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.