l’opposé de S0x (respectivement de S0y) est un nombre sans dimension qui représente la variation de la topographie selon x (respectivement selon y). Il est plus communément appelé pente selon x (respectivement selon y).
Saint-Venant Savage Hutter @h @(Q) @Q @ 5Q2 g Q tel-00531377, version 3 - 1 Feb 2011 + =0 + ( + (h2)) = ghµ(I) @t @x @t @x 4h 2 Q | | Fig. 1.2: Adhémar Jean-Claude Barré de Saint-Venant (extrait de Debauve [1893, p.432]). Q Le système de Saint-Venant a été introduit dans un cadre unidimensionnel par l’in- I = d génieur des Ponts et Chaussées Adhémar Jean-Claude Barré de Saint-Venant (figure 1.2) h2 P/ dans un Compte Rendu à l’Académie des Sciences (voir de Saint Venant [1871]).Integral Pour over the layer of grains décrire l’écoulement de l’eau dans un canal dont la section est rectangulaire et le fond plat et horizontal, il propose le système suivant NS µ(I) t=0 ∂h ∂hu + =0 ∂t ∂x NS µ(I) t=2 ∂hu ∂ gh2 (1.2) + hu2 + =0 NS µ(I) t=4 ∂t ∂x 2 1 ) * SVSH µ(I) t=0 20 SVSH µ(I) t=2 SVSH µ(I) t=4
0.8
0.6 h(x,t)
0.4
0.2
0 0 1 2 3 4 5 x L. Satron, J. Hinch (DAMTP), A. Mangeney (IPGP), E. Larieu (IMFT) Shallow Water RIL NPRESS IN ARTICLE
MODEL + 13 18 e 1 (2006) xx Software & Modelling Environmental / al. et Pirulli M.
i.1.Smlaeu ogtdnleogto n aea otato a and (a) contraction lateral and elongation longitudinal Simultaneous 19. Fig. ogtdnlcnrcinadltrleogto (b). elongation lateral and contraction longitudinal
scon- is ) K of values 4 with anisotropy , 1 Table ( 2b hypothesis au ndwsoedrcini h aeta in than same the is direction downslope in value K the sidered, ae2,adapoiaeytesm sdi aeo isotropy, of case in used same the approximately and 2a, case u ncossoedrcini spsil ohave to possible is it direction cross-slope in but .9 hscnmdf the modify can This 1.49. pass K K or 0.32 act K K ¼ chute. the ¼ along mass the ¼ of ¼ width .0strepae a econsidered be can phases three s 1.00 t At ¼ v v u v v v u v Saint-Venant Savage Hutter 34 0 > v v u v ; 0 < 0 > ; > Chute Gerris Þ ð y þ x y v x v y v x v
v v u v v v u v 0 < v v u v ; 0 > 0 < ; > zone Transition y þ x y v x v y v x v
35 Þ ð v v u v 36 0 > v v u v ; 0 > 0 > plane Horizontal Þ ð y þ x y v x v .1s ncossoedi- cross-slope in s, 0.51 t of case in as chute, the Along eto hr r ohatv n asv el u elements but cells passive ¼ and active both are there rection r npeaec.T osdrin consider To prevalence. in are x in active and y in passive
Gray (from coefficient pressure earth cross-slope the of value The 17. Fig. ). 1999 al., et
i.1.Smlaeu ogtdnladltrleogto a n longitudinal and (a) elongation lateral and longitudinal Simultaneous 18. Fig. i.2.Fakslide. Frank 20. Fig. (b). contraction lateral and Saint-Venant Savage Hutter WIELAND, J. M. N. T. GRAY AND K. HUTTER 1999 84 M. Wieland, J. M. N.Gerris T. GrayMarina and Pirulli, K. Marie-Odile Hutter Bristeau Anne Mangeney, Claudio Scavia 2006
4 t = 3.1 4 t = 9.1 y 0 0 –4 –4
0 10 20 30 40 0 10 20 30 40
4 t = 4.1 4 t = 11.1 y 0 0 –4 –4
0 10 20 30 40 0 10 20 30 40
4 t = 5.1 4 t =13.2 y 0 0 –4 –4
0 10 20 30 40 0 10 20 30 40
4 t = 6.0 4 t =15.2 y 0 0 –4 –4
0 10 20 30 40 0 10 20 30 40
4 t = 7.0 4 t =17.2 y 0 0 –4 –4
0 10 20 30 40 0 10 20 30 40
4 t = 8.0 4 t =18.12 y 0 0
–4 –4 (exp.) 0 10 20 30 40 0 10 20 30 40 x x Figure 5. A comparison between the dimensionless actual avalanche (solid) boundary in experiment V05 (vestolen) and the computed (dashed) edge is plotted at a sequence of time steps in projected curvilinear coordinates (x, y). The vertical dashed lines at x = 17.5 and x = 21.5 indicate the position of the transition zone, with the 40 inclined channel to the left and the horizontal run-out plane to the right. In the bottom right panel the thickness distribution of the experimental avalanche is illustrated using 0.1 unit contours. experiment is shown in figure 2. This evolution is typical of all three experiments described in this paper. The granular material is released from the cap on the inclined section of the chute and rapidly spreads out in the downhill direction, so that when the avalanche front reaches the run-out plane the tail has barely moved from its initial position. As the avalanche flows through the transition zone the lateral confinement ceases and the granular material spreads out laterally. This produces a very strong nose and tail structure, in which the nose is spreading and the tail is being channelized by the topography. A single-image photogrammetric method has been used to extract the position of the avalanche boundary from a series of these photographs. A comparison of the Shallow Water Full 2D
Basilisk
▪ Basiliscus basiliscus is the latin name of the extraordinary Jesus Christ lizard, famous for its ability to run on the surface of water, a characteristic it shares with another well-known water-walker Gerris lacustris. Shallow water front Pouliquen 99 experiment
Phys. Fluids, Vol. 11, No. 7, July 1999 Shape of granular fronts down rough inclined planes 1957
