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Grid and Particle Based Methods for Complex Flows - the Way Forward

Grid and Particle Based Methods for Complex Flows - the Way Forward. Tim Phillips Cardiff University EPSRC Portfolio Partnership on Complex Fluids and Complex Flows. Dynamics of Complex Fluids 10 Years On. Grid-Based Methods.

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Grid and Particle Based Methods for Complex Flows - the Way Forward

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  1. Grid and Particle Based Methods for Complex Flows - the Way Forward Tim Phillips Cardiff University EPSRC Portfolio Partnership on Complex Fluids and Complex Flows Dynamics of Complex Fluids 10 Years On

  2. Grid-Based Methods • Finite difference, finite element, finite volume, spectral element methods • Traditionally based on macroscopic description • Characterised by the solution of large systems of algebraic equations (linear/nonlinear) • Upwinding or reformulations of the governing equations required for numerical stability e.g. SUPG, EEME, EVSS, D-EVSS, D-EVSS-G,log of conformation tensor, …

  3. fe triangular element fv triangular sub-cells fe vertex nodes (p, u, ) fe midside nodes (u, ) i , j + 2 fv vertex nodes () U T3 i , j + 1 V T2 T4 l P, xx, yy,  i - 2 , j i - 1 , j i , j i + 1 , j i + 2 , j T1 xy T5 i , j - 1 T6 i , j - 2 FE/FV spatial discretisation and median dual cell FE with 4 fv sub-cells for FE/FV FV control volume and MDC for FE/FV SLFV spatial discretisation Finite Volume Grid for SLFV

  4. SXPP, 4:1 planar contraction, salient corner vortex intensity and cell size - scheme, Re and We variation Salient corner vortex cell size Salient corner vortex intensity  = 1/9,  = 1/3,  = 0.15, q = 2.

  5. The eXtended pom-pom model parameters Data is of DSM LDPE Stamylan LD2008 XC43, Scanned from Verbeeten et. al. J Non-Newtonian Fluid mech. (2002) Dimensionless parameters are: For U=1 and where

  6. Backbone Stretch – Max We=3.15

  7. Dynamics of Polymer SolutionsMicroscopic Formulation • The stress depends on the orientation and degree of stretch of a molecule • Coarse-grained molecular model for the polymers is derived neglecting interactions between different polymer chains • Polymeric stress determined using the Kramers expression

  8. Q Dumbbell Models Two beads connected by a spring. The equation of motion of each bead contains contributions from the tension force in the spring, the viscous drag force, and the force due to Brownian motion. The dimensionless form of the Fokker-Planck equation for homogeneous flows is

  9. Force Laws

  10. General Form of the Dimensionless Fokker-Planck Equation Equivalent SDE (see Öttinger (1995)) where D(Q(t),t) = B(Q(t),t) ·BT(Q(t),t)

  11. Fokker-Planck v. Stochastic Simulations • Stochastic simulation techniques are CPU intensive, require large memory requirements and suffer from statistical noise in the computation of p (Chauvière and Lozinski (2003,2004)) • The competitiveness of Fokker-Planck techniques diminishes for flows with high shear-rates. • Fokker-Planck techniques are restricted to models with low-dimensional configuration space due to computational cost – but see recent work of Chinesta et al. on reduced basis function techniques.

  12. Micro-Macro Techniques • CONNFFESSIT – Laso and Ottinger • Variance reduction techniques • Lagrangian particle methods – Keunings • Method of Brownian configuration fields - Hulsen

  13. Method of Brownian Configuration Fields • Devised by Hulsen et al (1997) to overcome the problem of tracking particle trajectories • Based on the evolution of a number of continuous configuration fields • Dumbbell connectors with the same initial configuration and subject to same random forces throughout the domain are combined to form a configuration field • The evolution of an ensemble of configuration fields provides the polymer dynamics

  14. Semi-Implicit Algorithm for the FENE Model

  15. x e w RJ RB Two Dimensional Eccentrically Rotating Cylinder Problem y k = 4, N = 6, RB = 2.5, RJ = 1.0, e = 1.0, w = 0.5, • r = 1, • hs = 0.1, • hp = 0.8, • Dt = 0.01, • = 0.3, Nf = 10000. A

  16. Force Evolution results for the Eccentrically Rotating Cylinder Model Oldroyd B vs Hookean Fx Time Fy Torque Time Time

  17. FENE and FENE-P Modelsλ=1, ω=2, b=50

  18. FENE and FENE-P Modelsλ=3, ω=2, b=50

  19. Particle Based Methods • Lattice Boltzmann Method - characterised by a lattice and some rule describing particle motion. • Smoothed Particle Hydrodynamics – based on a Lagrangian description with macroscopic variables obtained using suitable smoothing kernels.

  20. D2Q9 Lattice • 9 velocity model. • Allows for rest particles. • Multi speed model. • Isotropic.

  21. Spinodal Decomposition(density ratio=1, viscosity ratio=3)

  22. t=2000 t=1500 t=4000 t=3000

  23. t=6000 t=8000 t=15000 t=10000

  24. t=20000 t=25000 t=30000

  25. Particle Methods for Complex Fluids • Extension of LBM – possibly using multi relaxation model by exploiting additional eigenvalues of the collision operator or in combination with a micro approach to the polymer dynamics. • Extension of SPH to include viscoelastic behaviour.

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