ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Viscoelastic Flow Simulation in OpenFOAMPresentation of the viscoelasticFluidFoam Solver

Jovani L. Faverofavero@enq.ufrgs.br / jovani.favero@yahoo.com.br

Universidade Federal do Rio Grande do Sul - Department of ChemicalEngineering

http://www.ufrgs.br/ufrgs/

February 26, 2009

1 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

1 Introduction

2 Problem Definition

3 Constitutive Models

4 DEVSS and Solution Procedure

5 Solver Implementation

6 Using the Solver

7 Some Results

8 Conclusion

2 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

What about Viscoelastic Flows?

Understanding and modeling of viscoelastic flows areusually the key step in the definition of the finalcharacteristics and quality of the finished products inmany industrial sectors, such as in food and syntheticpolymers industries.

The rheological response of viscoelastic fluids is quitecomplex, including combination of viscous and elasticeffects and highly non-linear viscous and elasticphenomena.

Characteristics: Strain rate dependent viscosity, presenceof normal stress differences in shear flows, relaxationphenomena and memory effects, including die swell.

3 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

What about Viscoelastic Flows?

Understanding and modeling of viscoelastic flows areusually the key step in the definition of the finalcharacteristics and quality of the finished products inmany industrial sectors, such as in food and syntheticpolymers industries.

The rheological response of viscoelastic fluids is quitecomplex, including combination of viscous and elasticeffects and highly non-linear viscous and elasticphenomena.

Characteristics: Strain rate dependent viscosity, presenceof normal stress differences in shear flows, relaxationphenomena and memory effects, including die swell.

3 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

What about Viscoelastic Flows?

Understanding and modeling of viscoelastic flows areusually the key step in the definition of the finalcharacteristics and quality of the finished products inmany industrial sectors, such as in food and syntheticpolymers industries.

The rheological response of viscoelastic fluids is quitecomplex, including combination of viscous and elasticeffects and highly non-linear viscous and elasticphenomena.

Characteristics: Strain rate dependent viscosity, presenceof normal stress differences in shear flows, relaxationphenomena and memory effects, including die swell.

3 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Die Swell, Weissemberg Effect ...

(Loading viscoelastic.mpg)

4 / 59

viscoelastic.mpgMedia File (video/mpeg)

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Why OpenFOAM?

Its a Open Source CFD Toolbox build with a flexible set ofefficient C++ modules.

Ability of dealing with:

Complex geometries;

Unstructured, non orthogonal and moving meshes;

Large variety of interpolation schemes;

Large variety of solvers for the linear discretized system;

Fully and easily extensible;

Data processing parallelization among others benefits.

5 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Why OpenFOAM?

Its a Open Source CFD Toolbox build with a flexible set ofefficient C++ modules.

Ability of dealing with:

Complex geometries;

Unstructured, non orthogonal and moving meshes;

Large variety of interpolation schemes;

Large variety of solvers for the linear discretized system;

Fully and easily extensible;

Data processing parallelization among others benefits.

5 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Why OpenFOAM?

Its a Open Source CFD Toolbox build with a flexible set ofefficient C++ modules.

Ability of dealing with:

Complex geometries;

Unstructured, non orthogonal and moving meshes;

Large variety of interpolation schemes;

Large variety of solvers for the linear discretized system;

Fully and easily extensible;

Data processing parallelization among others benefits.

5 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Why OpenFOAM?

Its a Open Source CFD Toolbox build with a flexible set ofefficient C++ modules.

Ability of dealing with:

Complex geometries;

Unstructured, non orthogonal and moving meshes;

Large variety of interpolation schemes;

Large variety of solvers for the linear discretized system;

Fully and easily extensible;

Data processing parallelization among others benefits.

5 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Why OpenFOAM?

Its a Open Source CFD Toolbox build with a flexible set ofefficient C++ modules.

Ability of dealing with:

Complex geometries;

Unstructured, non orthogonal and moving meshes;

Large variety of interpolation schemes;

Large variety of solvers for the linear discretized system;

Fully and easily extensible;

Data processing parallelization among others benefits.

5 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Why OpenFOAM?

Its a Open Source CFD Toolbox build with a flexible set ofefficient C++ modules.

Ability of dealing with:

Complex geometries;

Unstructured, non orthogonal and moving meshes;

Large variety of interpolation schemes;

Large variety of solvers for the linear discretized system;

Fully and easily extensible;

Data processing parallelization among others benefits.

5 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Why OpenFOAM?

Its a Open Source CFD Toolbox build with a flexible set ofefficient C++ modules.

Ability of dealing with:

Complex geometries;

Unstructured, non orthogonal and moving meshes;

Large variety of interpolation schemes;

Large variety of solvers for the linear discretized system;

Fully and easily extensible;

Data processing parallelization among others benefits.

5 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Viscoelastic Fluid Flow Formulation

The governing equations of laminar, incompressible andisothermal flow of viscoelastic fluids are the equations ofconservation of mass (continuity):

(U) = 0

momentum:

(U)

t+ (UU) = p + S + P

and a mechanical constitutive equation that describes therelation between the stress and deformation rate.

