From micro to macro in granular matter

Transcription

1 From micro to macro in granular matter Thomas Weinhart, A.R. Thornton, O. Bokhove, S. Luding University of Twente April 2, 213 DLR, April 2, 213 1/3

2 rmalized form as: mber can als Grains in nature and industry Nanoindentation M Kappl, Mainz (PiKo) Rotating blast-furnace chute D Tunuguntla, O Bokhove (Chorus) fills up the first pitches. with conical insert. A 27mm cylinde coil 9.5-mm-pitch 2: Screw pitches. first Tephra expulsion L Courtland (USF) 64 CHAPTER 5. FLOW IN VENDING MACHINE CANISTERS AND DOSAGE Screw 1: Standard 9.5-mm-pitch coil. A 27mm cylinder fills up the Vending machine canister K Imole (PARDEM) Screw 3: 9.5-mm-pitch coil with an insert designed to transport volume linearly increasing with c Dose Mass Average Dose M Figure 5.2: Schematics of the three dosing augers used mass is computed as the ded by the average

3 Outline 1 The Discrete Particle Method (DPM) 2 The coarse graining method 3 Micro-Macro transition for free surface flows 4 Conclusions and future work

4 The Discrete Particle Method (DPM) Particles have position r i, velocity v i, angular velocity ω i, diameter d i, mass m i Governed by Newtonian mechanics: fji d 2 r i m i dt 2 = d ω i f i, I i dt = t i ω j r j v j Contact forces and body forces: v i bij fi = fij + m i g i, t i = bij f ij j j r i ω i fij

5 Coarse graining (CG) Objective Define macroscopic fields such as mass density ρ, velocity V, stress σ,... based on particle data ( r i, v i, m i, f i,... ). The fields should satisfy mass and momentum balance exactly. Example: A static system of 5 fixed and 5 free particles.

6 Coarse graining: density ρ and mass balance A) We define the macro-density using a coarse-graining function φ, ρ( r) = n i=1 m iφ( r r i ).

7 Coarse graining: density ρ and mass balance A) We define the macro-density using a coarse-graining function φ, ρ( r) = n i=1 m iφ( r r i ). B) We define the velocity s.t. mass balance, is satisfied: ρ t + (ρ V ) =, V = p n, where p = ρ i=1 m i v i φ( r r i ).

8 Coarse graining: momentum balance C) We define the stress s.t. momentum balance, is satisfied: σ k = n ρ D V Dt = σ + t + ρ g, i=1 m i v i v i φ( r r i ), σ c = 1 fij r ij φ( r ( r i + s r ij )) ds contacts {i,j} 1 fik bik φ( r ( r i + s b ik )) ds, wall contacts {i,k} t = wall contacts {i,k} fik φ( r c ik ), with boundary interaction force density t. Weinhart, Thornton, Luding, Bokhove, GranMat (212) 14:289

9 A static system of 5 fixed and 5 free particles. 8 Density ρ for 2D-Gaussian CG function of width w = d/ Magnitude of stress σ 2 and boundary interaction force t 2.

10 Satisfying mass and momentum balance. Mass balance in a static system is trivial. Satisfying the momentum balance in a static system requires σ = t + ρ g. 11 = + Magnitudes of stress divergence σ 2 (left), boundary interaction force density t 2 (centre), and weight density ρ g (right).

11 Satisfying mass and momentum balance. Mass balance in a static system is trivial. Satisfying the momentum balance in a static system requires σ = t + ρ g. 11 = + Magnitudes of stress divergence σ 2 (left), boundary interaction force density t 2 (centre), and weight density ρ g (right). Important result Mass/mom. bal. is satisfied locally for any coarse-graining function.

12 MercuryCG statistical toolbox open source: www2.msm.ctw.utwente.nl/athornton/md/ svn repository: allows shared development and easy upgrading

13 MercuryCG statistical toolbox open source: www2.msm.ctw.utwente.nl/athornton/md/ svn repository: allows shared development and easy upgrading live statistics for simulations in MercuryDPM post-processed statistics for MercuryDPM and other codes easy loading into MATLAB data structure for post-processing

14 MercuryCG statistical toolbox open source: www2.msm.ctw.utwente.nl/athornton/md/ svn repository: allows shared development and easy upgrading live statistics for simulations in MercuryDPM post-processed statistics for MercuryDPM and other codes easy loading into MATLAB data structure for post-processing exact mass and momentum conservation, even at boundaries exact spatial and temporal averaging, radial averaging

15 MercuryCG statistical toolbox open source: www2.msm.ctw.utwente.nl/athornton/md/ svn repository: allows shared development and easy upgrading live statistics for simulations in MercuryDPM post-processed statistics for MercuryDPM and other codes easy loading into MATLAB data structure for post-processing exact mass and momentum conservation, even at boundaries exact spatial and temporal averaging, radial averaging many CG-functions: Gaussian, Heaviside, Polynomials (Lucy) many fields: temperature, displacement, fabric tensor, mixture drag, rotational dof,... * Grey items will be available in the next release

