ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionViscoelastic Flow Simulation in
OpenFOAMPresentation of the viscoelasticFluidFoam SolverJovani L.
[email protected] / [email protected]
Federal do Rio Grande do Sul - Department of
ChemicalEngineeringhttp://www.ufrgs.br/ufrgs/February 26, 20091 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusion1 Introduction2 Problem Denition3 Constitutive
Models4 DEVSS and Solution Procedure5 Solver Implementation6 Using
the Solver7 Some Results8 Conclusion2 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhat about Viscoelastic Flows?Understanding and
modeling of viscoelastic ows areusually the key step in the
denition of the nalcharacteristics and quality of the nished
products inmany industrial sectors, such as in food and
syntheticpolymers industries.The rheological response of
viscoelastic uids is quitecomplex, including combination of viscous
and elasticeects and highly non-linear viscous and
elasticphenomena.Characteristics: Strain rate dependent viscosity,
presenceof normal stress dierences in shear ows,
relaxationphenomena and memory eects, including die swell.3 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhat about Viscoelastic Flows?Understanding and
modeling of viscoelastic ows areusually the key step in the
denition of the nalcharacteristics and quality of the nished
products inmany industrial sectors, such as in food and
syntheticpolymers industries.The rheological response of
viscoelastic uids is quitecomplex, including combination of viscous
and elasticeects and highly non-linear viscous and
elasticphenomena.Characteristics: Strain rate dependent viscosity,
presenceof normal stress dierences in shear ows,
relaxationphenomena and memory eects, including die swell.3 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhat about Viscoelastic Flows?Understanding and
modeling of viscoelastic ows areusually the key step in the
denition of the nalcharacteristics and quality of the nished
products inmany industrial sectors, such as in food and
syntheticpolymers industries.The rheological response of
viscoelastic uids is quitecomplex, including combination of viscous
and elasticeects and highly non-linear viscous and
elasticphenomena.Characteristics: Strain rate dependent viscosity,
presenceof normal stress dierences in shear ows,
relaxationphenomena and memory eects, including die swell.3 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionDie Swell, Weissemberg Eect ...(Loading
viscoelastic.mpg)4 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhy OpenFOAM?Its a Open Source CFD Toolbox build
with a exible set ofecient 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 benets.5 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhy OpenFOAM?Its a Open Source CFD Toolbox build
with a exible set ofecient 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 benets.5 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhy OpenFOAM?Its a Open Source CFD Toolbox build
with a exible set ofecient 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 benets.5 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhy OpenFOAM?Its a Open Source CFD Toolbox build
with a exible set ofecient 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 benets.5 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhy OpenFOAM?Its a Open Source CFD Toolbox build
with a exible set ofecient 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 benets.5 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhy OpenFOAM?Its a Open Source CFD Toolbox build
with a exible set ofecient 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 benets.5 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionWhy OpenFOAM?Its a Open Source CFD Toolbox build
