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Vorticity dynamics and numerical ResolutionofNavier-Stokes Equations

Published online by Cambridge University Press:  15 April 2002

Matania Ben-Artzi ,
Dalia Fishelov  and
Shlomo Trachtenberg
Matania Ben-Artzi
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel. ([email protected])
Dalia Fishelov
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. ([email protected])
Shlomo Trachtenberg
Affiliation:
Department of Membrane and Ultrastructure Research, The Hebrew University-Hadassah Medical School, P.O. Box 12271, Jerusalem 91120, Israel. ([email protected])
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Abstract

We present a new methodology for the numerical resolution of the hydrodynamicsof incompressible viscid newtonian fluids. It is based on the Navier-Stokesequations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurationstypical to the motion of biological structures in viscous fluids.Although the method is applicable to three dimensions, we address herein detail only the two dimensional case. We provide numerical data forsome test cases to which we apply the computational scheme.

Keywords

Navier-Stokes equationsvorticity-streamfunctionnumerical algorithmvorticity boundary conditions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 2 , March 2001 , pp. 313 - 330
Copyright
© EDP Sciences, SMAI, 2001

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References

Anderson, C.R., Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows. J. Comp. Phys. 80 (1989) 72-97. CrossRef
Bell, J.B., Colella, P. and Glaz, H.M., A second-order projection method for the incompressible navier-stokes equations. J. Comp. Phys. 85 (1989) 257-283. CrossRef
M. Ben-Artzi, Vorticity dynamics in planar domains. (In preparation).
Ben-Artzi, M., Global solutions of two-dimensional navier-stokes and euler equations. Arch. Rat. Mech. Anal. 128 (1994) 329-358. CrossRef
Bjorstad, P., Fast numerical solution of the biharmonic dirichlet problem on rectangles. SIAM J. Numer. Anal. 20 (1983) 59-71. CrossRef
Chorin, A.J., Numerical solution of the navier-stokes equations. Math. Comp. 22 (1968) 745-762. CrossRef
Chorin, A.J., Vortex sheet approximation of boundary layers. J. Comp. Phys. 27 (1978) 428-442. CrossRef
A.J. Chorin and J.E. Marsden, A mathematical introduction to fluid mechanics. 2nd edn., Springer-Verlag, New York (1990).
Dean, E.J., Glowinski, R. and Pironneau, O., Iterative solution of the stream function-vorticity formulation of the stokes problem, application to the numerical simulation of incompressible viscous flow. Comput. Method Appl. Mech. Engrg. 87 (1991) 117-155. CrossRef
Dennis, S.C.R. and Quartapelle, L., Some uses of green's theorem in solving the navier-stokes equations. Internat. J. Numer. Methods Fluids 9 (1989) 871-890. CrossRef
E, W. and Liu, J.-G., Essentially compact schemes for unsteady viscous incompressible flows. J. Comp. Phys. 126 (1996) 122-138. CrossRef
E, W. and Liu, J.-G., Vorticity boundary condition and related issues for finite difference scheme. J. Comp. Phys. 124 (1996) 368-382. CrossRef
E, W. and Liu, J.-G., Finite difference methods for 3-d viscous incompressible flows in the vorticity-vector potential formulation on nonstaggered grids. J. Comp. Phys. 138 (1997) 57-82. CrossRef
Fishelov, D., Simulation of three-dimensional turbulent flow in non-cartesian geometry. J. Comp. Phys. 115 (1994) 249-266. CrossRef
Ghia, U., Ghia, K.N. and Shin, C.T., High-re solutions for incompressible flow using the navier-stokes equations and a multigrid method. J. Comp. Phys. 48 (1982) 387-411. CrossRef
R. Glowinski, Personal communication.
Gresho, P.M., Incompressible fluid dynamics: some fundamental formulation issues. Ann. Rev. Fluid Mech. 23 (1991) 413-453. CrossRef
Gresho, P.M. and Chan, S.T., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix, parts I-II. Internat. J. Numer. Methods Fluids 11 (1990) 587-659. CrossRef
K.E. Gustafson and J.A. Sethian (Eds.), Vortex methods and vortex motion. SIAM, Philadelphia (1991).
Hwang, R.R. and Yao, C-C., A numerical study of vortex shedding from a square cylinder with ground effect. J. Fluids Eng. 119 (1997) 512-518. CrossRef
Kelkar, K.M. and Patankar, S.V., Numerical prediction of vortex sheddind behind a square cylinder. Internat. J. Numer. Methods Fluids 14 (1992) 327-341. CrossRef
O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York (1963).
L.D. Landau and E.M. Lifshitz, Fluid mechanics, Chap. II, Sec. 15. Pergamon Press, New York (1959).
Leray, J., Etudes de diverses equations integrales non lineaires et des quelques problemes que pose l'hydrodynamique. J. Math. Pures Appl. 12 (1933) 1-82.
Lyn, D.A., Einav, S., Rodi, S. and Park, J.H., A laser-doppler velocometry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. J. Fluid Mech. 304 (1995) 285-319. CrossRef
Orszag, S.A. and Israeli, M., in Numerical simulation of viscous incompressible flows, M. van Dyke, W.A. Vincenti, J.V. Wehausen, Eds., Ann. Rev. Fluid Mech. 6 (1974) 281-318. CrossRef
T.W. Pan and R. Glowinski, A projection/wave-like equation method for the numerical simulation of incompressible viscous fluid flow modeled by the navier-stokes equations. Comput. Fluid Dynamics 9 (2000).
O. Pironneau, Finite element methods for fluids. John Wiley & Sons, New York (1989).
L. Quartapelle, Numerical solution of the incompressible Navier-Stokes equations. Birkhauser Verlag, Basel (1993).
Quartapelle, L. and Valz-Gris, F., Projection conditions on the vorticity in viscous incompressible flows. Internat. J. Numer. Methods Fluids 1 (1981) 129-144. CrossRef
Temam, R., Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II. Arch. Ration. Mech. Anal. 33 (1969) 377-385. CrossRef
R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam (1979).
Tezduyar, T.E., Liou, J., Ganjoo, D.K. and Behr, M., Solution techniques for the vorticity-streamfunction formulation of the two-dimensional unsteady incompressible flows. Internat. J. Numer. Methods Fluids 11 (1990) 515-539. CrossRef