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Journal of Scientific Computing

, Volume 35, Issue 2–3, pp 372–396 | Cite as

A Numerical Method for Free-Surface Flows and Its Application to Droplet Impact on a Thin Liquid Layer

  • Kensuke YokoiEmail author
Article

Abstract

We propose a simple and practical numerical method for free surface flows. The method is based various methods, the level set method of an interface capturing method, the THINC/WLIC (tangent of hyperbola for interface capturing/weighed line interface calculation) method of an interface tracking method, the CIP-CSL (constrained interpolation profile conservative semi-Lagrangian) method of a conservation equation solver, VSIAM3 (volume/surface integrated average based multi-moment method) of a fluid solver and the CSF (continuum surface force) model of a surface force model. The level set method and the THINC/WLIC method are combined by using a CLSVOF (coupled level set and volume-of-fluid) framework. The method is applied to Rayleigh-Taylor instability with surface tension force and droplet impact on a thin liquid layer (milk crown).

Keywords

Interface capturing Level set CLSVOF THINC WLIC Free-surface flow 

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References

  1. 1.
    Adalsteinsson, D., Sethian, J.A.: The fast construction of extension velocities in level set methods. J. Comput. Phys. 148, 2 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aulisa, E., Manservisi, S., Scardovelli, R., Zaleski, S.: Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry. J. Comput. Phys. 225, 2301 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100, 335 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method of the incompressible Navier-Stokes equations. J. Comput. Phys. 85, 257 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bellman, R., Pennington, R.H.: Effect of surface tension and viscosity on Taylor instability. Q. Appl. Methods 12, 12, 151 (1954) MathSciNetGoogle Scholar
  6. 6.
    Chang, Y.C., Hou, T.Y., Merriman, B., Osher, S.: A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comput. Phys. 124, 449–464 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Daly, B.J.: Numerical study of the effect of surface tension on interface instability. Phys. Fluids 12, 1340 (1969) zbMATHCrossRefGoogle Scholar
  8. 8.
    Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1967) Google Scholar
  9. 9.
    Enright, D., Fedkiw, R., Ferziger, J., Mitchell, I.: A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fedkiw, R., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gerlach, D., Tomar, G., Biswas, G., Durst, F.: Comparison of volume-of-fluid methods for surface tension-dominant two-phase flows. Int. J. Heat Mass Transf. 49, 740 (2006) CrossRefGoogle Scholar
  12. 12.
    Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 313–503 (1998) Google Scholar
  13. 13.
    Glimm, J., et al.: Front tracking in two and three dimensions. J. Comput. Mech. 7, 1 (1998) Google Scholar
  14. 14.
    Gueyffier, D., Zaleski, S.: Formation de digitations lors de l’impact d’une goutte sur un film liquide, Finger formation during droplet impact on a liquid film. C. R. Acad. Sci. Ser. IIB—Mech.–Phys.–Astron. 326, 839 (1998) Google Scholar
  15. 15.
    Harlow, F.H., Welch, E.: Numerical calculation of time-dependent viscous incompressible flow of fluids with free surface. Phys. Fluids 8, 2182 (1965) CrossRefGoogle Scholar
  16. 16.
    Harlow, F.H., Shannon, J.P.: The splash of a liquid drop. J. Appl. Phys. 38, 3855 (1967) CrossRefGoogle Scholar
  17. 17.
    Hirt, C.W., Nichols, B.D.: Volume of fluid (VOF) methods for the dynamic of free boundaries. J. Comput. Phys. 39, 201 (1981) zbMATHCrossRefGoogle Scholar
  18. 18.
    Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kang, M., Fedkiw, R., Liu, X.-D.: A boundary condition capturing method for multiphase incompressible flow. J. Sci. Comput. 15, 323–360 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kayafas, G., Jussim, E.: Stopping Time: The Photographs of Harold Edgerton, New Ed edn. Abrams, New York (2000) Google Scholar
  21. 21.
    Kim, J., Moin, P.: Applications of a fractional step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59, 308 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S., Zanetti, G.: Modeling merging and fragmentation in multiphase flows with SURFER. J. Comput. Phys. 113, 134–147 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Li, J.: Calcul d’interface affine par morceaux (piecewise linear interface calculation). C. R. Acad. Sci. Paris, Sér. IIb 320, 391–396 (1995) zbMATHGoogle Scholar
  24. 24.
    Li, J., Renardy, Y., Renardy, M.: Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys. Fluids 12, 269–282 (2000) CrossRefGoogle Scholar
  25. 25.
    Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Liu, X.-D., Fedkiw, R., Kang, M.: A boundary condition capturing method for Poisson’s equation on irregular domains. J. Comput. Phys. 160, 151–178 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    LeVeque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. SIGGRAPH 2004, ACM TOG 23, 457–462 (2004) CrossRefGoogle Scholar
  29. 29.