FIG. 2. ~a! Picture of the front illuminated by the laser sheet for material 4, u521°, h`59.5 mm. ~b! Forces on an elementary material slice. FIG. 1. ~a! Front profiles measured 50 cm from the outlet ~circles! and once it has propagated 1.5 m down the slope ~filled circles!. ~b! Position of the front with time showing the constant propagation velocity. System of beads valid for high inclination which indeed corresponds to their 1, u524.5°, h`56.7 mm. experimental conditions. However, for moderate inclination as in our problem, the constant friction coefficient approxi- mation is no longer valid as it does not predict the existence averaged equations for flow down inclined planes starting of steady uniform flow. In order to describe the shape of the from the 3D conservation equations can be found in Ref. 13. front in this regime we thus have introduced for m(h,u) in The last term in Eq. ~2! is usually simplified as follows. Eq. ~3! the empirical rheology given by Eq. ~1!. Using the shallowness assumption, the vertical normal pres- The third term in Eq. ~3! represents the spreading force sure is given by syy5rgy. Assuming that the two normal related to the gradient of thickness. The coefficient of pro- stresses are proportional, i.e., sxx5ksyy , the integral in Eq. portionality k between the two normal stresses is unfortu- ~2! can be computed and gives, after simplification, the fol- nately unknown for the dense granular flow of interest here. lowing equation for the thickness: In a very dilute regime corresponding to the kinetic regime, the pressure is isotropic as in fluids and k is equal to 1. dh Takahashi12 and Patton et al.1 have used this assumption. On tan~u!2m~h,u !2k 50. ~3! ` dx the other hand, in the quasi static regime for very slow de- formations, k can be related to the internal friction of the The first term in this equation represents the driven grav- material through a Mohr-Coulomb plasticity theory as shown ity force. The second term is the friction force exerted at the by Savage, Hutter and co-workers.13,14 However, in the in- rough bed. This term is crucial as it contains the information termediate regime k is unknown. In this paper the two dif- about the rheological behavior of granular flows. Savage and ferent values of k are tested when comparing the theoretical Hutter13 in their model have simply chosen for m(h,u)a predictions with the experiments. We will successively use 2 2 constant friction coefficient. This assumption seems to be k51, and k5 (11sin u1)/(12sin u1) corresponding to the
FIG. 3. Front profiles: comparison between experi- ments for three values of h` ~symbols! and theory 2 @solid line: k51; dashed line: k5 (11sin u1)/(1 2 2sin u1# for system of beads 2 and 4 ~see Ref. 9 for the characteristics of the beads!.
Downloaded 24 Jun 2010 to 134.157.34.22. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp Shallow water front Pouliquen 99 experiment
Edwards Gray 2014
Adepth-averagedµ(I)-rheology for shallow granular free-surface flows 32
1.0
3 0.8 10 2⇥
0.6 h h 1 0.4
0.2 0 10 1 0.0 ⇠
100 80 60 40 20 0 20 ⇠
Figure 3. Comparison between the non-dimensional front thickness h for the new depth-averaged viscous model (dashed line) and Forterre’s (2006) depth-averaged viscous model (solid line) in the moving coordinate ⇠ and for parameters ⇣1 =20.9 , ⇣ =29 and ⇣2 =32.76 .Thenewmodel has exactly the same solution as the inviscid case (Pouliquen1999b;Gray&Ancey2009)with awelldefinedfrontandagrainfreeregionfor⇠ > 0. Conversely Forterre’s (2006) model does not allow grain free regions, as the inset diagram shows, and the shape of the front is sensitively dependent on the precise way in which h 0as⇠ .Inthecaseillustratedherethe ! ! 1 3 exponential solution ✏ exp(r⇠), for ⇠ > 0withr = 1and✏ =10 ,ismatchedtoanumerical solution for Fr0 =1.02 and RF =113.18, in the region ⇠ < 0, at ⇠ =0.
Conversely, the front problem for Forterre’s (2006) depth-averaged rheology is a↵ected by the viscous term. Even though mass balance still implies that if h =0 at the front thenu ¯ = uF everywhere, the viscous term involves the gradient of hu¯,whichisnotconstant.Assumingthatfarupstreamthereissteady-uniform
flow, thenu ¯ is equal to the steady-uniform velocity,u ¯0,everywhere.Itfollowsthat for shape factor =1andflatbasaltopography,thescalings(6.8)implythe non-dimensional depth-averaged momentum balance (5.5) in the moving frame is dh 1 1 Fr2 d 1 dh h = h(tan ⇣ tan ⇣ ) + 0 , (6.10) d⇠ 2 1 1+ 1+ h3/2 R d⇠ h d⇠ ✓ ◆ F ✓ ◆ where the equivalent of the Reynolds number for this rheologyis
2 h0 RF = . (6.11) ⌫F Dynamic refinement using quadtrees
- Gerris is a finite volume code by Stéphane Popinet! one part of the code is a Navier Stokes solver - automatic mesh adaptation - Volume Of Fluid method for two phase flows - free on sourceforge 6 P.-Y. Lagr´ee,L. Staron and S. Popinet 6 P.-Y. Lagr´ee,L. Staron and S. Popinet u(H+), we obtain the velocity profile. u(H+), we obtain the velocity profile. In practice, we solve two ordinary di⇥erential equations (using a shooting method with In practice, we solve two ordinary di⇥erential equations (using a shooting method with Runge–KuttaRunge–Kutta di di⇥⇥erentiation),erentiation), and and we we determine determine using using Newton Newton iterations iterations the the value value of of ⇤0⇤ whichwhich allows allows the the velocity velocity profiles in in 0 0 3.3. Implementing Implementing the the viscosity in in the the Gerris Gerris flow flow solver solver 3.1.3.1. TheThe Gerris Gerris flow flow solver solver GerrisGerris is is an an open-source open-source solver solver for for the the solution solution of of incompressible incompressible fluid fluid motion motion using using the the finite-volumefinite-volume approach approach (Popinet (Popinet 2003, 2003, 2009). 