13/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory 1 Spectral Elements Method for free surface and viscoelastic flows Giancarlo Russo, supervised.

13/04/200713/04/2007 UK Numerical Analysis Day, Oxford CoUK Numerical Analysis Day, Oxford Computing Laboratory mputing Laboratory

11

Spectral Elements Method for Spectral Elements Method for free surface and viscoelastic free surface and viscoelastic

flowsflows

Giancarlo Russo, Giancarlo Russo, supervised bysupervised by

Prof. Tim PhillipsProf. Tim Phillips


Outline • Theoretical Issue: Theoretical Issue: existence and uniqueness of a solution for a steadystate die swell problem

From the free to the fixed boundarySetting up the problems with new variables and new operatorNecessary and sufficient conditions for a solution to exist Uniqueness

• Numerical Issue: Numerical Issue: Matrix-Logarithm approach and free surface’s tracking

The log-conformation representationChannel flow test Free surface tracking for die swell and filament stretching.


The ProblemThe Problem

0

0

2

(0, ) ( )

( , ( )) 0

( ( , ( ))) ( , ( )) 0

( ) ( , ( ))

(0)

t

die

p T F

u

T d

u y u y

u x x n

p x x I n T x x n

x u x xx

R

( 1)

( 2)

C

C

inin

inin

onon

onon

2, 21 2

{( , ) ((0, ) (0, ( ))} {( , ) ((0, ) (0,1)}

{ (0, ) : 0 ( ) , ( ) , ( ) }fix

die

From x y X x to x y X

Ad W X R x K D x K D x K

Domain change and set of admissible functions for the free surfaceDomain change and set of admissible functions for the free surface


New Variables New Variables

( , ( ) ), ( , ( ) )

( , ( ) )

u x x y p x x y

and x x y

( )

( )

1

( ( ) ) ( )

D xy

x x x y

x y x y

,

0,

2

p div T F in

div u in

T d

New Differential OperatorsNew Differential Operators New EquationsNew Equations

New Operators New Operators

1 2 1 2

( )1

( )

10

( )

:[ ( )] [ ( )

(3)

(4

(]

)

5)

T

div G div

G

with

D xy

xG

x

G H H

o

o

The modified problemThe modified problem

Compatibility conditions for existence and uniqueness of a solutionCompatibility conditions for existence and uniqueness of a solution

201 2

1 200

1 202 4

2 4

21 0( )

[ ( )] [ ( )]

1 22 0[ ( )]

[ ( )][ ( )]

sup , ( )

:sup , [ ( )]

Lu H H

HL

L

div u qdq q L

u

u du u H


Key steps for conditions (1) and (2) to be satisfiedKey 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 The proof of condition (1) in the case of the usual divergence lies on the application that the operator operator

2 20 0: ( ), with ( ) : ( )div S L S q L u qd

is an isomorphism, and the whole point is the application of the divergence theorem. 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.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: Using the hypotesis on the domain and the free surface, we can bound the following quantities:

1 1

1

2 2 2 2

2 2 2

( ) ( ) ( ) ( )[ ( )] [ ( )] 2 [ ( )] 2 [ ( )]

( ) ( ) ( ) ( )

1 1[ ( )] [ ( )]

( ) ( )

fix fix fix fix

fix

x xx x y y y yH H

y y yH

w wD x D x D x D xGw w yw yw yw yw

x x y x x x x y x x

Gw w wx x y x


So far we have proved that, picked a free surface map, the corresponding problem (with that 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 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 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 fixed point theorem we have to prove that the following operator

0

: , : , ( ) ( ) ( , (1 ))x

dieE Ad Ad E E OpEx x R u d a

Existence and uniqueness of a solution for the free Existence and uniqueness of a solution for the free boundary problem boundary problem

This operator has to be: This operator has to be: 1) CONTINOUS1) CONTINOUS 2)COMPACT2)COMPACT 3)A CONTRACTION3)A CONTRACTION

1) CONTINUITY: 1) CONTINUITY: consider the following sequence, supposing it converges strongly in the consider the following sequence, supposing it converges strongly in the

1, (0, )-normW X

1

2

0

in (0, )

in L (0, )

in C (0, )

weaklyN

weaklyN

N

u u H X

u uX

x x

u u X

,N Ad let’s say. There will be a corresponding sequence ulet’s say. There will be a corresponding sequence uNN satisfying the intial (weak) problem, and which will have the following satisfying the intial (weak) problem, and which will have the following properties: properties:

From the definition of the operator E, and taking the limit of the sequence From the definition of the operator E, and taking the limit of the sequence of weak problems, we can deduce that of weak problems, we can deduce that

,NE Ad E

namely E is continuous.namely E is continuous.


2)COMPACTNESS : 2)COMPACTNESS : the operator E is the composition of a continuous function and a compact embedding, the operator E is the composition of a continuous function and a compact embedding,

therefore is compact. More precisely: therefore is compact. More precisely: 1 0 0: ( ) ( , ) ( ) ( ) ( ,1) (0, ) ( )compact

fix fixE x Ad u x y H C u C X x

Existence and uniqueness of a solution for the free Existence and uniqueness of a solution for the free boundary problem IIboundary problem II

3)CONTRACTION : 3)CONTRACTION : We have to prove that the operator E is a contraction, namely: We have to prove that the operator E is a contraction, namely:

1, 1,2 1 2 1(0, ) (0, )( ) , with lim ( ) 0.

