Skip to main content
SpringerLink
Log in

Recent advances in the marker and cell method

  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Summary

In this article recent advances in the MAC method will be reviewed. The MAC technique dates back to the early sixties at the Los Alamos Laboratories and this paper starts with a historical review, and then a summary of related techniques. Improvements since the early days of MAC (and the Simplified MAC-SMAC) include automatic time-stepping, the use of the conjugate gradient method to solve the Poisson equation for the corrected velocity potential, greater efficiency through stripping out all particles (markers) other than those near the free surface, more accurate approximations of the free surface boundary conditions, the addition of a bounded high accuracy upwinding for the convected terms (thereby being able to solve higher Reynolds number flows), and a (dynamic) flow visualization facility. This article will concentrate, in the main, on a three-dimensional version of the SMAC method. It will show how to approximate curved boundaries by considering one configurational example in detail; the same will also be done for the free surface. The article will avoid validation, but rather focus on many of the examples and applications that the MAC method can solve from turbulent flows to rheology. It will conclude with some speculative comments on the future direction of the methodology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amsden, A. and Harlow, F. (1970).The SMAC method: a numerical technique for calculating incompressible fluid flows. Technical Report LA-4370 (Los Alamos National Laboratory).

  2. Armenio, V. (1997). An improved MAC method (SIMAC) for unsteady high-Reynolds free surface flows.Int. J. Numer. Meth. Fluids,24, 185–214.

    Article  MATH  Google Scholar 

  3. Baker, G.R. and Moore, D.W. (1989). The rise and distorsion of a two-dimensional gas bubble in an inviscid liquid.Phys. Fluids A,9, 1451–1459.

    Article  MathSciNet  Google Scholar 

  4. Batchelor, G. K. (1967).An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  5. Boulton-Stone, J. M. and Blake, J. R. (1993). Gas bubbles bursting at a free surface.J. Fluid Mech.,254, 437–466.

    Article  MATH  Google Scholar 

  6. Bousfield, D. W., Keunings, R. and Denn, M.M. (1986). Nonlinear analysis of the surface tension driven breakup of viscoelastic filaments.J. Non-Newtonian Fluid Mech.,21, 79–97.

    Article  Google Scholar 

  7. Bousfield, D.W., Keunings, R. and Denn M.M. (1988). Transient deformation of an inviscid inclusion in a viscoelastic extensional flow.J. Non-Newtonian Fluid Mech.,27, 205–221.

    Article  Google Scholar 

  8. Borue, V., Orszag, S.A. and Staroselsky, I. (1995). Interaction of surface waves with turbulence: direct numerical simulations of turbulent open-channel flow.J. Fluid Mech.,286, 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  9. Brocchini, M. and Peregrine, D.H. (2001). The dynamics of strong turbulence at free surfaces. Part I. Description.J. Fluid Mech.,449, 225–254.

    Article  MathSciNet  MATH  Google Scholar 

  10. Brocchini, M. and Peregrine, D.H. (2001). The dynamics of strong turbulence at free surfaces. Part 2. Free surface boundary conditions.J. Fluid Mech.,449, 255–290.

    Article  MathSciNet  Google Scholar 

  11. Castelo, A., Tomé, M.F., Cesar, C.N.L., McKee, S. and Cuminato, J.A. (2000). Freeflow-3D: and integrated simulation system for three-dimensional free surface flows.J. Comput. Visual. Science,2, 199–210.

    Article  Google Scholar 

  12. Chenn, I.-L., Glimm, J., McBryan, O., Plohr, B. and Yaniv, S. (1986). Front tracking for gas dynamics.J. Comput. Phys.,62, 83–110.

    Article  MathSciNet  Google Scholar 

  13. Cormenzana, J., Ledda, A., Laso, M. and Debbaut, B. (2001). Calculation of free surface flows using CONNFESSITT.J. Rheol.,45, 237–258.

