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Research Papers: Fundamental Issues and Canonical Flows

High-Order Solution of Viscoelastic Fluids Using the Discontinuous Galerkin Method

[+] Author and Article Information
Ehsan Mirzakhalili

School of Mechanical Engineering,
College of Engineering,
University of Tehran,
North Karegar Avenue,
P.O. Box 11155-4563,
Tehran, Iran
e-mail: e.mirzakhalili@ut.ac.ir

Amir Nejat

Assistant Professor
School of Mechanical Engineering,
College of Engineering,
University of Tehran,
North Karegar Avenue,
P.O. Box 11155-4563,
Tehran, Iran
e-mail: nejat@ut.ac.ir

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 20, 2014; final manuscript received October 4, 2014; published online November 20, 2014. Assoc. Editor: D. Keith Walters.

J. Fluids Eng 137(3), 031205 (Mar 01, 2015) (9 pages) Paper No: FE-14-1035; doi: 10.1115/1.4028779 History: Received January 20, 2014; Revised October 04, 2014; Online November 20, 2014

In this paper, the high-order solution of a viscoelastic fluid is investigated using the discontinuous Galerkin (DG) method. The Oldroyd-B model is used to describe the viscoelastic behavior of the fluid flow. The high-order accuracy of the applied DG method is verified for a Newtonian benchmark problem with an exact solution. Next, the same algorithm is utilized to solve the viscoelastic flow by separating the stress tensor into the stress due to the Newtonian solvent and the stress due to the solved viscoelastic polymers. The high-order accuracy of the solution for viscoelastic flow is demonstrated by solving the planar Poiseuille flow. Then, the planar contraction problem is simulated as a benchmark for the viscoelastic flow. The obtained results are in good agreement with the results in the literature for both creeping and inertial flow when high-order polynomials were used even on coarse meshes.

