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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. [email protected] / [email protected]
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.
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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