    Article  Google Scholar 

  14. Craik, A.D., Latham, R.C., Fawkes, M.J. and Gribbon, W.F. (1981). The circular hydraulic jump.J. Fluid Mech.,112, 347–362.

    Article  Google Scholar 

  15. Crochet, M.J. and Keunings, R. (1982). Finite element analysis of die-swell of a highly elastic fluid.J. Non-Newtonian Fluid Mech.,10, 339–356.

    Article  MATH  Google Scholar 

  16. Cruickshauk, J.O. and Munson, B.R. (1981). Viscous-fluid buckling of plane and axisymmetric jets.J. Fluid Mech.,113, 221–239.

    Article  Google Scholar 

  17. Cruickshank, J.O. (1988). Low-Reynolds-number instabilities in staguating jet flows.J. Fluid Mech.,193, 111–127.

    Article  Google Scholar 

  18. Darip, P., Glimm, J., Lindquist, B., Maesumi, M. and McBryan, O. (1988).On the Simulation of Heterogeneous Petroleum Reservoirs, in Numerical Simulation in Oil recovery, edited by M. Wheeler, Springer Verlag, New York.

    Google Scholar 

  19. Deville, M.O. (1974). Numerical experiments on the MAC code for slow flow.J. Comput. Phys.,15, 362–374.

    Article  Google Scholar 

  20. Durbin, P.A. (1996). On the κ−ε stagnation point anomaly.J. Heat and Fluid Flow,17, 89–90.

    Article  Google Scholar 

  21. Fauci, L.J. and Peskin, C.S. (1988). A computational model of aquatic animal locomotion.J. Comput. Phys.,77, 85–108.

    Article  MathSciNet  MATH  Google Scholar 

  22. Ferreira, V.G., Tomé, M.F., Mangiavacchi, N., Castelo, A., Cuminato, J.A., Fortuna, A.O. and McKee, S. (2002). High order upwinding and the hydraulic jump.Int. J. Numer. Meth. Fluids,39, 549–583.

    Article  MATH  Google Scholar 

  23. Ferreira, V.G., Mangiavacchi N., Tomé, M.F., Castelo, A., Cuminato, J.A., and McKee, S. (2004). Numerical simulation of turbulent free surface flow with two-equation k-ε eddy-viscosity models.Int. J. Numer. Meth. Fluids,44, 347–375.

    Article  MATH  Google Scholar 

  24. Ferziger, J.H. (1987). Simulation of incompressible turbulent flows.J. Comput. Phys.,69, 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  25. Fogelson, A.L. and Peskin, C.S. (1988). A fast numerical method for solving the three-dimensional Stokes equations in the presence of gas suspended particles.J. Comput. Phys.,79, 50–69.

    Article  MathSciNet  MATH  Google Scholar 

  26. Glimm, J., Grove, J., Lindquist, B., McBryan, O. and Tryggvason, G. (1988). The bifurcation of tracked scalar waves.SIAM J. Sci. Stat. Comput.,9, 61–79.

    Article  MathSciNet  MATH  Google Scholar 

  27. Golafshani, M. (1988). A simple numerical technique for transient creep flows with free surfaces.Int. J. Numer. Meth. Fluids,8, 897–912.

    Article  Google Scholar 

  28. Harlow, F. and Welch, J.E. (1965). Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface.Phys. Fluids,8, 2182–2189.

    Article  Google Scholar 

  29. Harlow, F.H. and Welch, J.E. (1981). A fast ICE solution procedure for flows with largely invariant compressiblity.J. Comput. Phys.,40, 254–261.

    Article  Google Scholar 

  30. Hirsch, C. (1988).Numerical Computation of Internal and External Flows, Vols. 1 and 2, Wiley series in Numerical Methods in Engineering.

  31. Hirt, C.W. and Shannon, J.P. (1968). Free-surface stress conditions for incompressible-flow calculations.J. Comput. Phys.,2, 403–411.

    Article  MATH  Google Scholar 

  32. Hirt, C.W. and Cook, J.L. (1972). Calculating three-dimensional flows around structures and over rough terrain.J. Comput. Phys.,10, 324–340.