    Marella, S., Krishnan, S., Liu, H., Udaykumar, H.S.: Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations. J. Comput. Phys. 210, 1 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Noh, W.F., Woodward, P.: SLIC (simple line interface method). In: Lecture Notes in Physics, vol. 24, p. 330. Springer, Berlin (1976) Google Scholar
  31. 31.
    Osher, S., Sethian, J.A.: Front propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulation. J. Comput. Phys. 79, 12 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamics Implicit Surface. Applied Mathematical Sciences, vol. 153. Springer, New York (2003) Google Scholar
  33. 33.
    Pilliod, J.E., Puckett, E.G.: Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J. Comput. Phys. 199, 465–502 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Rider, W.J., Kothe, D.B.: Reconstructing volume tracking. J. Comput. Phys. 141, 112 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Rieber, M., Frohn, A.: A numerical study on the mechanism of splashing. Int. J. Heat Fluid Flow 20, 453 (1999) CrossRefGoogle Scholar
  36. 36.
    Rudman, M.: Volume-tracking method for interfacial flow calculations. Int. J. Numer. Methods Fluids 24, 679–691 (1997) CrossRefMathSciNetGoogle Scholar
  37. 37.
    Scardovelli, R., Zaleski, S.: Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–603 (1999) CrossRefMathSciNetGoogle Scholar
  38. 38.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  39. 39.
    Sussman, M., Smereka, P., Osher, S.: A level set approach for capturing solution to incompressible two-phase flow. J. Comput. Phys. 114, 146 (1994) zbMATHCrossRefGoogle Scholar
  40. 40.
    Sussman, M., Fatemi, E.: An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow. SIAM J. Sci. Comput. 20, 1165 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Sussman, M., Smereka, P.: Axisymmetric free boundary problems. J. Fluid Mech. 341, 269 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Sussman, M., Smith, K.M., Hussaini, M.Y., Ohta, M., Zhi-Wei, R.: A sharp interface method for incompressible two-phase flows. J. Comput. Phys. 221, 469 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Sussman, M., Puckett, E.G.: A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162, 301–337 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Unverdi, S.O., Tryggvason, G.: A front tracking method for viscous, incompressible multi-fluid flow. J. Comput. Phys. 100, 25 (1992) zbMATHCrossRefGoogle Scholar
  45. 45.
    van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. 13, 631 (1992) zbMATHCrossRefGoogle Scholar
  46. 46.
    Xiao, F., Yabe, T., Peng, X., Kobayashi, H.: Conservative and oscillation-less atmospheric transport schemes based on rational functions. J. Geophys. Res. 107, 4609 (2002) CrossRefGoogle Scholar
  47. 47.
    Xiao, F., Honma, Y., Kono, T.: A simple algebraic interface capturing scheme using hyperbolic tangent function. Int. J. Numer. Method. Fluid. 48, 1023 (2005) zbMATHCrossRefGoogle Scholar
  48. 48.
    Xiao, F., Ikebata, A., Hasegawa, T.: Numerical simulations of free-interface fluids by a multi integrated moment method. Comput. Struct. 83, 409–423 (2005) CrossRefMathSciNetGoogle Scholar
  49. 49.
    Xiao, F., Akoh, R., Ii, S.: Unified formulation for compressible and incompressible flows by using multi integrated moments II: multi-dimensional version for compressible and incompressible flows. J. Comput. Phys. 213, 31–56 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Xiao, F., Peng, X.D., Shen, X.S.: A finite-volume grid using multimoments for geostrophic adjustment. Mon. Weather Rev. 134, 2515 (2006) CrossRefGoogle Scholar
  51. 51.
    Yabe, T., Tanaka, R., Nakamura, T., Xiao, F.: An exactly conservative semi-Lagrangian scheme (CIP-CSL) in one dimension. Mon. Weather Rev. 129, 332–344 (2001) CrossRefGoogle Scholar
  52. 52.
    Yarin, A.L.: DROP IMPACT DYNAMICS: Splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38, 129–157 (2006) CrossRefMathSciNetGoogle Scholar
  53. 53.
    Yabe, T., Xiao, F., Utsumi, T.: Constrained interpolation profile method for multiphase analysis. J. Comput. Phys. 169, 556 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Yokoi, K., Xiao, F.: Mechanism of structure formation in circular hydraulic jumps: Numerical studies of strongly deformed free surface shallow flows. Physica D 161, 202 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Yokoi, K.: Numerical method for complex moving boundary problems in a Cartesian fixed grid. Phys. Rev. E 65, 055701(R) (2002) Google Scholar
  56. 56.
    Yokoi, K.: Numerical method for moving solid object in flows. Phys. Rev. E 67, 045701(R) (2003) CrossRefGoogle Scholar
  57. 57.
    Yokoi, K.: Efficient implementation of THINC scheme: a simple and practical smoothed VOF algorithm. J. Comput. Phys. 226, 1985 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Youngs, D.L.: Time-dependent multi-material flow with large fluid distortion. In: Morton, K.W., Baines, M.J. (eds.) Numerical Methods for Fluid Dynamics, vol. 24, pp. 273–285. Academic Press, New York (1982) Google Scholar
  59. 59.
    Zalesak, S.T.: Fully multi-dimensional flux corrected transport algorithm for fluid flow. J. Comput. Phys. 31, 335 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    Zhang, Y., Yabe, T.: Effect of ambient gas on three-dimensional breakup in coronet formation. CFD J. 8, 378–382 (1999) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of CaliforniaLos AngelesUSA

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