A Multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface

Transcription

1 A Multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface Damanik, H., Mierka, O., Ouazzi, A., Turek, S. (Lausanne, August 2013) Page 1

2 Motivation Polymer melts: One of industrial interests + Physically fascinating + Rheologically difficult - Numerically challenging - Highly accurate, robust numerical solver which represents the rheological nature is still challenging Page 2

3 Polymer as Viscoelastic model Viscoelastic fluid models (D. D. Joseph): Integral form τ t = 1 t t s WW 2 e WW Differential form: Upper-convected derivative More practical to implement than integral form Represent many viscoelastic models F s, t F(s, t) T dd t + u τ τ τ T = f τ Conformation tensor (τ), velocity (u), source (f τ ) Not able to capture high stress gradient at higher We number f τ can be Oldroyd-B, Giesekus, FENE, PTT, WM, Pom-Pom Page 3

4 Numerical Results Cutline of Stress_11 component at y = 1.0 We = 1.5 with LCR Old Formulation Vs Lcr We= 0.5 We= 1.5 stress_ ,00 0,20 0,40 0,60 0,80 1,00 x We = 0.5 with Old formulation Page 4

5 Log-conformation Reformulation Experience (Kupferman et. al): Stresses grow exponentially Conformation tensor looses positive properties during numerics t + u τ τ τ T = f τ ψ = Ω + B + Nτ 1 + u τ Ωτ τω 2Bτ = f τ τ = e ψ + u ψ Ωψ ψω 2B = g ψ Page 5

6 LCR based models LCR based viscoelastic fluid: Ability to capture high stress gradients at higher We number Positivity preserving by design τ = e ψ Numerically more stable with appropriate FEM ψ + u ψ Ωψ ψω 2B = g ψ LCR tensor (ψ), velocity (u), source (g ψ ) u = Ω + B + Nτ 1 g ψ can be Oldroyd-B, Giesekus, FENE, PTT, WM, Pom-Pom Page 6

7 Exemplary models Model OdB Gie FENE LPTT XPTT WM Pom LCR based viscoelastic fluid: ( I τ) ( τ) f g( ψ) 1 / λ 1/ λ (exp( ψ) I) 2 1/ λ ( I τ α( τ I) ) 1/ λ (f (R) τ αf (R) I) 1/ λ (1 + ε(tr( τ) 3))( I τ) 1/ λ (exp( ε(tr( τ) 3)))( I τ) 1/ λ( γ ) ( τ I) 1/ λ (f ( τ) 2α + ατ + ( α 1) I) b 1/ λ (exp( ψ) I) αexp( ψ)(exp( ψ) I) 1/ λ (f(r) αf(r)exp( ψ)) 1/ λ (1 + ε(tr(exp( ψ)) 3))(exp( ψ) I) 1/ λ (exp( ε(tr(exp( ψ)) 3)))(exp( ψ) I) 1/ λ( γ ) ( I exp( ψ)) 1/ λ (f ( ψ) 2α + αexp( ψ) + ( α 1)exp( ψ)) b 2 ) Relaxation time (λ) Page 7

8 Multiphysics flow model Navier-Stokes equation (u, p) ρ + u u = + σ s + 1 λ η pe ψ, u = 0 + Nonlinear viscosities σ s =2η s γ, Θ, p D, γ = tt D 2 + Temperature effects with Boussinesq and viscous dissipation (Θ) ρc p + Viscoelastic fluid models (ψ) ψ + u ψ Ωψ ψω 2B = g ψ + Multiphase flow with Level-Set equation φ + u Θ = k 1 2 Θ+k 2 D: D + u φ = 0 Page 8

9 Discretizations In Time: Second order Crank-Nicolson Can be adaptively applied In Space: Higher order finite element (Arnoldi) Inf-sup stable for velocity and pressure High order: good for accuracy Discontinuous pressure: good for solver & physics Edge oriented FEM for numerical stabilitation (Burman) Page 9

10 Discrete system Saddle point problem: u consists of all numerical variables except pressure Newton with multigrid as well-known solver Monolithic way of solving A B 0 B T u p = rhs u rhs p A consists of differential operators B is gradient operator Page 10

11 Newton iteration Newton for nonlinear system: Strongly coupled problem Automatic damping control ω n for each nonlinear step Black-box for many given viscoelastic models x n+1 = x n + ω n (xn ) 1 R(x n ) Quadratic convergence when iterative solutions are close Solution x n+1 = (u, p), Residual equation R(x n ) Black-box is made possible by divided difference technique (x n ) ii = R i x n + εe j R i x n + εe j 2ε Page 11

