08-10/08/2007 epfdc 2007, university of birmingham 1 flow of polymer solutions using the matrix...

08-10/08/200708-10/08/2007 EPFDC 2007, University of BirminghaEPFDC 2007, University of Birmingham m

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Flow of polymer solutions using Flow of polymer solutions using the matrix logarithm approach in the matrix logarithm approach in the spectral elements frameworkthe spectral elements framework

Giancarlo Russo, Giancarlo Russo, Prof.Prof. Tim Phillips Tim PhillipsCardiff School of MathematicsCardiff School of Mathematics


Outline

1)The model and the spectral framework 2) The log-conformation representation 3) Planar channel flow simulations 4) Flow past a cylinder simulation5) Some remarks about the code (discretization in time, upwinding, etc) 6) Things still to be sorted (hopefully SOON) and future work 7) References


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

( , ) ( , ) ( )

( , ) 0

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

b p w d T w l w

b q u

c T t d t u f u t

(1)

(2) 0

(3) ( ( ) ( )) (1 )T

uu u p F

tu

We u u u dt

Differential formulation Weak Formulation

f includes the UCD terms and b, c, d, and l are defined as follows :

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


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 The 2-D discretization process

, 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 solutions the Oldroyd-B model is often used as constitutive equation: often used as constitutive equation:

exp( )( ) ( ) 2 ( exp( ))

with

, ln

u B It We

We I

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 (see [2], [3]) : proposed by Fattal and Kupferman (see [2], [3]) :

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

1

2

2 12 1 21

2 1

11

22

( )

T T

T

T

R R R R

M MM R u R

MB R R

M

where the relative quantities are defined where the relative quantities 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 number, higher Weissenberg number, because advecting because advecting psi psi instead instead of of tau (or sigma) tau (or sigma) reduces the reduces the discrepancy in balancing discrepancy in balancing deformation through advection. deformation through advection.


11

1

1( ) ( )

1( ) ( )

, lnln( )

jn n n nj j j j j

a x b xt t We

a tt b

x Wea

xb We

VV

V

V

11

1

1( ) ( )

1[1 ( )] [ ]j jn n n

j j j j

j

j

a x b xt t Wea t a t

t bx We x

ax

b We

V VV

V V

V

The The exampleexample of a 1-D toy problem of a 1-D toy problem

Comparing the one dimensional problems on the left we can have an idea of how the use Comparing the one dimensional problems on the left we can have an idea of how the use of logarithmic transformation weakens the stability constraint. of logarithmic transformation weakens the stability constraint.

- a(x) plays the role of u (convection)a(x) plays the role of u (convection)

- b(x) is grad u instead (exponential growth)b(x) is grad u instead (exponential growth)


Results from the log-conformation channel flow: Re =1, beta=0.167, We =5, Results from the log-conformation channel flow: Re =1, beta=0.167, We =5, Parabolic Inflow/Outflow, 2 Elements, N=6Parabolic Inflow/Outflow, 2 Elements, N=6


Matrix-Logarithm Approach(left, usual OLD-B right) : Flow past a cylinder(1:1): Re =0. 1, We Matrix-Logarithm Approach(left, usual OLD-B right) : Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); =5, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p);

OLDROYD-B,(I, velocity)OLDROYD-B,(I, velocity)


Matrix-Logarithm Approach (left): Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Matrix-Logarithm Approach (left): Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); OLDROYD-Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); OLDROYD-

B,(I, velocity)B,(I, velocity)


Matrix-Logarithm Approach: Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Matrix-Logarithm Approach: Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); OLDROYD-Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); OLDROYD-

B,(II, pressure and pressure gradient)B,(II, pressure and pressure gradient)


Remarks about the code Remarks about the code

1

1 1 1

1

0(1)

Re( ) (2)

(3)(1 )

nN N

n T n nN N N N N N N N

n nN N N N N

D u

C E u D p g Bt

S B u h

In order to use (1) to eliminate the velocity, In order to use (1) to eliminate the velocity, we have to premultiply by the inverse of H , we have to premultiply by the inverse of H , and then by D; so we’ll finally obtain the and then by D; so we’ll finally obtain the following equation to recover the pressure. following equation to recover the pressure. The matrix acting on the pressure now is The matrix acting on the pressure now is known as the UZAWA (U) operator. known as the UZAWA (U) operator.

Re( ) , is the Helmoltz operator;N NC E H

t

1 1 1 1 1T n n

N N N N N N N ND H D p D H g D H B

-OIFS 1 (Operator Integration Factor Splitting, 1OIFS 1 (Operator Integration Factor Splitting, 1stst order) is used to discretize the material derivative of order) is used to discretize the material derivative of velocity; Euler ( 1velocity; Euler ( 1stst order ) for the stress in the const. eq.; order ) for the stress in the const. eq.;- LUST ( Local Upwinding Spectral Technique) is used ( see [1] ); - LUST ( Local Upwinding Spectral Technique) is used ( see [1] ); - To invert H, a Schur complement method is used to reduce the size of the problem, then a direct LU - To invert H, a Schur complement method is used to reduce the size of the problem, then a direct LU factorization is performed.factorization is performed.- To invert U, since is symmetric, Preconditioned Conjugate Gradient methods is used; - To invert U, since is symmetric, Preconditioned Conjugate Gradient methods is used; - The constitutive equation is solved via BiConjugate Gradient Stabilized (non-symm);- The constitutive equation is solved via BiConjugate Gradient Stabilized (non-symm);

UU


Future Work

- Fix the stress profiles and then calculate the correct drag ; - Testing the log-conformation method for higher We and different geometries;- Find a general expression of the constitutive equation to apply the matrix logarithm method to a broader class of constitutive models (i.e. XPP and PTT ) in order to simulate polymer melts flows;- Eventually join the free surface “wet” approach with the log-conformation method in a SEM framework to investigate the extrudate swell and the filament stretchingproblems


[1] OWENS R.G., CHAUVIERE C., PHILLIPS T.N., A locally upwinded spectral

technique for viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, 108:49-71, 2002.

[2] 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.

[3] 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.

[4] VAN OS R. Spectral Element Methods for predicting the flow of polymer solutions and melts, Ph.D. thesis, The University of Wales, Aberystwyth, 2004.

References References

08-10/08/2007 epfdc 2007, university of birmingham 1 flow of polymer solutions using the matrix...

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