Numerical Simulation of Powder Flow

Transcription

1 Numerical Simulation of Powder Flow Stefan Turek, Abderrahim Ouazzi Institut für Angewandte Mathematik, Univ. Dortmund Models for granular flow Numerical aspects Discussion and outlook p. 1/27

2 Motivation of our Resarch Preserve the high efficiency of special PDE solvers for flow problems p. 2/27

3 Motivation of our Resarch Preserve the high efficiency of special PDE solvers for flow problems High (guaranteed) accuracy for user-specific quantities! With minimal #d.o.f.s! Via robust solvers with optimal numerical complexity! Exploiting the huge sequential/parallel GFLOP/s rates! Realization for complex CFD? p. 2/27

4 CFD Software FEATFLOW Continuum mechanical description, for example ρ Du Dt S = µ = p + S + ρg, u = 0 p ǫ + D(u) D(u), µ = 2 sinφ Finite Elemente methods, iterative solvers Nonlinear models for granular flow based on generalized Navier-Stokes equations S = ν(µ, p, D(u))D(u) p. 3/27

5 Application to Granular Flow Pharmaceutical Industry, Food Processing, Soil Mechanics The behaviour of granular flow is different from that of fluids since it does not exhibit viscosity and the dominant interaction between particles is friction. [Nevertheless, numerical methods are similar.] We do not examine flow of large grains but smaller-sized bulk powders where a continuum approach may be more advantageous. [ generalized Navier-Stokes equations] p. 4/27

6 Regimes of Powder Flow (cf. Tardos) Stick-Slip Regime Uncertain Boundary Slow, Frictional Flow Regime (Quasi-Static Regime) Intermediate Flow Regime Increasing Fluctuations of (stress and) strain rate Rapid Granular Flow Regime f Fluid-like behavior n (n<1) Static Regime Dimensionless Shear Rate, * = [d p /g] 1/2 p. 5/27

7 Static Regime Assumption: The flow is slow enough so that any movement between 2 static states can be neglected. Advantages: static equilibrium equation can be applied theory predicts the onset of flow Disdvantages: no flow field can be predicted no flow rates result from theory Method of Slices (Jenkins 1895) It is (still) used for the design of silos! p. 6/27

8 Slow, Frictional Regime General equations of flow result from: adding inertial effects to the static equations considerations of continuity using a yield condition invoking a "flow rule" (Most) Important from industrial point of view! First model: Schaeffer (1987) p. 7/27

9 Intermediate + Rapid-Granular Regimes Intermediate: Both inter-particle friction and collisional energies are important. More research needed! Rapid-Granular: Short particle-particle contacts are important while frictional forces are neglected; particles behave as molecules in a dense gas. Equations of flow result from a Kinetic Theory type model: conservation of mass conservation of momentum conservation of energy p. 8/27

10 Characteristics of Slow Granular Flow Open a silo hopper and...??? The drag force for Schaeffer and Bingham flow acting on a cylinder is independent of the grain velocity, contrary to Stokes flow Stokes Bingham Schaeffer Velocity 0.7 Schaeffer Cylinder diameter When mechanical ploughs replaced draught animals, it was observed that ploughing at greater speeds does not require greater forces! (Schaeffer) p. 9/27

11 General Powder Flow Model Conservation of mass (u is the velocity vector) Dρ Dt = ρ t + (ρu) = 0 For an incompressible material the bulk density, ρ, is a constant: Equation of motion u = 0 ρ Du Dt = T + ρg with T = S + pi ("stress field" T, deviatoric part S, pressure p) p. 10/27

12 Constitutive Equations The constitutive equation is devoted to correlate between the deviatoric tensor S and the velocity u, through the rate of deformation D = 1 2 ( u + T u), and to assure the closure of the equations. Newtonian law Power law S = 2ν o D S = 2ν( D 2 )D, ν(z) = z r 2 1, r > 1 Schaeffer s law (1987): For a powder a constitutive equation was first introduced by Schaeffer (1987), which has to obey a von Mises yield condition; S = 2p sin φ, and Levy s flow rule; S = λd. We use this correlation to obtain the constitutive equation S = 2p sinφ D D p. 11/27

