Skip to main content
SpringerLink
Log in

Numerical analysis of the impact of two droplets with a liquid film using an incompressible SPH method

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

In this study, a truly incompressible smoothed particle hydrodynamics (SPH) algorithm combined with an effective surface tension model is extended to simulate the dynamic process of multiple droplets impacting on a liquid film in 2D and 3D. This approach uses a pressure Poisson equation to satisfy the incompressibility constraints, and the Navier–Stokes equations are solved in a Lagrangian form using a fractional-step projection method. The mathematical model is first validated by the simulations of several fluid impact phenomena in comparison with those obtained by other numerical methods. Then the interesting phenomena of two 2D droplets impacting successively on a rigid solid/liquid film are numerically predicted and compared with the corresponding experimental results. Next, the fluid mechanics of two 2D droplets impinging simultaneously on a thin liquid film are numerically investigated. The effects of the impact velocity and the two droplets’ horizontal spacing on the collision behavior are discussed in detail. Lastly, the splashing phenomenon of a 3D droplet impacting on a thin liquid film is simulated. All numerical results obtained are in agreement with the available data.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Yarin AL (2006) Drop impact dynamics: splashing, spreading, receding, bouncing. Ann Rev Fluid Mech 38:159–192

    Article  ADS  MathSciNet  Google Scholar 

  2. Josserand C, Zaleski S (2003) Droplet splashing on a thin liquid film. Phys Fluids 15:1650–1657

    Article  ADS  Google Scholar 

  3. Rocco G, Coppola G, de Luca L (2010) Simulation of drop impact on a thin liquid film by means of the VOF method. Aerotec Missili Spazio 89:25–35

    Google Scholar 

  4. Wang AB, Chen ChCh (2000) Splashing impact of a single drop onto very thin liquid films. Phys Fluids 12:2155–2158

    Article  ADS  Google Scholar 

  5. Xie H, Koshizuka S, Oka Y (2004) Modeling of a single drop impact onto liquid film using particle method. Int J Numer Methods Fluids 45:1009–1023

    Article  MATH  Google Scholar 

  6. Fujimoto H, Ito S, Takezaki I (2002) Experimental study of successive collision of two water droplets with a solid. Exp Fluids 33:500–502

    Article  Google Scholar 

  7. Roisman IV, Prunet-Foch B, Tropea C, Vignes-Adler M (2002) Multiple drop impact onto a dry solid substrate. J Colloid Interface Sci 256:396–410

    Article  Google Scholar 

  8. Hirt CW, Nicholls BD (1981) Volume of fluid (VOF) method for dynamics of free boundaries. J Comput Phys 39:201–221

    Article  ADS  MATH  Google Scholar 

  9. Osher S, Sethian JA (1988) Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79:12–49

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Liao Y, Lin S (2012) A meshless method using the Takagi–Sugeno fuzzy model. J Eng Math 72:159–175

    Article  MATH  MathSciNet  Google Scholar 

  11. Wu N, Tsay T (2009) Applicability of the method of fundamental solutions to 3-D wave-body interaction with fully nonlinear free surface. J Eng Math 63:61–78

    Article  MATH  MathSciNet  Google Scholar 

  12. Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 83:1013–1024

    Article  ADS  Google Scholar 

  13. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. R Astron Soc, Monthly Notices 181:375–389

    ADS  MATH  Google Scholar 

  14. Xu X, Ouyang J, Jiang T, Li Q (2012) Numerical simulation of 3D-unsteady viscoelastic free surface flows by improved smoothed particle hydrodynamics method. J Non-Newtonian Fluid Mech 177:109–120

    Article  Google Scholar 

  15. Liu J (2011) Meshless study of dynamic failure in shells. J Eng Math 71:205–222

    Article  MATH  Google Scholar 

  16. Wang SLN (2011) A large-deformation Galerkin SPH method for fracture. J Eng Math 71:305–318

    Article  MATH  Google Scholar 

  17. Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110:399–406

    Article  ADS  MATH  Google Scholar 

  18. Xu X, Ouyang J, Yang B, Liu Z (2013) SPH simulations of three-dimensional non-Newtonian free surface flows. Comput Methods Appl Mech Eng 256:101–116

    Article  ADS  MathSciNet  Google Scholar 

  19. Cummins SJ, Rudman M (1999) An SPH projection method. J Comput Phys 152(2):584–607

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Shao S, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26:787–800

    Article  ADS  Google Scholar 

  21. Shao S (2012) Incompressible smoothed particle hydrodynamics simulation of multifluid flows. Int J Numer Methods Fluids 69:1715–1735

    Article  MATH  Google Scholar 

  22. Shadloo MS, Zainali A, Sadek SH, Yidiz M (2011) Improved incompressible smoothed particle hydrodynamics method for simulating flow around bluff bodies. Comput Methods Appl Mech Eng 200:1008–1020

    Article  ADS  MATH  Google Scholar 

  23. Lee ES, Moulinec C, Xu R, Violeau D, Laurence D, Stansby P (2008) Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. J Comput Phys 227:8417–8436

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Xu R, Stansby P, Laurence D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. J Comput Phys 228:6703–6725

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Nugent S, Posch HA (2000) Liquid drops and surface tension with smoothed particle applied mechanics. Phys Rev E 62:4968–4975

    Article  ADS  Google Scholar 

  26. Melean Y, Sigalotti LDG, Hasmy A (2004) On the SPH tensile instability in forming viscous liquid drops. Comput Phys Commun 157:191–200

    Article  ADS  Google Scholar 

  27. Melean Y, Sigalotti LDG (2005) Coalescence of colliding van der Waals liquid drops. Int J Heat Mass Transf 48:4041–4061