16 Free-surface flows: A) flow through contraction Glass particles flowing through a contraction from: Vreman et al., J. Fluid Mech. 578 (27)

17 Free-surface flows: B) impingement on inclined plane Sand particles impacting an inclined plane from: Johnson, Gray, JFM 675 (211) 87

18 The granular shallow-layer equations Depth- and width-averaged mass/mom. balance for shallow flow: h t + hū t (hū) + ( hαū 2 + K ) x 2 gh2 cos θ x =, = gh(sin θ µ ū cos θ), ū Closure relations are required for velocity shape factor α= u2, ū 2 normal stress ratio K = σxx σ zz, and bed friction µ = tt t n at z = b.

19 Objective Find closure relations µ(h, ū), α(h, ū) and K(h, ū) using small DPM simulations of steady uniform flow. g z x DPM of steady uniform chute flow over fixed-particle base, periodic in x- and y-direction, varying inclination θ and particle number N.

20 Parameters of the DPM Scaled s.t. diameter d = 1, mass m = 1, gravity g = 1. Linear elastic, dissipative and frictional contact forces: f n ij fij = f n ij n + f t ij t, = k n δ ij γ n v n ij, f t ij = min( k t δ t ij + γ t v t ij, µ c f n ij ), Collision time t c =.5 d/g, restitution r =.88, friction µ c =.5. Integration (Velocity-Verlet) until t = 2 d/g. DLR, April 2, /3

21 Three regimes: arresting, steady, accellerating h Demarkation line h stop (θ) between arrested and steady flow, fit to h stop (θ) = Ad tan(θ 2) tan(θ) tan(θ) tan(θ 1 ). fit according to Pouliquen, Forterre, JFM, 22 θ

22 Choosing the right coarse-graining scale For h = 3, θ = 28, we average over x [, 2], y [, 1], t [2, 25] (dt = t c /2) for several coarse graining widths w: ρ w= z/d the larger w, the less fluctuations the larger w, the more artificial smoothing. Weinhart, Hartkamp, Thornton, Luding, Phys. Fl. (212), submitted

23 Choosing the right time-interval For h = 3, θ = 28, we integrate over t [2, 2 + T ] (dt = t c /2) for coarse graining widths w =.1: ρ T= z/d the larger T, the less fluctuations we choose T = 5.

24 Choosing the right coarse-graining scale.6.55 ν.5.45 w =.5 w = z/d.6.55 ν Volume fraction ν = ρ/ρ p for varying w. w/d sub-particle scale particle scale z = 1.35d z = 2.4d z = 2.56d z = 15.65d z = 27.53d w =.5d w = 1d Two scale-independent ranges exist: sub-particle scale.5d < w <.1d (shows oscillations) particle scale.6d < w < d (only valid in bulk) Here, we choose w =.1d.

25 Kinetic stress σ k xx and granular temp. T g z σ k xx w=.5 σ k xx w=1 σ k xx w=1 σ k xx w= σ k xx w/d σ xx k Kinetic stress is scale-dependent on the particle scale [1], σ k xx = N mv i=1 ixv ixw i with v i = v i V ( r). The kinetic-theory definition is scale-independent, σ k xx = N i=1 mv ixv ixw i with v i = v i V ( r i ). One can show σ k xx ρ γ 2 w 2 3 σk xx. [1] Glasser, Goldhirsch, Phys Fl. 13, 47 (21) w/d

26 Closure for the velocity shape factor α = u2 ū 2 Bagnold velocity profile in bulk, quadratic near base (z < b h stop (θ)/h), linear near surface (z > h 5).

27 Closure for the velocity shape factor α = u2 ū 2 α θ N = 8 N = 6 N = 4 N = Shape factor α(h, θ) = α(h, tan 1 (µ)) from simulations (markers) and fit (lines).

28 Closure for the normal stress ratio and bed friction N 3 mgcosθ lxly 25 2 σ xx σ zz σ yy σ xz σ zx σαβ Normal stress ratio K = σxx σ zz 1. Friction µ = σxz σ zz = tan θ. z

29 Closure for the normal stress ratio and bed friction N 3 mgcosθ lxly 25 2 σ xx σ zz σ yy σ xz σ zx σαβ Normal stress ratio K = σxx σ zz 1. Friction µ = σxz σ zz = tan θ. But note: Stress is slightly asymmetric, oscillating at the boundary, and anisotropic.* *see W, Hartkamp, Thornton, Luding, Phys. Fl. (212) submitted z

30 Closure for the bed friction µ = σ xz σ zz µ (F +γ)/h ( µ(h, ū) = tan(θ 1 ) + (tan(θ 2 ) tan(θ 1 )) 1 + β ) 1 h. Ad F + γ

31 The Micro-Macro transition We closed the granular shallow-layer equations, h t = hū x, hū = ( h α(h, ū) ū 2 K(h, ū) + gh 2 cos θ ) t x 2 + gh(sin θ µ(h, ū) ū cos θ), ū for a given set of microscopic and geometric parameters. Objective Study the dependence of the closure laws on the parameters M (the micro-macro transition).