2009). Gerris Gerris uses uses the the Volume-of-Fluid Volume-of-Fluid (VOF) (VOF) methodmethod to to describe describe variable variable density two-phase two-phase flows. flows. In In this this method method the the Navier–Stokes Navier–Stokes equations are written as equations are written as u =0, · u =0, ⇥ ⇤u + u u =· p + (2 D), ⇥ ⇤⇤ut + u· u = p + · (2 D), ⇤t ⇤·c + (c u )=0, · ⇤⇤tc ⇥ · ⇤t + ⇥ (cu)=0, ⇥ = c⇥1 +· (1 c)⇥2, ⇥ = c⇥1 + (1 c)⇥2, = c 1 + (1 c) 2, = c 1 + (1 c) 2, where the volume fraction c(x, y, t) enables the tracking of the position of the interface. whereD is the the deformation volume fraction tensorc (x,u y,+ t) enablesuT )/2. Gerris the tracking uses a second-order of the position staggered-in-time of the interface. T Ddiscretisationis the deformation combined tensor with ( a time-splittingu+ Projectionu )/2. projectionGerris uses Method method. a second-order This gives staggered-in-time the following discretisationtime-stepping combined scheme with a time-splitting projection method. This gives the following time-stepping scheme cn+ 1 cn 1 2 2 + (c u )=0, c tc n n n+ 1 n 1 · 2 2 u un t + (cnun)=0, ⇥n+ 1 ⇥ t + un+ 1 un+ 1 ·= ( n+ 1 D ) pn 1 , (3.1) 2 2 · 2 · 2 ⇥ 2 u un ⇥n+ 1 ⇤ ⇥ t + un+ 1 utn+ 1⌅ = ( n+ 1 D ) pn 1 , (3.1) 2 un+1 = u 2 · ( 2 pn+ 1 · pn 21 ),⇥ 2 (3.2) ⇥ n+ 1 2 2 ⇤ 2t ⌅ un+1 = u ( pn+ 1 pn 1 ), (3.2) ⇥ nu+n1+1 =0. 2 2 (3.3) · 2 Combining equations (3.2) and (3.3) ofu then+1 above=0. set results in the following Poisson(3.3) · Combiningequation multigrid equations (3.2) solver and (3.3) for of Laplacien the above set of results pressure in the following Poisson equation t t p 1 = u + p 1 . (3.4) n+ 2 n 2 · ⇥n+ 1 · ⇥ ⇥n+ 1 ⇧ t2 ⌃ ⇧ 2t ⌃ pn+ 1 = u + pn 1 . (3.4) The momentum equation · ⇥ (3.1)1 can be2 reorganised · ⇥ as ⇥ 1 2 ⇧ n+ 2 ⌃ ⇧ n+ 2 ⌃ ⇥n+ 1 u The momentum2 equation (3.1) can be reorganisedn as u⇥ ( n+ 1 D )=⇥n+ 1 un+ 1 un+ 1 pn 1 , (3.5) t · 2 ⇥ 2 t 2 · 2 2 ⇥ 1 nimplicit+ 2 for u* ⌥ un where the velocityu advection( 1 termD )=u ⇥ 1 1 u 1 isu estimated1 u by1 meansp of1 the, Bell-(3.5) ⇥ n+ n+n2 + n+ 2 n+ n+ n n+ 1 t 1 · 2 ⇥ · 2 t u 2 · 2 2 1 Colella-Glaz2 second-order unsplit upwind scheme (Popinetn 2003; Bell et al. 1989). Note T u ( n+ 1 u )= n+⌥ 1 un+ 1 un+ 1 pn 1 + un n+ 1 . where the t velocity 2 advection · 2 term un+ 1 2un + 1t is estimated2 · by2 means of 2 the2 Bell- 2 2 · 2 Colella-Glaz second-order unsplit upwind scheme (Popinet 2003; Bell et al. 1989). Note VOF reconstruction cn+ 1 cn 1 2 2 + (c u )=0 t · n n implementation in Gerris flow solver? µ2 µ1 µ(I)=µ1 + I0/I +1 ui,j + uj,i D = D D Dij = 2 ij ij 2 constructionp of a viscosity based on the D2 invariant and redefinition of I µ(I) ⌘ = min(⌘max, max p, 0 ) I = dp2D / ( p /⇢). p2D 2 ✓ 2 ◆ | | - the «min» limits viscosity to a large value! p - always flow, even slow Boundary Conditions: no slip and P=0 at the interface implementation in Gerris flow solver? µ2 µ1 µ(I)=µ1 + I0/I +1 ui,j + uj,i D = D D Dij = 2 ij ij 2 constructionp of a viscosity based on the D2 invariant and redefinition of I µ(I) ⌘ = min(⌘max, max p, 0 ) I = dp2D / ( p /⇢). p2D 2 ✓ 2 ◆ | | p u u =0, + u u = p + (2 D)+ g, · t · · Boundary Conditions: no slip and P=0 at the interface implementation in Gerris flow solver? µ2 µ1 µ(I)=µ1 + I0/I +1 ui,j + uj,i D = D D Dij = 2 ij ij 2 constructionp of a viscosity based on the D2 invariant and redefinition of I µ(I) ⌘ = min(⌘max, max p, 0 ) I = dp2D / ( p /⇢). p2D 2 ✓ 2 ◆ | | p u u =0, + u u = p + (2 D)+ g, · t · · @c + (cu)=0, ⇢ = c⇢ + (1 c)⇢ , ⌘ = c⌘ + (1 c)⌘ @t r · 1 2 1 2 The granular fluid is covered by a passive light fluid (it allows for a zero pressure boundary condition at the surface, bypassing an up to now difficulty which was to impose this condition on a unknown moving boundary). Boundary Conditions: no slip and P=0 at the top no velocity p=0 small density fluid small viscosity y g u(y) @c + (cu)=0, ⇢ = c⇢ + (1 c)⇢ , ⌘ = c⌘ + (1 c)⌘ @t r · 1 2 1 2 The granular fluid is covered by a passive light fluid (it allows for a zero pressure boundary condition at the surface, bypassing an up to now difficulty which was to impose this condition on a unknown moving boundary). Boundary Conditions: no slip and P=0 at the top Test of the code: «Bagnold» avalanche kind of Nusselt film solution! “Half Poiseuille” Contact Dynamic simulation Lydie Staron Test of the code: «Bagnold» avalanche H grains y g u(y) µ(I) = tan 3.5 gerris u(– y)– Bagnold 3 µ(I) gerris tan(α) D2*sqrt(2)– du(– y)/d– y– bag 2.5 – – y) p gerris 3 – – 3/2 – p( 2 H y p(y) u = I↵ gd cos ↵ 1 1 , 2 3 d3 H – y and r ✓ ◆ – y)/d ⇣ ⌘ – u( 1.5 – y), d – u( 1 y v =0,p= gH 1 cos . 0.5 H 0 0 0.2 0.4 0.6 0.8 1 y– Test of the code: «Bagnold» avalanche slip velocity p=0 H grains y g u(y) µ(I) = tan gerris 14 Bagnold 12 10 2 H3 y 3/2 u = I↵ gd cos ↵ 3 1 1 , 8 3 r d H – u(1) ✓ ⇣ ⌘ ◆ 6 4 y v =0,p= gH 1 cos . 2 H 0 0.3 0.35 0.4 0.45 0.5 0.55 α The sand pit problem: quickly remove the bucket of sand http://www.mylot.com/w/photokeywords/pail.aspx Granular Column Collapse E. Lajeunesse A. Mangeney-Castelnau and J. P. Vilotte PoF 204 The sand pit problem: quickly remove the bucket of sand http://www.mylot.com/w/photokeywords/pail.aspx Granular Column Collapse aspect ratio a = H0/R0 = H0/L0 H0 t =0 L0 H t = L The sand pit problem: quickly remove the bucket of sand Collapse of columns a=0.37 Contact Dynamic simulation Lydie Staron Collapse of columns a=0.90 Contact Dynamic simulation Lydie Staron Collapse of columns simulation Gerris µ(I) page 8 j’ai refaitCollapse les of figures columns of de aspect run-out ratio 0.5 avec set key bottom et d´ecall´eta courbe comparison of Discrete Simulation Contact un poil vers le basMethod `achaque and Navier Stokes fois, gerris, 6.26 shape entre at time 0 et 0.6 pour mieux voir 0, 1, 2, 3, 4 and position of the front of the avalanche as function of time (time measured with and space with aH ) Comparer `a H0/g 0 2.2 1 DCM NS µ(I) t=0 NS µ(I) t=1 NS µ(I) t=2 2 NS µ(I) t=3 0.8 NS µ(I) t=4 1.8 0.6 1.6 x(t) h(x,t) 0.4 1.4 0.2 1.2 NS µ(I) DCM 1 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 t x Figure 3: a =0.5 4.5 1 DCM NS µ(I) t=0 NS µ(I) t=1 4 NS µ(I) t=2 NS µ(I) t=3 0.8 NS µ(I) t=4 3.5 3 0.6 2.5 x(t) h(x,t) 0.4 2 1.5 0.2 1 NS µ(I) DCM 0.5 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 t x Figure 4: a =1.42 14 0.6 DCM NS µ(I) t=0 NS µ(I) t=1 