Some recent models and simulations for granular flows

Transcription

1 Some recent models and simulations for granular flows François Bouchut LAMA, CNRS & Université Paris-Est all collaborations with Anne Mangeney Institut de Physique du Globe de Paris HYP2016, Aachen, August Models and simulations for granular flows 1

2 Introduction : granular materials Where they come from Granular materials are involved in several processes on the Earth s surface : erosion in mountainous and coastal areas, landslides, volcanos, seismic events... Granular materials are involved in industrial processes (construction, food...) Definition : A granular material is made of grains that can have individually an impact on their neighbourhood. It means that they have non negligible size (from 1µm to 1m). The grains are much larger than molecules. Only the collective behaviour is considered. Main consequence : A granular material can behave as a solid or as a fluid. Think of a sandpile. Introduction Models and simulations for granular flows 2

3 Introduction : granular materials Several notions Granular materials can be dry (in aerial environment) or wet (immersed in a fluid, like water for example) Granular materials can be monodisperse (all grains of the same size/nature) or polydisperse (different sizes/nature) Granular materials can be dilute, dense, or jammed/blocked. A reference B. Andreotti, Y. Forterre, O. Pouliquen, Granular media, between fluid and solid, Cambridge University Press, Introduction Models and simulations for granular flows 3

4 Main issues for the description of granular materials Many issues are unsolved for the description of granular materials Establishing a model for the continuous description of a granular material that is able to describe the static and flowing behaviours correctly (at least from the dense to the jammed regimes) and has a good mechanical/mathematical structure Correctly describing the behaviour of wet granular materials (examples : erosion processes in rivers, quicksand, landslides in the event of heavy rain...) Describing the static/flowing transition dynamics (example : start of a snow avalanche) Establishing thin-layer models These are described in the ANR prospective study MathsInTerre, 2013 Introduction Models and simulations for granular flows 4

5 Outline 1 Introduction 2 Viscoplastic models 3 Dilatancy in granular/fluid mixtures Introduction Models and simulations for granular flows 5

6 Viscoplastic models Viscoplastic models Models and simulations for granular flows 6

7 Viscoplastic materials Viscoplastic materials are characterized by Nonlinear relation between stress and strain rate (non-newtonian fluid) Irreversible deformations inducing dissipation Examples of such materials : snow, mud, lava, grains in a surrounding fluid (wet sand). Behaviour can be fluid or solid Viscoplastic models Models and simulations for granular flows 7

8 Modeling of viscoplastic materials Momentum equation for an incompressible (div u = 0) viscoplastic material : where the stress σ is decomposed as tu + u u div σ = f, σ = p Id +σ, tr σ = 0. The scalar p is the pressure and σ is the deviatoric stress. It is related to the strain rate Du = ( u + ( u) t )/2 by a nonlinear relation (that defines the rheology). For a Bingham fluid σ = 2νDu + κ Du Du, for some constant yield stress κ. Multivalued meaning : σ (t, x) κ when Du = 0 (solid behaviour). µ(i )-rheology (Jop, Forterre, Pouliquen 06) : pressure and rate dependent yield stress well-adapted to dry granular materials σ = µ(i )p Du Du, with I = 2d Du / p p/ρ s, d diameter of the grains, ρ s density of the solid. No justification from microscopic (anelastic) laws, just scaling laws! Viscoplastic models Models and simulations for granular flows 8

9 Simulation of incompressible viscoplastic materials Simulation of incompressible viscoplastic materials Augmented Lagrangian method (Temam, Glowinski, Roquet-Saramito, Ionescu...) Regularization method (Temam, Papanastasiou, Frigaard-Nouar...) σ Du = 2νDu + κ p ε2 + Du. 2 Proof of convergence of the regularization method (for a Bingham fluid) : Glowinski-Lions-Trémolières (1981), at rate ε if ν > 0. Bouchut-Eymard-Prignet (2014), at rate ε if ν = 0. Viscoplastic models Models and simulations for granular flows 9