6 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Viscoelastic Fluid Flow Formulation

The governing equations of laminar, incompressible andisothermal flow of viscoelastic fluids are the equations ofconservation of mass (continuity):

(U) = 0

momentum:

(U)

t+ (UU) = p + S + P

and a mechanical constitutive equation that describes therelation between the stress and deformation rate.

6 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Viscoelastic Fluid Flow Formulation

The governing equations of laminar, incompressible andisothermal flow of viscoelastic fluids are the equations ofconservation of mass (continuity):

(U) = 0

momentum:

(U)

t+ (UU) = p + S + P

and a mechanical constitutive equation that describes therelation between the stress and deformation rate.

6 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Viscoelastic Fluid Flow Formulation

where S are the solvent contribution to stress:

S = 2SD

S is the solvent viscosity and D is the deformation rate tensor:

D =1

2(U + [U]T )

The extra elastic contribution, corresponding to the polymericpart P , is obtained from the solution of an appropriateconstitutive differential equation.

7 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Viscoelastic Fluid Flow Formulation

where S are the solvent contribution to stress:

S = 2SD

S is the solvent viscosity and D is the deformation rate tensor:

D =1

2(U + [U]T )

The extra elastic contribution, corresponding to the polymericpart P , is obtained from the solution of an appropriateconstitutive differential equation.

7 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Viscoelastic Fluid Flow Formulation

where S are the solvent contribution to stress:

S = 2SD

S is the solvent viscosity and D is the deformation rate tensor:

D =1

2(U + [U]T )

The extra elastic contribution, corresponding to the polymericpart P , is obtained from the solution of an appropriateconstitutive differential equation.

7 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Important Definitions

1 Upper Convective Derivative of a generic tensor A:A =

D

DtA

hUT A

i [A U]

or for symmetric tensors:A =

D

DtA [A U] [A U]T

2 Lower Convective Derivative of a generic tensor A:

A =D

DtA + [U A] +

hA UT

i3 Gordon-Schowalter Derivative of a generic tensor A:

A =

D

DtA [UT A] [A U] + (A D + D A)

where: DDt A =t A + U A

8 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Important Definitions

1 Upper Convective Derivative of a generic tensor A:A =

D

DtA

hUT A

i [A U]

or for symmetric tensors:A =

D

DtA [A U] [A U]T

2 Lower Convective Derivative of a generic tensor A:

A =D

DtA + [U A] +

hA UT

i

3 Gordon-Schowalter Derivative of a generic tensor A:A =

D

DtA [UT A] [A U] + (A D + D A)

where: DDt A =t A + U A

8 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Important Definitions

1 Upper Convective Derivative of a generic tensor A:A =

D

DtA

hUT A

i [A U]

or for symmetric tensors:A =

D

DtA [A U] [A U]T

2 Lower Convective Derivative of a generic tensor A:

A =D

DtA + [U A] +

hA UT

i3 Gordon-Schowalter Derivative of a generic tensor A:

A =

D

DtA [UT A] [A U] + (A D + D A)

where: DDt A =t A + U A

8 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Important Definitions

1 Upper Convective Derivative of a generic tensor A:A =

D

DtA

hUT A

i [A U]

or for symmetric tensors:A =

D

DtA [A U] [A U]T

2 Lower Convective Derivative of a generic tensor A:

A =D

DtA + [U A] +

hA UT

i3 Gordon-Schowalter Derivative of a generic tensor A:

A =

D

DtA [UT A] [A U] + (A D + D A)

where: DDt A =t A + U A

8 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Important Definitions

1 Upper Convective Derivative of a generic tensor A:A =

D

DtA

hUT A

i [A U]

or for symmetric tensors:A =

D

DtA [A U] [A U]T

2 Lower Convective Derivative of a generic tensor A:

A =D

DtA + [U A] +

hA UT

i3 Gordon-Schowalter Derivative of a generic tensor A:

A =

D

DtA [UT A] [A U] + (A D + D A)

where: DDt A =t A + U A

8 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

Maxwell linear:

PK + KPKt

= 2PK D

UCM and Oldroyd-B:

PK + K PK = 2PK D

where K and PK are the relaxation time and polymerviscosity coefficient at zero shear rate, respectively.

9 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

Maxwell linear:

PK + KPKt

= 2PK D

UCM and Oldroyd-B:

PK + K PK = 2PK D

where K and PK are the relaxation time and polymerviscosity coefficient at zero shear rate, respectively.