    Article  Google Scholar 

  33. Hirt, C.W. and Nichols, B.D. (1981). Volume of Fluid (VOF) method for the dynamics of free boundaries.J. Comput. Phys. 39, 201–225.

    Article  MATH  Google Scholar 

  34. Hirt, C.W. (1988).Flow3D Users Manual, Flow Science Inc.

  35. Hoffman, G. (1975). Improved form of the low Reynolds number κ−ε turbulence model.Phys. Fluids,18, 309–312.

    Article  Google Scholar 

  36. Keunings, R. (1986). An algorithm for the simulation of transient viscoelastic flows with free surfaces.J. Comput. Phys.,62, 199–220.

    Article  MathSciNet  MATH  Google Scholar 

  37. Keunings, R. and Bousfield, D.W. (1987). Analysis of surface tension driven leveling in horizontal viscoelastic films.J. Non-Newtonian Fluid Mech.,22, 219–233.

    Article  MATH  Google Scholar 

  38. Kolte, M.I., Rasmussen, H.K. and Hassager, O. (1997). Transient filament stretching rheometer. 2. Numerical simulation.Rheol. Acta,36, 285–302.

    Google Scholar 

  39. Launder, B.E. and Spalding, D.B. (1974). The numerical computation of turbulent flows.Int. J. Numer. Meth. Fluids,15, 127–146.

    Google Scholar 

  40. Lemos, C. (1996). Higher-order schemes for free surface flows with arbitrary configurations.Int. J. Numer. Meth. Fluids,23, 545–566.

    Article  MATH  Google Scholar 

  41. LeVeque, R.J. (1992).Numerical Methods for Conservation Laws, Lectures in Mathematics, ETH, Zurich.

    MATH  Google Scholar 

  42. Liang, Y., Ozetkin, A. and Neti, S. (1999). Dynamics of viscoelastic jets of polymeric liquid extrudate.J. Non-Newtonian Fluid Mech.,81, 105–132.

    Article  MATH  Google Scholar 

  43. Mangiavacchi, N. Castelo, A., Tomé, M.F., Cuminato, J.A., Bambozzi de Oliveira, M.L. and McKee, S. A numerical technique for including surface tension effects for axisymmetric and planar flows using the GENSMAC method.SIAM J. Sci. Comput. (to appear).

  44. Mäntylä, M. (1988).An Introduction to Solid Modeling. Computer Science Press, Rockville.

    Google Scholar 

  45. Markham, G. and Proctor, M.V. (1983). Modifications to the two-dimensional incompressible fluid flow code ZUNI to provide enhanced performance. C.E.G.B. report TPRD/L/0063/M82.

  46. McQueen, J.F. and Rutter, P. (1981). Outline description of a recently implemented fluid flow code ZUNI. C.E.G.B. report LM/PHYS/258.

  47. Melville, W.K., (1996). The role of surface-wave breaking in air-sea interaction.Annual Rev. Fluid Mech.,28, 279–323.

    Article  MathSciNet  Google Scholar 

  48. Miyata, H. and Nishimura, S. (1985). Finite-difference simulation of nonlinear waves generated by ships of arbitrary three-dimensional configuration.J. Comput. Phys.,60, 391–436.

    Article  MathSciNet  MATH  Google Scholar 

  49. Miyata, H. (1986). Finite-difference simulation of breaking waves.J. Comput. Phys.,65, 179–214.

    Article  MATH  Google Scholar 

  50. Moretti, G. (1987). Computation of flows with shocks.Annual Rev. Fluid Mech.,19, 313–337.

    Article  Google Scholar 

  51. Nichols, B.D. and Hirt, C.W. (1971). Improved free surface boundary conditions for numerical incompressible flow calculations.J. Comput. Phys.,8, 434.

