Skip to main content
Log in

Simulation of single mode Rayleigh–Taylor instability by SPH method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A smoothed particle hydrodynamics (SPH) solution to the Rayleigh–Taylor instability (RTI) problem in an incompressible viscous two-phase immiscible fluid with surface tension is presented. The present model is validated by solving Laplace’s law, and square bubble deformation without surface tension whereby it is shown that the implemented SPH discretization does not produce any artificial surface tension. To further validate the numerical model for the RTI problem, results are quantitatively compared with analytical solutions in a linear regime. It is found that the SPH method slightly overestimates the border of instability. The long time evolution of simulations is presented for investigating changes in the topology of rising bubbles and falling spike in RTI, and the computed Froude numbers are compared with previous works. It is shown that the numerical algorithm used in this work is capable of capturing the interface evolution and growth rate in RTI accurately.

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

Access this article

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. Rayleigh L (1883) Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc Lond Math Soc 14(1): 170–177

    Article  MathSciNet  MATH  Google Scholar 

  2. Taylor G The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc R Soc Lond A 201:192–196 (1950)

  3. Waddell JT, Niederhaus CE, Jacobs JW (2001) Experimental study of Rayleigh–Taylor instability: low Atwood number liquid systems with single-mode initial perturbations. Phys Fluids 13: 1263

    Article  Google Scholar 

  4. Andrews MJ, Dalziel SB (2010) Small Atwood number Rayleigh–Taylor experiments. Philos Trans R Soc A 368(1916): 1663

    Article  MATH  Google Scholar 

  5. Piriz AR, Cortazar OD, Cela JJL, Tahir NA (2006) The Rayleigh–Taylor instability. Am J Phys 74: 1095

    Article  Google Scholar 

  6. Mikaelian KO (1993) Effect of viscosity on Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys Rev E 47(1): 375

    Article  Google Scholar 

  7. Youngs DL (1984) Numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Phys D 12(1–3): 32–44

    Article  Google Scholar 

  8. Pullin D (1982) Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability. J Fluid Mech 119(1): 507–532

    Article  MathSciNet  MATH  Google Scholar 

  9. Bell JB, Marcus DL (1992) A second-order projection method for variable-density flows* 1. J Comput Phys 101(2): 334–348

    Article  MathSciNet  MATH  Google Scholar 

  10. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension* 1. J Comput Phys 100(2): 335–354

    Article  MathSciNet  MATH  Google Scholar 

  11. Puckett EG, Almgren AA, Bell JB, Marcus DL, Rider WJ (1997) A high-order projection method for tracking fluid interfaces in variable density incompressible flows* 1. J Comput Phys 130(2): 269–282

    Article  MATH  Google Scholar 

  12. Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Annu Rev Fluid Mech 31(1): 567–603

    Article  MathSciNet  Google Scholar 

  13. Tryggvason G (1988) Numerical simulations of the Rayleigh–Taylor instability. J Comput Phys 75(2): 253–282

    Article  MathSciNet  MATH  Google Scholar 

  14. Tryggvason G, Bunner B, Ebrat O, Tauber W (1998) Computations of multiphase flows by a finite difference/front tracking method. I. Multi-fluid flows. Lecture series—Von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode

  15. Ding H, Spelt PDM, Shu C (2007) Diffuse interface model for incompressible two-phase flows with large density ratios. J Comput Phys 226(2): 2078–2095

    Article  MATH  Google Scholar 

  16. Oevermann M, Klein R, Berger M, Goodman J (2000) A projection method for two-phase incompressible flow with surface tension and sharp interface resolution. Konrad-Zuse-Zentrum für Informationstechnik, Berlin

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. 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(2): 610–624

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu XY, Adams NA (2007) An incompressible multi-phase SPH method. J Comput Phys 227(1): 264–278

    Article  MATH  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(7): 787–800

    Article  Google Scholar 

  21. 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(22): 8380–8393

    Article  MathSciNet  MATH  Google Scholar 

  22. Shadloo MS, Zainali A, Yildiz M, Suleman A (2012) A robust weakly compressible SPH method and its comparison with an incompressible SPH. Int J Numer Methods Eng 89: 939–956

    Article  MathSciNet  MATH  Google Scholar 

  23. Shadloo MS, Yildiz M (2011) Numerical modeling of Kelvin–Helmholtz instability using smoothed particle hydrodynamics. Int J Numer Methods Eng 87: 988–1006

