• Special Column on the 2020 Spheric Harbin International Workshop (Guest Editors A-Man Zhang, Shi-Ping Wang, Peng-Nan Sun)
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New instability and mixing simulations using SPH and a novel mixing measure


This paper assesses the ability of smoothed particle hydrodynamics (SPH) to simulate mixing of two-phase flows and their transition to instabilities under different flow regimes. A new measure for quantification of the degree of mixing between phases in a Lagrangian framework is also developed. The method is validated using the lid-driven cavity and two-phase Poiseuille flow cases. The velocity along the centre of the cavity is compared with results from the literature, whilst commercial volume-of-fluid code STAR-CCM+ provides a benchmark for the mixing and different mixing measures are considered. The velocity of two-phase Poiseuille flow along the channel is compared to the analytical solution, and the appearance of interfacial instabilities with perturbation theory. This is the first time SPH has been used to investigate the onset and development of these instabilities. In particular, it is able to model the deforming shape of the interface, which is not given by analytical studies, while also offering improved predictions over conventional mesh-based computational fluid dynamics simulations.

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  1. [1]

    Chang S., Zhou C., Golchert B. Eulerian approach for multiphase flow simulation in a glass melter [J]. Applied Thermal Engineering, 2005, 25(17–18): 3083–3103.

    Google Scholar 

  2. [2]

    Kunz R. F., Boger D. A., Stinebring D. R. et al. A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction [J]. Computers and Fluids, 2000, 29(8): 849–875

    MATH  Google Scholar 

  3. [3]

    Gingold R. A., Monaghan J. J. Smoothed particle hydrodynamics: theory and application to non-spherical stars [J]. Monthly Notices of the Royal Astronomical Society, 1977, 181(3): 375–389.

    MATH  Google Scholar 

  4. [4]

    Lucy L. B. A numerical approach to the testing of the fission hypothesis [J]. The Astronomical Journal, 1977, 82: 1013–1024.

    Google Scholar 

  5. [5]

    Hughes J. P., Graham D. I. Comparison of incompressible and weakly-compressible SPH models for free-surface water flows [J]. Journal of Hydraulic Research, 48(Sup1): 2010, 105–117.

    Google Scholar 

  6. [6]

    Wang Z. B., Chen R., Wang H. et al. An overview of smoothed particle hydrodynamics for simulating multiphase flow [J]. Applied Mathematical Modelling, 2016, 40(23): 9625–9655.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    Liu M., Zhang Z. Smoothed particle hydrodynamics (SPH) for modeling fluid-structure interactions [J]. Science China Physics, Mechanics and Astronomy, 2019, 62(8): 984701.

    MathSciNet  Google Scholar 

  8. [8]

    Monaghan J. Implicit SPH drag and dusty gas dynamics [J]. Journal of Computational Physics, 1977, 138(2): 801–820.

    MATH  Google Scholar 

  9. [9]

    Colagrossi A., Landrini M. Numerical simulation of interfacial flows by smoothed particle hydrodynamics [J]. Journal of computational physics, 2003, 191(2): 448–475.

    MATH  Google Scholar 

  10. [10]

    Lind S., Stansby P., Rogers B. D. Incompressible-compressible flows with a transient discontinuous interface using smoothed particle hydrodynamics (SPH) [J]. Journal of Computational Physics, 2016, 309: 129–147.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    Hu X. Y., Adams N. A. A multi-phase SPH method for macroscopic and mesoscopic flows [J]. Journal of Computational Physics, 2006, 213(2): 844–861.

    MathSciNet  MATH  Google Scholar 

  12. [12]

    Fourtakas G., Rogers B. Modelling multi-phase liquidsediment scour and resuspension induced by rapid flows using smoothed particle hydrodynamics (SPH) accelerated with a graphics processing unit (GPU) [J]. Advances in Water Resources, 2016, 92: 186–199.

    Google Scholar 

  13. [13]

    Zubeldia E. H., Fourtakas G., Rogers B. D. et al. Multiphase SPH model for simulation of erosion and scouring by means of the shields and Drucker-Prager criteria [J]. Advances in Water Resources, 2018, 117: 98–114.

    Google Scholar 

  14. [14]

    Manenti S., Sibilla S., Gallati M. et al. SPH simulation of sediment flushing induced by a rapid water flow [J]. Journal of Hydraulic Engineering, ASCE, 2011, 138(3): 272–284.

    Google Scholar 

  15. [15]

    Kwon J., Monaghan J. J. A novel SPH method for sedimentation in a turbulent fluid [J]. Journal of Computational Physics, 2015, 300: 520–532.

    MathSciNet  MATH  Google Scholar 

  16. [16]

    Kwon J., Monaghan J. J. Sedimentation in homogeneous and inhomogeneous fluids using SPH [J]. International Journal of Multiphase Flow, 2015, 72: 155–164.