    Article  MATH  Google Scholar 

  52. Nichols, B.D., Hirt, C.W. and Hotchkins, R.S. (1988).SOLA-VOF: A solution algorithm for transient fluid flow with multiple free boundaries. Technical Report LA-8355 (Los Alamos Scientific Laboratory).

  53. Osher, S. and Sethian, J.A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations.J. Comput. Phys.,79 (1), 1–49.

    Article  MathSciNet  Google Scholar 

  54. Owens, R.G. and Phillips, T.N. (2002).Computational Rheology. Imperial College Press.

  55. Pan, Y. and Banerjee, S. (1995). A numerical study of free surface turbulence in channel flow.Phys. Fluids,7, 1649–1664.

    Article  MATH  Google Scholar 

  56. Peskin, C.S. (1977). Numerical analysis of blood flow in the heart.J. Comp. Phys.,25, 220–252.

    Article  MathSciNet  MATH  Google Scholar 

  57. Pracht, W.E. (1971). A Numerical method for calculating transient creeping flows.J. Comput. Phys.,7, 46–60.

    Article  MATH  Google Scholar 

  58. Pracht, W.E. (1975). Calculating three-dimensional fluid flows at all speeds with an Eulerian-Lagrangian computing mesh.J. Comput. Phys.,17, 132–159.

    Article  Google Scholar 

  59. Ryan, M.E. and Dutta, A. (1981). A finite Difference simulation of Extrudate Swell. Proceedings of the2nd World Congr. Chem. Eng.,VI, 277–281, Montreal.

    Google Scholar 

  60. Santos, F.L.P., Mangiavacchi, N., Castelo, A., Tomé, M.F., Cuminato, J.A. and McKee, S. A novel technique for free surface 2D multiphase flows. Submitted for publication.

  61. Sarpkaya, T. (1996). Vorticity, free-surface and surfactants.Annual Rev. Fluid Mech.,28, 88–128.

    Article  Google Scholar 

  62. Sethian, J.A. (1995).Theory. Algorithmns, and Applications of Level Set Methods for Propagating Interfaces. Acta Numerica, C.U.P., Cambridge, UK.

    Google Scholar 

  63. Sethian, J.A. (1996).Level Set Methods: Evolving Interfaces in Geometry. Fluid Mechanics, Computer Vision and Material Sciences, C. U. P., Cambridge, UK.

    MATH  Google Scholar 

  64. Sicilian, J.M. and Hirt, C.W. (1984). An efficient computation scheme for tracking contaminant concentrations in fluid flows.J. Comput. Phys.,56, 428–447.

    Article  MATH  Google Scholar 

  65. Sousa, F.S., Mangiavacchi, N., Nonato, L.G., Castelo, A., Tomé, M.F., Ferreira, V.G., Cuminato, J.A. and McKee, S. A front tracking finite difference method for solve 3D Navier-Stokes equations for multi-fluid flows with free surfaces.J. Comput. Phys. (to appear).

  66. Sussman, M. and Smereka, P. (1997). Axisymmetric free boundary problems.J. Fluid Mech.,341, 269–294.

    Article  MathSciNet  MATH  Google Scholar 

  67. Sussman, M. and Puckett, E.G. (2000). A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows.J. Comput. Phys.,162, 301–337.

    Article  MathSciNet  MATH  Google Scholar 

  68. Tanner, R.I. (1970). A theory of die-swell.J. Polymer Science,8, 2067–2078.

    Google Scholar 

  69. Tsai, T.M. and Miksis, M.J. (1994). Dynamics of a drop in a constricted capillary tube.J. Fluid Mech.,274, 197–217.

    Article  MathSciNet  MATH  Google Scholar 

  70. Tsai, W.-T. and Yue, D.K.P. (1996). Computations of nonlinear free-surface flows.Annual Rev. Fluid Mech.,28, 249–278.

    MathSciNet  Google Scholar 

  71. Tsai, W.-T. (1998). A numerical study of the evolution and structure of a turbulent shear layer under a free surface.J. Fluid Mech.,354, 239–276.

