Deterministic vortex methods for the incompressible Navier-Stokes equations

  • Giovanni Russo
Section IV: SPH and Analysis/Error Evaluation
Part of the Lecture Notes in Physics book series (LNP, volume 395)


The vorticity formulation of the Navier-Stokes equations is described in connection to a free-Lagrangian approach for the treatment of vorticity diffusion. The position of the vortices is updated according to the fluid velocity, suitably reconstructed from the vortex distribution. The strengths satisfy a diffusion equation discretized on the irregular grid of the vortices. Its implementation is based on the discretization of the Laplacian on the particle grid and is obtained with the use of a Voronoi diagram that is updated in O(N) operations at each time step. The scheme for the diffusion term is conservative and there is an energy estimate (total “energy” associated with vortices decreases). The comparison to the exact solution shows a second order spatial convergence. The problem of an elastic flag in a fluid is under investigation. Preliminary results are shown.


Voronoi Diagram Smooth Particle Hydrodynamic Point Vortex Vortex Method Vortex Dipole 
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  1. Anderson, C. and Greengard, C. (1985): “On Vortex Methods”, SIAM J. Math. Anal. 22, pp. 413–440.Google Scholar
  2. Birdsall, C.K. and Langdon, A.B. (1985): Plasma Physics via Computer Simulation (McGraw Hill, New York).Google Scholar
  3. Börgers, C. and Peskin, C.S. (1987): “A Lagrangian Fractional Step Method for the Incompressible Navier-Stokes Equations on a Periodic Domain”, J. Comput. Phys. 70, pp. 397–438.Google Scholar
  4. Chang, Chien-Chang (1988): “Random Vortex Methods for the Navier-Stokes Equations”, J. Comput. Phys. 76, pp. 281–300.Google Scholar
  5. Chorin, A.J. (1973): “Numerical Study of Slightly Viscous Flow”, J. Fluid Mech. 57, pp. 785–796.Google Scholar
  6. Chorin, A.J. (1980): “Vortex Methods and Boundary Layer Instability”, SIAM J. Sci. Stat. Comput. 1, pp. 1–21.Google Scholar
  7. Cottet, G.H. (1984): “Convergence of a Vortex in Cell Method for the Two-Dimensional Euler Equation”, Rap. Int. n. 108, Centre de Mathématique Appliquées, Ecole Polytechnique, Palaiseau.Google Scholar
  8. Cottet, G.H., Mas-Gallic, S. and Raviart, P.A. (1986): “Vortex Method for the Incompressible Euler and Navier-Stokes Equations”, Proceeding of the Workshop on Computational Fluid Dynamics and Reacting Gas Flow, I.M.A., September 1986.Google Scholar
  9. Fauci, L.J. and Peskin, C.S. (1988): “A Computational Model of Aquatic Animal Locomotion”, J. Comput. Phys. 77, pp. 85–108.Google Scholar
  10. Fishelov, D. (1990): “A New Vortex Scheme for Viscous Flow”, J. Comput. Phys. 86, pp. 211–224.Google Scholar
  11. Goodman, J. (1987): “Convergence of the Random Vortex Method”, Comm. Pure Appl. Math. 40, pp. 189–220.Google Scholar
  12. Greengard, C. and Rokhlin, V. (1981): “A Fast Algorithm for Particle Simulations”, J. Comput. Phys. 73, pp. 325–348.Google Scholar
  13. Hockney, R.W. and Eastwood, J.W. (1981): Computer Simulations using Particles (McGraw-Hill, New York).Google Scholar
  14. Krasny, R. (1986): “On Singularity Formation in a Vortex Sheet and the Point Vortex Approximation”, J. Fluid Mech. 167, pp. 65–93.Google Scholar
  15. McKracken, M.F. and Peskin, C. (1980): “A Vortex Method for the Blood Flow Through the Heart Valves”, J. Comput. Phys. 35, pp. 183–205.Google Scholar
  16. Perlman, M. (1985): “On the Accuracy of Vortex Methods”, J. Comput. Phys. 59, pp. 200–223.Google Scholar
  17. Russo, G. (1990): “Deterministic Diffusion of Particles”, Comm. Pure Appl. Math. 43, in press.Google Scholar
  18. Russo, G., “Deterministic Vortex Methods for the Navier-Stokes Equations”, submitted to J. Comput. Phys. Google Scholar
  19. Yih, Chia-Shun (1979): Fluid Mechnics (West River Press, Ann Arbor, Michigan)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Giovanni Russo
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York

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