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Implicit Methods for Simulating Low Reynolds Number Free Surface Flows: Improvements on MAC-Type Methods

  • José A. Cuminato
  • Cassio M. Oishi
  • Rafael A. Figueiredo
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 1)

Abstract

This paper is concerned with describing the main improvements introduced to the MAC (Marker-And-Cell) method for the numerical simulation of low Reynolds number free surface flows, namely: a stable implicit treatment of the pressure boundary condition for projection methods, a semi-implicit method based on the Crank–Nicolson (C–N) discretization of the momentum equations, a more accurate method for moving the massless particles representing the free surface and a viscoelastic model based on the Pom-Pom constitutive law, are discussed. Low Reynolds number free surface flows appear in a number of important industrial processes in the oil, food, cosmetic and medical industries and their simulation present a challenge for explicit MAC-type methods due to their parabolic time step constraint. The simulation of moving boundary problems presents a number of difficulties for a numerical method. For the semi-implicit (C–N) MAC method the main difficulty appears in applying the projection method to uncouple velocity and pressure, this is in addition to other difficulties of correctly imposing the boundary conditions on the free surface and the free surface representation itself.

Keywords

Navier–Stokes equations Viscoelastic fluid flows  Free surface MAC scheme Implicit strategy Jet buckling Extrudate swell MAC method review Non-Newtonian fluids 

Notes

Acknowledgments

The authors would like to acknowledge the financial support of FAPESP (projects nos. 2013/07375-0, 2011/09194-7, 2009/15892-9) and CNPq (projects nos. 305447/2010-6 , 473589/2013-3).

