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Viscoelastic fluids widely exist in nature and industrial production, and the study of their complex rheological properties has important academic value and application significance. In this work, an improved smoothed particle hydrodynamics (SPH) method is proposed to numerically simulate the viscoelastic flow based on the eXtended Pom-Pom (XPP) model. In order to improve the accuracy of the calculation, a kernel gradient correction discrete format without kernel derivative calculation is adopted. In order to prevent fluid particles from penetrating the solid wall, an enhanced boundary processing technology is proposed. To eliminate the tensile instability, an artificial stress is coupled into the momentum equation of conservation. Based on the XPP model, the viscoelastic Poiseuille flow and the viscoelastic droplet impacting solid wall problem are simulated by using the improved SPH method. The effectiveness and advantages of the improved SPH method are verified by comparing the SPH solutions with the solutions from the analytical method or finite difference method. The convergence of the improved SPH method is further evaluated by using several different particle sizes. On this basis, the influences of rheological parameters such as Reyonlds number Re, Weissenberg number Wi, solvent viscosity ratio β, anisotropy parameter α, relaxation time ratio γ and molecular chain arm number Q on the flow process are analyzed in depth. For the viscoelastic Poiseuille flow, the bigger the value of Re, Wi, and α, the larger the steady-state velocity is; the larger the value of γ and Q, the smaller the steady-state velocity is; the larger the value of β, the weaker the velocity overshoot is, but it does not affect the steady-state velocity. For the viscoelastic droplet problem, the larger the value of Re and Wi, the faster the droplet spreads; the larger the value of β, the weaker the droplet shrinkage behavior is, but it does not affect the final spreading width of droplet; the larger the value of α, the larger the droplet’s spreading width is; the larger the value of γ is, the stronger the droplet shrinkage behavior is; the larger the value of Q, the weaker its influence on the droplet’s spread width is. The improved SPH method in this paper can effectively describe the complex rheological properties and the free surface variation characteristics of viscoelastic fluid based on XPP model.
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Keywords:
- smoothed particle hydrodynamics /
- eXtended Pom-Pom model /
- Poiseuille flow /
- droplet
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图 1 黏弹性Poiseuille流的几何区域
Figure 1. Geometric region of viscoelastic Poiseuille flow.
图 2 基于XPP模型的黏弹性Poiseuille流的SPH模拟 (Re = 2, Wi = 1, β = 0.1, α = 0.01, γ = 0.9, Q = 4) (a) 速度分布图; (b) 空间点1—3速度u随时间的变化; (c) 空间点1—3弹性剪切应力τxy随时间的变化
Figure 2. SPH simulation of viscoelastic Poiseuille flow based on XPP model (Re = 2, Wi = 1, β = 0.1, α = 0.01, γ = 0.9, Q = 4): (a) Velocity profile; (b) time change of velocity u at points 1 to 3; (c) time change of elastic shear stress τxy at points 1 to 3.
图 3 本文改进SPH数值解与传统SPH数值解和解析解的比较
Figure 3. Comparison of improved SPH solutions with original SPH solutions and analytical solutions.
图 4 利用传统SPH和改进SPH方法得到的u数值解与解析解的L-2范数误差随时间变化的比较
Figure 4. Comparison of the time change of L-2 norm error obtained based on original SPH solutions and improved SPH solutions.
图 5 利用不同粒子初始间距Δx得到的SPH数值解 (a) 空间点2处速度u随时间的变化; (b) 空间点1处弹性剪切应力τxy随时间的变化
Figure 5. SPH solutions obtained by different initial particle spacings Δx : (a) Time change of velocity u at point 2; (b) time change of elastic shear stress τxy at point 1.
图 6 不同流变参数下空间点2处速度u随时间的变化情况 (a) Re; (b) Wi; (c) β; (d) α; (e) γ; (f) Q
Figure 6. Time change of velocity u at point 2 under different rheological parameters: (a) Re; (b) Wi; (c) β; (d) α; (e) γ; (f) Q
图 7 不同流变参数下空间点1处弹性剪切应力τxy随时间的变化情况 (a) Re; (b) Wi; (c) β; (d) α; (e) γ; (f) Q
Figure 7. Time change of elastic shear stress τxy at point 1 under different rheological parameters: (a) Re; (b) Wi; (c) β; (d) α; (e) γ; (f) Q.
图 8 XPP黏弹性液滴产生张力不稳定性的SPH结果
Figure 8. SPH results of a XPP viscoelastic droplet with tensile instability.
图 9 液滴撞击固壁问题的计算模型
Figure 9. Computational model of droplet impacting solid wall problem.
图 10 基于XPP模型的液滴撞击固壁问题的SPH模拟(Re = 5, Wi = 1, β = 0.1, α = 0.01, γ = 0.9, Q = 4) (a) T = 1.55; (b) T = 1.85; (c) T = 2.35; (d) T = 2.85; (e) T = 3.45; (f) T = 5.00
Figure 10. SPH simulation of droplet impacting solid wall problem based on XPP model (Re = 5, Wi = 1, β = 0.1, α = 0.01, γ = 0.9, Q = 4): (a) T = 1.55; (b) T = 1.85; (c) T = 2.35; (d) T = 2.85; (e) T = 3.45; (f) T = 5.00.