NS (I) t=2 12 µ 0.5 NS µ(I) t=3 NS µ(I) t=4 10 0.4 8 0.3 x(t) h(x,t) 6 0.2 4 0.1 2 NS µ(I) DCM 0 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 t x Figure 5: a =6.26 8 page 8 j’ai refait les figures de run-out avec set key bottom et d´ecall´eta courbe un poil vers le bas `achaque fois, 6.26 entre 0 et 0.6 pour mieux voir Comparer `a 2.2 1 DCM NS µ(I) t=0 NS µ(I) t=1 NS µ(I) t=2 2 NS µ(I) t=3 0.8 NS µ(I) t=4 1.8 0.6 1.6 x(t) Collapse of columns simulationh(x,t) Gerris µ(I) 0.4 1.4 0.2 1.2 NS µ(I) DCM 1 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 t x Collapse of columns of aspect ratio 1.42 comparison of Discrete Simulation Contact Method and Navier Stokes Figuregerris, shape at 3: time a =0.5 0, 1, 2, 3, 4 and position of the front of the avalanche as function of time (time measured with H 0 /g and space with aH 0 ) 4.5 1 DCM NS µ(I) t=0 NS µ(I) t=1 4 NS µ(I) t=2 NS µ(I) t=3 0.8 NS µ(I) t=4 3.5 3 0.6 2.5 x(t) h(x,t) 0.4 2 1.5 0.2 1 NS µ(I) DCM 0.5 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 t x Figure 4: a =1.42 14 0.6 DCM NS µ(I) t=0 NS µ(I) t=1 NS (I) t=2 12 µ 0.5 NS µ(I) t=3 NS µ(I) t=4 10 0.4 8 0.3 x(t) h(x,t) 6 0.2 4 0.1 2 NS µ(I) DCM 0 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 t x Figure 5: a =6.26 8 page 8 j’ai refait les figures de run-out avec set key bottom et d´ecall´eta courbe un poil vers le bas `achaque fois, 6.26 entre 0 et 0.6 pour mieux voir Comparer `a 2.2 1 DCM NS µ(I) t=0 NS µ(I) t=1 NS µ(I) t=2 2 NS µ(I) t=3 0.8 NS µ(I) t=4 1.8 0.6 1.6 x(t) h(x,t) 0.4 1.4 0.2 1.2 NS µ(I) DCM 1 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 t x Figure 3: a =0.5 4.5 1 DCM NS µ(I) t=0 NS µ(I) t=1 4 NS µ(I) t=2 NS µ(I) t=3 0.8 NS µ(I) t=4 3.5 3 0.6 2.5 x(t) Collapse of columns simulationh(x,t) Gerris µ(I) 0.4 2 1.5 0.2 1 NS µ(I) DCM 0.5 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 t x Collapse of columns of aspect ratio 6.26 comparison of Discrete Simulation Contact Method and Navier StokesFigure gerris, shape 4:at timea =1.42 0, 1, 2, 3, 4 and position of the front of the avalanche as function of time (time measured with H 0 /g and space with aH 0 ) 14 0.6 DCM NS µ(I) t=0 NS µ(I) t=1 NS (I) t=2 12 µ 0.5 NS µ(I) t=3 NS µ(I) t=4 10 0.4 8 0.3 x(t) h(x,t) 6 0.2 4 0.1 2 NS µ(I) DCM 0 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 t x Figure 5: a =6.26 8 Collapse of columns simulation Gerris µ(I) a =0.5 DCM vs Gerris µ(I) Collapse of columns simulation Gerris µ(I) a =1.42 DCM vs Gerris µ(I) Collapse of columns simulation Gerris µ(I) a =6.6 DCM vs Gerris µ(I) Basilisk Full 2D ▪ Basiliscus basiliscus is the latin name of the extraordinary Jesus Christ lizard, famous for its ability to run on the surface of water, a characteristic it shares with another well-known water-walker Gerris lacustris. Collapse of columns simulation Gerris µ(I) DCM vs Gerris µ(I) Collapse of columns simulation Gerris µ(I) DCM vs Gerris µ(I) Collapse of columns simulation Gerris µ(I) DCM vs Gerris µ(I) Collapse of columns simulation Gerris µ(I) DCM vs Gerris µ(I) Collapse of columns simulation Gerris µ(I) at the tip, a=6.6 t=1.33 2 2.66 DCM vs Gerris µ(I) These figures can be reproduced using lydie/0.9/tas.gfs and lydie/9.1/tas.gfs.We should also do this better using Lydie’s results directly. These simulations were repeated for a ranging from 0.25 to 60. In order to estimate the influence of numerical integration errors, the spatial resolution was also varied from 32/29 to 32/ 212.Figure7illustratestheevolutionofthenormalisedfinaldeposit extent as a function of aspect ratio a.Well-definedpowerlawdependenciesareobservedwithexponents of 1 and 2/3 respectively. The transition between the two regimes occursfora ≈ 7.Thisisalargeraspect ratio than that observed in experiments or discrete-grain simulations (a ≈ 2). Recovering a tran- sition for smaller a Collapsewould require either a larger of prefactor columns for the linear regimeorasmallerpref-simulation µ(I) actor for the power-law regime. For example, while the prefactor of 3.5 for the power-law regime Gerris is close to that of Staron and Hinch (2005) (3.25), the prefactor for the linear regime is only 1.85 compared to 2.5 for Staron and Hinch. This may suggest that themobilityofthetipofcolumns is under-estimated by the continuum model which could be explained by the limitations dis- cussed previously. Note also the good convergence of the results with spatial resolution. 64 Normalised final deposit 32 extent as a function of aspect 16 ratio a. ! 8 0 )/R 0 -R 4 Well-defined power law inf (R 2 dependencies with exponents 1 1.85*x of 1 and 2/3 respectively. 3.5*x**(2./3.) 9 levels 0.5 10 levels 11 levels 12 levels 0.25 0.5 1 2 4 8 16 32 64 2 a Figure 7. Normalised final deposit extent as a function of aspect ratio.Thedifferent sets of points cor- respond to the spatial resolutions given in the legend. The points which are obvious outliers correspond to simulations which were still running when I generated the figure. This figure can be reproduced using the scaling.plot gnuplot script. We recover the experimental scaling 1 [Lajeunesse et al. 04] and [Staron et al. 05]. Figure 8 gives the evolution of the maximum thickness of the final deposit as a function of the initial aspect ratio. Linear dependence is observed for a ! 0.65, a power law with an expo-Differences between the values of the 0 nent of ≈/L 0.35 for 0.65 FIG. 4. Sequence of scaled profiles h͑x,t͒/Li of three different granular heaps of same initial aspect ratio a =3.2 but obtained with different masses or bead sizes. ͑a͒ t=0, ͑b͒ t =0.5c, ͑c͒ t=c, ͑d͒ t=2c, ͑e͒ t =3c, and ͑f͒ final deposit. The plain line profiles correspond to M =650 g, Li =5.3 cm, and d =1.15 mm. The plain line with circles profiles correspond to M =650 g, L =5.3 cm, and d=3 mm. 103302-5 Granular slumping on a horizontal surface Phys. Fluids 17,103302͑2005͒ i The dotted line profiles correspond to M =162.5 g, Li =2.6 cm, and d =1.15 mm. The different curves are barely distinguishable. 