Grid and Particle Based Methods for Complex Flows - the Way

Forward

Tim PhillipsCardiff 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

T3T2

T1

T6

T5

T4l

fe triangular elementfv 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 , j i , ji - 1 , j i + 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

g q r

0.0038946 72006 1 7 0.3

0.05139 15770 1 5 0.3

0.50349 3334 2 3 0.15

4.5911 300.8 10 1.1 0.03

Data is of DSM LDPE Stamylan LD2008 XC43, Scanned from Verbeeten et. al. J Non-Newtonian Fluid mech. (2002)

Dimensionless parameters are:

/i i i pg %

For U=1 and where

We q 1/r

0.0038946 0.067567 1 0.142857 0.3

0.05139 0.195259 1 0.2 0.3

0.50349 0.404442 2 0.333333 0.15

4.5911 0.332732 10 0.909091 0.03

0.3i

iq

sii

bi

Backbone Stretch – Max We=3.15

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

)(QQFI

Dumbbell Models

2

2)(

2

1)(

2

1)(

QQFQt

tc

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

Hookean FENE FENE-P

21 /Q bQ

21 /Q bQ

3

)()()(R

dQQQfQf

Q

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)

1, , ) , , ,

2t t t t t

t

Q A Q Q D Q QQ Q Q

, ,d t t t dt t t d t Q A Q B Q W

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

jj

jjjjj W

tt

btQ

tQtQttQtQ

/)(1

)(

2

1)()()()(

21

jj

jjjjj

jjj

Wt

tbtQ

tQtQttQt

tQtQbtQ

t

/)(1

)(

2

1)()()()(

2

1

)()(/)(14

11

211

11

2

Two Dimensional Eccentrically Rotating Cylinder Problem

RJ

RB

ex

y

= 1,s = 0.1, p = 0.8, t = 0.01, = 0.3,Nf = 10000.

k = 4,N = 6,RB = 2.5,RJ = 1.0,e = 1.0, = 0.5,

A

Force Evolution results for the Eccentrically Rotating Cylinder Model

Oldroyd B vs Hookean

0 5 10 15

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Time0 5 10 15

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 5 10 15

0.0

0.1

0.2

0.3

0.4

Time

Time

Fx

Fy Torque

FENE and FENE-P Modelsλ=1, ω=2, b=50

FENE and FENE-P Modelsλ=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.