Numerical Simulation of Three-Dimensional Free Surface Flows Using the K–BKZ–PSM Integral Constitutive Equation †
Abstract
:1. Introduction
2. Governing Equations
- continuity equation:
- linear momentum equations:
- stress tensor :
3. Numerical Method
- ■
- Fluid entrance: Inflow — I,
- ■
- Fluid exit: Outflow — O,
- ■
- Rigid boundaries: Boundary — B,
- ■
- Empty cells: Empty — E,
- ■
- Free surface cells: Surface — S,
- ■
- Full cells: Full — F.
- Step 1 — Calculation of and
- Step 2 — Calculation of the extra stress tensor and free surface update
- (a)
- Set and ;
- (b)
- , where .
4. Semi-Analytical Solution
- Step 1:
- Set an interval such that .
- Step 2:
- Determine the zero for taking , where is a small value ( is the tolerance for the error). We carefully selected the value of to ensure the attainment of a semi-analytical solution accurate to six significant digits. Using Gauss–Laguerre quadrature in Equation (33), obtain . Using Equation (39), obtain the value of using Simpson 1/3 quadrature.
- Step 3:
5. Results
5.1. Confined Pipe Flows
- Diameter ;
- ;
- Number of deformation fields ;
- ;
- Geometry: × × ;
- Meshes (number of cells in the x, y and z directions): (), (), (), () and ().
5.2. Free Surface Flows
- Pipe dimension: ; (see Figure 7a);
- Pipe diameter ;
- Number of deformation fields ;
- C1 − , , ;
- C2 − , , .
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
K–BKZ | Kaye–Bernstein, Kearsley, Zapas |
PSM | Papanastasiou, Scriven, Macosko |
EVSS | Elastic–Viscous Split Stress |
UCM | Upper-convected Maxwell |
PTT | Phan–Thien–Tanner |
FENE-P | Finite Extensible Nonlinear Elastic Peterlin |
HDPE | High-Density Polyethylene |
LDPE | Low-Density Polyethylene |
FEM | Finite element Method |
FDM | Finite Difference Method |
PDE | Partial Differential Equation |
ALE | Arbitrary Lagrangian Eulerian |
FBC | Free Boundary Condition |
References
- Clermont, J.-R.; Normandin, M. Numerical simulation of extrudate swell for Oldroyd-B fluids using the stream-tube analysis and a streamline approximation. J. Non-Newton. Fluid Mech. 1993, 50, 193–215. [Google Scholar] [CrossRef]
- Tomé, M.F.; Castelo, A.; Afonso, A.M.; Alves, M.A.; Pinho, F.T. Application of the log-conformation tensor to three-dimensional time-dependent free surface flows. J. Non-Newton. Fluid Mech. 2012, 175–176, 44–54. [Google Scholar] [CrossRef]
- Mompean, G.; Thais, L.; Tomé, M.F.; Castelo, A. Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models. Comput. Fluids 2011, 44, 68–78. [Google Scholar] [CrossRef]
- Tomé, M.F.; Paulo, G.S.; Pinho, F.T.; Alves, M.A. Numerical solution of the PTT constitutive equation for unsteady three-dimensional free surface flows. J. Non-Newton. Fluid Mech. 2010, 165, 247–262. [Google Scholar] [CrossRef]
- Béraudo, C.; Fortin, A.; Coupez, T.; Demay, Y.; Vergnes, B.; Agassant, J.F. A finite element method for computing the flow of multi-mode viscoelastic fluids: Comparison with experiments. J. Non-Newton. Fluid Mech. 1998, 75, 1–23. [Google Scholar] [CrossRef]
- Mu, Y.; Zhao, G.; Wu, X.; Zhai, J. Modeling and simulation of three-dimensional planar contraction flow of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive models. Appl. Math. Comput. 2012, 218, 8429–8443. [Google Scholar] [CrossRef]
- Hulsen, M.A.; van der Zanden, J. Numerical simulation of contraction flows using a multi-mode Giesekus model. J. Non-Newton. Fluid Mech. 1991, 38, 183–221. [Google Scholar] [CrossRef]