9 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

White-Metzner (WM):

PK + K (IID) PK = 2PK (IID)D

where: (IID) = =

2D : D

Larson:

PK (IID) =PK

1 + aK IID;K (IID) =

K1 + aK IID

Cross:

PK (IID) =PK

1 + (kIID)1m ;K (IID) =

K

1 + (LIID)1n

Carreau-Yasuda:

PK (IID) = PK [1 + (kIID)a]

m1a ;K (IID) = K

[1 + (LIID)

b] n1

b

10 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

White-Metzner (WM):

PK + K (IID) PK = 2PK (IID)D

where: (IID) = =

2D : D

Larson:

PK (IID) =PK

1 + aK IID;K (IID) =

K1 + aK IID

Cross:

PK (IID) =PK

1 + (kIID)1m ;K (IID) =

K

1 + (LIID)1n

Carreau-Yasuda:

PK (IID) = PK [1 + (kIID)a]

m1a ;K (IID) = K

[1 + (LIID)

b] n1

b

10 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

White-Metzner (WM):

PK + K (IID) PK = 2PK (IID)D

where: (IID) = =

2D : D

Larson:

PK (IID) =PK

1 + aK IID;K (IID) =

K1 + aK IID

Cross:

PK (IID) =PK

1 + (kIID)1m ;K (IID) =

K

1 + (LIID)1n

Carreau-Yasuda:

PK (IID) = PK [1 + (kIID)a]

m1a ;K (IID) = K

[1 + (LIID)

b] n1

b

10 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

White-Metzner (WM):

PK + K (IID) PK = 2PK (IID)D

where: (IID) = =

2D : D

Larson:

PK (IID) =PK

1 + aK IID;K (IID) =

K1 + aK IID

Cross:

PK (IID) =PK

1 + (kIID)1m ;K (IID) =

K

1 + (LIID)1n

Carreau-Yasuda:

PK (IID) = PK [1 + (kIID)a]

m1a ;K (IID) = K

[1 + (LIID)

b] n1

b

10 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

White-Metzner (WM):

PK + K (IID) PK = 2PK (IID)D

where: (IID) = =

2D : D

Larson:

PK (IID) =PK

1 + aK IID;K (IID) =

K1 + aK IID

Cross:

PK (IID) =PK

1 + (kIID)1m ;K (IID) =

K

1 + (LIID)1n

Carreau-Yasuda:

PK (IID) = PK [1 + (kIID)a]

m1a ;K (IID) = K

[1 + (LIID)

b] n1

b

10 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

Giesekus:

PK + K PK + K

KPK

(PK . PK ) = 2PK D

FENE-P:1 + 3(13/L2K ) + KPK tr(PK )L2K

K +K PK = 2 1(1 3/L2K )PK D

FENE-CR:L2K + KPK tr(PK )(L2K 3)

K +K PK = 2L2K + KPK tr(PK )

(L2K 3)

PK D

11 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

Giesekus:

PK + K PK + K

KPK

(PK . PK ) = 2PK D

FENE-P:1 + 3(13/L2K ) + KPK tr(PK )L2K

K +K PK = 2 1(1 3/L2K )PK D

FENE-CR:L2K + KPK tr(PK )(L2K 3)

K +K PK = 2L2K + KPK tr(PK )

(L2K 3)

PK D11 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Kinetic Theory Models

Giesekus:

PK + K PK + K

KPK

(PK . PK ) = 2PK D

FENE-P:1 + 3(13/L2K ) + KPK tr(PK )L2K

K +K PK = 2 1(1 3/L2K )PK D

FENE-CR:L2K + KPK tr(PK )(L2K 3)

K +K PK = 2L2K + KPK tr(PK )

(L2K 3)

PK D11 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Network Theory of Concentrated Solutions andMelts Models

Phan-Thien-Tanner linear (LPTT):(1 +

KKPK

tr(PK )

)PK + K

PK = 2PK D

Phan-Thien-Tanner exponential (EPTT):

exp

(KKPK

tr(PK )

)PK + K

PK = 2PK D

Feta-PTT:(1 +

KK ()

PK ()tr(PK )

)PK + K ()

PK = 2PK ()D

where:

PK () =PK

1 + A

II

2k

2PK

affb ; K () = K1 + KK IPK

12 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Network Theory of Concentrated Solutions andMelts Models

Phan-Thien-Tanner linear (LPTT):(1 +

KKPK

tr(PK )

)PK + K

PK = 2PK D

Phan-Thien-Tanner exponential (EPTT):

exp

(KKPK

tr(PK )

)PK + K

PK = 2PK D

Feta-PTT:(1 +

KK ()

PK ()tr(PK )

)PK + K ()

PK = 2PK ()D

where:

PK () =PK

1 + A

II

2k

2PK

affb ; K () = K1 + KK IPK

12 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Network Theory of Concentrated Solutions andMelts Models

Phan-Thien-Tanner linear (LPTT):(1 +

KKPK

tr(PK )

)PK + K

PK = 2PK D

Phan-Thien-Tanner exponential (EPTT):

exp

(KKPK

tr(PK )

)PK + K

PK = 2PK D

Feta-PTT:(1 +

KK ()

PK ()tr(PK )

)PK + K ()

PK = 2PK ()D

where:

PK () =PK

1 + A

II

2k

2PK

affb ; K () = K1 + KK IPK

12 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Pom-Pom model:Evolution of Orientation:

SPK + 2[D : SPK ]SPK +

1

OBK

[SPK

1

3I

]= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PKK

(32PK SPK I )

13 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Pom-Pom model:Evolution of Orientation:

SPK + 2[D : SPK ]SPK +

1

OBK

[SPK

1

3I

]= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PKK

(32PK SPK I )

13 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Pom-Pom model:Evolution of Orientation:

SPK + 2[D : SPK ]SPK +

1

OBK

[SPK

1

3I

]= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PKK

(32PK SPK I )

13 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Double-equation eXtended Pom-Pom (DXPP) model:Evolution of Orientation:

S PK + 2[D : SPK ]SPK +

1OBK

2PK

h3K

4PK

SPK SPK + (1 K 3K 4PK

I SS )SPK (1K )

3Ii

= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PKK

(32PK SPK I )

14 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Double-equation eXtended Pom-Pom (DXPP) model:Evolution of Orientation:

S PK + 2[D : SPK ]SPK +

1OBK

2PK

h3K

4PK

SPK SPK + (1 K 3K 4PK

I SS )SPK (1K )

3Ii

= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PKK

(32PK SPK I )

14 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Double-equation eXtended Pom-Pom (DXPP) model:Evolution of Orientation:

S PK + 2[D : SPK ]SPK +

1OBK

2PK

h3K

4PK

SPK SPK + (1 K 3K 4PK

I SS )SPK (1K )

3Ii

= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PKK

(32PK SPK I )

14 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Single-equation eXtended Pom-Pom (SXPP) model:Viscoelastic stress:

PK + ()

1 PK =2PK D

K

Relaxation time tensor:

()1 =1

OBK

[KOBKPK

PK + f ()1I +

OBKPK

(f ()1 1)1PK

]

Extra function:

1

OBKf ()1 =

2

SK

(1 1

)+

2

OBK 2

(1 K

2KI

32PK

)Backbone stretch and stretch relaxation time:

=

1 +

KI 3PK

, SK = OSK e(1), =

2

q

15 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Single-equation eXtended Pom-Pom (SXPP) model:Viscoelastic stress:

PK + ()

1 PK =2PK D

K

Relaxation time tensor:

()1 =1

OBK

[KOBKPK

PK + f ()1I +

OBKPK

(f ()1 1)1PK

]Extra function:

1

OBKf ()1 =

2

SK

(1 1

)+

2

OBK 2

(1 K

2KI

32PK

)

Backbone stretch and stretch relaxation time:

=

1 +

KI 3PK

, SK = OSK e(1), =

2

q

15 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Single-equation eXtended Pom-Pom (SXPP) model:Viscoelastic stress:

PK + ()

1 PK =2PK D

K

Relaxation time tensor:

()1 =1

OBK

[KOBKPK

PK + f ()1I +

OBKPK

(f ()1 1)1PK

]Extra function:

1

OBKf ()1 =

2

SK

(1 1

)+

2

OBK 2

(1 K

2KI

32PK

)Backbone stretch and stretch relaxation time:

=

1 +

KI 3PK

, SK = OSK e(1), =

2

q

15 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Single-equation eXtended Pom-Pom (SXPP) model:Viscoelastic stress:

PK + ()

1 PK =2PK D

K

Relaxation time tensor:

()1 =1

OBK

[KOBKPK

PK + f ()1I +

OBKPK

(f ()1 1)1PK

]Extra function:

1

OBKf ()1 =

2

SK

(1 1

)+

2

OBK 2

(1 K

2KI

32PK

)Backbone stretch and stretch relaxation time:

=

1 +

KI 3PK

, SK = OSK e(1), =

2

q

15 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Double Convected Pom-Pom (DCPP) model:Evolution of Orientation:

1

2

S PK +

2

SPK

+(1)[2D : SPK ]SPK +

1

OBK 2PK

SPK

I

3

= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PK

(1 )K(32PK SPK I )

16 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Double Convected Pom-Pom (DCPP) model:Evolution of Orientation:

1

2

S PK +

2

SPK

+(1)[2D : SPK ]SPK +

1

OBK 2PK

SPK

I

3

= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PK

(1 )K(32PK SPK I )

16 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Reptation theory / tube Models

Double Convected Pom-Pom (DCPP) model:Evolution of Orientation:

1

2

S PK +

2

SPK

+(1)[2D : SPK ]SPK +

1

OBK 2PK

SPK

I

3

= 0

Evolution of the backbone stretch:

D (PK )

Dt= PK [D : SPK ] +

1

SK[PK 1]

SK = OSK e(PK1), =

2

q, PK q

Viscoelastic stress:

PK =PK

(1 )K(32PK SPK I )

16 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Multimode form

The value of P is obtained by the sum of the K modes:

P =n

K=1

PK

17 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Multimode form

The value of P is obtained by the sum of the K modes:

P =n

K=1

PK

17 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

HWNP

Was used the DEVSS methodology. The momentum equationis rewritten as:

(U)

t+(UU) (S +)(U) = p+P(U)

where is a positive number. The value of depend of themodel parameters, but = PK usually is a good choise.