10 Simulation of the µ(i ) rheology Simulation of the µ(i ) rheology Lagrée-Staron-Popinet (2011) regularization and tracking of free surface Chauchat-Médale (2014) regularization, fixed domain Ionescu-Mangeney-Bouchut-Roche (2015) Augmented Lagrangian, ALE for the free surface Lusso-Ern-Bouchut-Mangeney-Farin-Roche (2015) Regularization + ALE Viscoplastic models Models and simulations for granular flows 10

11 Well-posedness The Bingham model is well-posed (monotone operator, falls into the general theory of Brézis) The µ(i )-rheology is ill-posed (Barker-Schaeffer-Bohorquez-Gray 2015) Question : how to modify the µ(i ) model to make it well-posed? Do we have to reintroduce a regularization at the grain size? Do we have to introduce compressible effects? Viscoplastic models Models and simulations for granular flows 11

12 Compressible viscoplastic models? Compressible viscoplastic models are yet to be settled Malek-Rajagopal 2010 Take for example a simple law tρ + div(ρu) = 0, t(ρu) + div(ρu u) div σ = ρf. σ = p 0(ρ) + κ(ρ) Du Du. When κ depends on ρ, is the model well-posed (in 1d)? Viscoplastic models Models and simulations for granular flows 12

13 Dilatancy in granular/fluid mixtures Dilatancy in granular/fluid mixtures Models and simulations for granular flows 13

14 Dilatancy in granular/fluid mixtures We consider a dry granular material. The only non Newtonian effect that we take into account is dilatancy. Figures from Les milieux granulaires, B. Andreotti, Y. Forterre, O. Pouliquen (a) The material dilates when deformed (b) The material contracts when deformed Relation between deformation and dilation/contraction! Dilatancy in granular/fluid mixtures Models and simulations for granular flows 14

15 Dilatancy We consider a wet granular material. Figures from Les milieux granulaires, B. Andreotti, Y. Forterre, O. Pouliquen (a), (b) When a dense packing of sand is deformed, the granular material dilates, and water is sucked When a loose packing of sand is deformed, the granular material contracts, and water is expelled (ex : landslide, quicksand sables mouvants ). Dilatancy in granular/fluid mixtures Models and simulations for granular flows 15

16 Thin layer approximations Basis of classical thin layer approximations : Saint Venant (shallow water) model derived from incompressible Euler equations above an almost flat topography Savage, Hutter (1989) take into account the bottom friction and main curvature, Submerged flows models proposed by Iverson, Denlinger (2001), Pitman, Le (2005), Pailha, Pouliquen (2009), Iverson, George (2014) Difficulties : Complex rheology not taken into account in the above expansions, but only dilatancy effects. Benefits : Physics of diphasic mixtures is involved. Dilatancy laws are not obvious to set. Replace viscoplatic laws by boundary friction Less space variables is better for simulation In this talk : Use asymptotic expansions as far as possible without introducing heuristic rules. Keep the mathematical structure to avoid unphysical effects. In particular, keep an energy balance. Dilatancy in granular/fluid mixtures Models and simulations for granular flows 16

17 Jackson s two-phase mixture model Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, J. Fluid Mech. (2016) We consider a wet granular material. The features to take into account are buoyancy force interphase friction force (drag) solid (granular) pressure and pore (fluid) pressure, both non hydrostatic because of the relative interphase velocity Dilatancy : the granular material dilates if its volume fraction is larger than a critical value, or contracts in the opposite case. In the first case the granular friction increases (stiffening of the granular matrix). In the second case the granular friction diminishes (fluidisation). Law proposed by Roux and Radjai (1998). Dilatancy in granular/fluid mixtures Models and simulations for granular flows 17

