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Cartesian Grid Methods for Fluid Flow in Complex Geometries

  • Randall J. Leveque
  • Donna Calhoun
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 124)

Abstract

Biological fluid dynamics typically involves geometrically complicated structures which are often deforming in time. We give a brief overview of some approaches based on using fixed Cartesian grids instead of attempting to use a grid which conforms to the boundary. Both finite-difference and finite-volume methods are discussed, as well as a combined approach which has recently been used for computing incompressible flow using the streamfunction-vorticity formulation of the incompressible Navier-Stokes equations.

Key words

Cartesian grids fluid dynamics advection-diffusion immersed boundary method immersed interface method 

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References

  1. [1]
    D.J. Acheson, Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series, Clarendon Press, 1990.Google Scholar
  2. [2]
    A.S. Almgren, J.B. Bell, P. Colella, and T. Marthaler, A Cartesian grid projection method for the incompressible Euler equations in complex geometries, SIAM J. Sci. Comput., 18 (1997), pp. 1289–1309.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    C. Anderson, Vorticity Boundary Conditions and Boundary Vorticity Generation for Two-dimensional Viscous Incompressible Flows, J. Comput. Phys., 80 (1989), pp. 72–97.CrossRefzbMATHGoogle Scholar
  4. [4]
    M. Berger and R.J. Leveque, An adaptive Cartesian mesh algorithm for the Euler equations in arbitrary geometries. AIAA paper AIAA-89-1930, 1989.Google Scholar
  5. [5]
    —, Stable boundary conditions for Cartesian grid calculations, Computing Systems in Engineering, 1 (1990), pp. 305–311.Google Scholar
  6. [6]
    R.P. Beyer, A computational model of the cochlea using the immersed boundary method, PhD thesis, University of Washington, 1989.Google Scholar
  7. [7]
    —, A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98 (1992), pp. 145–162.Google Scholar
  8. [8]
    R.P. Beyer and R.J. LeVeque, Analysis of a one-dimensional model for the immersed boundary method, SIAM J. Num. Anal., 29 (1992), pp. 332–364.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Calhoun, A Cartesian grid method for solving the streamfunction-vorticity equations in irregular geometries, PhD thesis, University of Washington, 1999.Google Scholar
  10. [10]
    D. Calhoun and R.J. LeVeque, Solving the advection-diffusion equation in irregular geometries, J. Comput. Phys., 156 (2000), pp. 1–38MathSciNetGoogle Scholar
  11. [11]
    A. Cheer and M. Koehl, Fluid flow through filtering appendages of insects, IMA Journal of Mathematics Applied in Medicine and Biology, 4 (1987), pp. 185–199.CrossRefGoogle Scholar
  12. [12]
    —, Paddles and Rakes: Fluid Flow through ristled Appendages of Small Organisms, J. Theor. Biol., 129 (1987), pp. 17–39.CrossRefGoogle Scholar
  13. [13]
    A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), pp. 745–762.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    —, Numerical study of slightly viscous flow, J. Fluid Mech., 75 (1973), pp. 785–796.MathSciNetCrossRefGoogle Scholar
  15. [15]
    A.J. Chorin and J.E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, 1979.Google Scholar
  16. [16]
    M. Coutanceau and R. Bouard, Experimental determination of the main features of the vicous flow in the wake of a circular cylinder in uniform translation. Part I. Steady flow, J. Fluid Mech., 79 (1977), pp. 231–256.CrossRefGoogle Scholar
  17. [17]
    —, Experimental determination of the main features of the vicous flow in the wake of a circular cylinder in uniform translation. Part II. Unsteady flow, J. Fluid Mech., 79 (1977), pp. 257–272.CrossRefGoogle Scholar
  18. [18]
    M.S. Day, P. Colella, M.J. Lijewski, C.A. Rendleman, and D.L. Marcus, Embedded boundary algorithms for solving the Poisson equation on complex domains. Preprint LBNL-41811, Lawrence Berkeley Lab, 1998.Google Scholar
  19. [19]
    D. De Zeeuw and K. Powell, An adaptively-refined Cartesian mesh solver for the Euler equations, J. Comput. Phys., 104 (1993), pp. 56–68.CrossRefzbMATHGoogle Scholar
  20. [20]
    H.V. der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631–644.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    R. Dillon, L. Fauci, A. Fogelson, and D. Gaver, Modeling Biofilm Processes Using the Immersed Boundary Method, J. Comput. Phys., 129 (1996), pp. 57–73.CrossRefzbMATHGoogle Scholar
  22. [22]
    R. Dillon, L. Fauci, and D. Gaver, A Microscale Model of Bacterial Swimming, Chemotaxis and Substrate Transport, J. Theor. Biol., 177 (1995), pp. 325–340.CrossRefGoogle Scholar