18 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

HWNP

Was used the DEVSS methodology. The momentum equationis rewritten as:

(U)

t+(UU) (S +)(U) = p+P(U)

where is a positive number. The value of depend of themodel parameters, but = PK usually is a good choise.

18 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

HWNP

Was used the DEVSS methodology. The momentum equationis rewritten as:

(U)

t+(UU) (S +)(U) = p+P(U)

where is a positive number. The value of depend of themodel parameters, but = PK usually is a good choise.

18 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solving the problem

The procedure used to solve the problem of viscoelastic fluidflow can be summarized in 4 steps for each time step:

1 With an initial known velocity field U, a given pressure p and

stress , the momentum equation is implicitly solved for each

component of the velocity vector resulting in U. The pressure

gradient and the stress divergent are calculated explicitly with

values of the previous step.2

With the news velocity values U it is estimated the new

pressure field p using an equation for the pressure and makes

the correction of velocity field to satisfy the continuity equation,

resulting in U. The PISO algorithm is used.3 With the corrected velocity field U is made the calculation of

the stress tensor field using a constitutive equation desired.4 For more accurate solutions to transient flow the steps 1, 2 and

3 can be iterate in a same time step.

19 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solving the problem

The procedure used to solve the problem of viscoelastic fluidflow can be summarized in 4 steps for each time step:

1 With an initial known velocity field U, a given pressure p and

stress , the momentum equation is implicitly solved for each

component of the velocity vector resulting in U. The pressure

gradient and the stress divergent are calculated explicitly with

values of the previous step.2 With the news velocity values U it is estimated the new

pressure field p using an equation for the pressure and makes

the correction of velocity field to satisfy the continuity equation,

resulting in U. The PISO algorithm is used.3

With the corrected velocity field U is made the calculation of

the stress tensor field using a constitutive equation desired.4 For more accurate solutions to transient flow the steps 1, 2 and

3 can be iterate in a same time step.

19 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solving the problem

The procedure used to solve the problem of viscoelastic fluidflow can be summarized in 4 steps for each time step:

1 With an initial known velocity field U, a given pressure p and

stress , the momentum equation is implicitly solved for each

component of the velocity vector resulting in U. The pressure

gradient and the stress divergent are calculated explicitly with

values of the previous step.2 With the news velocity values U it is estimated the new

pressure field p using an equation for the pressure and makes

the correction of velocity field to satisfy the continuity equation,

resulting in U. The PISO algorithm is used.3 With the corrected velocity field U is made the calculation of

the stress tensor field using a constitutive equation desired.4

For more accurate solutions to transient flow the steps 1, 2 and

3 can be iterate in a same time step.

19 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solving the problem

The procedure used to solve the problem of viscoelastic fluidflow can be summarized in 4 steps for each time step:

1 With an initial known velocity field U, a given pressure p and

stress , the momentum equation is implicitly solved for each

component of the velocity vector resulting in U. The pressure

gradient and the stress divergent are calculated explicitly with

values of the previous step.2 With the news velocity values U it is estimated the new

pressure field p using an equation for the pressure and makes

the correction of velocity field to satisfy the continuity equation,

resulting in U. The PISO algorithm is used.3 With the corrected velocity field U is made the calculation of

the stress tensor field using a constitutive equation desired.4 For more accurate solutions to transient flow the steps 1, 2 and

3 can be iterate in a same time step.

19 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solving the problem

The procedure used to solve the problem of viscoelastic fluidflow can be summarized in 4 steps for each time step:

1 With an initial known velocity field U, a given pressure p and

stress , the momentum equation is implicitly solved for each

component of the velocity vector resulting in U. The pressure

gradient and the stress divergent are calculated explicitly with

values of the previous step.2 With the news velocity values U it is estimated the new

pressure field p using an equation for the pressure and makes

the correction of velocity field to satisfy the continuity equation,

resulting in U. The PISO algorithm is used.3 With the corrected velocity field U is made the calculation of

the stress tensor field using a constitutive equation desired.4 For more accurate solutions to transient flow the steps 1, 2 and

3 can be iterate in a same time step.

19 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solver structure

The solver was structured as:

1 viscoelasticFluidFoam.C = the main file of the solver.

2 createFields.C = to read the fields and create theviscoelastic model.

3 viscoelasticModels/viscoelasticLaws/anyModel .C = toread viscoelastic properties and solve the constitutiveequation.

20 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solver structure

The solver was structured as:

1 viscoelasticFluidFoam.C = the main file of the solver.

2 createFields.C = to read the fields and create theviscoelastic model.

3 viscoelasticModels/viscoelasticLaws/anyModel .C = toread viscoelastic properties and solve the constitutiveequation.

