FEM techniques for nonlinear fluids

Transcription

1 FEM techniques for nonlinear fluids From non-isothermal, pressure and shear dependent viscosity models to viscoelastic flow A. Ouazzi, H. Damanik, S. Turek Institute of Applied Mathematics, LS III, TU Dortmund IAMCS Workshop: Complex Fluid Dynamics March 22 25, 2010 KAUST, Thuwal, Saudi Arabia Page 1

2 Complex fluid dynamics Page 2

3 Behavior of dense granular material Axial flow experiment in the Couette device: spherical glass beads, 0.1 mm in diameter. Axial flow device Rotating Cylinder Normal Sterss Sensor Stationary outer wall The powder can transit from the quasi-static to the intermediate regime as the shearing rate is increased Pressure and shear dependent viscosity Page 3

4 Solidification Heat transfer for solidification Release of latent heat due the phase-change Crystallization over a large temperature range Due to the friction, the temperature increases Enthalpy method with friction Page 4

5 Non-Newtonian phenomena Effects due to normal stresses Effects due to elongational viscosity The drag reduction phenomenon Differential models Page 5

6 Governing equations Generalized Navier-Stokes equations ρ ( ) t + u u σ + p = ρf(x, y, Θ), u =0, ρ c p ( t + u ) Θ (k Θ) D(u) :σ = ρg(θ), σ = σ s + σ p, D(u) = 1 2 ( u +( u) T ). Viscous stress Elastic stress Viscoelastic flow models Page 6

7 Quasi-Newtonian models Viscous stress Power law model Powder flow in the quasi-static and intermediate regimes η s (z, p, Θ) = 2 p [sin φz b cos φz r 1 2 ] Non-isothermal model (φ is angle of internal friction, r > 1) η s (z, p, Θ) =η 0 e a 1 + a 2 a 3 +Θ (b 1 + b 2 z) r 2 1 (a i,b j are material parameters, r > 1) Page 7

8 Enthalpy model The source term The latent heat function Conservation of heat k(θ) := H L (Θ) = g(θ) = ( ) t + u H L (Θ) 0, Θ < Θ s ( ) t + u Θ (k(θ) Θ) D(u) :σ =0, k ρ(c p + L/(Θ L Θ s )), Θ s Θ < Θ L L Θ Θ s Θ L Θ s, L, Θ > Θ L k ρc p, Θ s Θ < Θ L otherwise Page 8

9 Constitutive models Elastic stress (Oldroyd/Maxwell/Jeffreys) Upper/Lower convective derivative Problems Blow up phenomena for time dependent problems High Weissenberg Number Problem (HWNP!) Page 9

10 HWNP Different highly developed models Oldroyd A/B, Maxwell A/B, Jeffreys Phan-Thien Tanner, Phan-Thien, Giesekus Different numerical methods FEM, FVM, FDM, DEVSS,DG, SUPG HWNP remains Kinetic energy for two different We numbers Zoom shows oscillation..!! Reformulation Problem reformulation Page 10

11 Conformation tensor reformulation Conformation Tensor (Oldroyd-B) (Lee & Xu) Using the identity Change of variable Conformation tensor reformulation Rate type expression Integral expression positive definite of exponential type Positivity preserving discretizations Page 11

12 LCR formulation Conformation reformulation (Fattal & Kupferman) The diagonalizing transformation Transformation and decomposition of velocity gradient The symmetric part The anti-symmetric part Page 12

13 LCR formulation Conformation reformulation (Fattal & Kupferman) New conformation tensor reformulation Log Conformation Reformulation (LCR) Change of variable Positivity preserving via LCR Page 13

14 LCR equations ρ u t + ρ u u (2η s(d II, p, Θ)D(u)) + p exp(σ lc )=ρf(x, y, Θ), u =0, Θ t + u Θ (k(θ) Θ) 2η str(d(u) 2 ) D(u) : exp(σ lc )=0, ( ) ( ) t + u σ lc W σ lc σ lc W 2 G 1 We exp( σlc )= η p We 2 I. Page 14

15 Variational formulations Standard Navier-Stokes New non-symmetric bilinear forms due to LCR Page 15

16 Variational formulations New nonlinear tensor variational form due to LCR Energy equation with friction Source terms Page 16

17 Problem formulation Set Find such that K = [ Ã B B T 0 ] Typical saddle point problem! Page 17

18 Compatibility conditions Compatibility condition for existance and uniqueness What about LCR? Page 18

19 Compatibility conditions for LCR The `NEW non-symmetric bilinear forms due to LCR c(τ,v)= Ω exp(τ) : D(v) dω β 2 exp(τ) 0,Ω v 1,Ω β 2 τ 0,Ω v 1,Ω τ T PD, v [H 1 0 (Ω)] 2 Page 19

20 FEM Discretization High order approximations for velocity-stress-temperaturepressure Advantages: Inf-sup stable for velocity and pressure High order: good for accuracy Discontinuous pressure: good for solver Disadvantage Stabilization for same approximation spaces for stress-velocity a single d.o.f. belongs to four elements Compatibility condition between the stress and velocity spaces via EO-FEM! Page 20

21 EO-FEM Edge-oriented stabilization for Same finite element interpolation velocity and stress convective dominated problem Efficient Newton-type and multigrid solvers can be easily applied! Page 21

22 Higher order nonconforming FEM Larger FE space which allows the approximation of singular sloution d.o.f.s belong to at most two elements which is good for parallelism Coupling of different polynomial order Mortar condition: test space order at slave side E 1 v h K2 L E,k ds = E 1 E No hanging nodes E v h K1 L E,k ds, 0 k<2 Research in progress! Page 22

23 Nonlinear solver Newton with damping results in the solution of the form Inexact Newton The Jacobian matrix is approximated using finite differences Typical saddle point problem! Page 23

24 Linear solver Monolithic multgrid solver Standard geometric multigrid approach Full restrictions and prolongations Local MPSC via Vanka-like smoother Coupled Monolithic Multigrid Solver! Page 24

25 Powder flow Experimental and numerical results for dry, frictional powder flows in the quasi-static and intermediate regimes The numerical method do not introduce errors! Page 25

26 Non-isothermal flow Solidification Enthalpy method for binary alloy solidification The condition of zero velocity in solid regions is accounted with Temperature dependent viscosity Fictitious boundary Viscous dissipation due to flow Page 26

27 Viscoelastic flow: lip vortex growth Reentrant corner singularities: 4 to1 contraction (Oldroyd-B) The numerical method reproduce the lip vortex in contraction flow! Page 27

28 Viscoelastic benchmark Planar flow around cylinder (Oldroyd-B) The numerical method is quantitatively validated Page 28

29 Viscoelastic benchmark Axial stress w.r.t. X-curved: Oldroyd-B vs. Giesekus The lack of pointwise mesh convergence! Page 29

30 Solvers M-FEM Newton-Multgrid solution Oldroyd-B vs. Giesekus Oldroyd-B Giesekus Stable Newton-multigrid solver! Page 30

31 Summary New numerical and algorithmic tools are available using Monolithic Finite Element Method (M-FEM) Log Conformation Reformulation (LCR) Edge Oriented stabilization (EO-FEM) Fast Multigrid Solver with local MPSC smoother for the simulation of nonlinear fluids from non-isothermal and shear-dependent models to viscoelastic flow Advantages No CFL-condition restriction due to the ful coupling Positivity preserving Large order and local adaptivity Page 31

32 Complex Fluid dynamics Page 32