viscoelasticFluidFoam

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 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 = 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 = 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

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