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Dynamics of high-Deborah-number entry flows: a numerical study

Published online by Cambridge University Press:  13 April 2011

A. M. AFONSO ,
P. J. OLIVEIRA ,
F. T. PINHO  and
M. A. ALVES
A. M. AFONSO
Affiliation:
Departamento de Engenharia Química, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal
P. J. OLIVEIRA
Affiliation:
Departamento de Engenharia Electromecânica, Unidade de Materiais Texteis e Papeleiros, Universidade da Beira Interior, 6201-001 Covilhã, Portugal
F. T. PINHO
Affiliation:
Departamento de Engenharia Mecânica, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal
M. A. ALVES*
Affiliation:
Departamento de Engenharia Química, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal
*
†Email address for correspondence: mmalves@fe.up.pt
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Abstract

High-elasticity simulations of flows through a two-dimensional (2D) 4 : 1 abrupt contraction and a 4 : 1 three-dimensional square–square abrupt contraction were performed with a finite-volume method implementing the log-conformation formulation, proposed by Fattal & Kupferman (J. Non-Newtonian Fluid Mech., vol. 123, 2004, p. 281) to alleviate the high-Weissenberg-number problem. For the 2D simulations of Boger fluids, modelled by the Oldroyd-B constitutive equation, local flow unsteadiness appears at a relatively low Deborah number (De) of 2.5. Predictions at higher De were possible only with the log-conformation technique and showed that the periodic unsteadiness grows with De leading to an asymmetric flow with alternate back-shedding of vorticity from pulsating upstream recirculating eddies. This is accompanied by a frequency doubling mechanism deteriorating to a chaotic regime at high De. The log-conformation technique provides solutions of accuracy similar to the thoroughly tested standard finite-volume method under steady flow conditions and the onset of a time-dependent solution occurred approximately at the same Deborah number for both formulations. Nevertheless, for Deborah numbers higher than the critical Deborah number, and for which the standard iterative technique diverges, the log-conformation technique continues to provide stable solutions up to quite (impressively) high Deborah numbers, demonstrating its advantages relative to the standard methodology. For the 3D contraction, calculations were restricted to steady flows of Oldroyd-B and Phan-Thien–Tanner (PTT) fluids and very high De were attained (De ≈ 20 for PTT with ϵ = 0.02 and De ≈ 10000 for PTT with ϵ = 0.25), with prediction of strong vortex enhancement. For the Boger fluid calculations, there was inversion of the secondary flow at high De, as observed experimentally by Sousa et al. (J. Non-Newtonian Fluid Mech., vol. 160, 2009, p. 122).

