# Vorticity dynamics and numerical resolution of Navier-Stokes equations

Matania Ben-Artzi; Dalia Fishelov; Shlomo Trachtenberg

- Volume: 35, Issue: 2, page 313-330
- ISSN: 0764-583X

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topBen-Artzi, Matania, Fishelov, Dalia, and Trachtenberg, Shlomo. "Vorticity dynamics and numerical resolution of Navier-Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.2 (2001): 313-330. <http://eudml.org/doc/194052>.

@article{Ben2001,

abstract = {We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data for some test cases to which we apply the computational scheme.},

author = {Ben-Artzi, Matania, Fishelov, Dalia, Trachtenberg, Shlomo},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Navier-Stokes equations; vorticity-streamfunction; numerical algorithm; vorticity boundary conditions; vorticity streamfunction; incompressible viscid Newtonian fluids; vorticity projection method},

language = {eng},

number = {2},

pages = {313-330},

publisher = {EDP-Sciences},

title = {Vorticity dynamics and numerical resolution of Navier-Stokes equations},

url = {http://eudml.org/doc/194052},

volume = {35},

year = {2001},

}

TY - JOUR

AU - Ben-Artzi, Matania

AU - Fishelov, Dalia

AU - Trachtenberg, Shlomo

TI - Vorticity dynamics and numerical resolution of Navier-Stokes equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2001

PB - EDP-Sciences

VL - 35

IS - 2

SP - 313

EP - 330

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

LA - eng

KW - Navier-Stokes equations; vorticity-streamfunction; numerical algorithm; vorticity boundary conditions; vorticity streamfunction; incompressible viscid Newtonian fluids; vorticity projection method

UR - http://eudml.org/doc/194052

ER -

## References

top- [1] C.R. Anderson, Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows. J. Comp. Phys. 80 (1989) 72–97. Zbl0656.76034
- [2] J.B. Bell, P. Colella and H.M. Glaz, A second-order projection method for the incompressible navier-stokes equations. J. Comp. Phys. 85 (1989) 257–283. Zbl0681.76030
- [3] M. Ben-Artzi, Vorticity dynamics in planar domains. (In preparation).
- [4] M. Ben-Artzi, Global solutions of two-dimensional navier-stokes and euler equations. Arch. Rat. Mech. Anal. 128 (1994) 329–358. Zbl0837.35110
- [5] P. Bjorstad, Fast numerical solution of the biharmonic dirichlet problem on rectangles. SIAM J. Numer. Anal. 20 (1983) 59–71. Zbl0561.65077
- [6] A.J. Chorin, Numerical solution of the navier-stokes equations. Math. Comp. 22 (1968) 745–762. Zbl0198.50103
- [7] A.J. Chorin, Vortex sheet approximation of boundary layers. J. Comp. Phys. 27 (1978) 428–442. Zbl0387.76040
- [8] A.J. Chorin and J.E. Marsden, A mathematical introduction to fluid mechanics. 2nd edn., Springer-Verlag, New York (1990). Zbl0712.76008MR1058010
- [9] E.J. Dean, R. Glowinski and O. Pironneau, 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. Zbl0760.76044
- [10] S.C.R. Dennis and L. Quartapelle, Some uses of green’s theorem in solving the navier-stokes equations. Internat. J. Numer. Methods Fluids 9 (1989) 871–890. Zbl0695.76017
- [11] W. E and J.-G. Liu, Essentially compact schemes for unsteady viscous incompressible flows. J. Comp. Phys. 126 (1996) 122–138. Zbl0853.76045
- [12] W. E and J.-G. Liu, Vorticity boundary condition and related issues for finite difference scheme. J. Comp. Phys. 124 (1996) 368–382. Zbl0847.76050
- [13] W. E and J.-G. Liu, 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. Zbl0901.76046
- [14] D. Fishelov, Simulation of three-dimensional turbulent flow in non-cartesian geometry. J. Comp. Phys. 115 (1994) 249–266. Zbl0812.76063
- [15] U. Ghia, K.N. Ghia and C.T. Shin, High-re solutions for incompressible flow using the navier-stokes equations and a multigrid method. J. Comp. Phys. 48 (1982) 387–411. Zbl0511.76031
- [16] R. Glowinski, Personal communication.
- [17] P.M. Gresho, Incompressible fluid dynamics: some fundamental formulation issues. Ann. Rev. Fluid Mech. 23 (1991) 413–453. Zbl0717.76006
- [18] P.M. Gresho and S.T. Chan, 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. Zbl0712.76035
- [19] K.E. Gustafson and J.A. Sethian (Eds.), Vortex methods and vortex motion. SIAM, Philadelphia (1991). Zbl0748.76010MR1095601
- [20] R.R. Hwang and C-C. Yao, A numerical study of vortex shedding from a square cylinder with ground effect. J. Fluids Eng. 119 (1997) 512–518.
- [21] K.M. Kelkar and S.V. Patankar, Numerical prediction of vortex sheddind behind a square cylinder. Internat. J. Numer. Methods Fluids 14 (1992) 327–341. Zbl0746.76066
- [22] O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York (1963). Zbl0121.42701MR155093
- [23] L.D. Landau and E.M. Lifshitz, Fluid mechanics, Chap. II, Sec. 15. Pergamon Press, New York (1959). Zbl0146.22405MR108121
- [24] J. Leray, Etudes de diverses equations integrales non lineaires et des quelques problemes que pose l’hydrodynamique. J. Math. Pures Appl. 12 (1933) 1–82. Zbl0006.16702
- [25] D.A. Lyn, S. Einav, S. Rodi and J.H. Park, 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.
- [26] S.A. Orszag and M. Israeli, 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. Zbl0295.76016
- [27] 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).
- [28] O. Pironneau, Finite element methods for fluids. John Wiley & Sons, New York (1989). Zbl0712.76001MR1030279
- [29] L. Quartapelle, Numerical solution of the incompressible Navier-Stokes equations. Birkhauser Verlag, Basel (1993). Zbl0784.76020MR1266843
- [30] L. Quartapelle and F. Valz-Gris, Projection conditions on the vorticity in viscous incompressible flows. Internat. J. Numer. Methods Fluids 1 (1981) 129–144. Zbl0465.76028
- [31] R. Temam, 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. Zbl0207.16904
- [32] R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam (1979). Zbl0426.35003MR603444
- [33] T.E. Tezduyar, J. Liou, D.K. Ganjoo and M. Behr, Solution techniques for the vorticity-streamfunction formulation of the two-dimensional unsteady incompressible flows. Internat. J. Numer. Methods Fluids 11 (1990) 515–539. Zbl0711.76020

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