20 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solver structure

The solver was structured as:

1 viscoelasticFluidFoam.C = the main file of the solver.

2 createFields.C = to read the fields and create theviscoelastic model.

3 viscoelasticModels/viscoelasticLaws/anyModel .C = toread viscoelastic properties and solve the constitutiveequation.

20 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Solver structure

The solver was structured as:

1 viscoelasticFluidFoam.C = the main file of the solver.

2 createFields.C = to read the fields and create theviscoelastic model.

3 viscoelasticModels/viscoelasticLaws/anyModel .C = toread viscoelastic properties and solve the constitutiveequation.

20 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Main file: viscoelasticFluidFoam.C

Beginning file

#include "fvCFD.H" 1#include "viscoelasticModel.H"

// //5

int main(int argc, char argv[]){

# include "setRootCase.H"10

# include "createTime.H"# include "createMesh.H"# include "createFields.H"# include "initContinuityErrs.H"

15// //

Info

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Main file: viscoelasticFluidFoam.C

// Pressurevelocity PISO corrector loop 31for (int corr = 0; corr < nCorr; corr++){

// Momentum predictor35

tmp UEqn(

fvm::ddt(U)+ fvm::div(phi, U) visco.divTau(U) 40

);

UEqn().relax();

solve(UEqn() == fvc::grad(p)); 45

p.boundaryField().updateCoeffs();volScalarField rUA = 1.0/UEqn().A();U = rUAUEqn().H();UEqn.clear(); 50phi = fvc::interpolate(U) & mesh.Sf();adjustPhi(phi, U, p);

// Store pressure for underrelaxationp.storePrevIter(); 55

22 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Main file: viscoelasticFluidFoam.C

// Nonorthogonal pressure corrector loop 56for (int nonOrth=0; nonOrth

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Main file: viscoelasticFluidFoam.C

visco.correct(); 81}

runTime.write();85

Info

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Linear Phan-Thien-Tanner model: LPTT .C

Beginning file

#include "LPTT.H" 1#include "addToRunTimeSelectionTable.H"

// //5

namespace Foam{

// Static Data Members //10

defineTypeNameAndDebug(LPTT, 0);addToRunTimeSelectionTable(viscoelasticLaw, LPTT, dictionary);

// Constructors //15

// from componentsLPTT::LPTT(

const word& name,const volVectorField& U, 20const surfaceScalarField& phi,const dictionary& dict

):

viscoelasticLaw(name, U, phi), 25

25 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Linear Phan-Thien-Tanner model: LPTT .C

tau 26(

IOobject(

"tau" + name, 30U.time().timeName(),U.mesh(),IOobject::MUST READ,IOobject::AUTO WRITE

), 35U.mesh()

),rho (dict.lookup("rho")),etaS (dict.lookup("etaS")),etaP (dict.lookup("etaP")), 40epsilon (dict.lookup("epsilon")),lambda (dict.lookup("lambda")),zeta (dict.lookup("zeta"))

{}45

// Member Functions //

tmp LPTT::divTau(volVectorField& U) const{ 50

26 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Linear Phan-Thien-Tanner model: LPTT .C

// dimensionedScalar etaPEff = (1 + 1/epsilon )etaP ; 51dimensionedScalar etaPEff = etaP ;

return( 55

fvc::div(tau /rho , "div(tau)") fvc::laplacian(etaPEff/rho , U, "laplacian(etaPEff,U)")+ fvm::laplacian( (etaPEff + etaS )/rho , U, "laplacian(etaPEff+etaS,U)")

);} 60

void LPTT::correct(){

// Velocity gradient tensor 65volTensorField L = fvc::grad( U() );

// Convected derivate termvolTensorField C = tau & L;

70// Twice the rate of deformation tensorvolSymmTensorField twoD = twoSymm( L );

// Stress transport equation 75

27 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Linear Phan-Thien-Tanner model: LPTT .C

tmp tauEqn 76(

fvm::ddt(tau )+ fvm::div(phi(), tau )== 80etaP / lambda twoD+ twoSymm( C ) zeta / 2 ( (tau & twoD) + (twoD & tau ) ) fvm::Sp( epsilon / etaP tr(tau ) + 1/lambda , tau )

); 85

tauEqn().relax();solve(tauEqn);

}90

// //

} // End namespace Foam 95

End file

28 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Important functions

1 divTau(U) = is the coupled term between momentumand constitutive models.

2 correct() = solve the constitutive model.

29 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Important functions

1 divTau(U) = is the coupled term between momentumand constitutive models.

2 correct() = solve the constitutive model.

29 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Running a case

A case is organized as follow:

The time directories containing individual files of data forparticular fields. The data can be: either, initial valuesand boundary conditions that the user must specifyto define the problem; or, results written to file byOpenFOAM. The files U, p and a file tau+ < name >are needed (< name > correspond to name ofeach individual mode, e.g. first, second ...).

A system directory for setting parameters associatedwith the solution procedure itself. It contains at leastthe following 3 files: controlDict, fvSchemes, fvSolution.