103302-6 Lajeunesse, Monnier, and Homsy Phys. Fluids 17,103302͑2005͒ FIG. 4. Sequence of scaled profiles h͑x,t͒/Li of three different granular heaps of same initial aspect ratio a when a increases. Second, pressure gradients, which scale as =3.2 but obtained with different 0ga, are likely to be small compared to friction forces at masses or bead sizes. ͑a͒ t=0, ͑b͒ t small a but become important at large a. The change of =0.5c, ͑c͒ t=c, ͑d͒ t=2c, ͑e͒ t =3c, and ͑f͒ final deposit. The plain power-law exponents observed for aϷ3 might therefore be line profiles correspond to M interpreted as the transition between small a flows dominated =650 g, Li =5.3 cm, and d =1.15 mm. The plain line with by friction and large a flows where vertical acceleration and circles profiles correspond to M pressure gradient effects become predominant. Note also that =650 g, Li =5.3 cm, and d=3 mm. the mass conservation expressions are different in the rect- The dotted line profiles correspond angular and the axisymmetric geometries, which may ac- size, the granular material internal friction angle, or the sub- to MB.=162.5 Deposit g, Li =2.6 morphology cm, and d =1.15 mm. The different curves are count for the different power-law exponents for large a. strate roughness exert an influence on the spreading dynam- barely distinguishable. The evolution of the deposit shape with a was quantita-The deposit height exhibits two different regimes char- ics. tively investigated by measuring the scaled deposit heightacterized by changes of power-law exponents depending on The results in Fig. 4, together with dimensional analysis, the range of a see Fig. 6 b . For aՇ0.7, all the data fall on H /L and the scaled runout length ⌬L/L =͑L −L ͒/L , ͓ ͑ ͔͒ f i i f thei samei line independent of the flow geometry: H /L a. strongly suggest that c is the characteristic time scale in our f i Ϸ where Hf and Lf are respectively the deposit height and This is of course a trivial consequence of the fact that Hf experiments. To verify this, we compared the flow dynamics length. The results are plotted in Fig. 6. In order to compare =Hi for the truncated cone deposits observed in this range for different values of a by tracking the position L͑t͒ of the the scaling laws observed in the rectangular channel toof thesea. front ͑or granular pile foot͒ in all experimental runs. The observed in axisymmetric geometries, we have added theFor aտ0.7, two different behaviors are observed de- results are illustrated in Fig. 5, where we display the normal- data set for axisymmetric collapses from Lajeunesse etpending al.10 on the flow geometry. In the axisymmetric geom- ized distance traveled by the pile foot L−L /L as a func- etry, the scaled deposit height roughly saturates at a value of ͑ i͒ i In both geometries and despite rather complex flow dynam- 10 tion of the scaled time t/ for three different values of a. the order of 0.74. In the rectangular channel, it increases as c ics, the scaled deposit height and runout collapse ona quite1/3. Interestingly this latter result is recovered in two recent Interestingly, all runs exhibit the same time evolution regard- simple power-law curves summarized below: 2D numerical investigations of the collapse of a granular size, the granularless material of internal the value friction of angle,a. After or the a sub- transientB. Deposit acceleration morphology phase column of disks using contact dynamics.16,17 The similarity strate roughness exert an influence on the spreading dynam- • In the axisymmetric geometry: lasting approximately 0.8c, the foot of theThe heap evolution moves of the at deposit a shape with a was quantita- between the experiments and the numerical simulations ͑per- ics. tively investigated by measuring the scaled deposit height formed without a wall͒ strongly suggests that the differences The results innearly Fig. 4, together constant with spreading dimensional velocity analysis, V for about 2c. Most of Hf /Li and the scaled runout length ⌬L/Li =͑Lf −Li͒/Li, between the evolution of Hf /Li in the axisymmetric geom- strongly suggest thatthe totalc is the distance characteristic traveled time scale by in the our foot of the heap is covered H a a Շ 0.74, where Hf and Lf are respectively thef deposit height and etry and that in the rectangular channel are not an experimen- experiments. To verifyduring this, this we compared time interval. the flow Finally, dynamics thelength. flow The front results decelerates are plotted in Fig. 6. In= order to compare Li ͭ0.74 a տ 0.74, ͮ tal artifact due to the friction at the wall of the rectangular for different values of a by tracking the position L͑t͒ of the the scaling laws observed in the rectangular channel to these and comes to rest in a time on the order of 0.6c. The total channel but have their origin in the geometry itself. front ͑or granular pile foot͒ in all experimentalE Lajeunesse runs. The J. B. observedMonnier and in axisymmetric G. M. Homsy geometries,PoF 05 we have added the duration of the flow is therefore of the order of 3 for all No model has yet provided a fully satisfactory explana- results are illustrated in Fig. 5, where we display the normal- data set for axisymmetricc collapses from Lajeunesse et al.10 tion of the slumping dynamics that has revealed the physical ized103302-7 distance Granular traveledvalues slumping by the of on a pilea horizontal. The foot surface͑ sameL−Li͒/ behaviorLi as a func- is observedIn both geometries in the semiaxi- Phys. and Fluidsdespite17,103302 rather͑2005 complex͒ flow dynam-Շ FIG. 5.