JFM classification

Complex Fluids: Complex Fluids Low-Reynolds-number flows: Low-Reynolds-number flows Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 677 , 25 June 2011 , pp. 272 - 304
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Aboubacar, M. & Webster, M. F. 2001 A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows. J. Non-Newtonian Fluid Mech. 98, 83–106.CrossRefGoogle Scholar
Afonso, A., Oliveira, P. J., Pinho, F. T., Alves, M. A. 2009 The log-conformation tensor approach in the finite-volume method framework. J. Non-Newtonian Fluid Mech. 157, 55–65.CrossRefGoogle Scholar
Alves, M. A., Oliveira, P. J. & Pinho, F. T. 2003 a Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. J. Non-Newtonian Fluid Mech. 110, 45–75.Google Scholar
Alves, M. A., Oliveira, P. J. & Pinho, F. T. 2003 b A convergent and universally bounded interpolation scheme for the treatment of advection. Intl J. Numer. Meth. Fluids 41, 47–75.CrossRefGoogle Scholar
Alves, M. A., Pinho, F. T. & Oliveira, P. J. 2000 Effect of a high-resolution differencing scheme on finite-volume predictions of viscoelastic flows. J. Non-Newtonian Fluid Mech. 93, 287–314.CrossRefGoogle Scholar
Alves, M. A., Pinho, F. T. & Oliveira, P. J. 2005 Visualizations of Boger fluid flows in a 4 : 1 square–square contraction. AIChE J. 51 (11), 2908–2922.CrossRefGoogle Scholar
Alves, M. A., Pinho, F. T. & Oliveira, P. J. 2008 Viscoelastic flow in a 3D square/square contraction: Visualizations and simulations. J. Rheol. 52, 1347–1368.CrossRefGoogle Scholar
Alves, M. A. & Poole, R. J. 2007 Divergent flow in contractions. J. Non-Newtonian Fluid Mech. 144, 140–148.CrossRefGoogle Scholar
Baaijens, F. T. P. 1998 Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newtonian Fluid Mech. 79, 361–385.Google Scholar
Belblidia, F., Keshtiban, I. J. & Webster, M. F. 2006 Stabilised computations for viscoelastic flows under compressible implementations. J. Non-Newtonian Fluid Mech. 134, 56–76.CrossRefGoogle Scholar
Bird, R. B., Dotson, P. J. & Johnson, N. L. 1980 Polymer solution rheology based on a finitely extensible bead-spring chain model. J. Non-Newtonian Fluid Mech. 7, 213–235.CrossRefGoogle Scholar
Boger, D. V. 1987 Viscoelastic flows through contractions. Annu. Rev. Fluid Mech. 19, 157–182.Google Scholar
Cable, P. J. & Boger, D. V. 1978 a A comprehensive experimental investigation of tubular entry flow of viscoelastic fluids. Part I. Vortex characteristics in stable flow. AIChE J. 24, 869–879.Google Scholar
Cable, P. J. & Boger, D. V. 1978 b A comprehensive experimental investigation of tubular entry flow of viscoelastic fluids. Part II. The velocity field in stable flow. AIChE J. 24, 992–999.CrossRefGoogle Scholar
Cable, P. J. & Boger, D. V. 1979 A comprehensive experimental investigation of tubular entry flow of viscoelastic fluids. Part III. Unstable flow. AIChE J. 25, 152–159.CrossRefGoogle Scholar
Chiba, K., Sakatani, T. & Nakamura, K. 1990 Anomalous flow patterns in viscoelastic entry flow through a planar contraction. J. Non-Newtonian Fluid Mech. 36, 193–203.Google Scholar
Chiba, K., Yoshida, I., Sako, S. & Mori, N. 2004 Anomalous entry flow patterns in the transition regime to global flow instability generated after vortex enhancement. J. Soc. Rheol. Japan 32–35, 303–311.CrossRefGoogle Scholar
Coronado, O. M., Arora, D., Behr, M. & Pasquali, M. 2007 A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation. J. Non-Newtonian Fluid Mech. 147, 189–199.Google Scholar
El Hadj, M. & Tanguy, P. A. 1990 A finite element procedure coupled with the method of characteristics for simulation of viscoelastic fluid flow. J. Non-Newtonian Fluid Mech. 36, 333–349.CrossRefGoogle Scholar
Fattal, R. & Kupferman, R. 2004 Constitutive laws of the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123, 281–285.CrossRefGoogle Scholar
Fattal, R. & Kupferman, R. 2005 Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newtonian Fluid Mech. 126, 23–37.CrossRefGoogle Scholar
Fortin, M. & Esselaoui, D. 1987 A finite element procedure for viscoelastic flows. Intl. J. Numer. Meth. Fluids 7, 1035–1052.CrossRefGoogle Scholar
