Abstract
In this paper, the spreading process of two XPP model droplets impacting on a plate in sequence at low Reynolds number is numerically simulated by using an improved smoothed particle hydrodynamics (I-SPH) method. The I-SPH method is a coupled approach which uses the traditional SPH (TSPH) method near the boundary domain and uses a kernel-gradient-corrected SPH method in the interior of fluid flow for the reason of remedying the accuracy and stability of TSPH. Meanwhile, an artificial stress term and a periodic density re-initialization technique are presented to eliminate the tensile instability and restrain pressure oscillation, respectively. A new boundary treatment is also adopted. The ability and merit of proposed I-SPH method combined with other techniques are first illustrated by simulating three typical examples. Subsequently, the deformation phenomena of two viscoelastic droplets impacting and spreading on a plate in sequence are numerically investigated. Particularly, the influences of the falling time interval, Weissenberg number and other rheological parameters on the deformation process are studied respectively. All numerical results agree well with the available data.
Similar content being viewed by others
References
Tomé MF, Castelo A, Ferreira VG, McKee S, Walters K (2007) Die-swell, splashing drop and a numerical technique for solving the Oldroyd-B model for axisymmetric surface flows. J Non-Newtonian Flu Mech 141:148–166
Lunkad Siddhartha F, Buwa Vivek V, Nigam KDP (2007) Numerical simulations of drop impact and spreading on horizontal and inclined surfaces. Chem Eng Sci 62:7214–7224
Tomé MF, Mangiavacchi N, Castelo A, Cuminato JA, McKee S (2002) A finite difference technique for simulating unsteady viscoelastic free surface flows. J Non-Newtonian Fluid Mech 106:61–106
Jiang T, Ouyang J, Yang B, Ren J (2010) The SPH method for simulating a viscoelastic drop impact and spreading on an inclined plate. Comput Mech 45:573–583
Vebeeten WMH, Peters GWM, Baaijens FPT (2001) Differential constitutive equations for polymer melts: The extended Pom-Pom model. J Rheol 45:823–843
McKee S, Tomé MF, Ferreira VG, Cuminato JA, Castelo A, Sousa FS, Mangiavacchi N (2008) The MAC method. Comput Fluids 37:907–930
Hirt CW, Nicholls BD (1981) Volume of fluid (VOF) method for dynamics of free boundaries. J Comput Phys 39:201–221
Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49
Li Q, Ouyang J, Yang B, Jiang T (2011) Modelling and simulation of moving interfaces in gas-assisted injection moulding process. Appl Math Model 35:257–275
Li S, Qian D, Liu WK, Belytschko T (2000) A mesh-free contact-detection algorithm. Comput Methods Appl Mech Eng 190:3271–3292
Li S, Liu WK (2002) Mesh-free particle methods and their applications. Appl Mech Rev 54:1–34
Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a mesh-free particle method. World Scientific, Singapore
Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17:25–76
Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110:399–406
Flebbe O, Munzel S, Herold H et al (1994) Smoothed particle hydrodynamics-physical viscosity and the simulation of accretion disks. Astrophys J 431:754–760
Morris JP, Fox PJ, Zhu Y (1997) Modeling low Reynolds number incompressible flows using SPH. J Comput Phys 136:214–226
Cleary PW, Monaghan JJ (1999) Conduction modeling using smoothed particle hydrodynamics. J Comput Phys 148:227–264
Cummins SJ, Rudman M (1999) An SPH projection method. J Comput Phys 152:584–607
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
Monaghan JJ, Kocharyan A (1995) SPH simulation of multi-phase flow. Comput Phys Commun 87:225–235
Colagrossi A, Landrini M (2003) Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J Comput Phys 191:448–475
Hu X, Adams N (2006) A multi-phase SPH method for macroscopic and mesoscopic flows. J Comput Phys 213:844–861
Monaghan JJ (2002) SPH compressible turbulence. Mon Not R Astron Soc 335:843–852