⌬ ScaledL distancea traveleda 3, by the pile foot ͓L͑t͒−Li͔/Li as a function tion of the scaled time t/c for three different values of a. ics, the scaled deposit height and runoutϰ collapse on quite origin of the different power-law exponents reported above. symmetric setup, which is consistent with the observations of of t/c. ͑a͒ a=0.6,1/2M =470 g, Li =102 mm. ͑b͒ a=2.4, M =560 g, Li Interestingly, all runs exhibit the9 same time evolution regard- simple power-law curves summarizedL below:i ͭa a տ 3.ͮ This is what motivates the investigation of the internal flow =56 mm. ͑c͒ a=16.7, M =170 g, Li =10 mm. less of the valueLube of a. Afteret al. a, transientwho reported acceleration a total phase flow duration on the order structure reported in the next section. • In the axisymmetric geometry: lasting approximatelyof a 0.8 fewc, the multiples foot of the of heapc for moves axisymmetric at a collapses. • In the rectangular channel: nearly constant spreading velocity V for about 2c. Most of the total distance traveled by the foot of the heap is covered a a Շ 0.74, IV. INTERNAL FLOW STRUCTURE Hf Hf a a Շ 0.7, during this time interval.Downloaded Finally, 02 the Jul flow 2010 front to decelerates 195.83.220.187. Redistribution= subject to AIPϰ license or copyright; see http://pof.aip.org/pof/copyright.jsp Li ͭ0.74 a տ 0.74, ͮ 1/3 Li ͭa a տ 0.7, ͮ We used two different tools in order to probe the internal and comes to rest in a time on the order of 0.6c. The total structure of the slumping granular mass. First, the shape and duration of the flow is therefore of the order of 3c for all ⌬L a a Շ 3, evolution of the flowing layer were investigated by calculat- values of a. The same behavior is observed in the semiaxi- ⌬L a a Շ 3, ϰ ϰ 2/3 ing the intensity difference between two consecutive images. symmetric setup, which is consistent with the observations of 1/2 Li ͭa a տ 3.ͮ 9 Li ͭa a տ 3.ͮ Lube et al., who reported a total flow duration on the order The result is then thresholded so as to distinguish between of a few multiples of c for axisymmetric collapses. • In the rectangular channel: Note that the power-law exponent of the runout observed in static regions, which appear as black pixels, and the flowing our rectangular channel is identical to the one reported by layer, where the motion of the beads appears as white pixels. 11 Downloaded 02 Jul 2010 to 195.83.220.187. Redistribution subject to AIP licenseFIG. 6. or Scaled copyright; runout ⌬L see/Li ͑ ahttp://pof.aip.org/pof/copyright.jsp͒ andBalmforth scaled deposit and Kerswell for a narrow channel. In practice, one needs to evaluate the noise caused by light- height Hf /Li ͑b͒ as functions of a. Circles and triangles correspond to experiments performed in the 2DLet channel us first comment on the behavior of the runout dis- ing fluctuations or intrinsic vibrations of camera and appara- working respectively with glass beads of diameter d =1.15 mm or d=3 mm. Crosses correspondplayed to the data in Fig. 6͑a͒. Two different regimes are observed de- tus. The intensity difference between two consecutive images set of axisymmetric collapses from103302-4 Lajeunessepending Lajeunesse,et al. on the Monnier, range and Homsy of a. For small a, namely aՇ3, ⌬L Phys./L Fluids 17of,103302 the͑2005 pile͒ at rest shows that lighting fluctuations are negli- ͑Ref. 10͒. i increases linearly with a for both flow geometries. In other gible compared to the signal generated by the flowing beads, words, the runout ⌬L increases linearly with Hi, a result that provided the time interval between the two images is larger is easily obtained from dimensional analysis.9 It is also for than about ⌬t=5 ms. We chose to work with ⌬t=10 ms, this range of a that shallow-water equations show good which turned out to be long enough to achieve a good signal- 13,14 agreement with the experimental data. FIG. 2. Three sequences of imagesto-noise corresponding to ratio, and small enough compared to the character- d=1.15 mm beads spreading in the semiaxisymmetric For aտ3, the scaled runout does not vary linearlysetup. The first with image ofa each sequenceistic corresponds slumping to time scale c. Typical sequences of image dif- the moment where the gate is being lifted, the time but follows a power law whose exponent dependsinterval between on the the following imagesferences is ⌬t=ͱHi /g, observed for different initial aspect ratios a in both except for the last image taken at the very end of the flow geometry as summarized above. Two differentflow when the mecha- heap is at rest. ͑a͒ Regimeflow 1 a=0.6, geometriesM are displayed in Figs. 7 and 8. =100 g, Li =39 mm, ⌬t=49 ms. ͑b͒ Regime 1 a=2.4, nisms are likely to account for this crossover. First,M =400 g, L verticali =39 mm, ⌬t=98 ms. ͑c͒ RegimeWe 2 alsoa measured the velocity field in the flowing layer =3.6, M =600 g, Li =39 mm, ⌬t=120 ms. acceleration, which is negligible at low a, becomes important using a particle image velocimetry ͑PIV͒ algorithm based on Downloaded 02 Jul 2010 to 195.83.220.187. