Transcript 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.

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
T3
T4
fe triangular element
fv triangular sub-cells
T2
fe vertex nodes (p, u,  )
l
fe midside nodes (u,  )
T1
T5
fv vertex nodes ( )
T6
FE with 4 fv sub-cells for FE/FV
FV control volume and MDC for FE/FV
i,j+2
SLFV spatial discretisation
U
i,j+1
V
P, xx, yy,
xy
Finite Volume Grid for SLFV

i-2,j
i-1,j
i,j
i,j-1
i,j-2
i+1,j
i+2,j
SXPP, 4:1 planar contraction, salient
corner vortex intensity and cell size scheme, Re and We variation
Salient corner vortex intensity
Salient corner vortex cell size
 = 1/9,  = 1/3,  = 0.15, q = 2.
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:

We
For U=1 and where
1/r
q
%i  gi i / p

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
i 
0.3
qi
i 
si
bi
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
  I  QF(Q)
Dumbbell Models
Q
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
2



1 (c)
1


  (t )Q  F (Q)  
2
t
2
  2 Q

Force Laws
Hookean
FENE
FENE-P
Q
Q
2
1 Q / b
Q
1  Q2 / b
f (Q) 
 f (Q) (Q)dQ
R3
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)
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

Q(t j ) 
1
t
Q (t j 1 )  Q(t j )   (t j )Q(t j ) 
W j
 t 
2
2 1  Q (t j ) / b 




1
t
1 
Q(t j 1 )  Q(t j )
2
 4 1  Q (t j 1 ) / b 
Q(t j ) 
1
1
t
  (t j 1 )Q (t j 1 )   (t j )Q(t j ) 
W j
 t 
2
2 
2 1  Q (t j ) / b 

Two Dimensional Eccentrically Rotating
Cylinder Problem
y
A
x
e
w
RJ
RB
k = 4,
N = 6,
RB = 2.5,
RJ = 1.0,
e = 1.0,
w = 0.5,
r
s
p
t

Nf
= 1,
= 0.1,
= 0.8,
= 0.01,
= 0.3,
= 10000.
Force Evolution results
for the Eccentrically
Rotating Cylinder
Model
0.4
0.3
Fx
0.2
0.1
0.0
0
Oldroyd B vs Hookean
Fy
5
10
15
10
15
Time
4.5
0
4.0
-1
3.5
-2
3.0
-3
2.5
Torque -4
2.0
-5
1.5
-6
1.0
-7
0.5
-8
-9
0.0
0
5
10
Time
15
0
5
Time
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=1500
t=2000
t=3000
t=4000
t=6000
t=10000
t=8000
t=15000
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.