with a exible set ofecient 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 benets.5 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionViscoelastic Fluid Flow FormulationThe governing
equations of laminar, incompressible andisothermal ow of
viscoelastic uids are the equations ofconservation of mass
(continuity): (U) = 0momentum:(U)t + (UU) = p + S + Pand a
mechanical constitutive equation that describes therelation between
the stress and deformation rate.6 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionViscoelastic Fluid Flow FormulationThe governing
equations of laminar, incompressible andisothermal ow of
viscoelastic uids are the equations ofconservation of mass
(continuity): (U) = 0momentum:(U)t + (UU) = p + S + Pand a
mechanical constitutive equation that describes therelation between
the stress and deformation rate.6 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionViscoelastic Fluid Flow FormulationThe governing
equations of laminar, incompressible andisothermal ow of
viscoelastic uids are the equations ofconservation of mass
(continuity): (U) = 0momentum:(U)t + (UU) = p + S + Pand a
mechanical constitutive equation that describes therelation between
the stress and deformation rate.6 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionViscoelastic Fluid Flow Formulationwhere S are the
solvent contribution to stress:S = 2SDS is the solvent viscosity
and D is the deformation rate tensor:D = 12(U + [U]T)The extra
elastic contribution, corresponding to the polymericpart P, is
obtained from the solution of an appropriateconstitutive dierential
equation.7 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionViscoelastic Fluid Flow Formulationwhere S are the
solvent contribution to stress:S = 2SDS is the solvent viscosity
and D is the deformation rate tensor:D = 12(U + [U]T)The extra
elastic contribution, corresponding to the polymericpart P, is
obtained from the solution of an appropriateconstitutive dierential
equation.7 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionViscoelastic Fluid Flow Formulationwhere S are the
solvent contribution to stress:S = 2SDS is the solvent viscosity
and D is the deformation rate tensor:D = 12(U + [U]T)The extra
elastic contribution, corresponding to the polymericpart P, is
obtained from the solution of an appropriateconstitutive dierential
equation.7 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionImportant Denitions1Upper Convective Derivative of
a generic tensor A:A = DDtA hUT Ai[A U]or for symmetric tensors:A =
DDtA [A U] [A U]T2Lower Convective Derivative of a generic tensor
A:A = DDtA + [U A] +hA UTi3Gordon-Schowalter Derivative of a
generic tensor A:
A = DDtA [UT A] [A U] + (A D +D A)where: DDtA = tA + U A8 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionImportant Denitions1Upper Convective Derivative of
a generic tensor A:A = DDtA hUT Ai[A U]or for symmetric tensors:A =
DDtA [A U] [A U]T2Lower Convective Derivative of a generic tensor
A:A = DDtA + [U A] +hA UTi3Gordon-Schowalter Derivative of a
generic tensor A:
A = DDtA [UT A] [A U] + (A D +D A)where: DDtA = tA + U A8 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionImportant Denitions1Upper Convective Derivative of
a generic tensor A:A = DDtA hUT Ai[A U]or for symmetric tensors:A =
DDtA [A U] [A U]T2Lower Convective Derivative of a generic tensor
A:A = DDtA + [U A] +hA UTi3Gordon-Schowalter Derivative of a
generic tensor A:
A = DDtA [UT A] [A U] + (A D +D A)where: DDtA = tA + U A8 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionImportant Denitions1Upper Convective Derivative of
a generic tensor A:A = DDtA hUT Ai[A U]or for symmetric tensors:A =
DDtA [A U] [A U]T2Lower Convective Derivative of a generic tensor
A:A = DDtA + [U A] +hA UTi3Gordon-Schowalter Derivative of a
generic tensor A:
A = DDtA [UT A] [A U] + (A D +D A)where: DDtA = tA + U A8 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionImportant Denitions1Upper Convective Derivative of
a generic tensor A:A = DDtA hUT Ai[A U]or for symmetric tensors:A =
DDtA [A U] [A U]T2Lower Convective Derivative of a generic tensor
A:A = DDtA + [U A] +hA UTi3Gordon-Schowalter Derivative of a
generic tensor A:
A = DDtA [UT A] [A U] + (A D +D A)where: DDtA = tA + U A8 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsMaxwell linear:PK + KPKt =
2PKDUCM and Oldroyd-B:PK + KPK = 2PKDwhere K and PK are the
relaxation time and polymerviscosity coecient at zero shear rate,
respectively.9 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsMaxwell linear:PK + KPKt =
2PKDUCM and Oldroyd-B:PK + KPK = 2PKDwhere K and PK are the
relaxation time and polymerviscosity coecient at zero shear rate,
respectively.9 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsWhite-Metzner (WM):PK +
K(IID)PK = 2PK(IID)Dwhere: (IID) = =2D : DLarson:PK(IID) = PK1 +
aKIID; K(IID) = K1 + aKIIDCross:PK(IID) = PK1 + (kIID)1m; K(IID) =
K1 + (LIID)1nCarreau-Yasuda:PK(IID) = PK [1 + (kIID)a]m1a; K(IID) =
K_1 + (LIID)bn1b10 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsWhite-Metzner (WM):PK +
K(IID)PK = 2PK(IID)Dwhere: (IID) = =2D : DLarson:PK(IID) = PK1 +
aKIID; K(IID) = K1 + aKIIDCross:PK(IID) = PK1 + (kIID)1m; K(IID) =
K1 + (LIID)1nCarreau-Yasuda:PK(IID) = PK [1 + (kIID)a]m1a; K(IID) =
K_1 + (LIID)bn1b10 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsWhite-Metzner (WM):PK +
K(IID)PK = 2PK(IID)Dwhere: (IID) = =2D : DLarson:PK(IID) = PK1 +
aKIID; K(IID) = K1 + aKIIDCross:PK(IID) = PK1 + (kIID)1m; K(IID) =
K1 + (LIID)1nCarreau-Yasuda:PK(IID) = PK [1 + (kIID)a]m1a; K(IID) =
K_1 + (LIID)bn1b10 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsWhite-Metzner (WM):PK +
K(IID)PK = 2PK(IID)Dwhere: (IID) = =2D : DLarson:PK(IID) = PK1 +
aKIID; K(IID) = K1 + aKIIDCross:PK(IID) = PK1 + (kIID)1m; K(IID) =
K1 + (LIID)1nCarreau-Yasuda:PK(IID) = PK [1 + (kIID)a]m1a; K(IID) =
K_1 + (LIID)bn1b10 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsWhite-Metzner (WM):PK +
K(IID)PK = 2PK(IID)Dwhere: (IID) = =2D : DLarson:PK(IID) = PK1 +
aKIID; K(IID) = K1 + aKIIDCross:PK(IID) = PK1 + (kIID)1m; K(IID) =
K1 + (LIID)1nCarreau-Yasuda:PK(IID) = PK [1 + (kIID)a]m1a; K(IID) =
K_1 + (LIID)bn1b10 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsGiesekus:PK + KPK + KKPK(PK.
PK) = 2PKDFENE-P:__1 +3(13/L2K) + KPKtr (PK)L2K__K+KPK = 2 1(1
3/L2K)PKDFENE-CR:__L2K + KPKtr (PK)(L2K 3)__K+KPK = 2__L2K + KPKtr
(PK)(L2K 3)__PKD11 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsGiesekus:PK + KPK + KKPK(PK.
PK) = 2PKDFENE-P:__1 +3(13/L2K) + KPKtr (PK)L2K__K+KPK = 2 1(1
3/L2K)PKDFENE-CR:__L2K + KPKtr (PK)(L2K 3)__K+KPK = 2__L2K + KPKtr
(PK)(L2K 3)__PKD11 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionKinetic Theory ModelsGiesekus:PK + KPK + KKPK(PK.
PK) = 2PKDFENE-P:__1 +3(13/L2K) + KPKtr (PK)L2K__K+KPK = 2 1(1
3/L2K)PKDFENE-CR:__L2K + KPKtr (PK)(L2K 3)__K+KPK = 2__L2K + KPKtr
(PK)(L2K 3)__PKD11 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionNetwork Theory of Concentrated Solutions andMelts
ModelsPhan-Thien-Tanner linear (LPTT):_1 + KKPKtr (PK)_PK + K
PK = 2PKDPhan-Thien-Tanner exponential (EPTT):exp_KKPKtr (PK)_PK
+ K
PK = 2PKDFeta-PTT:_1 + KK()PK() tr (PK)_PK + K()
PK = 2PK()Dwhere:PK () = PK1 +AII2k2PKab ; K() = K1 + KKIPK12 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionNetwork Theory of Concentrated Solutions andMelts
ModelsPhan-Thien-Tanner linear (LPTT):_1 + KKPKtr (PK)_PK + K
PK = 2PKDPhan-Thien-Tanner exponential (EPTT):exp_KKPKtr (PK)_PK
+ K
PK = 2PKDFeta-PTT:_1 + KK()PK() tr (PK)_PK + K()
PK = 2PK()Dwhere:PK () = PK1 +AII2k2PKab ; K() = K1 + KKIPK12 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionNetwork Theory of Concentrated Solutions andMelts