12 Multigrid iteration Multigrid for linearized system: Full-Vanka for strongly coupled Jacobian in local system Full prolongation Black-box for many given viscoelastic models u l+1 p l+1 = u l p l " + " ω A u Ω i kk Ωi B T Ω 0 i patch Ω i 1 defu Ωi def p Ωi Page 12

13 Numerical examples 1 Flow around cylinder: Mod. Gie FENE-P FENE-C WM-Lr WM-Cr WM-Ca LPPT/ XPPT Par α=0.01 α = 0, L2=100 α = 1, L2=100 l=0.01 k=0.01, l=0.01, m=0.01, n=0.01 a = 0.95, b = 0.95, k= 0.01, l = 0.01, m = 0.01, n = 0.01 Pom ε=0.01 α=0.01, ν = 0.2, r = 1 Page 13

14 Numerical examples 1 Flow around cylinder: Lev. Oldroyd-B Giesekus FENE-P FENE-CR LPTT R [5/1] [5/1] [5/1] [5/1] [5/1] R [5/1] [4/1] [5/1] [5/1] [5/1] R [3/1] [3/1] [3/1] [3/1] [3/1] R [2/2] [2/2] [2/2] [2/2] [2/2] R [2/2] [2/2] [2/2] [2/2] [2/2] XPTT WM-Larson WM-Cross WM-Carreau Pom-Pom R [5/1] [5/1] [5/1] [5/1] [4/1] R [5/1] [4/1] [5/1] [5/1] [5/1] R [3/1] [3/1] [3/1] [3/1] [3/1] R [2/2] [2/2] [2/2] [2/2] [2/2] R [2/2] [2/2] [2/2] [2/2] [2/2] Newton-multigrid behaviour for We=0.1 Page 14

15 Numerical examples 1 Flow around cylinder: We Oldroyd-B Giesekus FENE-P FENE-CR LPTT [2] [2] [2] [2] [2] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [4] [3] [4] [4] [3] [3] [3] [4] [4] [3] [3] [3] [4] [4] [3] [5] [5] [3] [3] [3] XPTT WM-Larson WM-Cross WM-Carreau Pom-Pom [2] [2] [2] [2] [4] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [4] [4] [4] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [4] Moderate number of nonlinear steps for all models Page 15

16 Numerical examples 1 Flow around cylinder: We=1.2 Different drag behaviour of different models at increasing We number Page 16

17 Numerical examples 2 Rising bubble in viscoelastic fluid: Material 1: Viscoelastic fluid described by the Oldroyd-B model Material 2: Newtonian fluid Page 17

18 Numerical examples 2 Rising bubble in viscoelastic fluid: A better visualisation from the data before. cheating bubbles Multiphase flow in a cylindrical coordinate system is ongoing Page 18

19 Numerical examples 3 3D flow around a sphere: An LCR based FEM solver for 3D viscoelastic flow Tests with Oldroyd-B for We=0.3 and 0.6 We=0.3 We=0.6 Page 19

20 Numerical examples 4 3D flow around cylinder: An LCR based FEM solver for 3D viscoelastic flow Tests with Oldroyd-B Invariance in z-direction for We=1.6, agreement with Sahin et. al Page 20

21 Numerical examples 5 Polymer stretching: A 2D+1 membrane model (Sollogoub et. Al) ee = 0 e 2μμ + 2μ tt D I = ee + U τ τ τ T = 1 λ f Level set-fem +(U )φ = 0 Source: Schöppner, Wibbeke 2012 Page 21

22 Numerical examples 5 Polymer stretching: A 2D+1 membrane model from Sollogoub et. al 0,25 0,2 0,15 0,1 0, ,5 1 L2 Sollogoub L3 1,5 1 0, ,5 1 L2 Sollogoub L3 Page 22

23 Conclusion We have presented: LCR-based viscoelastic models Higher order FEM discretizations Black-box Newton-multigrid solver Numerical examples: o 2D benchmark flow around cylinder o Rising bubble surrounded by viscoelastic fluid o 3D solver for LCR-based viscoelastic models o Polymer stretching We would like in the future: 3D viscoelatic multiphase Collection of different viscoelastic models FBM and viscoelastic integral model Page 23

24 Thank you for listening! Page 24