18 Jackson s 3d model of solid/fluid mixture Jackson s model can be written as follows. Two mass equations t(ρ sϕ) + (ρ sϕv) = 0, t(ρ f (1 ϕ)) + (ρ f (1 ϕ)u) = 0, with ρ s, ρ f the (constant) solid/fluid mass densities, ϕ the solid volume fraction, v the solid velocity, and u the fluid velocity. Two momentum equations ρ sϕ( tv + v v) = T s ϕ p fm + f + ρ sϕg, ρ f (1 ϕ)( tu + u u) = T fm + ϕ p fm f + ρ f (1 ϕ)g, where g is gravity, f is the drag force with ν f f = β(u v), β = 18ν f ρ f ϕ d 2 the fluid viscosity and d the diameter of the grains. T s, T fm are the solid and fluid stress tensors, taken as T s = p sid + e T s, T fm = p fm Id + e T fm, with e T k small far from shearing boundary layers. Dilatancy in granular/fluid mixtures Models and simulations for granular flows 18

19 Jackson s 3d model of solid/fluid mixture Unknowns in Jackson s model : 5 unknowns : ϕ, v (vector), u (vector), p s, p fm 4 equations (2 vectors) A constitutive equation is required to close the system : dilatancy law Dilatancy in granular/fluid mixtures Models and simulations for granular flows 19

20 Dilatancy The only non Newtonian effects that we take into account result from dilatancy. Qualitative explanation of dilatancy effects (loose) Granular medium contracts when deformed Water must be expelled (dense) Granular medium dilates when deformed Water must be sucked Increasing pore pressure Decreasing pore pressure Liquefaction Stiffening of the granular matrix p fm p fm Pailha and Pouliquen, 2009; Y. Forterre, Seville 2014 Dilatancy in granular/fluid mixtures Models and simulations for granular flows 20

21 Modeling of dilatancy effects Compression/dilatation of the solid phase : : dilatancy angle, Roux and Radjai, 1998 : shear strain rate T s xz T s zz Pailha and Pouliquen, 2009 Closure equation : : compression : dilatation Impact of the dilatancy angle on the Coulomb friction force : Dilatancy in granular/fluid mixtures Models and simulations for granular flows 21

22 Energy balance With a slighly modified Roux-Radjai closure, Jackson s system has a relevant energy equation «t ρ sϕ v ρ f (1 ϕ) u 2 2 (g X )`ρ sϕ + ρ f (1 ϕ) + ρ sϕe c + ρ sϕ v 2 2 v + ρ f (1 ϕ) u 2 2 u (g X )`ρ sϕv + ρ f (1 ϕ)u «+p fm`ϕv + (1 ϕ)u + Tfm e u + T sv + ρ sϕe cv = (p s p c) v + e T s : v + e T fm : u + f (v u). with X the space position, and the closure is taken as v = K p γ(p c(ϕ) p s) with p c(ϕ) the critical pressure, and e c(ϕ) the critical specific internal energy defined by de c/dϕ = p c/(ρ sϕ 2 ). The right-hand side of the energy balance equation is nonpositive The closure law can be interpreted as a compressible bulk viscoplastic rheology with γ = Dv, Dv = ( v + ( v) t )/2. p s = p c(ϕ) v K p γ, Dilatancy in granular/fluid mixtures Models and simulations for granular flows 22

23 A two-phase two-layer model with dilatancy Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina (2015). Two thin layers : one layer of solid/fluid mixture (volume fraction 0 < ϕ < 1) treated via Jackson s model, and one layer of pure fluid above h f b h m Fig.: Two-layer configuration The fluid can enter or get out of the mixture when the granular medium dilates or contracts In contrast with all previously proposed models, we have h f > 0, and two unknown velocities (one for the solid and one for the fluid) instead of one. Suitable boundary conditions have to be set. Dilatancy in granular/fluid mixtures Models and simulations for granular flows 23

