Vorticity dynamics and numerical resolution of Navier-Stokes equations
Ben-Artzi, Matania ; Fishelov, Dalia ; Trachtenberg, Shlomo
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001), p. 313-330 / Harvested from Numdam

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.

Publié le : 2001-01-01
Classification:  35Q30,  65M06,  76D17
@article{M2AN_2001__35_2_313_0,
     author = {Ben-Artzi, Matania and Fishelov, Dalia and Trachtenberg, Shlomo},
     title = {Vorticity dynamics and numerical resolution of Navier-Stokes equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {35},
     year = {2001},
     pages = {313-330},
     mrnumber = {1825701},
     zbl = {0987.35122},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2001__35_2_313_0}
}
Ben-Artzi, Matania; Fishelov, Dalia; Trachtenberg, Shlomo. Vorticity dynamics and numerical resolution of Navier-Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) pp. 313-330. http://gdmltest.u-ga.fr/item/M2AN_2001__35_2_313_0/

[1] C.R. Anderson, Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows. J. Comp. Phys. 80 (1989) 72-97. | Zbl 0656.76034

[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. | Zbl 0681.76030

[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. | Zbl 0837.35110

[5] P. Bjorstad, Fast numerical solution of the biharmonic dirichlet problem on rectangles. SIAM J. Numer. Anal. 20 (1983) 59-71. | Zbl 0561.65077

[6] A.J. Chorin, Numerical solution of the navier-stokes equations. Math. Comp. 22 (1968) 745-762. | Zbl 0198.50103

[7] A.J. Chorin, Vortex sheet approximation of boundary layers. J. Comp. Phys. 27 (1978) 428-442. | Zbl 0387.76040

[8] A.J. Chorin and J.E. Marsden, A mathematical introduction to fluid mechanics. 2nd edn., Springer-Verlag, New York (1990). | MR 1058010 | Zbl 0712.76008

[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. | Zbl 0760.76044

[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. | Zbl 0695.76017

[11] W. E and J.-G. Liu, Essentially compact schemes for unsteady viscous incompressible flows. J. Comp. Phys. 126 (1996) 122-138. | Zbl 0853.76045

[12] W. E and J.-G. Liu, Vorticity boundary condition and related issues for finite difference scheme. J. Comp. Phys. 124 (1996) 368-382. | Zbl 0847.76050

[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. | Zbl 0901.76046

[14] D. Fishelov, Simulation of three-dimensional turbulent flow in non-cartesian geometry. J. Comp. Phys. 115 (1994) 249-266. | Zbl 0812.76063

[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. | Zbl 0511.76031

[16] R. Glowinski, Personal communication.

[17] P.M. Gresho, Incompressible fluid dynamics: some fundamental formulation issues. Ann. Rev. Fluid Mech. 23 (1991) 413-453. | Zbl 0717.76006

[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. | Zbl 0712.76035

[19] K.E. Gustafson and J.A. Sethian (Eds.), Vortex methods and vortex motion. SIAM, Philadelphia (1991). | MR 1095601 | Zbl 0748.76010

[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. | Zbl 0746.76066

[22] O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York (1963). | MR 155093 | Zbl 0121.42701

[23] L.D. Landau and E.M. Lifshitz, Fluid mechanics, Chap. II, Sec. 15. Pergamon Press, New York (1959). | MR 108121 | Zbl 0146.22405

[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. | Zbl 0006.16702

[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. | Zbl 0295.76016

[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). | MR 1030279 | Zbl 0712.76001

[29] L. Quartapelle, Numerical solution of the incompressible Navier-Stokes equations. Birkhauser Verlag, Basel (1993). | MR 1266843 | Zbl 0784.76020

[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. | Zbl 0465.76028

[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. | Zbl 0207.16904

[32] R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam (1979). | MR 603444 | Zbl 0426.35003

[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. | Zbl 0711.76020