13 Generalized Navier-Stokes Equations The generalized incompressible Navier-Stokes problem reads: ρ Du Dt = p + (ν(p, D II)D) + ρg, u = 0 If we define the pseudo viscosity ν(, ) as function of D II (u) = 1 2 D : D and p, we obtain a full range of different nonlinear flow models : Power law defined for Bingham law defined for ν(z, p) = ν o z r 2 1 ν(z, p) = ν o z 1 2 Schaeffer s law (including the pressure) defined for ν(z, p) = 2 sin φ pz other laws with pressure-dependent viscosity... p. 12/27

14 Mathematical Challenges I Mathematical Analysis: For pressure-dependent viscosity: Hron/Malek/Necas/Rajagopal, Schaeffer Discretization method: LBB-stable FEM discretization? Uniqueness of the pressure? Inflow profiles? Operator-Splitting (a la Chorin, Van Kan)? Nonlinear solver: Newton techniques for this highly nonlinear problem? Decoupling of pressure and velocity? Linear multigrid solver: New types of saddle-point problems? p. 13/27

15 Mathematical Challenges II 1. Discretization method: The approximation of incompressible velocity fields is subject to the so-called LBB-condition min q h Q h max v h V h (q h, v h ) q h 0 v h 0 β > 0, (1) Discrete Korn s inequality v H 1 (τ) c( v 2 L + D(v) 2 2 L ) (2) τ τ h 2. Convection-dominated transport specially at high Reynolds number (Re = 1 ν ) 3. Boundary condition p. 14/27

16 Spatial Discretization Quadrilateral Rannacher-Turek Stokes Element Q 1 := { q ψ 1 T : q span < 1, x, y, x2 y 2 > } M4 P1 M1 M P2 M2 E1 E2 The degree of freedom are determined by the nodal functionals F a Γ := Γ 1 Advantage Γ {F (a,b) Γ ( ),Γ T h }, with vdγ or F b Γ := v(m Γ ) (m Γ midpoint of edge Γ) (3) Stable and efficient for incompressible flow. Compact data structures. P4 M3 P3 Disadvantage Not satisfying classical discrete Korn s inequality Numerical oscillations for convection dominated problem p. 15/27

17 Classical Stabilization Methods FEM streamline diffusion Ñ(u)v,w = N(u)v,w + N sd(u)v,w, where (4) N sd (u)v,w = (u v)(u w) dx. (5) T T h δ T T FEM upwinding Ñ(u)v,w = l k Λ l Γ lk u n lk dγ (6) [1 λ lk (u)(v(m k ) v(m l ))]w(m l ) Based on the local Reynolds number Re τ = u τ h h ν, we can either define 8 δ τ = δ h τ u Ω 2Re τ 1 + Re τ, λ lk (u h ) = >< >: δ Re τ 1 + δ Re τ if Re τ 0 1 2(1 δ Re τ ) otherwise (7) p. 16/27

18 New Edge-oriented FEM Stabilization Based only on the smoothness of the discrete solution we have developed the following jump term edge E max(γνh E, γ h 2 E) E [ u][ v]dσ withγ, γ [0.0001,0.1] (8) Only one generic stabilization takes care of all instabilities 1. due to Korn s inequality 2. due to convection dominated flow for medium and high Reynolds number, even for pure transport 3. due to problems with variable viscosity Independent of the local Reynolds number p. 17/27

19 Standing vortex Re = Cutline of the x-velocity component for the Standing Vortex problem Exact_solution eo_64x64 upw_64x64 sd_64x Exact_solution eo_128x128 upw_128x128 sd_128x Exact_solution eo_256x256 upw_256x256 sd_256x D presentation of the norm of the velocity (VECT MAG) for the Standing Vortex problem: edge-oriented (left) vs. streamline-diffusion (middle) and central (right) Upwind: significant smearing effects EO: preserves perfectly the solution with high accuracy p. 18/27