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp a the granular mass spreads through avalanching of the profiles h͑x,t͒ are scaled with respect to Li and the time flanks, producing either truncated cone deposits for a interval between two consecutive profiles is scaled with re- Ͻ0.74 Fig. 3 a or conical deposits for aϾ0.74 Fig. 3 b . ͓ ͑ ͔͒ ͓ ͑ ͔͒ spect to the free-fall time of the granular column c =ͱHi /g. A transition towards a different flow regime is observed The three profiles are identical at each time. This observation when a is increased. This second flow regime is illustrated in demonstrates that, for fixed granular material and substrate Fig. 3͑c͒: upon release, the upper part of the granular mass properties, the flow dynamics and the final deposit morphol- descends, conserving its shape while the foot of the pile propagates along the channel. Along the deposit an inflection ogy do not depend on the volume of granular material re- FIG. 7. Sequence of successive imagepoint differences separates cor- a steep sloped from a large, almost flat re- leased but only depend on a. The range of substrate and responding to experiments performedgion. in the semiaxi- material properties ͑including the bead size͒ explored in this symmetric setup. ͑a͒ a=0.6, Li =39 mm,FigureM =100 4 showsg, and the time evolution of the profiles of three paper is too restricted to evaluate their influence. Note how- 10 d=1.15 mm: t=0.5c, c,2c,3c,4differentc, and 6granularc. ͑b͒ a heaps of the same initial aspect ratio a ever that the observations of Lajeunesse et al. and Balm- =2.4, Li =39 mm, M =400 g, and d=1.15=3.2 but mm: obtainedt=0.5c, with different masses or bead sizes. The forth and Kerswell11 indicate that properties such as the bead c,2c,3c, 3.5c, and 4c. ͑c͒ a=3.6, Li =39 mm, M =600 g, and d=1.15 mm: t=0.5c, c,2c,3c, 3.5c, and 4c. FIG. 3. Same as Fig. 2 but in the rectangular channel. ͑a͒ Regime 1, a=0.6, M =470 g, Li =102 mm, ⌬t =80 ms. b Regime 1, a=2.4, M =560 g, L =56 mm, Downloaded 02 Jul 2010 to 195.83.220.187. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp ͑ ͒ i ⌬t=117 ms. ͑c͒ Regime 2, a=16.7, M =170 g, Li =10 mm, ⌬t=130 ms. Downloaded 02 Jul 2010 to 195.83.220.187. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp • good quantitative behaviour! • test another case? • A well know experimental result: Hagen Beverloo constant discharge law! • To o l : contact dynamics for discrete simulation and continuum «µ(I) rheology»! • the Hour Glass: discrete versus continuum simulations The Hour Glass XVIII century watchkeeping catalogue de l’exposition à la Conquête des Mers, Hospice Comtesse Lille 1983 The Hour Glass 2D model computed with DCM ! ~ 90x90 grains Nedderman D u gD Nedderman • Hagen 1852 Beverloo 1961 constant discharge law mass flow rate W = C g(D kd)5 in 3D Gotthilf Hagen 1797-1884 C 0.6 k 1.5 d no influence of the hight ! nor the width! influence of D, d and , ! so by dimensional analysis: D u gD W = C g(D kd)3 in 2D • A well know experimental result: Hagen Beverloo constant discharge law! • Tool: contact dynamics for discrete simulation! ••PresentationProblem: of continuum «µ(I) rheology»! • ImplementationSimulate the hour in a glassNavier with Stokes discrete FV VOF and! continuum • exampletheories of! simulations DCM versus NS: collapse of columns! ••thetry Hour to recover Glass: discretethe Beverloo versus 1961continuum Hagen simulations 1852 law from discrete and continuum simulations • Flow in a Hourglass Discharge from Hoppers simulation DCM • Flow in a Hourglass Discharge from Hoppers simulation Navier Stokes µ(I) • Flow in a Hourglass Discharge from Hoppers simulation discrete vs continuum • comparing Torricelli Evangelista Torricelli 1608 1647 0.9 µ(I) plastic flow y = 0.9 - 0.053 t 0.6 Newtonian flow Volume Torricelli's flow 0.3 0.0 0 2 4 6 8 10 12 14 16 18 time viscosity of the Newtonian flow extrapolated from the µ(I) near the orifice • Flow in a Hourglass Discharge from Hoppers discrete vs continuum (a shift) Beverloo (1961)! 2 — — Hagen (1852) Q 3 — Q — L µ2 µ1 µ(I)=µ1 + — I0/I +1 LWidth of the aperture 3 NO influence of initial filling height 4 FIG. 3: Volume V¯ of material remaining in the silo as a func- tion of time for L¯ =0.1875 and filling heights H¯ =1.5, H¯ =2.5, H¯ =3.5andH¯ =4.5 for a µ(I)plasticflow (µs =0.32, µ =0.28 and I0 =0.40). Inset: Berverloo scalings corresponding to each case. We observe that the flow rate remains constant through- out the discharge of the plastic silo over a large range of FIG. 2: a- Volume V¯ of material remaining in the silo as outlet size L¯. Varying L¯ between 0.0625 (ie 16 computa- ¯ a function of time for an outlet size L =0.125 and filling tion cells for 258 computations cells in W) and 0.28125FIG. (ie 5: Volume of material V¯ remaining in the silo as a func- tion of time t¯ for L¯ =0.1875 and filling height H¯ =4.5, for height H¯ =0.9inthecaseofaplasticflow(µs =0.32, µ = 72 computation cells for 258 computations cells ina W),µ(I)plasticflowwithµs =0.10, µs =0.20, and µs =0.32 0.28 and I =0.40) and in the case of a Newtonian flow ¯ 0 ¯ ( µ =0.28 and I0 =0.40). Inset: Flow rate Q as a function 1 3 we measure the flow rate Q, and search for a relation ¯ ¯ of outlet size L for µs =0.10 and µs =0.32. ( =0.01⇤g 2 W 2 ); b- Berveloo scaling obtained for L varying satisfyingFIG. the 4: shape Pressure of field the in the Beverloo early stage scaling: of the discharge for between 0.0625 and 0.28125. two granular plastic silos of initial filling heights H¯ =1.5 (right) and H¯ =4.5 (left), outlet L¯ =0.1875 and friction co- e⌅cient µs =0.32 ( µ =0.28 and I0 =03 .40). The color scale is identical on bothQ¯ = images:C L¯ the highestk 2 bound (red color) is (2) ¯ silo in the course of time for the same system. We observe set to P =1.3; the pressure jump between two isolines is 0.15. a linear evolution throughout the discharge for the granu- where C and k are constants. For a continuum silo, the lar fluid, revealing a constant flow rate as in real discrete numerical value of k is expected to be zero, in contrast granular silos. We measure the value of the viscosity to granular silos where the grain diameter imposes a vol- in