    Article  MathSciNet  MATH  Google Scholar 

  24. Hoover WG (1998) Isomorphism linking smooth particles and embedded atoms. Phys A 260(3): 244–254

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  26. Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82: 1013–1024

    Article  Google Scholar 

  27. Fatehi R, Manzari M (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  MathSciNet  MATH  Google Scholar 

  28. Rafiee A, Thiagarajan KP (2009) An SPH projection method for simulating fluid-hypoelastic structure interaction. Comput Methods Appl Mech Eng 198(33–36): 2785–2795

    Article  MATH  Google Scholar 

  29. Hashemi MR, Fatehi R, Manzari MT (2011) SPH simulation of interacting solid bodies suspended in a shear flow of an Oldroyd-B fluid. J Non-Newton Fluid Mech 166: 1239–1252

    Article  Google Scholar 

  30. Monaghan JJ (2005) Smoothed particle hydrodynamics. Rep Prog Phys 68: 1703

    Article  MathSciNet  Google Scholar 

  31. Randles PW, Libersky LD (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Comput Methods Appl Mech Eng 139(1): 375–408

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9): 1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  33. Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1): 195–227

    Article  MathSciNet  MATH  Google Scholar 

  34. Jun S, Liu WK, Belytschko T (1998) Explicit reproducing kernel particle methods for large deformation problems. Int J Numer Methods Eng 41(1): 137–166

    Article  MATH  Google Scholar 

  35. Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38(10): 1655–1679

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu WK, Jun S, Sihling DT, Chen Y, Hao W (1997) Multiresolution reproducing kernel particle method for computational fluid dynamics. Int J Numer Methods Fluids 24(12): 1391–1415

    Article  MathSciNet  MATH  Google Scholar 

  37. Chen JK, Beraun JE (2000) A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems. Comput Methods Appl Mech Eng 190(1): 225–239

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Morris JP (2000) Simulating surface tension with smoothed particle hydrodynamics. Int J Numer Methods Fluids 33(3): 333–353

    Article  MATH  Google Scholar 

  40. Yildiz M, Rook RA, Suleman A (2009) SPH with the multiple boundary tangent method. Int J Numer Methods Eng 77(10): 1416–1438

    Article  MathSciNet  MATH  Google Scholar 

  41. Ginzburg I, Wittum G (2001) Two-phase flows on interface refined grids modeled with VOF, staggered finite volumes, and spline interpolants. J Comput Phys 166(2): 302–335

    Article  MATH  Google Scholar 

  42. Lafaurie B, Nardone C, Scardovelli R, Zaleski S, Zanetti G (1994) Modelling merging and fragmentation in multiphase flows with SURFER. J Comput Phys 113(1): 134–147

    Article  MathSciNet  MATH  Google Scholar 

  43. Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. Dover, New York

    MATH  Google Scholar 

  44. Guermond JL, Quartapelle L (2000) A projection FEM for variable density incompressible flows. J Comput Phys 165(1): 167–188

    Article  MathSciNet  MATH  Google Scholar 

  45. Ramaprabhu P, Dimonte G, Andrews MJ (2005) A numerical study of the influence of initial perturbations on the turbulent Rayleigh–Taylor instability. J Fluid Mech 536(1): 285–319

    Article  MATH  Google Scholar 

  46. Goncharov VN (2002) Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys Rev Lett 88(13): 134502

    Article  Google Scholar 

  47. Abarzhi SI, Nishihara K, Glimm J (2003) Rayleigh–Taylor and Richtmyer–Meshkov instabilities for fluids with a finite density ratio. Phys Lett A 317(5–6): 470–476

    Article  MATH  Google Scholar 

  48. Scorer RS (1957) Experiments on convection of isolated masses of buoyant fluid. J Fluid Mech 2(6): 583–594

    Article  Google Scholar 

  49. Ramaprabhu P, Dimonte G, Young YN, Calder AC, Fryxell B (2006) Limits of the potential flow approach to the single-mode Rayleigh–Taylor problem. Phys Rev E 74(6): 066308

    Article  MathSciNet  Google Scholar 

  50. Wilkinson JP, Jacobs JW (2007) Experimental study of the single-mode three-dimensional Rayleigh–Taylor instability. Phys Fluids 19: 124102

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Yildiz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shadloo, M.S., Zainali, A. & Yildiz, M. Simulation of single mode Rayleigh–Taylor instability by SPH method. Comput Mech 51, 699–715 (2013). https://doi.org/10.1007/s00466-012-0746-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-012-0746-2

Keywords

Navigation