W X W X x XE E g x g x

If we write If we write

1

0

( ,1) ( ,1)u

u dyy

and combine with and combine with (OpE) (OpE) we obtainwe obtain

1,2 1

2 1 (0, )( )

W XL

u uE E C

y y

Finally, expanding the continuity equation and since G is invertible, we write Finally, expanding the continuity equation and since G is invertible, we write

1,

( )

2 1 2 2 1 1, , 2 1 (0, )

2 1( )

( ) ( )( , ) ( , )

( ) ( )L

Y Y

K Rdie x W XL

u u D x u D x uK y x y y x y C

y y x y x y

We remark that the y-component of the velocity field eventually vanishes when we We remark that the y-component of the velocity field eventually vanishes when we approach the total relaxation stress configuration. approach the total relaxation stress configuration.


Setting up the spectral approximation: the weak formulation for the Oldroyd-B Setting up the spectral approximation: the weak formulation for the Oldroyd-B modelmodel

( , ) ( , ) ( ),

( , ) 0,

( , ) 2 ( , ) ( , )

b p w d T w l w

b q u

c T t d T u h t

2 1 20

2 4 2 4

2 4 1 2

1 2

: ( ) [ ( )] , ( , ) ( ) ,

:[ ( )] [ ( )] , ( , ) : ,

:[ ( )] [ ( )] , ( , ) : ,

:[ ( )] , ( ) .

b L H b r v v rd

c L L c S s S sd

d L H d S u S ud

l H l u F u

r r

% %% %r r% %rr r

2 1 2 2 40( , , ) ( ) [ ( )] [ ( )]bdryp u T L H L

r %We look for 2 1 2 2 40( , , ) ( ) [ ( )] [ ( )]q w t L H L

r %such that for all

the following equations are satisfied :

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 :

Remark: 1 2[ ( )]bdryH 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) .

(5)

(6)


The 1-D discretization processThe 1-D discretization process(note: all the results are obtained for N=5 and an error tolerance of 10°-05 in the CG routine)(note: all the results are obtained for N=5 and an error tolerance of 10°-05 in the CG routine)

• The spectral (Lagrange) basis :The spectral (Lagrange) basis :

2(1 ) ( )( ) , 0,

( 1) ( )( )N

iN i i

Lh i N

N N L

0

1

1

0

( ) ( ), 1, 2

( ) ( ), 1

( ) ( ), , 1, 2

i

i

i

Nk kN i

i

Nk kN i

i

Nkl klN i

i

u u h k

p p h k

h k l

• Approximating the solution: replacing velocity, pressure and stress by the following expansions and the integral by Approximating the solution: replacing velocity, pressure and stress by the following expansions and the integral by a Gaussian quadrature on the Gauss-Lobatto-Legendre nodes, namely the roots of L’(x), a Gaussian quadrature on the Gauss-Lobatto-Legendre nodes, namely the roots of L’(x), (5)(5) becomes a linear becomes a linear system: system:

( ) , ( 3) (7) 12xu x e u u


The 2-D discretization process IThe 2-D discretization process I

, 0

1

, 1

, 0

( , ) ( ) ( ), 1, 2

( , ) ( ) ( ), 1

( , ) ( ) ( ), , 1, 2

i

i

i

Nk kN ij j

i j

Nk kN ij j

i j

Nkl klN ij j

i j

u u h h k

p p h h k

h h k l

• The 2-D spectral (tensorial) expansion :The 2-D spectral (tensorial) expansion :


To model the dynamics of polymer To model the dynamics of polymer solutions the Oldroyd-B model is often solutions the Oldroyd-B model is often used as constitutive equation: used as constitutive equation:

1( ) ( ) 2 ( exp( ))

with

ln

u B It We

The Oldroyd-B model and the log-conformation representationThe Oldroyd-B model and the log-conformation representation A new equivalent constitutive equation is A new equivalent constitutive equation is

proposed by Fattal and Kupferman: proposed by Fattal and Kupferman:

( ( ) ( )) (1 )TWe u u u dt

1

2

11

22

2 12 1 21

2 1

( )

T

T

T

T

R R

M R u R

MB R R

M

R R

M M

Where the Where the relative relative quantities quantities are defined are defined as follows: as follows:

The main aim of this new The main aim of this new approach is the chance of approach is the chance of modelling flows with much modelling flows with much higher Weissenberg higher Weissenberg number, because it looks number, because it looks like the oscillations due to like the oscillations due to the use of polynomials to the use of polynomials to approximate exponential approximate exponential behaviours are deeply behaviours are deeply reduced. reduced.


First results from the log-conformation channel flow: Re =1, We =5, Parabolic First results from the log-conformation channel flow: Re =1, We =5, Parabolic Inflow/Outflow, 2 Elements, N=6Inflow/Outflow, 2 Elements, N=6


In the figure we remark the approach we are going to follow to track the free surfaces in the die swell 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 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 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 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 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.approach has been proposed by Webster & al. in a finite differences context.

Free surface problems: a complete “wet” approachFree surface problems: a complete “wet” approach


Future Work •Analyze the unsteady free surface die swell problem Testing the log-conformation method for higher We and different geometriesImplementing the free surface “wet” approach in a SEM framework for the die swell and the filament stretchingEventually join the latter with the log-conformation method for the constitutive equation


[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: 281-285.

[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 free-surface techniques (CM and VOF), J. Non-Newtonian Fluid Mechanics, 137 (2006): 81-102.

References References

13/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory 1 Spectral Elements Method for free surface and viscoelastic flows Giancarlo Russo, supervised.

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