  23. [23]
    W. E and J. Liu, Vorticity Boundary Condition and Related Issues for Finite Difference Schemes, J. Comput. Phys., 124 (1996), pp. 368–382.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    L. Fauci and C.S. Peskin, A computational model of aquatic animal locomotion, J. Comput. Phys., 77 (1988), pp. 85–108.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    L.J. Fauci, Interaction of oscillating filaments — a computational study, J. Comput. Phys., 86 (1990), pp. 294–313.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A. Fogelson and J. Keener, Immersed interface methods for Neumann and related problems in two and three dimensions, to appear, SIAM J. Sci. Comput.Google Scholar
  27. [27]
    A.L. Fogelson, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 56 (1984), pp. 111–134.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    —, Mathematical and computational aspects of blood clotting, in Proceedings of the 11th IMACS World Congress on System Simulation and Scientific Computation, Vol. 3, B. Wahlstrom, ed., North Holland, 1985, pp. 5–8.Google Scholar
  29. [29]
    A.L. Fogelson and C.S. Peskin, Numerical solution of the three dimensional stokes equations in the presence of suspended particles, in Proc. SIAM Conf. Multi-phase Flow, SIAM, June 1986.Google Scholar
  30. [30]
    B. Fornberg, A numerical study of steady viscous flow past a circular cylinder, J. Fluid Mech., 98 (1980), pp. 819–855.CrossRefzbMATHGoogle Scholar
  31. [31]
    H. Forrer, Boundary Treatments for Cartesian-Grid Methods, PhD thesis, ETH-Zurich, 1997.Google Scholar
  32. [32]
    H. Forrer and M. Berger, Flow simulations on Cartesian grids involving complex moving geometries, in Proc. 7’th Intl. Conf. on Hyperbolic Problems, R. Jeltsch, ed., Birkhauser Verlag, 1998, pp. 315–324.Google Scholar
  33. [33]
    H. Forrer and R. Jeltsch, A higher-order boundary treatment for Cartesian-grid methods, J. Comput. Phys., 140 (1998), pp. 259–277.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    R. Glowinski, T.-S. Pan, and J. Periaux, A fictitious domain method for Dirich-let problem and applications, Comp. Meth. Appl. Mech. Eng., 111 (1994), pp. 283–303.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    —, A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comp. Meth. Appl. Mech. Eng., 112 (1994), pp. 133–148.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    T.Y. Hou, Z. Li, H. Zhao, and S. Osher, A hybrid method for moving interface problems with application to the Hele-Shaw flow, J. Comput. Phys., 134 (1997), pp. 236–252.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    M. Israeli, On the Evaluation of Iteration Parameters for the Boundary Vorticity, Studies in Applied Mathematics, LI (1972), pp. 67–71.Google Scholar
  38. [38]
    H. Johansen and P. Colella, A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. Comput. Phys., 147 (1998), pp. 60–85.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    M. Koehl, Fluid flow through hair-bearing appendages: Feeding, smelling, and swimming at low and intermediate Reynolds numbers., in Biological Fluid Dynamics, C.P. Ellington and T.J. Pedley, eds., Vol. 49, Soc. Exp. Biol. Symp, 1995, pp. 157–182.Google Scholar
  40. [40]
    R.J. Leveque, clawpack software. http://www.amath.washington.edu/~rj1/clawpack.html.Google Scholar
  41. [41]
    —, Cartesian grid methods for flow in irregular regions, in Num. Meth. Fl. Dyn. III, K.W. Morton and M.J. Baines, eds., Clarendon Press, 1988, pp. 375–382.Google Scholar
  42. [42]
    —, High resolution finite volume methods on arbitrary grids via wave propagation, J. Comput. Phys., 78 (1988), pp. 36–63.Google Scholar
  43. [43]
    —, Numerical Methods for Conservation Laws, Birkhauser-Verlag, 1990.Google Scholar
  44. [44]
    —, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys., 131 (1997), pp. 327–353.Google Scholar
  45. [45]
    R.J. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp. 1019–1044.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    —, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997), pp. 709–735.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    Z. Li, The Immersed Interface Method — A Numerical Approach for Partial Differential Equations with Interfaces, PhD thesis, University of Washington, 1994.Google Scholar
  48. [48]
    —, A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35 (1998), pp. 230–254.Google Scholar
  49. [49]
    —, The immersed interface method using a finite element formulation, Applied Numer. Math., 27 (1998), pp. 253–267.Google Scholar