- Goublomme, A.; Crochet, M.J. Numerical prediction of extrudate swell of a high-density polyethylene: Further results. J. Non-Newton. Fluid Mech. 1993, 47, 281–287. [Google Scholar] [CrossRef]
- Park, H.J.; Mitsoulis, E. Numerical simulation of circular entry flows of fluid M1 using an integral constitutive equation. J. Non-Newton. Fluid Mech. 1992, 42, 301–314. [Google Scholar] [CrossRef]
- Dupont, S.; Crochet, M.J. The vortex growth of a K.B.K.Z. fluid in an abrupt contraction. J. Non-Newton. Fluid Mech. 1988, 29, 81–91. [Google Scholar] [CrossRef]
- Papanastasiou, A.C.; Scriven, L.E.; Macosko, C.W. An Integral Constitutive Equation for Mixed Flows: Viscoelastic Characterization. J. Rheol. 1983, 27, 387–410. [Google Scholar] [CrossRef]
- Bernstein, B.; Kearsley, E.A.; Zapas, L.J. A Study of Stress Relaxation with Finite Strain. Trans. Soc. Rheol. 1963, 7, 391–410. [Google Scholar] [CrossRef]
- Kaye, A. Non-Newtonian Flow in Incompressible Fluids. Aerosp. Eng. Rep. 1963, 134. [Google Scholar]
- Mitsoulis, E. Extrudate swell of Boger fluids. J. Non-Newton. Fluid Mech. 2010, 165, 812–824. [Google Scholar] [CrossRef]
- Mitsoulis, E. 50 Years of the K-BKZ Constitutive Relation for Polymers. ISRN Polym. Sci. 2013, 2013, 1–22. [Google Scholar] [CrossRef]
- Luo, X.-L.; Mitsoulis, E. An efficient algorithm for strain history tracking in finite element computations of non-Newtonian fluids with integral constitutive equations. Int. J. Numer. Methods Fluids 1990, 11, 1015–1031. [Google Scholar] [CrossRef]
- Ansari, M.; Alabbas, A.; Hatzikiriakos, S.G.; Mitsoulis, E. Entry Flow of Polyethylene Melts in Tapered Dies. Int. Polym. Process. 2010, 25, 287–296. [Google Scholar] [CrossRef]
- Chai, M.S.; Yeow, Y.L. Modelling of fluid M1 using multiple-relaxation-time constitutive equations. J. Non-Newton. Fluid Mech. 1990, 35, 459–470. [Google Scholar] [CrossRef]
- Luo, X.-L.; Mitsoulis, E. A numerical study of the effect of elongational viscosity on vortex growth in contraction flows of polyethylene melts. J. Rheol. 1990, 34, 309–342. [Google Scholar] [CrossRef]
- Mitsoulis, E.; Malamataris, N.A. The free (open) boundary condition with integral constitutive equations. J. Non-Newton. Fluid Mech. 2012, 177–178, 97–108. [Google Scholar] [CrossRef]
- Olley, P.; Spares, R.; Coates, P.D. A method for implementing time-integral constitutive equations in commercial CFD packages. J. Non-Newton. Fluid Mech. 1999, 86, 337–357. [Google Scholar] [CrossRef]
- Araújo, M.S.B.; Fernandes, C.; Ferrás, L.L.; Tuković, Ž.; Jasak, H.; Nóbrega, J.M. A stable numerical implementation of integral viscoelastic models in the OpenFOAM® computational library. Comput. Fluids 2018, 172, 728–740. [Google Scholar] [CrossRef]
- Luo, X.-L.; Mitsoulis, E. Memory Phenomena in Extrudate Swell Simulations for Annular Dies. J. Rheol. 1989, 33, 1307–1327. [Google Scholar] [CrossRef]
- Goublomme, A.; Draily, B.; Crochet, M.J. Numerical prediction of extrudate swell of a high-density polyethylene. J. Non-Newton. Fluid Mech. 1992, 44, 171–195. [Google Scholar] [CrossRef]
- Ganvir, V.; Lele, A.; Thaokar, R.; Gautham, B.P. Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method. J. Non-Newton. Fluid Mech. 2009, 156, 21–28. [Google Scholar] [CrossRef]
- Ahmed, R.; Liang, R.F.; Mackley, M.R. The experimental observation and numerical prediction of planar entry flow and die swell for molten polyethylenes. J. Non-Newton. Fluid Mech. 1995, 59, 129–153. [Google Scholar] [CrossRef]
- Tomé, M.F.; Castelo, A.; Ferreira, V.G.; McKee, S. A finite difference technique for solving the Oldroyd-B model for 3D-unsteady free surface flows. J. Non-Newton. Fluid Mech. 2008, 154, 179–206. [Google Scholar] [CrossRef]
