Recent Advances in Computational Fluid Dynamics pp 484-507 | Cite as

# Penalty Finite Element Applications to Flow Problems

## Abstract

This paper focuses on the appropriate elements to analyze two-dimensional internal and external flow problems by the penalty finite element method. The governing equations are the incompressible Navier-Stokes equations. Four different elements were tested to study some two-dimensional flow problems, such as Couette flow, square cavity flow, circular eddy flow, and flow past a circular cylinder. The four elements used are T6, T3M, Q4, and Q4M respectively. The first two are triangular elements, and the remains are quadrilateral elements.

It was found that for the low Reynolds number linear cases, there are no major computational differences for the four elements due to the flow simplicity. However, only the Q4 element can accurately predict the flow structures of the high Reynolds number flows. For the flow past a circular cylinder, the salient characteristics of Karman vortex streets are vividly simulated, such as the instability, vortex shedding, and pressure oscillation. It is further observed that the time integrating parameter will play an important role in the flow simulation. The velocity field will be most accurately predicted while the pressure is not, when the central (Crank-Nicolson) difference scheme is adopted. On the other hand, the reversed conjecture is true when backward difference scheme is used instead.

### Keywords

Vortex Vorticity Advection Incompressibility## Preview

Unable to display preview. Download preview PDF.

### References

- 1.D. L. Young, C. M. Huang, and Q. H. Lin, “Application of finite element method to 2-D circulating flows”, Proceedings of CWC Symposium, Taipei, R.O.C., pp. 45–56, 1986.Google Scholar
- 2.T. R. J. Hughes, W. K. Liu, and A. Brooks, “Review of finite element analysis of incompressible viscous flows by the penalty function formulation”, J. Comp. Phys., Vol 30, No 1, pp. 1–60, 1979.CrossRefMATHADSMathSciNetGoogle Scholar
- 3.C. H. Wang, “Simulation of viscous incompressible flows by the finite element Analysis”, MS Thesis, National Taiwan University, 103pp. 1986.Google Scholar
- 4.C. M. Huang, “Solutions of viscous incompressible flows by the penalty finite element method”, MS Thesis, National Taiwan University, 86pp. 1985.Google Scholar
- 5.G. Dhatt, and G. Hubert, “A study of penalty elements for incompressible laminar flows”, Int. J. Numer. Meth. Fluids, Vol. 6, pp. 1–19, 1986.CrossRefMATHMathSciNetGoogle Scholar
- 6.F. N. Van De Vosse, A. Segal, A. A. Van Steenhoven, and J. D. Janssen, “A finite element approximation of the unsteady two-dimensional Navier-Stokes equations”, I. J. Numer. Methods Fluids, Vol. 6, pp. 427–443, 1986.CrossRefMATHADSGoogle Scholar
- 7.O. C. Zienkiewicz, The Finite Element Method, 3rd Ed. McGraw-Hill Book Company, New York, 1977.MATHGoogle Scholar
- 8.M. Nallasamy, and K. Krishna Prasad, “On cavity flow at high Reynolds Numbers”, J. Fluid Mech. 79, 391–414, 1977.CrossRefMATHADSGoogle Scholar
- 9.O. R. Burggraf, “Analytical and numerical studies of the structure of steady separated flows”, J. Fluid Mech., Vol. 24, Part 1, pp. 113–151, 1966.CrossRefADSGoogle Scholar
- 10.T. Y. Hu, “Analytic solutions of some two-dimensional viscous incompressible flow problems”, MS Thesis, National Taiwan University, 46pp. 1987.Google Scholar
- 11.G. Martinez, “Caracteristiques dynamiques et thermiques de I’ecoulement autour d’un cylindre circulaire a nombres de Reynolds moderes.”. These D. I. -I. N. P. Toulouse, 1979.Google Scholar
- 12.M. Kawaguti, “Numerical solution of the Navier-Stokes equations for the flow around a circular cylinder at Reynolds number 40”, J. Phys. Soc. Japan, 8, pp. 747–757, 1953.CrossRefADSMathSciNetGoogle Scholar
- 13.V. A. Patel, “Karman vortex street behind a circular cylinder by the series truncation method”, J. Comput Phys., Vol. 28,pp. 14–42, 1978.CrossRefADSGoogle Scholar
- 14.S. Taneda, “Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers”, J. Phys. Soc. Japan, 11, pp. 302–307, 1956.CrossRefADSGoogle Scholar
- 15.M. Braza, P. Chassaing, and H. Ha Minh, “Numerical study and physical analysis of the pressure and velocity fields in near wake of a circular cylinder”, J. Fluid Mech., Vol. 165,pp. 79130, 1986.CrossRefMathSciNetGoogle Scholar
- 16.R. L. Sani, M. S. Engelman, P. M. Gresho, and M. Bercovier, “Consistent vs reduced integration penalty methods for incompressible media using several old and new elements”, Int. J. Numer. Methods Fluids, Vol. 3, pp. 25–42, 1982.MathSciNetGoogle Scholar
- 17.Q. H. Lin, “Finite element analysis to the Navier-Stokes equations”, MS Thesis, National Taiwan University, 87pp. 1986.Google Scholar
- 18.G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, UK, pp. 255–263, 1967.MATHGoogle Scholar