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A corrected WCSPH scheme with improved interface treatments for the viscous/viscoelastic two-phase flows

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Abstract

In this work, a robust corrected weakly compressible smoothed particle hydrodynamics scheme combined with several improved interface treatments (CWCSPH-IT) is proposed to investigate viscous or viscoelastic two-phase flows of high density ratios. The proposed viscoelastic two-phase CWCSPH-IT model is mainly derived from three aspects: (a) a combined elastic stress approximation in the momentum equation based on the weighted idea is proposed; (b) the transition region near the interface of two fluids is regarded as a weak polymer fluid by approximating the relaxation time with a Shepard interpolation which handles the discontinuity of elastic stress near the interface; and (c) a new mixed boundary treatment is proposed to treat the wall boundary in a viscoelastic flow. Moreover, a weighted color gradient scheme is adopted to handle the surface tension near the phase interface, and a particle shifting technique is used to regularize particle distributions and enhance numerical stability. Several two-phase benchmark problems are simulated to test the validity and reliability of the proposed CWCSPH-IT. Subsequently, the proposed particle scheme is adopted to simulate the droplet rising in viscous/viscoelastic matrix fluid with high density ratios. Further, the stable dimple shape of a polymer drop falling in a low-density matrix, and the complex overshooting, oscillation and breakup phenomena of a drop in a high-viscoelasticity shear fluid are numerically investigated. The comparison between the SPH and other reliable reference results demonstrates the ability of the present viscoelastic two-phase scheme in modeling complex viscous or viscoelastic multi-phase problems.

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References

  1. Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids, fluid dynamics, vol 1, 2nd edn. Wiley, New York

    Google Scholar 

  2. Chinyoka T (2002) Numerical simulation of stratified flows and droplet deformation in 2D shear flow of Newtonian and viscoelastic fluids. Ph.D. thesis, Virginia Polytechnic Institute and State University

  3. Giesekus H (1983) Stress behaviour in simple shear flow as predicted by a new constitutive model for polymer fluids. J Non-Newtonian Fluid Mech 12:367–374

    Article  Google Scholar 

  4. Shonibare OY (2017) Numerical simulation of viscoelastic multiphase flows using an improved two-phase flow solver. Ph.D. thesis, Michigan Technological University Press, Michigan, American

  5. Li Q, Ouyang J, Yang BX, Jiang T (2011) Modelling and simulation of moving interfaces in gas-assisted injection moulding process. Appl Math Model 35:257–275

    Article  MathSciNet  MATH  Google Scholar 

  6. Liu YJ, Liao TY, Joseph DD (1995) A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid. J Fluid Mech 304:321–342

    Article  Google Scholar 

  7. Gupta A, Vincenzi D (2019) Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence. J Fluid Mech 870:405–418

    Article  MathSciNet  MATH  Google Scholar 

  8. Aggarwal N, Sarkar K (2007) Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear. J Fluid Mech 584:1–21

    Article  MathSciNet  MATH  Google Scholar 

  9. Afkhami S, Yue P, Renardy Y (2009) A comparison of viscoelastic stress wakes for two-dimensional and three-dimensional Newtonian drop deformations in a viscoelastic matrix under shear. Phys Fluids 21:072106

    Article  MATH  Google Scholar 

  10. Figueiredo RA, Oishi CM, Afonso AM, Tasso IVM, Cuminato JA (2016) A two-phase solver for complex fluids: Studies of the Weissenberg effect. In J Multi Flow 84:98–115

    Article  MathSciNet  Google Scholar 

  11. Wang ZC, Dong SC, Triantafyllou MS, Constantinides Y, Karniadakis GE (2019) A stabilized phase-field method for two-phase flow at high Reynolds number and large density/viscosity ratio. J Comput Phys 397:108832

    Article  MathSciNet  MATH  Google Scholar 

  12. Wagner AJ, Giraud L, Scott CE (2000) Simulations of a cusped bubble rising in a viscoelastic fluid with a new numerical method. Comput Phys Commun 129:227–231

