Transcript Slide 1
Spectral Elements Method for
free surface and viscoelastic
flows
Giancarlo Russo, supervised by
Prof. Tim Phillips
13/04/2007
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Outline
• Theoretical Issue: existence and uniqueness of a solution for a
steadystate die swell problem
From the free to the fixed boundary
Setting up the problems with new variables and new operator
Necessary and sufficient conditions for a solution to exist
Uniqueness
• Numerical Issue: Matrix-Logarithm approach and free surface’s
tracking
The log-conformation representation
Channel flow test
Free surface tracking for die swell and filament stretching.
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The Problem
p T F
u 0
T 2 d
u (0, y ) u ( y )
u ( x, ( x)) n 0
( p ( x, ( x))) I n T ( x, ( x ))n 0
( x) ut ( x, ( x))
x
(0) Rdie
0
(C1)
(C 2)
Domain change and set of admissible functions for the free surface
From {( x, y ) ((0, X ) (0, ( x))}to fix {( x, y ) ((0, X ) (0,1)}
Ad { W 2, (0, X ) : 0 Rdie ( x) K , D ( x) K1 , D 2 ( x) K 2 }
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The modified problem
New Differential Operators
New Variables
div G o div(3)
o G T (4)
with
D ( x)
1
y
( x)
G
1
0
( x)
G :[ H 1 ()]2 [ H 1 ()]2 (5)
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p div T F ,in
div u 0,in
T 2 d
D ( x)
y
x
x
( x)
y
1
( ( x) y ) ( x) y
u( x, ( x) y), p( x, ( x) y)
and ( x, ( x) y)
New Operators
New Equations
Compatibility conditions for existence and uniqueness of a solution
div u qd
sup
u
u[ H 01 ( )]2
1
[ H 01 ( )]2
: u d
sup
[ L2 ( )]4
2
u
q
L20 ( )
[ H 01 ( )]2
, q L20 ()
, u [ H 01 ()]2
[ L2 ( )]4
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Key steps for conditions (1) and (2) to be satisfied
The proof of condition (1) in the case of the usual divergence lies on the application that the
operator
div : S L20 (),withS q L20 () : ( u)qd
is an isomorphism, and the whole point is the application of the divergence theorem.
To prove (2), which involves the gradient, we have to apply the Poincare’ inequality
instead. What we need then for these conditions to hold for our operators defined in (3)
and (4) is for G, as defined in (5), to be bounded.
Using the hypotesis on the domain and the free surface, we can bound the following
quantities:
Gwx
2
H
1
wx
2
H
1
fix
[
w D ( x)
w D ( x)
D ( x)
D ( x)
(
ywy )]2 [ (
ywy )]2 2 [ x (
ywy )] 2 [ x (
ywy )]
x ( x)
y
(
x
)
x
x
(
x
)
y
x
(
x
)
fix
fix
fix
Gwy
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2
H
1
fix
[
1
1
(
wy )]2 [ (
wy )]2
x ( x)
y ( x)
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Existence and uniqueness of a solution for the free
boundary problem
So far we have proved that, picked a free surface map, the corresponding problem (with
that surface) has got a unique solution; now we have to prove that such a map always
exists, which means that the Cauchy problem (C1)-(C2) has a unique solution. Applying
the Schauder’s fixed point theorem we have to prove that the following operator
x
E : Ad Ad ,E : a ,E ( x) ( x) Rdie u( ,1)d(OpE )
0
This operator has to be:
1) CONTINOUS
2)COMPACT
3)A CONTRACTION
1,
1) CONTINUITY: consider the following sequence, supposing it converges strongly in the W (0, X )-norm
N Ad ,
let’s say. There will be a corresponding sequence uN satisfying the intial (weak)
weakly
problem, and which will have the following properties:
u
uin H 1 (0, X )
N
From the definition of the operator E, and taking the limit of the sequence
of weak problems, we can deduce that
EN Ad E,
u N weakly u
in L2 (0, X )
x
x
u N uin C0 (0, X )
namely E is continuous.
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Existence and uniqueness of a solution for the free
boundary problem II
2)COMPACTNESS :
the operator E is the composition of a continuous function and a compact embedding,
therefore is compact. More precisely:
compact
E : ( x) Ad u( x, y) H 1 ( fix )
C 0 ( fix ) u( ,1) C 0 (0, X ) ( x)
3)CONTRACTION :
We have to prove that the operator E is a contraction, namely:
E2 E1
W 1, (0, X )
u ( ,1)
If we write
1
0
E2 E1
u2 u1
y
y
W
L ()
1,
(0, X )
C
g ( x) 2 1
W 1, (0, X )
, with lim g ( x) 0.
x X
u
( ,1)dy and combine with (OpE) we obtain
y
u2 u1
y
y
L ( )
Finally, expanding the continuity equation and
since G is invertible, we write
D2 ( x) u2Y
D1 ( x) u1Y
K
y
( x, y )
y
( x, y)
2 ( x)
y
1 ( x)
y
CK , Rdie, x 2 1
W 1, (0, X )
L ( )
We remark that the y-component of the velocity field eventually vanishes when we
approach the total relaxation stress configuration.