Guénette, R., Fortin, A., Kane, A. & Hétu, J.-F. 2008 An adaptive remeshing strategy for viscoelastic fluid flow simulations. J. Non-Newtonian Fluid Mech. 153, 34–45.CrossRefGoogle Scholar
Hassager, O. 1988 Working group on numerical techniques. (Proceedings of the Vth Workshop on Numerical Methods in Non-Newtonian Flow) J. Non-Newtonian Fluid Mech. 29, 2–5.Google Scholar
van Heel, A. P. G., Hulsen, M. A. & van den Brule, B. H. A. A. 1998 On the selection of parameters in the FENE-P model. J. Non-Newtonian Fluid Mech. 75, 253–271.CrossRefGoogle Scholar
Howell, J. S. 2009 Computation of viscoelastic fluid flows using continuation methods. J. Comput. Appl. Maths 225, 187–201.Google Scholar
Hulsen, M. A., Fattal, R. & Kupferman, R. 2005 Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms. J. Non-Newtonian Fluid Mech. 127, 27–39.Google Scholar
Kane, A., Guénette, R. & Fortin, A. 2009 A comparison of four implementations of the log-conformation formulation for viscoelastic fluid flows. J. Non-Newtonian Fluid Mech. 164, 45–50.CrossRefGoogle Scholar
Keunings, R. 1989 Simulation of viscoelastic flow. In Computer Modeling for Polymer Processing (ed. Tucker, C. L.), pp. 404–469. Hanser.Google Scholar
Keunings, R. 2004 Micro-macro methods for the multiscale simulation of viscoelastic flow using molecular models of kinetic theory. Rheol. Rev. 2, 67–98.Google Scholar
Kim, J. M., Kim, C., Kim, J. H., Chung, C., Ahn, K. H. & Lee, S. J. 2005 High-resolution finite element simulation of 4 : 1 planar contraction flow of viscoelastic fluid. J. Non-Newtonian Fluid Mech. 129, 23–37.CrossRefGoogle Scholar
Koppol, A. P., Sureshkumar, R., Abedijaberi, A. & Khomami, B. 2009 Anomalous pressure drop behavior of mixed kinematics flow of viscoelastic polymer solutions: a multiscale simulation approach. J. Fluid Mech. 631, 231–253.CrossRefGoogle Scholar
Kwon, Y. 2004 Finite element analysis of planar 4 : 1 contraction flow with the tensor-logarithmic formulation of differential constitutive equations. Korea-Austral. Rheol. J. 16, 183–191.Google Scholar
Kwon, Y. 2006 Numerical analysis of viscoelastic flows in a channel obstructed by an asymmetric array of obstacles. Korea-Austral. Rheol. J. 18, 161–167.Google Scholar
Lee, J. S., Dylla-Spears, R., Teclemariam, N.-P. & Muller, S. J. 2007 Microfluidic four-roll mill for all flow types. Appl. Phys. Lett. 90, 074103.CrossRefGoogle Scholar
Lielens, G., Keunings, R. & Legat, V. 1999 The FENE-L and FENE-LS closure approximations to the kinetic theory of finitely extensible dumbbells. J. Non-Newtonian Fluid Mech. 87, 179–196.CrossRefGoogle Scholar
McKinley, G. H., Raiford, W. P., Brown, R. A. & Armstrong, R. C. 1991 Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions. J. Fluid Mech. 223, 411–456.CrossRefGoogle Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.CrossRefGoogle Scholar
Nigen, S. & Walters, K. 2002 Viscoelastic contraction flows: comparison of axisymmetric and planar configurations. J. Non-Newtonian Fluid Mech. 102, 343–359.CrossRefGoogle Scholar
Oldroyd, J. G. 1950 On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541.Google Scholar
Oliveira, P. J. 2001 Time-dependent simulations of shear-thinning elastic flows through contractions. In Proc. of ASME, IMECE, 11–16 November 2001, New York, NY.CrossRefGoogle Scholar
Oliveira, P. J. & Pinho, F. T. 1999 Plane contraction flows of upper convected Maxwell and Phan-Thien–Tanner fluids as predicted by a finite-volume method. J. Non-Newtonian Fluid Mech. 88, 63–88.CrossRefGoogle Scholar
Oliveira, P. J., Pinho, F. T. & Pinto, G. A. 1998 Numerical simulation of non-linear elastic flows with a general collocated finite-volume method. J. Non-Newtonian Fluid Mech. 79, 1–43.CrossRefGoogle Scholar
Owens, R. G. & Phillips, T. N. 2002 Computational Rheology. Imperial College Press.CrossRefGoogle Scholar
Pan, T. W. & Hao, J. 2007 Numerical simulation of a lid-driven cavity viscoelastic flow at high Weissenberg numbers. C. R. Acad. Sci. Paris I 344, 283–286.Google Scholar