24 Two-phase two-layer model with dilatancy : boundary conditions At the bottom we put non penetration conditions a solid Coulomb friction law u n = 0, v n = 0, v (T sn) τ = tan δ eff (Tsn) n, v and a Navier friction condition for the fluid phase (T fm n) τ = k b u. At the free surface we assume no tension for the fluid together with the kinematic condition T f N X = 0, N t + u f N X = 0, where N = (N t, N X ) is a time-space normal to the free surface. Dilatancy in granular/fluid mixtures Models and simulations for granular flows 24

25 Two-phase two-layer model with dilatancy : boundary conditions At the interface, we have the kinematic condition for the solid phase Ñ t + v ÑX = 0, where we denote by Ñ = (Ñt, ÑX ) a time-space upward normal to the interface. The total fluid mass is conserved, Ñ t + u f ÑX = (1 ϕ )(Ñt + u ÑX ) V f where ϕ is the value of the solid volume fraction at the interface (the limit is taken from the mixture side). The term V f defines the fluid mass that is transferred from the mixture to the fluid-only layer. The conservation of the total momentum is written ρ f V f (u u f ) + (T s + T fm )Ñ X = T f Ñ X. The energy balance through the interface yields the stress transfer condition T s Ñ X = ρ «2 f Ñ X (u u f ) 2 ÑX «! Ñ X ϕ + (T fm Ñ X ) Ñ X p 2 f m Ñ 1 ϕ X. These conditions are completed by a Navier fluid friction condition Tfm + T f Ñ X = k i (u f u) τ. 2 τ Dilatancy in granular/fluid mixtures Models and simulations for granular flows 25

26 Two-phase two-thin-layer model with dilatancy Velocity equations ρ s ϕ( tv x + v x xv x ) = ϕg cos θ(ρ s x(b + h m) + ρ f xh f ) (ρ s ρ f )g cos θ hm 2 x ϕ + (1 ϕ) xpe f ϕ(ρ sgn(v x s ρ f )g cos θh m (pf e ) b ) tan δ eff h m + β(u x v x ) ϕρ sg sin θ(1, 0) t, ρ f ( tu x + u x xu x ) = ρ f g cos θ x(b + h m + h f ) 1 ϕ h m xpf e (1 ϕ)h m + h f k bu x + βh m(u x v x ) (1 ϕ)h m + h f ρ f g sin θ(1, 0) t, Contains the average gradient of excess pore pressure xp e f. Fluid and solid pressures are expressed as p f = ρ f g cos θ(b + h m + h f z) + p e f, p s = ϕ(ρ s ρ f )g cos θ(b + h m z) p e f. Additionally to the hydrostatic pressures, they contain the excess pressure p e f, that is related to the dilatancy law Φ. Dilatancy in granular/fluid mixtures Models and simulations for granular flows 26

27 Two-phase two-thin-layer model with dilatancy Our thin-layer averaged model has three scalar equations on h m, h f, ϕ instead of two : t( ϕh m) + x ( ϕh mv x ) = 0, t`(1 ϕ)hm + h f + x `(1 ϕ)h mu x + h f u x f = 0, t ϕ + v x x ϕ. = ϕ Φ. with Φ = K γ` ϕ ϕ c. It satisfies conservation of granular and fluid mass conservation of total momentum decrease of energy dilatancy with excess pore pressure, and fluidisation/stiffening of the granular material convergence to hydrostatic equilibrium in case of no external forces diffusion term on the relative velocity similar to the George-Iverson model We get a hyperbolic system of conservation laws. Numerical simulations in progress! Open problem : include viscoplastic rheology with yield stress Dilatancy in granular/fluid mixtures Models and simulations for granular flows 27

28 Conclusion The modeling of dry and wet granular materials faces several very difficult issues The used (incompressible) viscoplastic laws are not justified from microscopic laws They are ill-posed Need to write a geophysically, mechanically and mathematically relevant rheology for compressible viscoplastic materials with yield stress (to represent the static and flowing behaviours) and dilatancy We do not know the form of equations for which there would be well-posedness Need to derive associated thin-layer models Dilatancy in granular/fluid mixtures Models and simulations for granular flows 28