    Article  MATH  Google Scholar 

  28. Colagrossi A, Landrini M (2003) Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J Comput Phys 191:448–475

    Article  ADS  MATH  Google Scholar 

  29. Grenier N, Antuono M, Colagrossi A, Le Touzé D, Alessandrini B (2009) An Hamiltonian interface SPH formulation for multi-fluid and free surface flows. J Comput Phys 228:8380–8393

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Grenier N, Le Touzé D, Ferrant P, Vila JP (2008) Two-phase simulations using a volume fraction SPH scheme with a Riemann solver. Proceedings of the 3rd international SPHERIC workshop, Lausanne, Switzerland, pp 173–179

  31. Tartakovsky AM, Meakin P (2005) Modeling of surface tension and contact angles with smoothed particle hydrodynamics. Phys Rev E 72:1–9

    Article  Google Scholar 

  32. Tartakovsky AM, Meakin P (2005) A smoothed particle hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh–Taylor instability. J Comput Phys 207:610–624

    Article  ADS  MATH  Google Scholar 

  33. Morris JP (2000) Simulating surface tension with smoothed particle hydrodynamics. Int J Numer Methods Fluids 33:333–353

    Article  MATH  Google Scholar 

  34. Rogers B, Leduc J, Marongi JC, Leboeuf F (2009) Comparison and evaluation of multi-phase and surface tension models. Proc of 4th Int SPHERIC Workshop, Nantes, pp 30–37

  35. Leduc J, Marongiu JC, Leboeuf F, Lance M, Parkinson E (2009) Multiphase SPH: a new model based on acoustic Riemann solver. Proc of 4th Int SPHERIC Workshop, Nantes, pp 8–13

  36. Leduc J, Leboeuf F, Lance M. (2010) Improvement of multiphase model using preconditioned Riemann solvers. Proc of 5th Int SPHERIC Workshop, Manchester, pp 1–6

  37. Zhang MY (2010) Simulation of surface tension in 2D and 3D with smoothed particle hydrodynamics method. J Comput Phys 229:7238–7259

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Hu XY, Adams NA (2006) A multiphase SPH method for macroscopic and mesoscopic flows. J Comput Phys 213:844–861

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Hu XY, Adams NA (2007) An incompressible multi-phase SPH method. J Comput Phys 227:264–278

    Article  ADS  MATH  Google Scholar 

  40. Hu XY, Adams NA (2009) A constant-density approach for incompressible multi-phase SPH. J Comput Phys 228:2082–2091

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. Fatehi R, Manzari MT (2012) A consistent and fast weakly compressible smoothed particle hydrodynamics with a new wall boundary condition. Int J Numer Methods Fluids 68:905–921

    Article  MATH  MathSciNet  Google Scholar 

  42. Fatehi R, Manzari MT (2011) Error estimation in smoothed particle hydrodynamics and a new scheme for second derivatives. Comput Math Appl 61:482–498

    Article  MATH  MathSciNet  Google Scholar 

  43. Hosseini SM, Feng JJ (2011) Pressure boundary conditions for computing incompressible flows with SPH. J Comput Phys 230:7473–7487

    Article  ADS  MATH  MathSciNet  Google Scholar 

  44. Van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Scientif Stat Comput 13:631–644

    Article  MATH  Google Scholar 

  45. Fang J, Owens RG, Tacher L, Parriaux A (2006) A numerical study of the SPH method for simulating transient viscoelastic free surface flows. J Non-Newtonian Fluid Mech 139:68–84

    Article  MATH  Google Scholar 

  46. Tome MF, Mangiavacchi N, Castelo A, Cuminato JA, McKee S (2002) A finite difference technique for simulating unsteady viscoelastic free surface flows. J Non-Newtonian Fluid Mech 106:61–106

    Article  MATH  Google Scholar 

  47. Vila JP (1999) On particle weighted methods and smoothed particle hydrodynamics. Math Models Methods Appl Sci 9:161–209

    Article  MATH  MathSciNet  Google Scholar 

  48. Monaghan JJ (1997) SPH and Riemann solvers. J Compuat Phys 136:298–307

    Article  ADS  MATH  MathSciNet  Google Scholar 

  49. Antuono M, Colagrossi A, Marrone S, Molteni D (2010) Free-surface flows solved by means of SPH schemes with numerical difiusive terms. Comput Phys Commun 181:532–549

    Article  ADS  MATH  MathSciNet  Google Scholar 

  50. Antuono M, Colagrossi A, Marrone S, Lugni C (2011) Propagation of gravity waves through an SPH scheme with numerical difiusive terms. Comput Phys Commun 182:866–877

    Article  ADS  MATH  Google Scholar 

  51. Marrone S, Antuono M, Colagrossi A, Colicchio G, Le Touzé D, Graziani G (2011) \(\delta \)-SPH model for simulating violent impact flows. Comput Methods Appl Mech Eng 200:1526–1542

    Article  ADS  MATH  Google Scholar 

Download references

Acknowledgments

The support of the National Basic Research Program of China (No. 2012CB025903) and that of the National Natural Science Foundation of China (No. 10871159) are fully acknowledged.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710129, China

    Xiaoyang Xu, Jie Ouyang, Tao Jiang & Qiang Li

Authors
  1. Xiaoyang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jie Ouyang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tao Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jie Ouyang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, X., Ouyang, J., Jiang, T. et al. Numerical analysis of the impact of two droplets with a liquid film using an incompressible SPH method. J Eng Math 85, 35–53 (2014). https://doi.org/10.1007/s10665-013-9634-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-013-9634-9

Keywords

Search

Navigation