A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows

Ahmed Elhanafy; Amr Guaily; Ahmed Elsaid

doi:doi:10.11648/j.mma.20160102.11

| Peer-Reviewed

A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows

Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid

Published in Mathematical Modelling and Applications (Volume 1, Issue 2)

Received: 10 September 2016 Accepted: 14 October 2016 Published: 21 October 2016

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.

Published in	Mathematical Modelling and Applications (Volume 1, Issue 2)
DOI	10.11648/j.mma.20160102.11
Page(s)	26-35
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Unsteady Incompressible Viscoelastic Flow, Oldroyd-B Model, Pressure Stabilization Technique, The DEVSS Method, Galerkin Least Squares

References

[1]	S. zou, X. Xu, J. Chen, X. Guo and Q. Wang, Benchmark numerical simulations of viscoelastic fluid flows with an efficient integrated lattice Boltzmann and finite volume scheme, Advances in mechanical engineering, Hindawi Publishing Corporation 2, 2014.
[2]	K. Kwak, C. Kiris, J. Chang, Computational challenges of viscous incompressible flows, Journal of computers and fluids. 34, 283-299, 2005.
[3]	G. X. Xu, E. Li, V. Tan and G. R. Liu, Simulation of steady and unsteady incompressible flow using gradient smoothing method (GSM), Journal of computers and structures 90-91, 131-144, 2012.
[4]	D. Choi, C. Merkle, The application of preconditioning in viscous flows, Journal of ComputPhys 105, 203–226, 1993.
[5]	E. Turkel, Preconditioned methods for solving the incompressible and low speed compressible equations, Journal of ComputPhys. 72, 277–375, 1987.
[6]	S. V. Patankar, Numerical heat transfer and fluid flow, New York: Hemisphere Publishing, 1980.
[7]	H. P. Langtangen, K. Mardal, R. Winther, Numerical methods for incompressible viscous flow, Journal of advances in water resources 25, 1125-1146, 2002.
[8]	R. Courant, Calculus of Variations and Supplementary Notes and Exercise. New York University, New York, 1956.
[9]	R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1984.
[10]	R. S. Marshall, J. C. Heinrich, and O. C. Zienkiewicz, Natural convection in a square enclosure by a finite element penalty function method using primitive fluid variables, Journal of Numerical Heat Transfer. 1, 315–330, 1987.
[11]	T. J. R. Hughes, W. K. Liu, and A. Brooks, Finite element analysis of incompressible viscous flows by the penalty function formulation, Journal of Computational Physics. 30, 1–60, 1979.
[12]	A. J. Chorin. A numerical method for solving incompressible viscous flow problems, Journal of Computational Physics, 212–26, 1967.
[13]	X. Li, X. Han, X. wang, Numerical modeling of viscoelastic flows using equal low-order finite elements, J. Comput. Methods Appl. Mech. Engrg. 199, 570-581, 2010.
[14]	S. R. Burdette, P. J. Coates, R. C. Armstrong, R. A. Brown, Calculations of viscoelastic flow through an axisymmetric corrugated tube using the explicitly elliptic momentum equation formulation, J. Non-Newtonian fluid Mech. 33, 1-23, 1989.
[15]	D. Rajagopalan, R. C. Armstrong, R. A. Brown, Finite element methods for calculation of steady viscoelastic flow using constitutive equations with a Newtonian viscosity, J. Non-Newtonian fluid Mech. 36, 159-192, 1990.
[16]	R. Guenette, M. Fortin, A new mixed finite element method for computing viscoelastic flow, J. Non-Newtonian fluid Mech. 60, 27-52, 1995.
[17]	T. J. R. Hughes, L. P. Franca and G. M. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/Least-squares method for advective-diffusive equations, Journal of computer methods in applied mechanics and engineering. 73, 173-189, 1989.
[18]	J. Hao, T. Pan, Simulation for high Weissenberg number viscoelastic flow by a finite element method, Applied mathematics letters 20, 988-993, 2007.
[19]	J. Su, J. Ouyang, X. Wang, B. Yang, W. Zhou, Lattice Boltzmann method for the simulation of viscoelastic fluid flows over a large range of Weissenberg number, J. Non-Newtonian fluid Mech. 194, 42-59, 2013.
[20]	A. Guily, Mathematical modeling and numerical simulation of viscoelastic liquids, PhD. Thesis, Faculty of graduate studies, Calgary University, 2010.
[21]	D. Kuzmin, J. Ha ̈ma ̈la ̈inen, Finite Element Methods for Computational Fluid Dynamics: A Practical Guide, SIAM, 267-270, 2015.
[22]	F. Brezzi, J. Pitkäranta, On the stabilization of finite element approximations of the Stokes problem, Efficient Solutions of Elliptic Systems (W. Hackbusch, ed.), Notes on Numerical Fluid Mechanics 10, 11-19, Vieweg, Braunschweig, 1984.
[23]	M. A. Hulsen, R. Fattal, R. Kupferman, Flow of viscoelastic fluids past a cylinder at high weissenberge number: stabilized simulations using matrix logarithms, J. Non-Newtonian fluid Mech. 127, 27-93, 2005.
[24]	Frank. P. T. Baaijens, Mixed finite element methods for viscoelastic flow analysis: a review, J. Non-Newtonian fluid Mech. 79, 361-385, 1998.
[25]	T. E. Tezduyar Stabilized finite element formulations for incompressible flow computations, Journal of Advances in applied mechanics 28, 1-44, 1991.
[26]	A. N. Brooks, T. J. R. Hughes, Streamline upwind/ petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Journal of computer methods in applied mechanics and engineering 32, 199-259, 1982.
[27]	K. Yapici, B. Karasozen, Y. Uludag, Finite volume simulation of viscoelastic laminar flow in a lid-driven cavity, J. Non-Newtonian fluid Mech. 164, 51-65, 2009.
[28]	P. Pakdel, S. H. Spiegelberg, G. H. McKinley, Cavity flows of elastic liquids: two-dimensional flows, Phys. Fluids 9 (11) 3123–3140, 1997.

