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NUMERICAL IMPLEMENTATION OF THE TWO-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATION

  • Received : 2015.05.18
  • Accepted : 2015.06.02
  • Published : 2015.06.25

Abstract

In this paper, we briefly review and describe a projection algorithm for numerically computing the two-dimensional time-dependent incompressible Navier-Stokes equation. The projection method, which was originally introduced by Alexandre Chorin [A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), pp. 745-762], is an effective numerical method for solving time-dependent incompressible fluid flow problems. The key advantage of the projection method is that we do not compute the momentum and the continuity equations at the same time, which is computationally difficult and costly. In the projection method, we compute an intermediate velocity vector field that is then projected onto divergence-free fields to recover the divergence-free velocity. Numerical solutions for flows inside a driven cavity are presented. We also provide the source code for the programs so that interested readers can modify the programs and adapt them for their own purposes.

Keywords

References

  1. A.S. Almgren, J.B. Bell, and W.G. Szymczak, A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput., 17(2) (1996), 358-369. https://doi.org/10.1137/S1064827593244213
  2. V. Barbu, Partial Differential Equations and Boundary Value Problems, Kluwer, Dordrecht, 1998.
  3. J.B. Bell, P. Colella, and H.M. Glaz, A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85(2) (1989), 257-283. https://doi.org/10.1016/0021-9991(89)90151-4
  4. A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762. https://doi.org/10.1090/S0025-5718-1968-0242392-2
  5. A.J. Chorin, J.E. Marsden, and J.E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer, New York, 1990.
  6. M.O. Deville, P.F. Fischer, and E.H. Mund, High-order Methods for Incompressible Fluid Flow, 9 Cambridge University Press, 2002.
  7. J.A. Escobar-Vargas, P.J. Diamessis, and C.F. Van Loan, The numerical solution of the pressure Poisson equation for the incompressible Navier-Stokes equations using a quadrilateral spectral multidomain penalty method, J. Comp. Phys.(submitted) (2011).
  8. S.R. Fulton, P.E. Ciesielski, and W.H. Schubert, Multigrid methods for elliptic problems: A review, Mon. Weather Rev., 114(5) (1986), 943-959. https://doi.org/10.1175/1520-0493(1986)114<0943:MMFEPA>2.0.CO;2
  9. M. Griebel, T. Dornseifer, and T. Neunhoeffer, Numerical Simulation in Fluid Dynamics: a Practical Introduction, SIAM, Philadelphia, 1997.
  10. J.L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195(44) (2006), 6011-6045. https://doi.org/10.1016/j.cma.2005.10.010
  11. J.L. Guermond and J. Shen, Quelques resultats nouveaux sur les methodes de projection, C. R. Acad. Sci. Paris, 333(12) (2001), 1111-1116. https://doi.org/10.1016/S0764-4442(01)02157-7
  12. J.L. Guermond and J. Shen, Velocity-correction projection methods for incompressible flows, SIAM J. Numer. Anal., 41(1) (2003), 112-134. https://doi.org/10.1137/S0036142901395400
  13. T. Guillet and R. Teyssier, A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries, J. Comput. Phys., 230(12) (2011), 4756-4771. https://doi.org/10.1016/j.jcp.2011.02.044
  14. M.M. Gupta and J. Zhang, High accuracy multigrid solution of the 3D convection-diffusion equation, Appl. Math. Comput., 113(2) (2000), 249-274. https://doi.org/10.1016/S0096-3003(99)00085-5
  15. K. Gustafson and K. Halasi, Cavity flow dynamics at higher Reynolds number and higher aspect ratio, J. Comput. Phys., 70(2) (1987), 271-283. https://doi.org/10.1016/0021-9991(87)90182-3
  16. F.H. Harlow and J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8(12) (1965), 2182-2189. https://doi.org/10.1063/1.1761178
  17. G.E. Karniadakis, M. Israeli, and S.A. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97(2) (1991), 414-443. https://doi.org/10.1016/0021-9991(91)90007-8
  18. J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59(2) (1985), 308-323. https://doi.org/10.1016/0021-9991(85)90148-2
  19. L. Landau and E.M. Lifshitz, Fluid mechanics, Course of Theoretical Physics 6 (2nd revised ed.), Pergamon Press (1987), 552.
  20. G. Markham and M.V. Proctor, Modifications to the two-dimensional incompressible fluid flow code ZUNI to provide enhanced performance, CEGB Report TPRD/L/0063/M82, 1983.
  21. S.A. Orszag, M. Israeli, and M. Deville, Boundary conditions for incompressible flows, J. Sci. Comput., 1(1) (1986), 75-111. https://doi.org/10.1007/BF01061454
  22. R. Peyret and T.D. Taylor, Computational Methods for Fluid Flow, Springer-Verlag, New York, 1985.
  23. P.J. Roache, Computational Fluid Dynamics, Hermosa publishers, Albuquerque, 1972.
  24. R. Temam, Sur l'approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires (II), Arch. Rational Mech. Anal., 33(5) (1969), 377-385. https://doi.org/10.1007/BF00247696
  25. L.J.P. Timmermans, P.D. Minev, and F.N. Van De Vosse, An approximate projection scheme for incompressible flow using spectral elements, Int. J. Numer. Meth. Fluids, 22(7) (1996), 673-688. https://doi.org/10.1002/(SICI)1097-0363(19960415)22:7<673::AID-FLD373>3.0.CO;2-O
  26. U. Trottenberg, C.W. Oosterlee, and A. Schuller, Multigrid, Academic press, London, 2000.
  27. J.J.I.M. Van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Comput., 7(3) (1986), 870-891. https://doi.org/10.1137/0907059
  28. J.E. Welch, F.H. Harlow, J.P. Shannon, and B.J. Daly, The MAC (Marker-and-Cell) Method - A computing technique for solving viscous, incompressible, transient fluid-flow problems involving free surfaces, Los Alamos Scientific Laboratory Report LA-3425, University of California, Los Alamos, 1966.
  29. Q. Zhang, A fourth-order approximate projection method for the incompressible Navier-Stokes equations on locally-refined periodic domains, Appl. Numer. Math., 77 (2014), 16-30. https://doi.org/10.1016/j.apnum.2013.10.009