References

  1. 1.
    Antonietti, P.F., Fadel, N.A., Verani, M.: Modelling and numerical simulation of the polymeric extrusion process in textile products. Commun. Appl. Ind. Math. 1, 1–13 (2010)MathSciNetGoogle Scholar
  2. 2.
    Baltussen, M.G.H.M., Verbeeten, W.M.H., Bogaerds, A.C.B., Hulsen, M.A., Peters, G.W.M.: Anisotropy parameter restrictions for the eXtended Pom-Pom model. J. Non-Newton. Fluid 165, 1047–1054 (2010)CrossRefMATHGoogle Scholar
  3. 3.
    Bonito, A., Picasso, M., Laso, M.: Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phy. 215(2), 691–716 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bonito, A., Clément, P., Picasso, M.: Viscoelastic flows with complex free surfaces: numerical analysis and simulation. Glowinski, R., Xu, J. (eds.) Handbook of Numerical Analysis, Numerical Methods for Non-Newtonian Fluids vol. 16, pp. 305–369 (2011)Google Scholar
  5. 5.
    Brown, D.L., Cortez, R., Minion, M.L.: Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168, 464–499 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Caboussat, A.: Numerical simulation of two-phase free surface flows. Arch. Comput. Meth. Eng. 12, 165–224 (2005)CrossRefMATHGoogle Scholar
  7. 7.
    Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Springer, New York (2000)Google Scholar
  8. 8.
    Ciarlet, P.G., Glowinsk, R., Lions, J.L.: Numerical methods for non-newtonian fluids. Handbook of Numerical Analysis, vol. 16, North-Holland, Amsterdam (2011)Google Scholar
  9. 9.
    Crochet, M.J., Keunings, R.: Finite element analysis of die-swell of a highly elastic fluids. J. Non-Newton. Fluid 10, 339–356 (1982)CrossRefMATHGoogle Scholar
  10. 10.
    Cruickshank, J.O.: Low-Reynolds-number instabilities in stagnating jet flows. J. Fluid Mech. 193, 111–127 (1988)CrossRefGoogle Scholar
  11. 11.
    Figueiredo, R.A., Oishi, C.M., Cuminato, J.A., Alves, M.A.: Three-dimensional transient complex free surface flows: numerical simulation of XPP fluid. J. Non-Newton. Fluid 195, 88–98 (2013)CrossRefGoogle Scholar
  12. 12.
    Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182–2189 (1965)CrossRefMATHGoogle Scholar
  13. 13.
    Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323 (1985)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Martins, F.P., Oishi, C.M., Sousa, F.S., Cuminato, J.A.: Numerical assessment of mass conservation on a MAC-type method for viscoelastic free surface flows. In: 6th European Congress on Computational Methods in Applied Sciences and Egineering (ECCOMAS 2012), vol. 1, pp. 6545–6562 (2012)Google Scholar
  15. 15.
    McKee, S., Tomé, M.F., Cuminato, J.A., Castelo, A., Ferreira, V.G.: Recent advances in the marker-and-cell method. Arch. Comput. Meth. Eng. 11, 107–142 (2004)Google Scholar
  16. 16.
    McKee, S., Tomé, M.F., Ferreira, V.G., Cuminato, J.A., Castelo, A., Sousa, F.S., Mangiavacchi, N.: MAC Method. Comput. Fluids 37, 907–930 (2008)Google Scholar
  17. 17.
    Oishi, C.M., Cuminato, J.A., Ferreira, V.G., Tomé, M.F., Castelo, A., Mangiavacchi, N., McKee, S.: A stable semi-implicit method for free surface flows. J. Appl. Mech. 73, 940–947 (2006)CrossRefMATHGoogle Scholar
  18. 18.
    Oishi, C.M., Cuminato, J.A., Yuan, J.Y., McKee, S.: Stability of numerical schemes on staggered grids. Numer. Linear Algebra Appl. 15, 945–967 (2008)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Oishi, C.M., Martins, F.P., Tomé, M.F., Alves, M.A.: Numerical simulation of drop impact and jet buckling problems using the eXtended Pom-Pom model. J. Non-Newton. Fluid 169, 91–103 (2012)CrossRefGoogle Scholar
  20. 20.
    Oishi, C.M., Martins, F.P., Tomé, M.F., Cuminato, J.A., McKee, S.: Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows. J. Non-Newton. Fluid 166, 165–179 (2011)CrossRefMATHGoogle Scholar
  21. 21.
    Oishi, C.M., Tomé, M.F., Cuminato, J.A., McKee, S.: An implicit technique for solving 3d low Reynolds number moving free surface flows. J. Comput. Phys. 227, 7446–7468 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002)CrossRefMATHGoogle Scholar
  23. 23.
    Quarteroni, A., Saleri, A., Veneziani, A.: Factorization methods for the numerical approximation of Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 188, 505–526 (2000)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Roberts, S.A., Rao, R.R.: Numerical simulations of mounding and submerging flows of shear-thinning jets impinging in a container. J. Non-Newton. Fluid 166, 1100–1115 (2011)CrossRefMATHGoogle Scholar
  25. 25.
    Russo, G., Phillips, T.N.: Numerical prediction of extrudate swell of branched polymer melts. Rheol. Acta. 49, 657–676 (2010)CrossRefGoogle Scholar
  26. 26.
    Tanner, R.I.: A theory of die-swell revisited. J. Non-Newton. Fluid 129, 85–87 (2005)CrossRefMATHGoogle Scholar
  27. 27.
    Tomé, M.F., Castelo, A., Afonso, A.M., Alves, M.A., Pinho, F.T.: Application of the log-conformation tensor to three-dimensional time-dependent free surface flows. J. Non-Newton. Fluid 175–176, 44–54 (2012)CrossRefGoogle Scholar
  28. 28.
    Tomé, M.F., Castelo, A., Ferreira, V.G., McKee, S.: A finite difference technique for solving the Oldroyd-B model for 3D-unsteady free surface flows. J. Non-Newton. Fluid 154, 159–192 (2008)CrossRefGoogle Scholar
  29. 29.
    Tomé, M.F., Castelo, A., Nóbrega, J.M., Carneiro, O.S., Paulo, G.S., Pereira, F.T.: Numerical and experimental investigations of three-dimensional container filling with Newtonian viscous Fluids. Comput. Fluids 90, 172–185 (2014)CrossRefGoogle Scholar
  30. 30.
    Ville, L., Silva, L., Coupez, T.: Convected level set method for the numerical simulation of fluid buckling. Internat. J. Numer. Methods Fluids 66, 324–344 (2011)CrossRefMATHGoogle Scholar
  31. 31.
    Xu, X., Ouyang, J., Yang, B., Liu, Z.: SPH simulations of three-dimensional non-Newtonian free surface flows. Comput. Methods Appl. Mech. Eng. 256, 101–116 (2013)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Yang, B., Ouyang, J., Wang, F.: Simulation of stress distribution near weld line in the viscoelastic melt mold filling process. J. Appl. Math. (2013)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • José A. Cuminato
    • 1
  • Cassio M. Oishi
    • 2
  • Rafael A. Figueiredo
    • 1
  1. 1.Instituto de Ciências Matemáticas e Computação (ICMC)Universidade de São PauloSão CarlosBrazil
  2. 2.Faculdade de Ciência e Tecnologia (FCT)Universidade Estadual Paulista Julio de Mesquita FilhoPresidente PrudenteBrazil

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