ModelsPhan-Thien-Tanner linear (LPTT):_1 + KKPKtr (PK)_PK + K
PK = 2PKDPhan-Thien-Tanner exponential (EPTT):exp_KKPKtr (PK)_PK
+ K
PK = 2PKDFeta-PTT:_1 + KK()PK() tr (PK)_PK + K()
PK = 2PK()Dwhere:PK () = PK1 +AII2k2PKab ; K() = K1 + KKIPK12 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsPom-Pom
model:Evolution of Orientation:SPK + 2[D : SPK]SPK + 1OBK_SPK 13I_=
0Evolution of the backbone stretch:D (PK)Dt = PK[D : SPK] + 1SK[PK
1]SK = OSKe(PK1), = 2q, PK qViscoelastic stress:PK = PKK(32PKSPK I
)13 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsPom-Pom
model:Evolution of Orientation:SPK + 2[D : SPK]SPK + 1OBK_SPK 13I_=
0Evolution of the backbone stretch:D (PK)Dt = PK[D : SPK] + 1SK[PK
1]SK = OSKe(PK1), = 2q, PK qViscoelastic stress:PK = PKK(32PKSPK I
)13 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsPom-Pom
model:Evolution of Orientation:SPK + 2[D : SPK]SPK + 1OBK_SPK 13I_=
0Evolution of the backbone stretch:D (PK)Dt = PK[D : SPK] + 1SK[PK
1]SK = OSKe(PK1), = 2q, PK qViscoelastic stress:PK = PKK(32PKSPK I
)13 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsDouble-equation
eXtended Pom-Pom (DXPP) model:Evolution of Orientation:SPK + 2[D :
SPK]SPK+1OBK 2PKh3K4PKSPK SPK + (1 K 3K4PKISS)SPK (1K)3 Ii =
0Evolution of the backbone stretch:D (PK)Dt = PK[D : SPK] + 1SK[PK
1]SK = OSKe(PK1), = 2q, PK qViscoelastic stress:PK = PKK(32PKSPK I
)14 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsDouble-equation
eXtended Pom-Pom (DXPP) model:Evolution of Orientation:SPK + 2[D :
SPK]SPK+1OBK 2PKh3K4PKSPK SPK + (1 K 3K4PKISS)SPK (1K)3 Ii =
0Evolution of the backbone stretch:D (PK)Dt = PK[D : SPK] + 1SK[PK
1]SK = OSKe(PK1), = 2q, PK qViscoelastic stress:PK = PKK(32PKSPK I
)14 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsDouble-equation
eXtended Pom-Pom (DXPP) model:Evolution of Orientation:SPK + 2[D :
SPK]SPK+1OBK 2PKh3K4PKSPK SPK + (1 K 3K4PKISS)SPK (1K)3 Ii =
0Evolution of the backbone stretch:D (PK)Dt = PK[D : SPK] + 1SK[PK
1]SK = OSKe(PK1), = 2q, PK qViscoelastic stress:PK = PKK(32PKSPK I
)14 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsSingle-equation
eXtended Pom-Pom (SXPP) model:Viscoelastic stress:PK + ()1 PK =
2PKDKRelaxation time tensor:()1= 1OBK_KOBKPKPK + f ()1I + OBKPK(f
()11)1PK_Extra function:1OBKf ()1= 2SK_1 1_+ 2OBK2_1
K2KI32PK_Backbone stretch and stretch relaxation time: =1 + KI3PK,
SK = OSKe(1), = 2q15 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsSingle-equation
eXtended Pom-Pom (SXPP) model:Viscoelastic stress:PK + ()1 PK =
2PKDKRelaxation time tensor:()1= 1OBK_KOBKPKPK + f ()1I + OBKPK(f
()11)1PK_Extra function:1OBKf ()1= 2SK_1 1_+ 2OBK2_1
K2KI32PK_Backbone stretch and stretch relaxation time: =1 + KI3PK,
SK = OSKe(1), = 2q15 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsSingle-equation
eXtended Pom-Pom (SXPP) model:Viscoelastic stress:PK + ()1 PK =
2PKDKRelaxation time tensor:()1= 1OBK_KOBKPKPK + f ()1I + OBKPK(f
()11)1PK_Extra function:1OBKf ()1= 2SK_1 1_+ 2OBK2_1
K2KI32PK_Backbone stretch and stretch relaxation time: =1 + KI3PK,
SK = OSKe(1), = 2q15 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsSingle-equation
eXtended Pom-Pom (SXPP) model:Viscoelastic stress:PK + ()1 PK =
2PKDKRelaxation time tensor:()1= 1OBK_KOBKPKPK + f ()1I + OBKPK(f
()11)1PK_Extra function:1OBKf ()1= 2SK_1 1_+ 2OBK2_1
K2KI32PK_Backbone stretch and stretch relaxation time: =1 + KI3PK,
SK = OSKe(1), = 2q15 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsDouble Convected
Pom-Pom (DCPP) model:Evolution of Orientation:1 2SPK + 2SPK+(1)[2D
: SPK]SPK+ 1OBK2PKSPK I3 = 0Evolution of the backbone stretch:D
(PK)Dt = PK[D : SPK] + 1SK[PK 1]SK = OSKe(PK1), = 2q, PK
qViscoelastic stress:PK = PK(1 )K(32PKSPK I )16 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsDouble Convected