32 Dependence of the basal roughness λ = 2d λ = 1/2d λ = 1/6d Increasing basal roughness by increasing base particle diameter λ.

33 Dependence of the basal roughness.55.5 µ λ = λ = 1/2 λ = 2/3 λ = 5/6 λ = 1 λ = (F +γ)/h Friction increases with increasing base particle diameter λ. Steady disordered flows fitted to F = β γ. h stop(θ;λ=1) h

34 Conclusions 1/2 Coarse-grained fields are defined s.t. mass and momentum balance holds exactly, even at external boundaries. - W, Thornton, Luding, Bokhove, From discrete particles to continuum fields near a boundary, Granular Matter (212) 14:289

35 Conclusions 1/2 Coarse-grained fields are defined s.t. mass and momentum balance holds exactly, even at external boundaries. - W, Thornton, Luding, Bokhove, From discrete particles to continuum fields near a boundary, Granular Matter (212) 14:289 Coarse-grained fields should be independent of CG width. For chute flows, two scales (sub-particle/particle) exist. Fluctuation velocity requires correction on particle scale. Objective description of stress tensor. - W, Hartkamp, Thornton, Luding, Coarse-grained local and objective continuum description of 3D granular flows, Phys. Fl. (212) submitted

36 Conclusions 1/2 Coarse-grained fields are defined s.t. mass and momentum balance holds exactly, even at external boundaries. - W, Thornton, Luding, Bokhove, From discrete particles to continuum fields near a boundary, Granular Matter (212) 14:289 Coarse-grained fields should be independent of CG width. For chute flows, two scales (sub-particle/particle) exist. Fluctuation velocity requires correction on particle scale. Objective description of stress tensor. - W, Hartkamp, Thornton, Luding, Coarse-grained local and objective continuum description of 3D granular flows, Phys. Fl. (212) submitted Micro-macro transition: Closure relations extracted from DEM. Closure depends on particle/ contact properties. - W, Thornton, Luding, Bokhove, Closure Relations for Shallow Granular Flows from Particle Simulations, Granular Matter (212) 14:531 - Thornton, W, Luding, Bokhove, Frictional dependence of shallow granular flows from discrete particle simulations, EPJ (212).

37 Conclusions 2/2 CG applied to mixtures (bidispersed flows): - Thornton, A.R., W, Luding, S., Bokhove, O., Modeling of particle size segregation: Calibration using the discrete particle method, Int. J. Mod. Phys. C 23, (212) - W, Luding, S., Thornton, A.R., From discrete particles to continuum fields in mixtures, Powders & Grains 213, Conference Proceedings (213), submitted

38 Conclusions 2/2 CG applied to mixtures (bidispersed flows): - Thornton, A.R., W, Luding, S., Bokhove, O., Modeling of particle size segregation: Calibration using the discrete particle method, Int. J. Mod. Phys. C 23, (212) - W, Luding, S., Thornton, A.R., From discrete particles to continuum fields in mixtures, Powders & Grains 213, Conference Proceedings (213), submitted CG applied to molecular flows - R. Hartkamp, A. Ghosh, T. Weinhart and S. Luding, A study of the anisotropy of stress in a fluid confined in a nanochannel, J. Chem. Phys. 137, (212) Download

39 Future Work (a) (c) (d) ψ.1 (b) y (m) C. G. Johnson and J. M. N. T. Gray y (m).5 y (m).5 x =.1 x =.35.1 h h (cm) h (cm) motion blur. The streamlines in this region, which follow the line of the shock, further resemble the experimental flow. The structure of these streams is visible in figure 8(c), which shows the values of the flow variables u (dashed line) and h (solid line) along a cross-section at x =.1 m, through the closed shock. Inside the shock (which occurs at y = ±.4 m), u.99 m s 1 as expected, and h 1.3 mm, close to the experimental Figure 8. Steady-state numerical solution for ζ = 26.7, ujet =.99 m s 1 and D = 15 mm. The axes correspond to the same regions shown in figures 2 and 4. In (a), the shading indicates supercritical flow (Fr > 1). The grey lines are streamlines, and the thick black line indicates the region of strongly converging flow velocity, an identifying feature of the shock. In (b), contours are of flow height, at intervals of 1 mm, with dark contours at intervals of 5 mm. Shading indicates the flowing region of material. Figures (c) and (d ) show cross-sectional plots of flow variables, for x =.1 m and x =.35 m, respectively. Downslope velocity u is indicated by a dashed line and flow depth h by a solid line..1 x (m) u (m s 1) u (m s 1) Where no closure rules are known (transient/boundary effects), a two-way multi-scale coupling is desired. 3 Validate contact model by comparing DPM to lab-scale experiments Validate closure laws by comparing FEM and DPM simulations 1