Vorticity dynamics and numerical Resolution of Navier-Stokes Equations

Matania Ben-Artzi; Dalia Fishelov; Shlomo Trachtenberg

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Ben-Artzi, Matania, Fishelov, Dalia, and Trachtenberg, Shlomo. "Vorticity dynamics and numerical Resolution of Navier-Stokes Equations." ESAIM: Mathematical Modelling and Numerical Analysis 35.2 (2010): 313-330. <http://eudml.org/doc/197531>.

@article{Ben2010,
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},
keywords = {Navier-Stokes equations; vorticity-streamfunction; numerical algorithm; vorticity boundary conditions.; vorticity streamfunction; incompressible viscid Newtonian fluids; vorticity boundary conditions; vorticity projection method},
language = {eng},
month = {3},
number = {2},
pages = {313-330},
publisher = {EDP Sciences},
title = {Vorticity dynamics and numerical Resolution of Navier-Stokes Equations},
url = {http://eudml.org/doc/197531},
volume = {35},
year = {2010},
}

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
DA - 2010/3//
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 boundary conditions; vorticity projection method
UR - http://eudml.org/doc/197531
ER -

References

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  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).  Zbl0712.76008
  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.  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 Fluids9 (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 Fluids11 (1990) 587-659.  Zbl0712.76035
  19. K.E. Gustafson and J.A. Sethian (Eds.), Vortex methods and vortex motion. SIAM, Philadelphia (1991).  Zbl0748.76010
  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 Fluids14 (1992) 327-341.  Zbl0746.76066
  22. O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York (1963).  Zbl0121.42701
  23. L.D. Landau and E.M. Lifshitz, Fluid mechanics, Chap. II, Sec. 15. Pergamon Press, New York (1959).  
  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 Dynamics9 (2000).  
  28. O. Pironneau, Finite element methods for fluids. John Wiley & Sons, New York (1989).  Zbl0665.73059
  29. L. Quartapelle, Numerical solution of the incompressible Navier-Stokes equations. Birkhauser Verlag, Basel (1993).  Zbl0784.76020
  30. L. Quartapelle and F. Valz-Gris, Projection conditions on the vorticity in viscous incompressible flows. Internat. J. Numer. Methods Fluids1 (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).  
  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 Fluids11 (1990) 515-539.  Zbl0711.76020

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