DOI: https://doi.org/10.26089/NumMet.v26r428

Modified axisymmetric contact smoothed particle hydrodynamics method with the possibility of modeling surface tension

Authors

  • Georgii D. Rublev

Keywords:

contact SPH
surface tension
axisymmetric SPH schemes
conservative SPH schemes
Riemann solver
ghost particles

Abstract

The conservation accuracy of total momentum and total energy is important in modeling many physical problems, including capillary phenomena, since non-physical acceleration can lead to incorrect deformation of free surfaces. Because the standard axisymmetric CSPH method of Parshikov does not satisfy the conservation laws, a modified axisymmetric CSPH method is considered to improve the accuracy of preserving the total momentum and the total energy. An external pressure boundary condition is proposed for the CSPH method. This boundary condition is used for surface tension modeling with the new axisymmetric CSPH method. The verification of the surface tension model is performed for three test cases: modeling the establishment of Laplace pressure in a water droplet, modeling small oscillations of a water droplet (comparing the period of small oscillations with Rayleighʼs formula), and modeling the effect of surface tension on a thin film (comparing the velocity of propagation of rims created by surface tension with Taylor-Culickʼs formula). A simulation of the development of Rayleigh-Plateau instability is conducted.


Downloads

Published

2025-10-23

Issue

Vol. 26 (2025): Issue 4.