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References

Šutalo, I. D., Bui, A., and Rudman, M., 2006, “The Flow of Non-Newtonian Fluids Down Inclines,” J. Non-Newtonian Fluid Mech., 136(1), pp. 64–75. [CrossRef]
Berli, C. L. A., and Olivares, M. L., 2008, “Electrokinetic Flow of Non-Newtonian Fluids in Microchannels,” J. Colloid Interface Sci., 320(2), pp. 582–589. [CrossRef] [PubMed]
Perkins, T. T., Smith, D. E., and Chu, S., 1997, “Single Polymer Dynamics in an Elongational Flow,” Science, 276(5321), pp. 2016–2021. [CrossRef] [PubMed]
Kim, H. J., Kim, J. T., Lee, K., and Choi, H. J., 2000, “Mechanical Degradation of Dilute Polymer Solutions Under Turbulent Flow,” Polymer, 41(21), pp. 7611–7615. [CrossRef]
Farid, M., and Abdul Ghani, A. G., 2004, “A New Computational Technique for the Estimation of Sterilization Time in Canned Food,” Chem. Eng. Process, 43(4), pp. 523–531. [CrossRef]
Crochet, M. J., and Walters, K., 1983, “Numerical Methods in Non-Newtonian Fluid Mechanics,” Annu. Rev. Fluid Mech., 15, pp. 241–260. [CrossRef]
Crochet, M. J., Davies, A. R., and Walters, K., 1984, Numerical Simulation of Non-Newtonian Flow, Elsevier, NY.
Owens, R. G., and Phillips, T. N., 2002, Computational Rheology, Imperial College, London, UK. [CrossRef]
Oldroyd, J., 1950, “On the Formulation of Rheological Equations of State,” Proc. R. Soc. London, Ser. A, 200(1063), pp. 523–541. [CrossRef]
Aboubacar, M., Matallah, H., and Webster, M. F., 2002, “Highly Elastic Solutions for Oldroyd-B and Phan-Thien/Tanner Fluids With a Finite Volume/Element Method: Planar Contraction Flows,” J. Non-Newtonian Fluid Mech., 103(1), pp. 65–103. [CrossRef]
Phillips, T. N., and Williams, A. J., 1999, “Viscoelastic Flow Through a Planar Contraction Using a Semi-Lagrangian Finite Volume Method,” J. Non-Newtonian Fluid Mech., 87(2–3), pp. 215–246. [CrossRef]
Phillips, T. N., and Williams, A. J., 2002, “Comparison of Creeping and Inertial Flow of an Oldroyd-B Fluid Through Planar and Axisymmetric Contractions,” J. Non-Newtonian Fluid Mech., 108(1–3), pp. 25–47. [CrossRef]
Alves, M. A., Oliveira, P. J., and Pinho, F. T., 2003, “Benchmark Solutions for the Flow of Oldroyd-B and PTT Fluids in Planar Contractions,” J. Non-Newtonian Fluid Mech., 110(1), pp. 45–75. [CrossRef]
Zhang, X. H., Ouyang, J., and Zhang, L., 2010, “Characteristic Based Split (CBS) Mesh-Free Method Modeling for Viscoelastic Flow,” Eng. Anal. Boundary Elem., 34(2), pp. 163–172. [CrossRef]
Malaspinasa, O., Fietier, N., and Devillea, M., 2010, “Lattice Boltzmann Method for the Simulation of Viscoelastic Fluid Flows,” J. Non-Newtonian Fluid Mech., 165(23–24), pp. 1637–1653. [CrossRef]
Baaijens, F. P. T., 1998, “Mixed Finite Element Methods for Viscoelastic Flow Analysis: A Review,” J. Non-Newtonian Fluid Mech., 79(2–3), pp. 361–385. [CrossRef]
Fiétier, N., and Deville., M. O., 2003, “Time-Dependent Algorithms for the Simulation of Viscoelastic Flows With Spectral Element Methods: Applications and Stability,” J. Comput. Phys., 186(1), pp. 93–121. [CrossRef]
Claus, S., and Phillips, T. N., 2013, “Viscoelastic Flow Around a Confined Cylinder Using Spectral/hp Element Methods,” J. Non-Newtonian Fluid Mech., 200, pp. 131–146. [CrossRef]
Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D., 2001, “Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems,” SIAM J. Numer. Anal., 39(5), pp. 1749–1779. [CrossRef]
Riviere, B., 2008, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM-Frontiers in Applied Mathematics, Houston, TX. [CrossRef]
Cockburn, B., and Shu, C. W., 2001, “Runge–Kutta Discontinuous Galerkin Methods for Convection Dominated Problems,” J. Sci. Comput., 16(3), pp. 173–261. [CrossRef]
Cockburn, B., Kanschat, G., Schotzau, D., and Schwab, C., 2002, “The Local Discontinuous Galerkin Method for the Stokes System,” SIAM J. Numer. Anal., 40(1), pp. 319–343. [CrossRef]
Hansbo, P., and Larson, M. G., 2002, “Discontinuous Galerkin Methods for Incompressible and Nearly Incompressible Elasticity by Nitsche's Method,” Comput. Methods Appl. Mech. Eng., 191(17), pp. 1895–1908. [CrossRef]
Schotzau, D., Schwab, C., and Toselli, A., 2003, “Mixed hp-DGFEM for Incompressible Flow,” SIAM J. Numer. Anal., 40(6), pp. 2171–2194. [CrossRef]
Nguyen, N. C., Peraire, J., and Cockburn, B., 2010, “A Hybridizable Discontinuous Galerkin Method for Stokes Flow,” Comput. Methods Appl. Mech. Eng., 199(9–12), pp. 582–597. [CrossRef]
Cockburn, B., Kanschat, G., and Schotzau, G., 2005, “A Locally Conservative LDG Method for the Incompressible Navier–Stokes Equations,” Math. Comput., 74(251), pp. 1067–1095. [CrossRef]