图 12 不同流变参数下液滴铺展宽度d(T)/d0随时间的变化 (a) Re; (b) Wi; (c) β; (d) α; (e) γ; (f) Q
Figure 12. Time change of droplet spread width d(T)/d0 under different rheological parameters: (a) Re; (b) Wi; (c) β; (d) α; (e) γ; (f) Q
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[1] Viezel C, Tomé M F, Pinho F T, McKee S 2020 J. Non-newton Fluid Mech. 285 104338
Google Scholar
[2] Li B, Chen L, Joo S 2021 Case Stud. Therm. Eng. 26 101109
Google Scholar
[3] Li S, Liu W K 2007 Meshfree Particle Methods (Springer Science & Business Media) p68
[4] Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
Google Scholar
[5] Lucy L B 1977 Astron. J. 82 1013
Google Scholar
[6] 马理强, 刘谋斌, 常建忠, 苏铁熊, 刘汉涛 2012 物理学报 61 244701
Google Scholar
Ma L Q, Liu M B, Chang J Z, Su T X, Liu H T 2012 Acta Phys. Sin. 61 244701
Google Scholar
[7] 邵绪强, 梅鹏, 陈文新 2021 物理学报 70 234701
Google Scholar
Shao X Q, Mei P, Chen W X 2021 Acta Phys. Sin. 70 234701
Google Scholar
[8] Macià F, Merino-Alonso P E, Souto-Iglesias A 2022 Comput. Methods Appl. Mech. Eng. 397 115045
Google Scholar
[9] Xu X, Dey M, Qiu M, Feng J J 2020 Appl. Math. Model. 83 719
Google Scholar
[10] Liu M B, Zhang Z L, Feng D L 2017 Comput. Mech. 60 513
Google Scholar
[11] Zhang C, Rezavand M, Hu X 2021 J. Comput. Phys. 429 110028
Google Scholar
[12] Liu W K, Jun S, Zhang Y F 1995 Int. J. Numer. Methods Fluid 20 1081
Google Scholar
[13] Liu M B, Liu G R 2006 Appl. Numer. Math. 56 19
Google Scholar
[14] Fang J, Parriaux A, Rentschler M, Ancey C 2009 Appl. Numer. Math. 59 251
Google Scholar
[15] Yang X, Liu M, Peng S 2014 Comput. Fluids 92 199
Google Scholar
[16] Antuono M, Sun P N, Marrone S, Colagrossi A 2021 Comput. Fluids 216 104806
Google Scholar
[17] Lyu H G, Sun P N 2022 Appl. Math. Model 101 214
Google Scholar
[18] Monaghan J J, Kajtar J B 2009 Comput. Phys. Commun. 180 1811
Google Scholar
[19] Morris J P, Fox P J, Zhu Y 1997 J. Comput. Phys. 136 214
Google Scholar
[20] Liu M B, Shao J R, Chang J Z 2012 Sci. China Technol. Sci. 55 244
Google Scholar
[21] Fang J, Owens R G, Tacher L, Parriaux A 2006 J. Non-newton Fluid Mech. 139 68
Google Scholar
[22] Hashemi M R, Fatehi R, Manzari M T 2011 J. Non-newton Fluid Mech. 166 1239
Google Scholar
[23] Xu X, Deng X L 2016 Comput. Phys. Commun. 201 43
Google Scholar
[24] Ozgen O, Kallmann M, Brown E 2019 Comput. Animat. Virtual Worlds 30 e1870
Google Scholar
[25] Vahabi M, Kamkari B 2019 Eur. J. Mech. B. Fluids 75 1
[26] King J R C, Lind S J 2021 J. Non-newton Fluid Mech. 293 104556
Google Scholar
[27] Verbeeten W M H, Peters G W M, Baaijens F P T 2001 J. Rheol. 45 823
Google Scholar
[28] O'connor J, Domínguez J M, Rogers B D, Lind S J, Stansby P K 2022 Comput. Phys. Commun. 273 108263
Google Scholar
[29] Jiang T, Ouyang J, Ren J L, Yang B, Xu X 2012 Comput. Phys. Commun. 183 50
Google Scholar
[30] Xu X, Yu P 2018 Comput. Mech. 62 963
Google Scholar
[31] Monaghan J J 2000 J. Comput. Phys. 159 290
Google Scholar
[32] Gray J P, Monaghan J J, Swift R P 2001 Comput. Methods Appl. Mech. Eng. 190 6641
Google Scholar
[33] Waters N D, King M J 1970 Rheol. Acta 9 345
Google Scholar
[34] Oishi C M, Martins F P, Tomé M F, Alves M A 2012 J. Non-newton Fluid Mech. 169 91
Google Scholar
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