- Rasmussen, H.K. Time-dependent finite-element method for the simulation of three-dimensional viscoelastic flow with integral models. J. Non-Newton. Fluid Mech. 1999, 84, 217–232. [Google Scholar] [CrossRef]
- Marín, J.M.R.; Rasmussen, H.K. Lagrangian finite element method for 3D time-dependent non-isothermal flow of K-BKZ fluids. J. Non-Newton. Fluid Mech. 2009, 162, 45–53. [Google Scholar] [CrossRef]
- Tomé, M.F.; Filho, A.C.; Cuminato, J.A.; Mangiavacchi, N.; Mckee, S. GENSMAC3D: A numerical method for solving unsteady three-dimensional free surface flows. Int. J. Numer. Methods Fluids 2001, 37, 747–796. [Google Scholar] [CrossRef]
- Castello, F.; Tomé, M.F.; César, C.N.L.; McKee, S.; Cuminato, J.A. Freeflow: An integrated simulation system for three-dimensional free surface flows. Comput. Vis. Sci. 2000, 2, 199–210. [Google Scholar] [CrossRef]
- Tomé, M.F.; Bertoco, J.; Oishi, C.M.; Araujo, M.S.B.; Cruz, D.; Pinho, F.T.; Vynnycky, M. A finite difference technique for solving a time strain separable K-BKZ constitutive equation for two-dimensional moving free surface flows. J. Comput. Phys. 2016, 311, 114–141. [Google Scholar] [CrossRef]
- Hulsen, M.A.; Peters, E.A.J.F.; van den Brule, B.H.A.A. A new approach to the deformation fields method for solving complex flows using integral constitutive equations. J. Non-Newton. Fluid Mech. 2001, 98, 201–221. [Google Scholar] [CrossRef]
- Rajagopalan, D.; Armstrong, R.C.; Brown, R.A. Finite element methods for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity. J. Non-Newton. Fluid Mech. 1990, 36, 159–192. [Google Scholar] [CrossRef]
- Quinzani, L.M.; McKinley, G.H.; Brown, R.A.; Armstrong, R.C. Modeling the rheology of polyisobutylene solutions. J. Rheol. 1990, 34, 705–748. [Google Scholar] [CrossRef]
- Mitsoulis, E. The numerical simulation of Boger fluids: A viscometric approximation approach. Polym. Eng. Sci. 1986, 26, 1552–1562. [Google Scholar] [CrossRef]
- López-Aguilar, J.E.; Tamaddon-Jahromi, H.R. Computational Predictions for Boger Fluids and Circular Contraction Flow under Various Aspect Ratios. Fluids 2020, 5, 85. [Google Scholar] [CrossRef]
- Satrape, J.V.; Crochet, M.J. Numerical simulation of the motion of a sphere in a Boger fluid. J. Non-Newton. Fluid Mech. 1994, 55, 91–111. [Google Scholar] [CrossRef]
, = 34,214 | |||
s, , Pa.s | |||
k | [s] | [Pa] | [Pa.s] |
1 | s | Pa | Pa.s |
2 | s | Pa | Pa.s |
3 | s | Pa | Pa.s |
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Bertoco, J.; Castelo, A.; Ferrás, L.L.; Fernandes, C. Numerical Simulation of Three-Dimensional Free Surface Flows Using the K–BKZ–PSM Integral Constitutive Equation. Polymers 2023, 15, 3705. https://doi.org/10.3390/polym15183705
Bertoco J, Castelo A, Ferrás LL, Fernandes C. Numerical Simulation of Three-Dimensional Free Surface Flows Using the K–BKZ–PSM Integral Constitutive Equation. Polymers. 2023; 15(18):3705. https://doi.org/10.3390/polym15183705
Chicago/Turabian StyleBertoco, Juliana, Antonio Castelo, Luís L. Ferrás, and Célio Fernandes. 2023. "Numerical Simulation of Three-Dimensional Free Surface Flows Using the K–BKZ–PSM Integral Constitutive Equation" Polymers 15, no. 18: 3705. https://doi.org/10.3390/polym15183705
APA StyleBertoco, J., Castelo, A., Ferrás, L. L., & Fernandes, C. (2023). Numerical Simulation of Three-Dimensional Free Surface Flows Using the K–BKZ–PSM Integral Constitutive Equation. Polymers, 15(18), 3705. https://doi.org/10.3390/polym15183705