    MathSciNet  Google Scholar 

  17. [17]

    Xenakis A., Lind S., Stansby P. et al. Landslides and tsunamis predicted by incompressible smoothed particle hydrodynamics (SPH) with application to the 1958 Lituya Bay event and idealized experiment [J]. Proceedings of the Royal Society A, 2017, 473(2199): 20160674.

    MathSciNet  MATH  Google Scholar 

  18. [18]

    Pu J. H., Huang Y., Shao S. et al. Three-Gorges Dam fine sediment pollutant transport: Turbulence SPH model simulation of multi-fluid flows [J]. Journal of Applied Fluid Mechanics, 2016, 9(1): 1–10.

    Google Scholar 

  19. [19]

    Monaghan J., Kocharyan A. SPH simulation of multiphase flow [J]. Computer Physics Communications, 1995, 87(1–2): 225–235.

    MATH  Google Scholar 

  20. [20]

    Szewc K., Pozorski J., Minier J. P. Spurious interface fragmentation in multiphase SPH [J]. International Journal for Numerical Methods in Engineering, 2015, 103(9): 625–649.

    MathSciNet  MATH  Google Scholar 

  21. [21]

    Tartakovsky A. M., Ferris K. F., Meakin P. Lagrangian particle model for multiphase flows [J]. Computer Physics Communications, 2009, 180(10): 1874–1881.

    MathSciNet  MATH  Google Scholar 

  22. [22]

    Robinson M. Turbulence and viscous mixing using smoothed particle hydrodynamics [D]. Doctoral Thesis, Melbourne, Australia: Monash University, 2009.

    Google Scholar 

  23. [23]

    Canelas R., Ricardo A., Ferreira R. et al. Hunting for Lagrangian coherent structures: SPH-LES turbulence simulations with wall-adapting local eddy viscosity (WALE) model [C]. Proceedings of the 11th International Smoothed Particle Hydrodynamics European Research Interest Community (SPHERIC) workshop, Munich, Germany, 2016.

  24. [24]

    Tóth B., Szabó K. Flow structure detection with smoothed particle hydrodynamics [C]. 9th International SPHERIC Workshop, Paris, France, 2014.

  25. [25]

    Dauch T. F., Ates C., Keller M. C. et al. Analyzing primary breakup in fuel spray nozzles by means of Lagrangian-coherent structures [C]. Proceedings of the 14th SPHERIC International Workshop, Exeter, UK, 2019.

  26. [26]

    Crespo A. J., Domínguez J. M., Rogers B. D. et al. DualSPHysics: Open-source parallel CFD solver based on smoothed particle hydrodynamics (SPH) [J]. Computer Physics Communications, 2015, 187: 204–216.

    MATH  Google Scholar 

  27. [27]

    Ghia U., Ghia K. N., Shin C. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method [J]. Journal of Computational Physics, 1982, 48(3): 387–411.

    MATH  Google Scholar 

  28. [28]

    Yih C. S. Instability due to viscosity stratification [J]. Journal of Fluid Mechanics, 1967, 27: 337–352.

    MATH  Google Scholar 

  29. [29]

    Yiantsios S. G., Higgins B. G. Linear stability of plane Poiseuille flow of two superposed fluids [J]. Physics of Fluids, 1988, 31(11): 3225–3238.

    MathSciNet  MATH  Google Scholar 

  30. [30]

    Charles M., Lilleleht L. An experimental investigation of stability and interfacial waves in co-current flow of two liquids [J]. Journal of Fluid Mechanics, 1965, 22: 217–224.

    Google Scholar 

  31. [31]

    Kao T. W., Park C. Experimental investigations of the stability of channel flows. Part 2. Two-layered co-current flow in a rectangular channel [J]. Journal of Fluid Mechanics, 1972, 52: 401–423.

    Google Scholar 

  32. [32]

    Quinlan N. J., Basa M., Lastiwka M. Truncation error in mesh-free particle methods [J]. International Journal for Numerical Methods in Engineering, 2006, 66(13): 2064–2085.

    MathSciNet  MATH  Google Scholar 

  33. [33]

    Pinarbasi A., Liakopoulos A. The effect of variable viscosity on the interfacial stability of two-layer Poiseuille flow [J]. Physics of Fluids, 1995, 7(6): 1318–1324.

    MATH  Google Scholar 

  34. [34]

    Valluri P., Náraigh L. Ó., Ding H. et al. Linear and nonlinear spatio-temporal instability in laminar two-layer flows [J]. Journal of Fluid Mechanics, 2010, 656: 458–480.

    MATH  Google Scholar 

  35. [35]

    Hooper A., Boyd W. Shear-flow instability at the interface between two viscous fluids [J]. Journal of Fluid Mechanics, 1983, 128: 507–528.

    MathSciNet  MATH  Google Scholar 

  36. [36]

    Hooper A. P. The stability of two superposed viscous fluids in a channel [J]. Physics of Fluids A: Fluid Dynamics, 1989, 1(7): 1133–1142.