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Setting up the spectral approximation: the weak formulation for the Oldroyd-B
model
r
1
( p, u , T%) L20 () [Hbdry
()]2 [ L2 ()]4
We look for
r
(q, w, %
t ) L20 () [H 1 ()]2 [ L2 ()]4
such that for all
the following equations are satisfied :
b( p, w) d (T , w) l ( w),
b(q, u ) 0,
c(T , t ) 2 d (T , u ) (h, t )
(5)
where h includes the UCD terms (which are approximated with a 1st order OIFS /Euler) and b,
c, d, and l are defined as follows :
r
r
b : L20 () [ H 1 ()]2 , b( r , v ) ( v ) rd ,
c : [ L2 ()]4 [ L2 ()]4 , c( S%, %
s ) S%: %
sd ,
r
r
d : [ L2 ()]4 [ H 1 ()]2 , d ( S%, u ) S%: ud ,
r
r
r
l : [ H 1 ()]2 , l (u ) F u .
(6)
Remark: [H ()] it simply means the velocity fields has to be chosen according to the
boundary conditions, which in the free surface case are the ones given in (4) .
1
bdry
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The 1-D discretization process
(note: all the results are obtained for N=5 and an error tolerance of 10°-05 in the CG routine)
•
The spectral (Lagrange) basis :
(1 2 ) LN ( )
hi ( )
, i 0, N
N ( N 1) LN (i )( i )
•
Approximating the solution: replacing velocity,
pressure and stress by the following expansions and
the integral by a Gaussian quadrature on the GaussLobatto-Legendre nodes, namely the roots of L’(x),
(5) becomes a linear system:
N
u ( ) uik hi ( ),
k
N
u ( x) ex , u(3) u(7) 12
k 1, 2
i 0
N 1
pNk ( ) pik hi ( ),
k 1
i 1
N
( ) ikl h ( ),
kl
N
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i 0
i
k , l 1, 2
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The 2-D discretization process I
•
The 2-D spectral (tensorial) expansion :
u Nk ( , )
pNk ( , )
Nkl ( , )
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N
u h ( )h ( ),
i , j 0
k
ij
j
i
N 1
p h ( )h ( ),
i , j 1
k
ij
N
i , j 0
kl
ij
i
j
k 1, 2
k 1
hi ( )h j ( ), k , l 1, 2
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The Oldroyd-B model and the log-conformation representation
To model the dynamics of polymer
solutions the Oldroyd-B model is often
used as constitutive equation:
We(
t
u (u )T (u )) (1 )d
A new equivalent constitutive equation is
proposed by Fattal and Kupferman:
t
(u ) ( ) 2 B
1
( I exp( ))
We
with
ln
The main aim of this new
approach is the chance of
modelling flows with much
higher Weissenberg
number, because it looks
like the oscillations due to
the use of polynomials to
approximate exponential
behaviours are deeply
reduced.
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1
R
Where the
relative
quantities
are defined
as follows:
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T
R
2
M R (u ) R
M
B R 11
T
T
R
M 22
T
R
R
M 1M 21
2 12
2 1
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First results from the log-conformation channel flow: Re =1, We =5, Parabolic
Inflow/Outflow, 2 Elements, N=6
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Free surface problems: a complete “wet” approach
In the figure we remark the approach we are going to follow to track the free surfaces in the die swell
and filament stretching problems: this is a completely “wet” approach, it means the values of the
fields in blue nodes, the ones on the free surfaces, are extrapolated from the values we have in
the interior nodes (the black ones) at each time step. After a certain number of timesteps we then
redistribute the nodes to avoid big gaps between the free surface nodes an the neighbours. This
approach has been proposed by Webster & al. in a finite differences context.
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•
Future Work
Analyze the unsteady free surface die swell problem
Testing the log-conformation method for higher We and different geometries
Implementing the free surface “wet” approach in a SEM framework for the die swell
and the filament stretching
Eventually join the latter with the log-conformation method for the constitutive
equation
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References
[1] ISTRATESCU V.I., Fixed Point Theory, An Introduction, Redidel Publishing
Company, 1981.
[2] CABOUSSAT A. Analysis and numerical simulation of free surface flows, Ph.D.
thesis, ´Ecole Polytechnique Federale de Lausanne, Lausanne, 2003.
[3] GERRITSMA M.I., PHILLIPS T.N., Compatible spectral approximation, for the
velocity-pressure-stress formulation of the Stokes problem, SIAM Journal of
Scientific Computing, 1999, 20 (4) : 1530-1550.
[4] FATTAL R.,KUPFERMAN R. Constitutive laws for the matrix logarithm of the
conformation tensor , Journal of Non-Newtonian Fluid Mechanics,2004, 123: 281285.
[5] FATTAL R.,KUPFERMAN R. Time-dependent simulation of viscoelastic flows at
high Weissenberg number using the log-conformation representation , Journal
of Non-Newtonian Fluid Mechanics,2005, 126: 23-37,
[6] HULSEN M.A.,FATTAL R.,KUPFERMAN R. Flow of viscoelastic fluids past a
cylinder at high Weissenberg number: stabilized simulations using matrix
logarithms, Journal of Non-Newtonian Fluid Mechanics,2005, 127: 27-39.
[7] VAN OS R. Spectral Element Methods for predicting the flow of polymer
solutions and melts, Ph.D. thesis, The University of Wales, Aberystwyth, 2004.
[8] WEBSTER M., MATALLAH H., BANAAI M.J., SUJATHA K.S., Computational
predictions for viscoelastic filament stretching flows: ALE methods and freesurface techniques (CM and VOF), J. Non-Newtonian Fluid Mechanics, 137 (2006):
81-102.
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