A constant directory that contains a full description ofthe case mesh in a subdirectory polyMesh and filesspecifying viscoelastic properties, e.g.viscoelasticProperties.

30 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Running a case

A case is organized as follow:

The time directories containing individual files of data forparticular fields. The data can be: either, initial valuesand boundary conditions that the user must specifyto define the problem; or, results written to file byOpenFOAM. The files U, p and a file tau+ < name >are needed (< name > correspond to name ofeach individual mode, e.g. first, second ...).

A system directory for setting parameters associatedwith the solution procedure itself. It contains at leastthe following 3 files: controlDict, fvSchemes, fvSolution.

A constant directory that contains a full description ofthe case mesh in a subdirectory polyMesh and filesspecifying viscoelastic properties, e.g.viscoelasticProperties.

30 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Running a case

A case is organized as follow:

The time directories containing individual files of data forparticular fields. The data can be: either, initial valuesand boundary conditions that the user must specifyto define the problem; or, results written to file byOpenFOAM. The files U, p and a file tau+ < name >are needed (< name > correspond to name ofeach individual mode, e.g. first, second ...).

A system directory for setting parameters associatedwith the solution procedure itself. It contains at leastthe following 3 files: controlDict, fvSchemes, fvSolution.

A constant directory that contains a full description ofthe case mesh in a subdirectory polyMesh and filesspecifying viscoelastic properties, e.g.viscoelasticProperties.

30 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Running a case

A case is organized as follow:

The time directories containing individual files of data forparticular fields. The data can be: either, initial valuesand boundary conditions that the user must specifyto define the problem; or, results written to file byOpenFOAM. The files U, p and a file tau+ < name >are needed (< name > correspond to name ofeach individual mode, e.g. first, second ...).

A system directory for setting parameters associatedwith the solution procedure itself. It contains at leastthe following 3 files: controlDict, fvSchemes, fvSolution.

A constant directory that contains a full description ofthe case mesh in a subdirectory polyMesh and filesspecifying viscoelastic properties, e.g.viscoelasticProperties.

30 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

One example of viscoelasticProperties file

31 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

One example of tau+ < name > file

32 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Test Geometry

A planar abrupt contraction with contraction ratio H/h of3.97 : 1 (upstream thickness of 2H = 0.0254[m] anddownstream thickness of 2h = 0.0064[m]) was chosen as testgeometry because of the availability of literature data forvalidation of the developed code.

Figure: Sketch of geometry and the boundary conditions.

33 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Flow Properties and models parameters

Q[cm3.s1] Uinlet [cm.s1]

[s1] [Kg .m3] Re De

252 3.875 48.4 803.87 0.56 1.45

Model Parameter [-] [s] P [Pa.s] S [Pa.s]

Giesekus 0.15 0.03 1.422 0.002LPTTS 0.25 0.03 1.422 0.002EPTTS 0.25 0.03 1.422 0.002FENE-P 6.0 0.04 1.422 0.002

FENE-CR 6.0 0.04 1.422 0.002Maxwell linear 0.03 1.422 0.002

Oldroyd-B 0.03 1.422 0.002

34 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Flow Properties and models parameters

Q[cm3.s1] Uinlet [cm.s1]

[s1] [Kg .m3] Re De

252 3.875 48.4 803.87 0.56 1.45

Model Parameter [-] [s] P [Pa.s] S [Pa.s]

Giesekus 0.15 0.03 1.422 0.002LPTTS 0.25 0.03 1.422 0.002EPTTS 0.25 0.03 1.422 0.002FENE-P 6.0 0.04 1.422 0.002

FENE-CR 6.0 0.04 1.422 0.002Maxwell linear 0.03 1.422 0.002

Oldroyd-B 0.03 1.422 0.002

34 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Mesh Properties

Figure: Mesh used.

Numbers of CVs xmin/h ymin/h

20700 0.0065 0.017

35 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Mesh Properties

Figure: Mesh used.

Numbers of CVs xmin/h ymin/h

20700 0.0065 0.017

35 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

36 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

36 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

37 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

37 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

38 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

38 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

39 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

39 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

40 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Contour plots

40 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Stream lines

41 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Stream lines

41 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Stream lines

41 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Numeric versus experimental dates

42 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Numeric versus experimental dates

42 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Giesekus / FENE-P / LPTTS

43 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Giesekus / FENE-P / LPTTS

43 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Giesekus / Maxwell linear / Oldroyd-B

44 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Giesekus / Maxwell linear / Oldroyd-B

44 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Giesekus / FENE-CR / EPTTS

45 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Giesekus / FENE-CR / EPTTS

45 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Deborah effect

46 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Deborah effect

46 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Multimode DCPP

= 12.4[s1]

Mode P [Pa.s] Ob[s] Os [s] [] q[]1 1.03x103 0.02 0.01 0.2 1.02 2.22x103 0.2 0.1 0.2 1.03 4.16x103 2.0 1.0 0.07 6.04 1.322x103 20.0 20.0 0.05 18.0

|PSD| =

(yy xx )2 + 42xy

47 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Multimode DCPP

= 12.4[s1]