Section

Methods and algorithms of computational mathematics and their applications

Author

Georgii D. Rublev

Dukhov Research Institute of Automatics

• Researcher

References

  1. L. B. Lucy, “A numerical approach to the testing of the fission hypothesis,” The Astronomical Journal. 82 (12), 1013-1024 (1977).
    doi 10.1086/112164
  2. R. A. Gingold, J. J. Monaghan, “Smoothed particle hydrodynamics: theory and application to non-spherical stars,” Monthly Notices of the Royal Astronomical Society. 181 (3), 375-389 (1977).
    doi 10.1093/mnras/181.3.375
  3. A. N. Parshikov, “Application of a solution to the Riemann problem in the SPH method,” Zh. Vychisl. Mat. Mat. Fiz. 39 (7), 1216-1225 (1999).
    https://www.mathnet.ru/links/5b072952639f95bbff3a74ed2d0873aa/zvmmf1657.pdf . Cited October 17, 2025 [in Russian].
  4. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction(Springer Berlin, Heidelberg, 2009).
    doi 10.1007/b79761
  5. M.-K. Li, A-M. Zhang, F.-R. Ming, P.-N. Sun, Y.-X. Peng, “An axisymmetric multiphase SPH model for the simulation of rising bubble,” Computer Methods in Applied Mechanics and Engineering. 366, 113039 (2020).
    doi 10.1016/j.cma.2020.113039
  6. D. Kurilovich, M. M. Basko, D. A. Kim, F. Torretti, R. Schupp, J. C. Visschers, J. Scheers, R. Hoekstra, W. Ubachs, O. O. Versolato, “Power-law scaling of plasma pressure on laser-ablated tin microdroplets,” Phys. Plasmas. 25, 012709 (2018).
    doi 10.1063/1.5010899
  7. C. S. Coleman, G. V. Bicknell, “Jets with entrained clouds -- I. Hydrodynamic simulations and magnetic field structure,” Monthly Notices of the Royal Astronomical Society. 214 (3), 337-355 (1985).
    doi 10.1093/mnras/214.3.337
  8. A. G. Petschek, L. D. Libersky, “Cylindrical smoothed particle hydrodynamics,” Journal of Computational Physics. 109 (1), 76-83 (1993).
    doi 10.1006/jcph.1993.1200
  9. M. Omang, S. Børve, J. Trulsen, “SPH in spherical and cylindrical coordinates,” Journal of Computational Physics. 213 (1), 391-412 (2006).
    doi 10.1016/j.jcp.2005.08.023
  10. A. N. Parshikov, S. A. Medin, I. I. Loukashenko, V. A. Milekhin, “Improvements in SPH method by means of interparticle contact algorithm and analysis of perforation tests at moderate projectile velocities,” International Journal of Impact Engineering. 24 (8), 779-796 (2000).
    doi 10.1016/S0734-743X(99)00168-2
  11. J. P. Morris, “Simulating surface tension with smoothed particle hydrodynamics,” International Journal for Numerical Methods in Fluids. 33 (3), 333-353 (2000).
    doi 10.1002/1097-0363(20000615)33: 3<333: : aid-fld11>3.0.co;2-7.
  12. X. Y. Hu, N. A. Adams, “A multi-phase SPH method for macroscopic and mesoscopic flows,” Journal of Computational Physics. 213 (2), 844-861 (2006).
    doi 10.1016/J.JCP.2005.09.001
  13. S. Adami, X. Y. Hu, N. A. Adams, “A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation,” Journal of Computational Physics. 229 (13), 5011-5021 (2010).
    doi 10.1016/j.jcp.2010.03.022
  14. J. U. Brackbill, D. B. Kothe, C. Zemach, “A continuum method for modeling surface tension,” Journal of Computational Physics. 100 (2), 335-354 (1992).
    doi 10.1016/0021-9991(92)90240-y
  15. Z.-B. Wang, R. Chen, H. Wang, Q. Liao, X. Zhu, S.-Z. Li, “An overview of smoothed particle hydrodynamics for simulating multiphase flow,” Applied Mathematical Modelling. 40 (23), 9625-9655 (2016).
    doi 10.1016/j.apm.2016.06.030
  16. S. Nugent, H. A. Posch, “Liquid drops and surface tension with smoothed particle applied mechanics,” Phys. Rev. E. 62 (4), 4968-4975 (2000).
    doi 10.1103/PhysRevE.62.4968
  17. A. Tartakovsky, P. Meakin, “Modeling of surface tension and contact angles with smoothed particle hydrodynamics,” Phys. Rev. E. 72 (2), 026301 (2005).
    doi 10.1103/PhysRevE.72.026301
  18. M. Kondo, J. Matsumoto, “Surface tension and wettability calculation using density gradient potential in a physically consistent particle method,” Computer Methods in Applied Mechanics and Engineering. 385, 114072 (2021).
    doi 10.1016/j.cma.2021.114072
  19. X.-L. Fang, A. Colagrossi, P.-P. Wang, A-M. Zhang, “An accurate and robust axisymmetric SPH method based on Riemann solver with applications in ocean engineering,” Ocean Engineering. 244, 110369 (2022).
    doi 10.1016/j.oceaneng.2021.110369
  20. G. D. Rublev, A. N. Parshikov, S. A. Dyachkov, “Improving approximation accuracy in Godunov-type smoothed particle hydrodynamics methods,” Applied Mathematics and Computation. 488, 129128 (2024).
    doi 10.1016/j.amc.2024.129128
  21. J. Michel, A. Vergnaud, G. Oger, C. Hermange, D. Le Touzé, “On particle shifting techniques (PSTs): Analysis of existing laws and proposition of a convergent multi-invariant law,” Journal of Computational Physics. 459, 110999 (2022).
    doi 10.1016/j.jcp.2022.110999
  22. S. Marrone, A. Colagrossi, D. Le Touzé, G. Graziani, “Fast free-surface detection and level-set function definition in SPH solvers,” Journal of Computational Physics. 229 (10), 3652-3663 (2010).
    doi 10.1016/j.jcp.2010.01.019
  23. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. VI: Fluid Mechanics(Nauka, Moscow, 1986;Pergamon, Oxford, 1987).
  24. J. W. Strutt (Rayleigh Lord, F.R.S.), “VI. On the capillary phenomena of jets,” Proceedings of the Royal Society of London. 29 (196-199), 71-97 (1879).
    doi 10.1098/rspl.1879.0015
  25. G. I. Taylor, “The dynamics of thin sheets of fluid II. Waves on fluid sheets,” Proceedings of the Royal Society of London. Series A. Mathematicaland Physical Sciences. 253 (1274), 296-312 (1959).
    doi 10.1098/rspa.1959.0195
  26. M. Dai, D. P. Schmidt, “Adaptive tetrahedral meshing in free-surface flow,” Journal of Computational Physics. 208 (1), 228-252 (2005).
    doi 10.1016/j.jcp.2005.02.012
  27. Rayleigh Lord, “On the instability of jets,” Proceedings of the London Mathematical Society. textbfs1-10 (1), 4-13 (1878).
    doi 10.1112/plms/s1-10.1.4
  28. A. N. Parshikov, S. A. Medin, G. D. Rublev, S. A. Dyachkov, “Numerical viscosity control in Godunov-like smoothed particle hydrodynamics for realistic flows modeling,” Physics of Fluids. 36 (1), 013101 (2024).
    doi 10.1063/5.0181276

License

Copyright (c) 2025 Г. Д. Рублев

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.