    Article  MATH  Google Scholar 

  13. Pillapakkam SB, Singh P (2001) A level-set method for computing solutions to viscoelastic two-phase flow. J Comput Phys 174:552–578

    Article  MATH  Google Scholar 

  14. Sostarecz M, Belmonte A (2003) Motion and shape of a viscoelastic drop falling through a viscous fluid. J Fluid Mech 497:235–252

    Article  MATH  Google Scholar 

  15. Yue PT, Feng JJ, Liu C, Shen J (2005) Viscoelastic effects on drop deformation in steady shear. J Fluid Mech 540:427–437

    Article  MATH  Google Scholar 

  16. Yue PT, Feng JJ, Liu C, Shen J (2005) Transient drop deformation upon startup of shear in viscoelastic fluids. Phys Fluid 17:123101

    Article  MATH  Google Scholar 

  17. Imaizumi Y, Kunugi T, Yokomine T, Kawara Z (2014) Viscoelastic fluid behaviors around a rising bubble via a new method of mesh deformation tracking. Chem Eng Sci 120:167–173

    Article  Google Scholar 

  18. You R, Borhan A, Haj-Hariri H (2008) A finite volume formulation for simulating drop motion in a viscoelastic two-phase system. J Non-Newtonian Fluid Mech 153:109–129

    Article  MATH  Google Scholar 

  19. Renardy Y (2008) Drop oscillations under simple shear in a highly viscoelastic matrix. Rheol Acta 47:89–96

    Article  Google Scholar 

  20. Lind SJ, Phillips TN (2010) The effects of viscoelasticity on a rising gas bubble. J Non-Newtonian Fluid Mech 165:852–865

    Article  MATH  Google Scholar 

  21. Mukherjee S, Sarkar K (2011) Viscoelastic drop falling through a viscous medium. Phys Fluids 23:013101

    Article  Google Scholar 

  22. Davoodi M, Norouzi M (2016) An investigation on the motion and deformation of viscoelastic drops descending in another viscoelastic media. Phys Fluids 28:103103

    Article  Google Scholar 

  23. Li J, Renardy YY, Renardy M (2000) Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys Fluids 12:269

    Article  MATH  Google Scholar 

  24. Hysing S, Turek S, Kuzmin D, Parolini N, Burman E, Ganesan S, Tobiska L (2009) Quantitative benchmark computations of two-dimensional bubble dynamics. Int J Numer Meth Fluids 60:1259–1288

    Article  MathSciNet  MATH  Google Scholar 

  25. Pillapakkam SB, Singh P, Blackmore D, Aubry N (2007) Transient and steady state of a rising bubble in a viscoelastic fluid. J Fluid Mech 589:215–252

    Article  MathSciNet  MATH  Google Scholar 

  26. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114:146–159

    Article  MATH  Google Scholar 

  27. Sussman M, Puckett EG (2000) A coupled level set and volume-of-fluid method for computing 3D and axisymmetric in compressible two-phase flows. J Comput Phys 162(2):301–337

    Article  MathSciNet  MATH  Google Scholar 

  28. Li Q (2016) Numerical simulation of melt filling process in complex mold cavity with insets using IB-CLSVOF method. Compt Fluids 132:94–105

    Article  MathSciNet  MATH  Google Scholar 

  29. Lorstad D, Fuchs L (2004) High-order surface tension VOF-model for 3D bubble flows with high density ratio. J Comput Phys 200:153–176

    Article  MATH  Google Scholar 

  30. Chinyoka T, Renardy YY, Renardy M, Khismatullin DB (2005) Two-dimensional study of drop deformation under simple shear for Oldroyd-B liquids. J Non-Newtonian Fluid Mech 130:45–56

    Article  MATH  Google Scholar 

  31. Oishi CM, Martins FP, Tome MF, Alves MA (2012) Numerical simulation of drop impact and jet buckling problems using the eXtended Pom-Pom model. J Non-Newtonian Fluid Mech 169–170:91–103

    Article  Google Scholar 

  32. Li S, Liu WK (2002) Mesh-free particle methods and their applications. Appl Mech Rev 54:1–34

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  34. Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Pub. Co. Inc., Singapore