    MathSciNet  MATH  Google Scholar 

  37. [37]

    Anturkar N. R., Papanastasiou T. C., Wilkes J. O. Linear stability analysis of multilayer plane Poiseuille flow [J]. Physics of Fluids A: Fluid Dynamics, 1990, 2(4): 530–541.

    MATH  Google Scholar 

  38. [38]

    Gómez-Gesteira M., Rogers B. D., Dalrymple R. A. et al. State-of-the-art of classical SPH for free-surface flows [J]. Journal of Hydraulic Research, 2010, 48(Suppl.1): 6–27.

    Google Scholar 

  39. [39]

    Monaghan J. J. Simulating free surface flows with SPH [J]. Journal of Computational Physics, 1994, 110(2): 399–406.

    MATH  Google Scholar 

  40. [40]

    Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree [J]. Advances in Computational Mathematics, 1995, 4(1): 389–396.

    MathSciNet  MATH  Google Scholar 

  41. [41]

    Dehnen W., Aly H. Improving convergence in smoothed particle hydrodynamics simulations without pairing instability [J]. Monthly Notices of the Royal Astronomical Society, 2012, 425(2): 1068–1082.

    Google Scholar 

  42. [42]

    Crespo A., Gómez-Gesteira M., Dalrymple R. A. Boundary conditions generated by dynamic particles in SPH methods [J]. Computers, Materials and Continua (CMC), 2007, 5(3): 173–184.

    MathSciNet  MATH  Google Scholar 

  43. [43]

    English A., Domínguez J., Vacondio R. et al. Correction for dynamic boundary conditions [C]. 14th SPHERIC International Workshop, Exeter, UK, 2019.

  44. [44]

    Skillen A., Lind S., Stansby P. K. et al. Incompressible smoothed particle hydrodynamics (SPH) with reduced temporal noise and generalised Fickian smoothing applied to body-water slam and efficient wave-body interaction [J]. Computer Methods in Applied Mechanics and Engineering, 2013, 265: 163–173.

    MathSciNet  MATH  Google Scholar 

  45. [45]

    Molteni D., Colagrossi A. A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH [J]. Computer Physics Communications, 2009, 180(6): 861–872.

    MathSciNet  MATH  Google Scholar 

  46. [46]

    Lo E. Y., Shao S. Simulation of near-shore solitary wave mechanics by an incompressible SPH method [J]. Applied Ocean Research, 2002, 24(5): 275–286.

    Google Scholar 

  47. [47]

    Sun P., Colagrossi A., Marrone S. et al. Detection of Lagrangian coherent structures in the SPH framework [J]. Computer Methods in Applied Mechanics and Engineering, 2016, 305: 849–868.

    MathSciNet  MATH  Google Scholar 

  48. [48]

    Lind S. J., Xu R., Stansby P. K. Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves [J]. Journal of Computational Physics, 2012, 231(4): 1499–1523.

    MathSciNet  MATH  Google Scholar 

  49. [49]

    Rogers B. D. Test 3: 2-D lid-driven cavity flow without gravity-2D SPH validation, http://spheric-sph.org/tests/test-3, 2020.

  50. [50]

    Bird R. B., Stewart W. E., Lightfoot E. N. Transport phenomena [M]. New York, USA: John Wiley and Sons, 2007.

    Google Scholar 

  51. [51]

    Rowlatt C. F., Lind S. J. Bubble collapse near a fluid-fluid interface using the spectral element marker particle method with applications in bioengineering [J]. International Journal of Multiphase Flow, 2017, 90: 118–143.

    MathSciNet  Google Scholar 

  52. [52]

    Tang H., Wrobel L. Modelling the interfacial flow of two immiscible liquids in mixing processes [J]. International Journal of Engineering Science, 2005, 43(15–16): 1234–1256.

    Google Scholar 

  53. [53]

    Yang G. Du B., Fan L. Bubble formation and dynamics in gas-liquid-solid fluidization-A review [J]. Chemical Engineering Science, 2007, 62(1–2): 2–27.

    Google Scholar 

  54. [54]

    Tang H., Wrobel L., Fan Z. Tracking of immiscible interfaces in multiple-material mixing processes [J]. Computational Materials Science, 2004, 29(1): 103–118.

    Google Scholar 

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The work was supported by the EPSRC and National Nuclear Laboratory (Grant No. 1961431).

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Corresponding author

Correspondence to Benedict D. Rogers.

Additional information

Biography: Georgina Reece, Ph. D. Candidate

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Reece, G., Rogers, B.D., Lind, S. et al. New instability and mixing simulations using SPH and a novel mixing measure. J Hydrodyn 32, 684–698 (2020). https://doi.org/10.1007/s42241-020-0045-x

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Key words

  • Smoothed particle hydrodynamics (SPH)
  • multi-phase
  • mixing
  • instability