Ellero M, Kröger M, Hess S (2002) Viscoelastic flows studied by smoothed particle dynamics. J Non-Newtonian Fluid Mech 105:35–51
Ellero M, Tanner RI (2005) SPH simulations of transient viscoelastic flows at low Reynolds number. J Non-Newtonian Fluid Mech 132:61–72
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
Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106
Li SF, Liu WK (1999) Reproducing kernel hierarchical partition of unity. part I-formulation and theory. Int J Numer Methods Eng 45:251–288
Li SF, Liu WK (1999) Reproducing kernel hierarchical partition of unity, part II-applications. Int J Numer Methods Eng 45:251– 288
Chen JK, Beraun JE (2000) A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems. Comput Methods Appl Mech Eng 190:225–239
Liu MB, Liu GR (2006) Restoring particle consistency in smoothed particle hydrodynamics. Appl Numer Math 56:19–36
Zhang GM, Batra RC (2004) Modified smoothed particle hydrodynamics method and its application to transient problems. Comput Mech 34:137–146
Batra RC, Zhang GM (2007) Search algorithm and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method. Comput Mech 40:531– 546
Zhang GM, Batra RC (2009) Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems. Comput Mech 43:321–340
Batra RC, Zhang GM (2008) SSPH basis functions for meshless methods, and comparison of solution of solutions with strong and weak formulations. Comput Mech 41:527–545
Liu MB, Xie WP, Liu GR (2005) Modeling incompressible flows using a finite particle method. Appl Math Model 29:1252–1270
Tao Jiang, Jie Ouyang, Jinlian Ren (2012) A mixed corrected symmetric SPH (MC-SSPH) method for computational dynamic problems. Comput Phys Commun 183:50–62
Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved SPH method: towards higher order convergence. J Comput Phys 225:1472–1492
Fetehi R, Manzari MT (2012) A consistent and fast weakly compressible smoothed particle hydrodynamics with a new wall boundary condition. Int J Num Methods Fluids 68:905–921
Ren JL, Ouyang J, Jiang T, Li Q (2011) Simulation of complex filling process based on the generalized Newtonian model using a corrected SPH scheme. Comput Mech 49:643–665
Wang W, Li XK, Han XH (2009) Equal low-order finite element simulation of the planar contraction flow for branched polymer melts. Polym Plast Technol Eng 48:1158–1170
Oishi CM, Martins FP, Tomé 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
Wendland H (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4:389–396
Fang J, Parriaux A, Rentschler M, Ancey C (2009) Improved SPH methods for simulating free surface flows of viscous fluids. Appl Numer Math 59:251–271
Monaghan JJ (2005) Smoothed particle hydrodynamics. Rep Prog Phys 68:1703–1759
Swegle JW, Hicks DL, Attaway SW (1995) Smoothed particle hydrodynamics stability analysis. J Comput Phys 116:123–134
Monaghan JJ (2000) SPH without a tensile instability. J Comput Phys 159:290–311
Gray JP, Monaghan JJ, Swift RP (2001) SPH elastic dynamics. Comput Methods Appl Mech Eng 190:6641–6662
Yildiz M, Rook RA, Suleman A (2009) SPH with the multiple boundary tangent method. Int J Numer Methods Eng 77:1416–1438
Xu R, Stansby P, Laurence D (2009) Accuracy and stability incompressible SPH (ISPH) based on the projection method and a new approach. J Comput Phys 228:6703–6725
Fujimoto H, Ito S, Takezaki I (2002) Experimental study of successive collision of two water droplets with a solid. Exp Fluids 33:500–502
Acknowledgments
The authors acknowledge the support from the Natural Science Fundamental Research Project of Jiangsu Colleges and Universities of China (No. 12KJD570001), and the Jiangsu Natural Science Foundation for Youths of China (No. BK20130436).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, T., Lu, LG. & Lu, WG. The numerical investigation of spreading process of two viscoelastic droplets impact problem by using an improved SPH scheme. Comput Mech 53, 977–999 (2014). https://doi.org/10.1007/s00466-013-0943-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-013-0943-7