20 Nonlinear Solver: Newton Iteration Let u l being the initial state, the (continuous) Newton method consists of finding u such that Ω 2ν(D II (u l ), p l )D(u) : D(v)dx ν(d II (u l ), p l )[D(u l ) : D(u)][D(u l ) : D(v)]dx Ω + Ω = 2 2 ν(d II (u l ), p l )[D(u l ) : D(v)]pdx fv Ω Ω 2ν(D II (u l ), p l )D(u l ) : D(v)dx, v, (9) where i ν(, );i = 1,2 is the partial derivative of ν related to the first and second variable, respectively. p. 19/27

21 New Linear Algebraic Problem Every Newton step consists of finding (u, p) as solution of the linear system { A(u l, p l )u + δ u A (u l, p l )u + Bp + δ p B (u l, p l )p = R u (u l, p l ), B T u = R p (u l, p l ), (10) where R u (, ) and R p (, ) denote the corresponding nonlinear residual terms for the momentum and continuity equations, and the matrix A (u l, p l ) and B (u l, p l ) are defined as follows, respectively A (u l, p l )u,v = Ω 2 1 ν(d II (u l ), p l )[D(u l ) : D(u)][D(u l ) : D(v)]dx. (11) B (u l, p l )p,v = Ω 2 2 ν(d II (u l ), p l )[D(u l ) : D(v)]pdx. (12) p. 20/27

22 Schaeffer Model (I) The nonlinear pseudo viscosity has the form ν(p, z) = Q(p)z r 2 1 New type of (nonsymmetric) saddle-point problems; ( Au B + δ M δp (ũ, p) = p B ) B T 0 The (linear) solution depends on the choice of imposing the uniqueness, since dim(null(m δp )) = 1 ν = exp(βp) ǫ+ D(u) β Newton Fixed Point /3 200/ /3 250/ /3 238/2 ν = βp ǫ+ D(u) Newton 12/13 405/2 12/10 522/2 12/10 581/2 Fixed Point p. 21/27

23 Konfigurationen Stokes Bingham Schaeffer Schaeffer_without_convection Schaeffer_with_convection p. 22/27

24 Schaeffer Model (II) Schaeffer s law: The nonstationary equations are linearly ill-posed according to Schaeffer. Flow rate: The quantitative prediction of the flow rate through silo hoppers is wrong by more than 100% Add compressibility: This approach is an effort to regularize the instability. p. 23/27

25 Compressible Powder Models (cf. Tardos) General equation of motion for a powder ρ Du Dt = p + [ Continuity equation Normality condition ρ t q(p,ρ) ( D D 1 1 n ui n ui)] + ρg, with + (ρu) = 0, and u = q(p,ρ) p D 1 n ui Yield condition q(p,ρ) is given by: Powder properties Non-cohesive Cohesive Incompressible p sinφ p sinφ + ccos φ [ ] Compressible p sin φ 2 p p sinφρ 1 β C (p ρ 1β ) 2 ρ 1 β ρ 1 β p. 24/27

26 Summary: More mathematical research is needed to understand and to analyse the models for granular and powder flows. Numerical analysis of discretization and solution techniques? The physically correct choice of pressure (mean)values is not clear. How to prescribe velocity profiles? [Or: Fluxes?] p. 25/27

27 Outlook I: Hypoplastic Model by Kolymbas Re [ u t S = Q(p) D(u) + (u )u] = p + S + f, u = 0 D(u) (Schaeffer/Power Law/etc.) + T (Kolymbas) T t + (u )T = [TW WT] + C 1 1 (TD DT) 2 +C 2 tr(td) I + C 3 trd 2 T + C 4 trd 2 +ν(d)[ D t 1. FEM discretization and solver? 2. Free boundary? Fluid-Structure interaction? + (u )D + DW WD] 3. Numerical falsification of material/flow models!? trt T2 p. 26/27

28 Outlook II: Particulate Flow Development of simulation tools for the understanding of: 1. Interaction of solid particles with flow ( many complex objects?) 2. Behaviour of nonlinear fluid models Fictitious Boundary Method on adapted meshes + Operator-Splitting! Many particles? Nonlinear fluids? Wide range of applications! p. 27/27