Cite This Article

Plain Text BibTeX RIS

APA Style

Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid. (2016). A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Mathematical Modelling and Applications, 1(2), 26-35. https://doi.org/10.11648/j.mma.20160102.11

Copy | Download

ACS Style

Ahmed Elhanafy; Amr Guaily; Ahmed Elsaid. A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Math. Model. Appl. 2016, 1(2), 26-35. doi: 10.11648/j.mma.20160102.11

Copy | Download

AMA Style

Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid. A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Math Model Appl. 2016;1(2):26-35. doi: 10.11648/j.mma.20160102.11

Copy | Download

@article{10.11648/j.mma.20160102.11,
  author = {Ahmed Elhanafy and Amr Guaily and Ahmed Elsaid},
  title = {A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows},
  journal = {Mathematical Modelling and Applications},
  volume = {1},
  number = {2},
  pages = {26-35},
  doi = {10.11648/j.mma.20160102.11},
  url = {https://doi.org/10.11648/j.mma.20160102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20160102.11},
  abstract = {The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows
AU  - Ahmed Elhanafy
AU  - Amr Guaily
AU  - Ahmed Elsaid
Y1  - 2016/10/21
PY  - 2016
N1  - https://doi.org/10.11648/j.mma.20160102.11
DO  - 10.11648/j.mma.20160102.11
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 26
EP  - 35
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20160102.11
AB  - The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.
VL  - 1
IS  - 2
ER  -

Copy | Download

Author Information

Ahmed Elhanafy

Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt
Amr Guaily

Engineering Mathematics & Physics Department, Faculty of Engineering, Cairo University, Giza, Egypt
Ahmed Elsaid

Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid. (2016). A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Mathematical Modelling and Applications, 1(2), 26-35. https://doi.org/10.11648/j.mma.20160102.11

Copy | Download

ACS Style

Ahmed Elhanafy; Amr Guaily; Ahmed Elsaid. A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Math. Model. Appl. 2016, 1(2), 26-35. doi: 10.11648/j.mma.20160102.11

Copy | Download

AMA Style

Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid. A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Math Model Appl. 2016;1(2):26-35. doi: 10.11648/j.mma.20160102.11

Copy | Download

@article{10.11648/j.mma.20160102.11,
  author = {Ahmed Elhanafy and Amr Guaily and Ahmed Elsaid},
  title = {A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows},
  journal = {Mathematical Modelling and Applications},
  volume = {1},
  number = {2},
  pages = {26-35},
  doi = {10.11648/j.mma.20160102.11},
  url = {https://doi.org/10.11648/j.mma.20160102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20160102.11},
  abstract = {The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows
AU  - Ahmed Elhanafy
AU  - Amr Guaily
AU  - Ahmed Elsaid
Y1  - 2016/10/21
PY  - 2016
N1  - https://doi.org/10.11648/j.mma.20160102.11
DO  - 10.11648/j.mma.20160102.11
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 26
EP  - 35
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20160102.11
AB  - The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.
VL  - 1
IS  - 2
ER  -

Copy | Download