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 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, …

FE/FV spatial discretisation and median dual cell FV control volume and MDC for FE/FV FE with 4 fv sub-cells for FE/FV T3T3 T2T2 T1T1 T6T6 T5T5 T4T4 l fe triangular element fv triangular sub-cells fe vertex nodes (p, u,  ) fe midside nodes (u,  ) fv vertex nodes (  ) Finite Volume Grid for SLFV i, j + 2 i, j - 2 i, j + 1 i + 2, ji - 2, ji, ji - 1, ji + 1, j i, j - 1 SLFV spatial discretisation U V P,  xx,  yy,  xy

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

The eXtended pom-pom model parameters  gqr  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 We  q  1/r 

Backbone Stretch – Max We=3.15

Dynamics of Polymer Solutions Microscopic 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

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. Q The dimensionless form of the Fokker-Planck equation for homogeneous flows is

Force Laws HookeanFENEFENE-P

General Form of the Dimensionless Fokker-Planck Equation Equivalent SDE (see Öttinger (1995)) where D(Q(t),t) = B(Q(t),t)  B T (Q(t),t)

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.

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

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

Semi-Implicit Algorithm for the FENE Model

Two Dimensional Eccentrically Rotating Cylinder Problem  RJRJ RBRB e x y  = 1,  s = 0.1,  p = 0.8,  t = 0.01, = 0.3, Nf = k = 4, N = 6, R B = 2.5, R J = 1.0, e = 1.0,  = 0.5, A

Force Evolution results for the Eccentrically Rotating Cylinder Model Oldroyd B vs Hookean Time FxFx FyFy Torque

FENE and FENE-P Models λ=1, ω=2, b=50

FENE and FENE-P Models λ=3, ω=2, b=50

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.

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

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

t=3000 t=1500 t=2000 t=4000

t=6000 t=15000 t=8000 t=10000

t=20000 t=25000 t=30000

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.