    Book  MATH  Google Scholar 

  35. Liu MB, Shao JR, Chang JZ (2012) On the treatment of solid boundary in smoothed particle hydrodynamics. Sci China 55:244–254

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  37. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  39. 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 13:68–84

    Article  MATH  Google Scholar 

  40. Colagrossi A, Nikolov G, Durante D, Marrone S, Souto-lglesias A (2019) Viscous flow past a cylinder close to a free surface: benchmarks with steady, periodic and metastable responses, solved by meshfree and mesh-based schemes. Comput Fluids 181:345–363

    Article  MathSciNet  MATH  Google Scholar 

  41. Sun PN, Colagrossi A, Marrone S, Zhang AM (2017) The δplus-SPH model: simple procedures for a further improvement of the SPH scheme. Comput Meth Appl Mech Eng 315:25–49

    Article  MathSciNet  MATH  Google Scholar 

  42. 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  MathSciNet  MATH  Google Scholar 

  43. Basa M, Quinlan NJ, Lastiwka M (2009) Robustness and accuracy of SPH formulations for viscous flow. Int J Numer Meth Fluids 60:1127–1148

    Article  MathSciNet  MATH  Google Scholar 

  44. Cleary PW, Savage G, Ha J, Prakash M (2014) Flow analysis and validation of numerical modelling for a thin walled high pressure die casting using SPH. Comput Part Mech 1:229–243

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. 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  MathSciNet  MATH  Google Scholar 

  47. Adami S, Hu X, Adams N (2010) A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation. J Comput Phys 229:5011–5021

    Article  MATH  Google Scholar 

  48. Monaghan JJ, Rafiee A (2013) A simple SPH algorithm for multi-fluid flow with high density ratios. Int J Numer Meth Fluids 71:537–561

    Article  MathSciNet  MATH  Google Scholar 

  49. Chen Z, Zong Z, Liu MB, Zou L, Li HT, Shu C (2015) An SPH model for multiphase flows with complex interfaces and large density differences. J Comput Phys 283:169–188

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang AM, Sun PN, Ming FR (2015) An SPH modeling of bubble rising and coalescing in three dimensions. Comput Meth Appl Mech Eng 294:189–209

    Article  MathSciNet  MATH  Google Scholar 

  51. Douillet-Grellier T, Leclaire S, Vidal D, Bertran F, Vuyst FD (2019) Comparison of multiphase SPH and LBM approaches for the simulation of intermittent flows. Comput Part Mech 6:695–720

    Article  Google Scholar 

  52. Tartakovsky AM, Panchenko A (2016) Pairwise force smoothed particle hydrodynamics model for multiphase flow: surface tension and contact line dynamics. J Comput Phys 305:1119–1146

    Article  MathSciNet  MATH  Google Scholar 

  53. Krimi A, Rezoug M, Khelladi S, Nogueira X, Deligant M, Ramirez L (2018) Smoothed Particle Hydrodynamics: a consistent model for interfacial multiphase fluid flow simulations. J Comput Phys 358:53–87

    Article  MathSciNet  MATH  Google Scholar 

  54. Lin YX, Liu GR, Wang GY (2019) A particle-based free surface detection method and its application to the surface tension effects simulation in smoothed particle hydrodynamics (SPH). J Comput Phys 383:196–206

    Article  MathSciNet  MATH  Google Scholar 

  55. Fourtakas G, Rogers BD (2016) Modelling multi-phase liquid-sediment scour and resuspension induced by rapid flows using smoothed particle hydrodynamics (SPH) accelerated with a graphics processing unit (GPU). Adv Water Resour 92:186–199

    Article  Google Scholar 

  56. Zhang ZL, Walayat K, Chang JZ, Liu MB (2018) Meshfree modeling of a fluid-particle two-phase flow with an improved SPH method. Int J Numer Methods Eng 116:530–569