Mode P [Pa.s] Ob[s] Os [s] [] q[]1 1.03x103 0.02 0.01 0.2 1.02 2.22x103 0.2 0.1 0.2 1.03 4.16x103 2.0 1.0 0.07 6.04 1.322x103 20.0 20.0 0.05 18.0

|PSD| =

(yy xx )2 + 42xy

47 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Multimode DCPP

= 12.4[s1]

Mode P [Pa.s] Ob[s] Os [s] [] q[]1 1.03x103 0.02 0.01 0.2 1.02 2.22x103 0.2 0.1 0.2 1.03 4.16x103 2.0 1.0 0.07 6.04 1.322x103 20.0 20.0 0.05 18.0

|PSD| =

(yy xx )2 + 42xy

47 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

PSD and velocity magnitude using DCPP model

48 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

PSD and velocity magnitude using DCPP model

48 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

PSD and velocity magnitude using DCPP model

48 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

PSD and velocity magnitude using DCPP model

48 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

LPDE flow

49 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

LPDE flow

50 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

A 3D capillary case: mesh

51 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

A 3D capillary case: Uz

52 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

A 3D capillary case: velocity magnitude

53 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

A 3D capillary case: stress magnitude

54 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

A 3D capillary case: stress magnitude

54 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Being Tested: viscoelasticFluidDyMFoam

Velocity magnitude:

(Loading magU.mpg)

55 / 59

magU.mpgMedia File (video/mpeg)

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Being Tested: viscoelasticFluidDyMFoam

Stress magnitude:

(Loading magTau.mpg)

56 / 59

magTau.mpgMedia File (video/mpeg)

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Being Tested: viscoelasticFluidDyMFoam

Mesh rearranged:

(Loading mesh.mpg)

57 / 59

mesh.mpgMedia File (video/mpeg)

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

About viscoelasticFluidFoam solver

Was showed a comparasion of Gisekus model with numericand experimental dates from literature.

An example using multimode DCPP, LPDE flow and 3Dcapillary flow.

A lot of models was tested: Giesekus, LPTT, EPTT,FENE-P, FENE-CR, Oldroyd-B, DCPP....

The results lead us to conclude that the solver leads toconsistent results.

Suggestions for future work: Test the other implementedmodels and more cases.

58 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

About viscoelasticFluidFoam solver

Was showed a comparasion of Gisekus model with numericand experimental dates from literature.

An example using multimode DCPP, LPDE flow and 3Dcapillary flow.

A lot of models was tested: Giesekus, LPTT, EPTT,FENE-P, FENE-CR, Oldroyd-B, DCPP....

The results lead us to conclude that the solver leads toconsistent results.

Suggestions for future work: Test the other implementedmodels and more cases.

58 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

About viscoelasticFluidFoam solver

Was showed a comparasion of Gisekus model with numericand experimental dates from literature.

An example using multimode DCPP, LPDE flow and 3Dcapillary flow.

A lot of models was tested: Giesekus, LPTT, EPTT,FENE-P, FENE-CR, Oldroyd-B, DCPP....

The results lead us to conclude that the solver leads toconsistent results.

Suggestions for future work: Test the other implementedmodels and more cases.

58 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

About viscoelasticFluidFoam solver

Was showed a comparasion of Gisekus model with numericand experimental dates from literature.

An example using multimode DCPP, LPDE flow and 3Dcapillary flow.

A lot of models was tested: Giesekus, LPTT, EPTT,FENE-P, FENE-CR, Oldroyd-B, DCPP....

The results lead us to conclude that the solver leads toconsistent results.

Suggestions for future work: Test the other implementedmodels and more cases.

58 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

About viscoelasticFluidFoam solver

Was showed a comparasion of Gisekus model with numericand experimental dates from literature.

An example using multimode DCPP, LPDE flow and 3Dcapillary flow.

A lot of models was tested: Giesekus, LPTT, EPTT,FENE-P, FENE-CR, Oldroyd-B, DCPP....

The results lead us to conclude that the solver leads toconsistent results.

Suggestions for future work: Test the other implementedmodels and more cases.

58 / 59

ViscoelasticFlow

Simulation inOpenFOAM

Jovani L.Favero

Outline

Introduction

ProblemDefinition

ConstitutiveModels

DEVSS andSolutionProcedure

Solver Imple-mentation

Using theSolver

Some Results

Conclusion

Acknowledgements

Special thanks are directed to three good friends, the

Professors Dr. Argimiro R. Secchi, Dr. Hrvoje Jasak

and Dr. Nilo S. M. Cardozo for their continued support

and guidance.

59 / 59

OutlineIntroductionProblem DefinitionConstitutive ModelsDEVSS and Solution ProcedureSolver ImplementationUsing the SolverSome ResultsConclusion