the vicinity of the outlet and find a roughly constant ume of exclusion reducing the e⇥ective size of the outlet. value during the discharge equal to ¯ =0.01 (normalized Imposing k = 0, we recover the Berverloo scaling with 1 3 by ⇤g 2 W 2 ). Using this value for the viscosity, we sim- a good accuracy,slight increase giving of theC flow=1 rate.4 (see is observed Figure for larger 2-b). Note ulate the discharge of a silo filled with Newtonian fluid; however thatfilling making height: approximating no assumption the discharge on by the an a fitting⌅ne pa- function over its full duration, we find a flow rate¯ the corresponding volume-vs-time evolution is shown in rameters,increase the best of 1.8% fit for givesH¯ =2k.5=0 and an.00938 increase = of 30..9%85 ford,where Figure 2-a, and is non-linear, suggesting a dependence on d¯=1/90 isH¯ the=4.5 grain compared diameter to the case used of H¯ =1 in.5. the µ(I)-rheology Measuring the flow rate Q¯ in the early stage of the FIG. 6: Pressure field in the early stage of the discharge for the height of material left in the silo, as in the case of an normalizeddischarge by W for. Although the di erent valuesnot completely of H¯ and di erent negligible, a granular plastic flow with friction coe⌅cient µ =0.10 we consideroutlet nevertheless dimension L¯, we that recover this the value Berveloo is scaling not physically (2), s hydrostatic pressure field. For comparison, we plot the ( µ =0.28 and I0 =0.40) (right) and for a Newtonian but with coe⌅cients varying slightly with the value of H¯ 1 2 solution of the (non-dimensional) Torricelli discharge for significant. flow of viscosity ¯ =0.01 (normalized by ⇤g 2 W 3 ) (left): in (Figure 3, inset). This weak influence of the initial filling both cases, no low-pressure cavity can develop. Outlet size an ideal fluid: height is maximum in the early stage of the discharge, L¯ =0.1875, filling height H¯ =1.5. The color scale is identical but vanishes at the end: the curves shown in Figure on both images and identical to Figure 4: the highest bound dh¯ 3 can be superimposed when shifted towards the final (red color) is set to P¯ =1.3; the pressure jump between two = L¯ 2h,¯ IV. INCREASINGstage of the discharge. THE FILLING HEIGHTisolines is 0.15. dt¯ Altogether, the flow rate during the discharge of con- p ¯ 2 tinuum granular material is very weakly a ected by the ¯ ¯ L To checkinitial that filling the height dischargeH¯ . By contrast, rate the remains pressure field constant V. INFLUENCE OF THE INTERNAL h(t¯)= H t¯ , strongly changes with the value of H¯ , as is visible on FRICTION p2 irrespective of the initial filling height,¯ we perform ✓ ◆ the two snapshots shown in Figure 4 for H =1.5 and p additionalH¯ simulations=4.5 in the early with stage ofH¯ the=1 discharge..5, 2 In.5, both 3.5 cases and 4.We5, suspect the deviation of the discharge rate from the where h¯ is the instantaneous height of material remain- with thehowever, same we rheological observe a low pressure parameters cavity in the as vicinity previouslyhydrostatic case to be the result of the non-newtonian na- ¯ of the outlet, suggesting that the discharge is a ected ture of the µ(I)-rheology, and more specifically of the ex- ing in the silo (normalized by W ) at time t (normalized (µs =0.32,by thisµ local=0 pressure.28 and conditionI0 =0 and.40). insensitive Figure to the 3 showsistence of a yield stress; accordingly, we expect the value by W/g). Torricelli’s discharge matches the onset of the volumemeanV¯ pressureleft in in the the silosilo. Comparison in the course with the of pressure time forof an the coe⌅cient of friction to have important repercus- field in a Newtonian silo shows that the existence of this sions on the discharge flow. Figure 5 shows the discharge the discharge of the Newtonian fluid provided we chose a outlet size L¯ =0.1875 for all four cases. We observe a ¯ ¯ p pressure cavity is contingent on the existence of a yield of a silo with H =4.5 and outlet L =0.1875 for di er- smaller numerical value for L¯ than that of the simulation linear evolutionstress (Figure of very 6). similar slope, suggesting thatent the values of the coe⌅cient of static friction µs =0.10, (L¯ =0.0714 instead of 0.125). Comparing the discharge discharge rate is essentially constant and independentµs =0.20 and µs =0.32: we observe that a smaller of the Newtonian fluid and the granular plastic fluid thus of the filling height. However, closer inspection shows points at the plastic property of the flow as responsible that the slope varies slightly during the discharge: for for the constant nature of the discharge rate in the latter H¯ =4.5 for instance, the initial flow rate has decreased case. of 1.4% halfway through the discharge. Moreover, a discrete vs continuum (at same rate) H1 y S2 S1 x y/d discrete vs continuum (at same rate) discrete vs continuum (at same rate) H1 y S2 S1 x y/d discrete vs continuum (at same rate) Rotating Drum Vidange de Benne Vidange de Benne Conclusion • µ(I) Rheology for granular flows?! • granular: ubiquitous! • Shallow Water: good tool for avalanches (geophysics)! • good qualitative behaviour (discr. / cont.)! • Collapse scaling - Beverloo scaling: µ(I) ! • Beverloo at same rate: velocity pressure superposed ! • but: coef. µ(I) depend on the geometry? Conclusion • discrete continus (like air water..) grains discretes Continus fluid , Navier Stokes simplified system : Saint Venant • lot of applications Conclusion ! • non local effects? Kamrin Boquet! • instabilities? • µ(I) ill posed: Barker, Schaeffer, Bohorquez, Gray…! • Shallow Water: extra term Edwards, Baker, Gray…! • Segregation: Gajjar, Gray… Thanks for attention Time for questions ?