    Article  MathSciNet  Google Scholar 

  57. Ellero M, Tanner RI (2005) SPH simulations of transient viscoelastic flows at low Reynolds number. J Non-Newtonian Fluid Mech 132:61–72

    Article  MATH  Google Scholar 

  58. Jiang T, Lu LG, Lu WG (2014) The numerical investigation of spreading process of two viscoelastic droplets impact problem by using an improved SPH scheme. Comput Mech 53:977–999

    Article  MathSciNet  Google Scholar 

  59. Ren JL, Jiang T, Lu WG, Li G (2016) An improved parallel SPH approach to solve 3D transient generalized Newtonian free surface flows. Comput Phys Comm 205:87–105

    Article  MathSciNet  MATH  Google Scholar 

  60. Zainali A, Tofighi N, Shadloo MS, Yildiz M (2013) Numerical investigation of Newtonian and non-Newtonian multiphase flows using ISPH method. Comput Methods Appl Mech Eng 254:99–113

    Article  MathSciNet  MATH  Google Scholar 

  61. Vahabi M, Kamkari B (2019) Simulating gas bubble shape during its rise in a confined polymeric solution by WC-SPH. Eur J Mech B Fluids 75:1–14

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  63. Belytschko T, Krongauz Y, Dolbow J, Gerlach C (1998) On the completeness of meshfree particle methods. Int J Numer Methods Eng 43:785–819

    Article  MathSciNet  MATH  Google Scholar 

  64. Bonet J, Loc TSL (1999) Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations. Comput Methods Appl Mech Eng 180:97–115

    Article  MathSciNet  MATH  Google Scholar 

  65. Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved SPH method: Towards higher order convergence. J Comput Phys 225:1472–1492

    Article  MathSciNet  MATH  Google Scholar 

  66. Zhang GM, Batra RC (2009) Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems. Comput Mech 43:321–340

    Article  MathSciNet  MATH  Google Scholar 

  67. Jiang T, Tang YS, Ren JL (2014) A corrected 3D parallel SPH method for simulating the polymer free surface flows based on the XPP model. CMES Comput Model Eng Sci 101:249–297

    MathSciNet  MATH  Google Scholar 

  68. Swegle JW, Hicks DL, Attaway SW (1995) Smoothed particle hydrodynamics stability analysis. J Comput Phys 116:123–134

    Article  MathSciNet  MATH  Google Scholar 

  69. Belytschko T, Guo Y, Liu WK, Xiao SP (2000) A unified stability analysis of meshless particle methods. Int J Numer Methods Eng 48:1359–1400

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11501495, 51779215, 11672259), the Postdoctoral Science Foundation of China (Grant Nos. 2014M550310, 2015M581869, 2015T80589), the Natural Science Foundation of Jiangsu Province (Grant No. BK20150436), the Jiangsu Government Scholarship for Overseas Studies (Grant No. JS-2017-227) and the Top-notch Academic Programs Project of Jiangsu High Education Institutions (Grant No. PPZY2015B109).

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Authors and Affiliations

  1. School of Mathematical Sciences, Yangzhou University, Yangzhou, 225002, China

    Tao Jiang, Yue Li, Jin-Lian Ren & Qiang Li

  2. LHEEA Lab. (ECN and CNRS), Ecole Centrale Nantes, Nantes, France

    Peng-Nan Sun

  3. Departmento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba, Brazil

    Jin-Yun Yuan

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  1. Tao Jiang
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Correspondence to Tao Jiang.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, and there is no professional or other personal interest of any kind in any product that could be construed as influencing the position presented in the manuscript entitled “A corrected WCSPH scheme with improved interface treatments for the viscous/viscoelastic two-phase flows.” Thank you very much. Sincerely yours, Tao Jiang, Yue Li, Pengnan Sun, Jinlian Ren, Qiang Li, Jinyun Yuan.

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Jiang, T., Li, Y., Sun, PN. et al. A corrected WCSPH scheme with improved interface treatments for the viscous/viscoelastic two-phase flows. Comp. Part. Mech. 9, 633–653 (2022). https://doi.org/10.1007/s40571-021-00435-9

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