    Article  MATH  Google Scholar 

  72. Tomé, M.F. and McKee, S. (1994). GENSMAC: A computational marker-and-cell method for free surface flows in general domains.J. Comput. Phys. 110, 171–186.

    Article  MATH  Google Scholar 

  73. Tomé, M.F., Duffy, B., and McKee, S. (1996). A numerical technique for solving unsteady non-Newtonian free surface flows.J. Non-Newtonian Fluid Mech.,62, 9–34.

    Article  Google Scholar 

  74. Tomé, M.F., Castelo, A., Cuminato, J.A. and McKee, S. (1996).GENSMAC3D: Implementation of the Navier-Stokes Equations and Boundary Conditions for 3D Free Surface Flows. Universidade de São Paulo, Departamento de Ciência de Computação e Estatística, Notas do ICMC, No. 29.

  75. Tomé, M.F. and McKee, S. (1999). Numerical simulation of viscous flow: buckling of planar jets.Int. J. Numer. Meth. Fluids,29, 705–718.

    Article  MATH  Google Scholar 

  76. Tomé, M.F. and McKee, S., Barratt, L., Jarvis, D.A. and Patrick, A.J. (1999). An experimental and numerical investigation of container filling: viscous liquids.Int. J. Numer. Meth. Fluids,31, 1333–1353.

    Article  MATH  Google Scholar 

  77. Tomé, M.F., Castelo, A., Cuminato, J.A., Murakami, J., Minghim, R. and Oliveira, C.F. and McKee, S. (2000). Numerical simulation of axisymmetric free surface flows.J. Comput. Phys.,157, 441–472.

    Article  MathSciNet  MATH  Google Scholar 

  78. Tomé, M.F., Castelo, A., Cuminato, J.A. and McKee, S. (2001). GENSMAC3D: A numerical method for solving unsteady three-dimensional free surfaces flows.Int. J. Numer. Meth. Fluids,37, 747–796.

    Article  MATH  Google Scholar 

  79. Tomé, M.F., Mangiavacchi, N., Cuminato, J.A., Castelo, A. and McKee, S. (2002). A numerical technique for solving unsteady viscoelastic free surface flows.J. Non-Newtonian Fluid Mech.,106, 61–106.

    Article  MATH  Google Scholar 

  80. Torrey, M.D., Mjolsness, R.C. and Stein, L.R. (1987).NASA-VOF3D: A three dimensional computer program for incompressible flows with free surfaces. Technical Report LA-11009-MS (Los Alamos National Laboratory).

  81. Varonos, A. and Bergeles, G. (1998). Development and assessment of a variable-order non-oscillatory scheme for convection term discretization.Int. J. Numer. Meth. Fluids,26, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  82. Welch, J.E., Harlow, F.H., Shannon, J.P. and Daly, B.J. (1965).The MAC method. Technical Report LA-3425 (Los Alamos National Laboratory)

  83. Yang, Z., and Shih, H. (1993). New time scale based κ−ε model for near-wall turbulence.AIAA J.,7, 1191–1198.

    Article  Google Scholar 

  84. Yao, M.W. and McKinley, G.H. (1998). Numerical simulation of extensional deformations of viscoelastic liquid bridges in filament stretching devices.J. Non-Newtonian Fluid Mech.,74, 47–88.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Strathclyde, Glasgow, UK

    S. McKee

  2. Department of Computer Science and Statistics, USP-University of São Paulo, São Carlos, SP, Brazil

    M. F. Tomé, J. A. Cuminato, A. Castelo & V. G. Ferreira

Authors
  1. S. McKee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. F. Tomé
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. A. Cuminato
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Castelo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. V. G. Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. McKee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McKee, S., Tomé, M.F., Cuminato, J.A. et al. Recent advances in the marker and cell method. Arch Computat Methods Eng 11, 107 (2004). https://doi.org/10.1007/BF02905936

Download citation

  • Received:

  • DOI: https://doi.org/10.1007/BF02905936

Keywords

Search

Navigation