Phan-Thien, N. 1978 A non-linear network viscoelastic model. J. Rheol. 22, 259–283.CrossRefGoogle Scholar
Phan-Thien, N. & Tanner, R. I. 1977 New constitutive equation derived from network theory. J. Non-Newtonian Fluid Mech. 2, 353–365.CrossRefGoogle Scholar
Rothstein, J. P. & McKinley, G. H. 1999 Extensional flow of a polystyrene Boger fluid through a 4 : 1 : 4 contraction/expansion. J. Non-Newtonian Fluid Mech. 86, 61–88.CrossRefGoogle Scholar
Rothstein, J. P. & McKinley, G. H. 2001 The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop. J. Non-Newtonian Fluid Mech. 98, 33–63.CrossRefGoogle Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.CrossRefGoogle Scholar
Sirakov, I., Ainser, A., Haouche, M. & Guillet, J. 2005 Three-dimensional numerical simulation of viscoelastic contraction flows using the Pom–Pom differential constitutive model. J. Non-Newtonian Fluid Mech. 126, 163–173.Google Scholar
Sousa, P. C., Coelho, P. M., Oliveira, M. S. N. & Alves, M. A. 2009 Three-dimensional flow of Newtonian and Boger fluids in square–square contractions. J. Non-Newtonian Fluid Mech. 160 (2–3), 122–139.CrossRefGoogle Scholar
Walters, K. & Webster, M. F. 2003 The distinctive CFD challenges of computational rheology, ECCOMAS Swansea 2001. Intl J. Numer. Meth. Fluids 45, 577–596.CrossRefGoogle Scholar
Walters, K., Webster, M. F. & Tamaddon-Jahromi, H. R. 2009 The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheology. Chem. Engng Sci. 64, 4632–4639.CrossRefGoogle Scholar
White, F. M. 1991 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.CrossRefGoogle Scholar
Yesilata, B., Öztekin, A. & Neti, S. 1999 Instabilities in viscoelastic flow through an axisymmetric sudden contraction. J Non-Newtonian Fluid Mech. 85, 35–62.CrossRefGoogle Scholar
Yoon, S. & Kwon, Y. 2005 Finite element analysis of viscoelastic flows in a domain with geometric singularities. Korea-Austral. Rheol. J. 17, 99–110.Google Scholar
Zhou, Q. & Akhavan, R. 2003 A comparison of FENE and FENE-P dumbbell and chain models in turbulent flow. J. Non-Newtonian Fluid Mech. 109, 115–155.CrossRefGoogle Scholar

Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 5 – Vortex merging and growth

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Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 5 – Vortex merging and growth

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Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 10 – Elastic vortex growth.

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Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 10 – Elastic vortex growth.

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Video 1.6 MB

Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 15 – Onset of third vortex

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Video 2.7 MB

Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 15 – Onset of third vortex

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Video 8.7 MB

Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 20 – Third vortex growth and vortex back-shedding

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Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 20 – Third vortex growth and vortex back-shedding

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Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 30 – Third vortex growth and vortex back-shedding (cont.).

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Video 1.7 MB

Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 30 – Third vortex growth and vortex back-shedding (cont.).

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Video 6.4 MB

Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 100 – Third vortex growth and vortex back-shedding (cont.).

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Video 1.4 MB

Afonso et al. supplementary material

Flow in a 4:1 planar contraction: De = 100 – Third vortex growth and vortex back-shedding (cont.).

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Video 2.5 MB