Girault, V., Riviere, B., and Wheeler, M. F., 2005, “A Discontinuous Galerkin Method With Non-Overlapping Domain Decomposition for the Stokes and Navier–Stokes Problems,” Math. Comput., 74, pp. 53–84. [CrossRef]
Shahbazi, K., Fischer, P. F., and Ethier, C. R., 2007, “A High-Order Discontinuous Galerkin Method for the Unsteady Incompressible Navier–Stokes Equations,” J. Comput. Phys., 222(1), pp. 391–407. [CrossRef]
Nguyen, N. C., Peraire, J., and Cockburn, B., 2011, “An Implicit High-Order Hybridizable Discontinuous Galerkin Method the Incompressible Navier–Stokes Equations,” J. Comput. Phys., 230(4), pp. 1147–1170. [CrossRef]
Aboubacar, M., and Webster, M. F., 2001, “A Cell-Vertex Finite Volume/Element Method on Triangles for Abrupt Contraction Viscoelastic Flows,” J. Non-Newtonian Fluid Mech., 98(2–3), pp. 83–106. [CrossRef]
Karniadakis, G. E., and Sherwin, S. J., 2005, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd ed., Oxford Science Publications, Oxford, UK. [CrossRef]
Mavriplis, D., Nastase, C., Shahbazi, K., Wang, L., and Burgess, N., 2009, “Progress in High-Order Discontinuous Galerkin Methods for Aerospace Applications,” AIAA Paper No. 2009-0601. [CrossRef]
Girault, V., and Wheeler, F., 2008, “Discontinuous Galerkin Methods,” Comput. Methods Appl. Sci., 16, pp. 3–26. [CrossRef]
Castillo, P., 2002, “Performance of Discontinuous Galerkin Methods for Elliptic PDEs,” SIAM J. Sci. Comput., 24(2), pp. 524–547. [CrossRef]
Deville, M. O., Fischer, P. F., and Mund, E. H., 2002, “High-Order Methods for Incompressible Fluid Flows,” Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK. [CrossRef]
Guermond, J. L., Minev, P., and Shen, J., 2006, “An Overview of Projection Methods for Incompressible Flows,” Comput. Methods Appl. Mech. Eng., 195(44–47), pp. 6011–6045. [CrossRef]
Karniadakis, G. E., Israeli, M., and Orszag, S. A., 1991, “High-Order Splitting Methods for Incompressible Navier–Stokes Equations,” J. Comput. Phys., 97(2), pp. 414–443. [CrossRef]
Hesthaven, J. S., and Warburton, T., 2008, Nodal Discontinuous Galerkin Methods—Algorithms, Analysis, and Applications, Springer, Germany.
Edussuriya, S. S., Williams, A. J., and Bailey, C., 2004, “A Cell-Centred Finite Volume Method for Modeling Viscoelastic Flow,” J. Non-Newtonian Fluid Mech., 117(1), pp. 47–61. [CrossRef]
Szego, G., 1939, Orthogonal Polynomials, Vol. 23, American Mathematical Society, Colloquium Publications, Providence, RI.
Shahbazi, K., 2005, “An Explicit Expression for the Penalty Parameter of the Interior Penalty Method,” J. Comput. Phys., 205(2), pp. 401–407. [CrossRef]
Kovasznay, L. S. G., 1948, “Laminar Flow Behind a Two-Dimensional Grid,” Proc. Cambridge Philos. Soc., 44(1), pp. 58–62. [CrossRef]
Kim, J. M., Kim, C., Kim, J. H., Chung, C., Ahn, K. H., and Lee, S. J., 2005, “High-Resolution Finite Element Simulation of 4:1 Planar Contraction Flow of Viscoelastic Fluid,” J. Non-Newtonian Fluid Mech., 129(1), pp. 23–37. [CrossRef]
Matallah, H., Townsend, P., and Webster, M. F., 1998, “Recovery and Stress-Sliptting Schemes for Viscoelastic Flows,” J. Non-Newtonian Fluid Mech., 75(2–3), pp. 139–166. [CrossRef]
Han, X. H., 2007, “Finite Element Modeling of Non-Isothermal Non-Newtonian Viscoelastic Flow in Mould Filling Process,” Ph.D. thesis, Dalian University of Technology, Dalian City, Liaoning Province, China.

Figures

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Fig. 1

Domain of the Kovasznay flow

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Fig. 2

Numerical solution of Kovasznay flow

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Fig. 3

Norm of the error for y-velocity component in the Kovasznay flow

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Fig. 4

The numerical solution of viscoelastic stresses for simple channel flow (exact solutions are shown by symbols)

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Fig. 5

The geometry of benchmark problem

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Fig. 6

The computational grid of benchmark problem consisting of 360 elements

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Fig. 7

Variation of corner vortex length with We number for the creeping flow

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Fig. 8

Length of the corner vortex for the creeping flow

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Fig. 9

Intensity of the corner vortex for the creeping flow

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Fig. 10

Variation of corner vortex length with We number for the inertial flow

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Fig. 11

Length of the corner vortex for the inertial flow

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