  50. [50]
    X.-D. Liu, R.P. Fedkiw, and M. Kang, A boundary condition capturing method for Poisson’s equation on irregular domains. CAM Report 99-15, UCLA Mathematics Department, 1999.Google Scholar
  51. [51]
    A. Mayo, The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Num. Anal., 21 (1984), pp. 285–299.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    A. Mayo and A. Greenbaum, Fast parallel iterative solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 101–118.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    A.A. Mayo and C.S. Peskin, An implicit numerical method for fluid dynamics problems with immersed elastic boundaries, Contemp. Math., 141 (1993), pp. 261–277.MathSciNetCrossRefGoogle Scholar
  54. [54]
    C.S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), pp. 220–252.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    —, Lectures on mathematical aspects of physiology, Lectures in Appl. Math., 19 (1981), pp. 69–107.MathSciNetGoogle Scholar
  56. [56]
    C.S. Peskin and D.M. McQueen, Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys., 37 (1980), pp. 113–132.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    —, Computer-assisted design of pivoting-disc prosthetic mitral valves, J. Thorac. Cardiovasc. Surg., 86 (1983), pp. 126–135.Google Scholar
  58. [58]
    —, Computer-assisted design of butterfly bileaflet valves for the mitral position, Scand. J. Thorac. Cardiovasc. Surg., 19 (1985), pp. 139–148.CrossRefGoogle Scholar
  59. [59]
    K. Powell, Solution of the Euler and Magnetohydrodynamic Equations on Solution-Adaptive Cartesian Grids. Von Karman Institute for Fluid Dynamics Lecture Series, 1996.Google Scholar
  60. [60]
    L. Quartapelle and F. Valz-griz, Projection conditions on the vorticity in viscous incompressible flows, Int. J. Numer. Methods. Fluids, 1 (1981), pp. 129–144.CrossRefzbMATHGoogle Scholar
  61. [61]
    J.J. Quirk, An alternative to unstructured grids for computing gas-dynamic flow around arbitrarily complex 2-dimensional bodies, Comput. Fluids, 23 (1994), pp. 125–142.CrossRefzbMATHGoogle Scholar
  62. [62]
    —, A Cartesian grid approach with hierarchical refinement for compressible flows. ICASE Report No. TR-94-51, NASA Langley Research Center, 1994.Google Scholar
  63. [63]
    J.M. Stockie and S.I. Green, Simulating the motion of flexible pulp fibres using the immersed boundary method, J. Comput. Phys., 147 (1998), pp. 147–165.CrossRefzbMATHGoogle Scholar
  64. [64]
    J.M. Stockie and B.T.R. Wetton, Stability analysis for the immersed fiber problem, SIAM J. Appl. Math., 55 (1995), pp. 1577–1591.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    D. Sulsky and J.U. Brackbill, A numerical method for suspension flow, J. Comput. Phys., 96 (1991), pp. 339–368.CrossRefzbMATHGoogle Scholar
  66. [66]
    A. Thom, The flow past circular cylinders at low speeds, Proc. Roy. Soc. A, 141 (1933), p. 651.CrossRefzbMATHGoogle Scholar
  67. [67]
    M. Titcombe and M.J. Ward, An asymptotic study of oxygen transport from multiple capillaries to skeletal muscle tissue, to appear, SIAM J. Appl. Math., 2000.Google Scholar
  68. [68]
    C. Tu and C.S. Peskin, Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 1361–1376.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    S.O. Unverdi and G. Tryggvason, Computations of multi-fluid flows, Physica D, 60 (1992), pp. 70–83.CrossRefzbMATHGoogle Scholar
  70. [70]
    —, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100 (1992), pp. 25–37.CrossRefzbMATHGoogle Scholar
  71. [71]
    Z.J. Wang, Vortex shedding and frequency selection in flapping flight. Submitted to the J. Fluid Mech., 1999.Google Scholar
  72. [72]
    A. Wiegmann, The Explicit Jump Immersed Interface Method and Interface Problems for Differential Equations, PhD thesis, University of Washington, 1998.Google Scholar
  73. [73]
    A. Wiegmann and K.P. Bube, The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 35 (1998), pp. 177–200.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    —, The explicit jump immersed interface method: Finite difference methods for pde with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), pp. 827–862.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    Z. Yang, A Cartesian grid method for elliptic boundary value problems in irregular regions, PhD thesis, University of Washington, 1996.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Randall J. Leveque
    • 1
  • Donna Calhoun
    • 2
  1. 1.Department of Applied MathematicsUniversity of WashingtonSeattle
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew York

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