Pom-Pom (DCPP) model:Evolution of Orientation:1 2SPK + 2SPK+(1)[2D
: SPK]SPK+ 1OBK2PKSPK I3 = 0Evolution of the backbone stretch:D
(PK)Dt = PK[D : SPK] + 1SK[PK 1]SK = OSKe(PK1), = 2q, PK
qViscoelastic stress:PK = PK(1 )K(32PKSPK I )16 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionReptation theory / tube ModelsDouble Convected
Pom-Pom (DCPP) model:Evolution of Orientation:1 2SPK + 2SPK+(1)[2D
: SPK]SPK+ 1OBK2PKSPK I3 = 0Evolution of the backbone stretch:D
(PK)Dt = PK[D : SPK] + 1SK[PK 1]SK = OSKe(PK1), = 2q, PK
qViscoelastic stress:PK = PK(1 )K(32PKSPK I )16 /
59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionMultimode formThe value of P is obtained by the
sum of the K modes:P =n
K=1PK17 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionMultimode formThe value of P is obtained by the
sum of the K modes:P =n
K=1PK17 / 59ViscoelasticFlowSimulation inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionHWNPWas 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 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionHWNPWas 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 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionHWNPWas 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 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolving the problemThe procedure used to solve the
problem of viscoelastic uidow can be summarized in 4 steps for each
time step:1With an initial known velocity eld U, a given pressure p
andstress , the momentum equation is implicitly solved for
eachcomponent of the velocity vector resulting in U. The
pressuregradient and the stress divergent are calculated explicitly
withvalues of the previous step.2With the news velocity values U it
is estimated the newpressure eld p using an equation for the
pressure and makesthe correction of velocity eld to satisfy the
continuity equation,resulting in U. The PISO algorithm is
used.3With the corrected velocity eld U is made the calculation
ofthe stress tensor eld using a constitutive equation desired.4For
more accurate solutions to transient ow the steps 1, 2 and3 can be
iterate in a same time step.19 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolving the problemThe procedure used to solve the
problem of viscoelastic uidow can be summarized in 4 steps for each
time step:1With an initial known velocity eld U, a given pressure p
andstress , the momentum equation is implicitly solved for
eachcomponent of the velocity vector resulting in U. The
pressuregradient and the stress divergent are calculated explicitly
withvalues of the previous step.2With the news velocity values U it
is estimated the newpressure eld p using an equation for the
pressure and makesthe correction of velocity eld to satisfy the
continuity equation,resulting in U. The PISO algorithm is
used.3With the corrected velocity eld U is made the calculation
ofthe stress tensor eld using a constitutive equation desired.4For
more accurate solutions to transient ow the steps 1, 2 and3 can be
iterate in a same time step.19 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolving the problemThe procedure used to solve the
problem of viscoelastic uidow can be summarized in 4 steps for each
time step:1With an initial known velocity eld U, a given pressure p
andstress , the momentum equation is implicitly solved for
eachcomponent of the velocity vector resulting in U. The
pressuregradient and the stress divergent are calculated explicitly
withvalues of the previous step.2With the news velocity values U it
is estimated the newpressure eld p using an equation for the
pressure and makesthe correction of velocity eld to satisfy the
continuity equation,resulting in U. The PISO algorithm is
used.3With the corrected velocity eld U is made the calculation
ofthe stress tensor eld using a constitutive equation desired.4For
more accurate solutions to transient ow the steps 1, 2 and3 can be
iterate in a same time step.19 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolving the problemThe procedure used to solve the
problem of viscoelastic uidow can be summarized in 4 steps for each
time step:1With an initial known velocity eld U, a given pressure p
andstress , the momentum equation is implicitly solved for
eachcomponent of the velocity vector resulting in U. The
pressuregradient and the stress divergent are calculated explicitly
withvalues of the previous step.2With the news velocity values U it
is estimated the newpressure eld p using an equation for the
pressure and makesthe correction of velocity eld to satisfy the
continuity equation,resulting in U. The PISO algorithm is
used.3With the corrected velocity eld U is made the calculation
ofthe stress tensor eld using a constitutive equation desired.4For
more accurate solutions to transient ow the steps 1, 2 and3 can be
iterate in a same time step.19 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolving the problemThe procedure used to solve the
problem of viscoelastic uidow can be summarized in 4 steps for each
time step:1With an initial known velocity eld U, a given pressure p
andstress , the momentum equation is implicitly solved for
eachcomponent of the velocity vector resulting in U. The
pressuregradient and the stress divergent are calculated explicitly
withvalues of the previous step.2With the news velocity values U it
is estimated the newpressure eld p using an equation for the
pressure and makesthe correction of velocity eld to satisfy the
continuity equation,resulting in U. The PISO algorithm is
used.3With the corrected velocity eld U is made the calculation
ofthe stress tensor eld using a constitutive equation desired.4For
more accurate solutions to transient ow the steps 1, 2 and3 can be
iterate in a same time step.19 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolver structureThe solver was structured
as:1viscoelasticFluidFoam.C = the main le of the
solver.2createFields.C = to read the elds and create
theviscoelastic model.3viscoelasticModels/viscoelasticLaws/anyModel
.C = toread viscoelastic properties and solve the
constitutiveequation.20 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolver structureThe solver was structured
as:1viscoelasticFluidFoam.C = the main le of the
solver.2createFields.C = to read the elds and create
theviscoelastic model.3viscoelasticModels/viscoelasticLaws/anyModel
.C = toread viscoelastic properties and solve the
constitutiveequation.20 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolver structureThe solver was structured
as:1viscoelasticFluidFoam.C = the main le of the
solver.2createFields.C = to read the elds and create
theviscoelastic model.3viscoelasticModels/viscoelasticLaws/anyModel
.C = toread viscoelastic properties and solve the
constitutiveequation.20 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionSolver structureThe solver was structured
as:1viscoelasticFluidFoam.C = the main le of the
solver.2createFields.C = to read the elds and create
theviscoelastic model.3viscoelasticModels/viscoelasticLaws/anyModel
.C = toread viscoelastic properties and solve the
constitutiveequation.20 / 59ViscoelasticFlowSimulation
inOpenFOAMJovani
L.FaveroOutlineIntroductionProblemDenitionConstitutiveModelsDEVSS
andSolutionProcedureSolver Imple-mentationUsing theSolverSome
ResultsConclusionMain le: viscoelasticFluidFoam.CBeginning
le#include "fvCFD.H" 1#include "viscoelasticModel.H"// //5int
main(int argc, char argv[]){# include "setRootCase.H"10# include
"createTime.H"# include "createMesh.H"# include "createFields.H"#
include "initContinuityErrs.H"15// //Info