Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-14T17:31:11.411Z Has data issue: false hasContentIssue false
  • English
  • Français

Collision modelling for the interface-resolved simulation of spherical particles in viscous fluids

Published online by Cambridge University Press:  23 August 2012

Tobias Kempe  and
Jochen Fröhlich
Tobias Kempe*
Affiliation:
Institut für Strömungsmechanik, Technische Universitt Dresden, Dresden, 01062, Germany
Jochen Fröhlich
Affiliation:
Institut für Strömungsmechanik, Technische Universitt Dresden, Dresden, 01062, Germany
*
†Email address for correspondence: tobias.kempe@tu-dresden.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper presents a model for particle–particle and particle–wall collisions during interface-resolving numerical simulations of particle-laden flows. The accurate modelling of collisions in this framework is challenging due to methodological problems generated by interface approach and contact as well as due to the greatly different time scales involved. To cope with this situation, multiscale modelling approaches are introduced avoiding excessive local grid refinement during surface approach and time step reduction during the surface contact. A new adaptive model for the normal forces in the phase of ‘dry contact’ is proposed, stretching the collision process in time to match the time step of the fluid solver. This yields a physically sound and robust collision model with modified stiffness and damping determined by an optimization scheme. Furthermore, the model is supplemented with a new approach for modelling the tangential force during oblique collisions which is based on two material parameters: a critical impact angle separating rolling from sliding and the friction coefficient for the sliding motion. The resulting new model is termed the adaptive collision model (ACM). All proposed sub-models only contain physical parameters, and virtually no numerical parameters requiring adjustment or tuning. The new model is implemented in the framework of an immersed boundary method but is applicable with any spatial and temporal discretization. Detailed validation against experimental data was performed so that a general and versatile model for arbitrary collisions of spherical particles in viscous fluids is now available.

JFM classification

Mathematical Foundations: Computational methods Multiphase and Particle-laden Flows: Multiphase flow Multiphase and Particle-laden Flows: Particle/fluid flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 709 , 25 October 2012 , pp. 445 - 489
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Apostolou, K. & Hrymak, A. 2008 Discrete element simulation of liquid-particle flows. Comput. Chem. Engng 32, 841–856.CrossRefGoogle Scholar
2. Apte, S. V., Martin, M. & Patankar, N. A. 2009 A numerical method for fully resolved simulation (FRS) of rigid particle-flow interactions in complex flows. J. Comput. Phys. 228, 2712–2738.CrossRefGoogle Scholar
3. Ardekani, A., Dabiri, S. & Rangel, R. 2008 Collision of multi-particle and general shape objects in a viscous fluid. J. Comput. Phys. 227, 10094–10107.Google Scholar
4. Ardekani, A. & Rangel, R. H. 2008 Numerical investigation of particle–particle and particle–wall collisions in a viscous fluid. J. Fluid Mech. 596, 437–466.Google Scholar
5. Barnocky, G. & Davis, R. 1988 Elastohydrodynamic collision and rebound of spheres: Experimental verification. Phys. Fluids 31, 1324–1329.CrossRefGoogle Scholar
6. Becker, V. & Briesen, H. 2008 Tangential-force model for interactions between bonded colloidal particles. Phys. Rev. E 78, 061404.CrossRefGoogle ScholarPubMed
7. Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.Google Scholar
8. Chan-Braun, C., García-Villalba, M. & Uhlmann, M. 2010 Numerical simulation of fully resolved particles in rough-wall turbulent open channel flow. In 7th International Conference on Multiphase Flows, Tampa, Florida, USA, http://conferences.dce.ufl.edu/icmf2010/.Google Scholar
9. Cox, R. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface. Small gap widths, including inertial effects. Chem. Engng Sci. 22, 1753–1777.CrossRefGoogle Scholar
10. Crowe, C. 2006 Multiphase Flow Handbook. CRC.Google Scholar
11. Crowe, C., Sommerfeld, M. & Tsuji, Y. 1998 Multiphase Flows with Droplets and Particles. CRC.Google Scholar
12. Cundall, P. & Strack, O. 1979 A discrete numerical model for granular assemblies. Geotechnique 29, 47–65.CrossRefGoogle Scholar
13. Dahmen, W. & Reusken, A. 2008 Numerik für Ingenieure und Naturwissenschaftler. Springer.Google Scholar
14. Davis, R., Rager, D. & Good, B. 2002 Elastohydrodynamic rebound of spheres from coated surfaces. J. Fluid Mech. 468, 107–119.CrossRefGoogle Scholar
15. Davis, R., Serayssol, J. & Hinch, E. 1986 The elastohydrodynamic collision of two spheres. J. Fluid Mech. 163, 479–497.Google Scholar
16. Diaz-Goano, C., Minev, P. D. & Nandakumar, K. 2003 A fictitious domain/finite element method for particulate flows. J. Comput. Phys. 192, 105–123.CrossRefGoogle Scholar
17. Donahue, C., Hrenya, C. & Davis, R. 2010a Stokes’s cradle: Newton’s cradle with liquid coating. Phys. Rev. Lett. 105, 034501.CrossRefGoogle ScholarPubMed
18. Donahue, C., Hrenya, C., Davis, R., Nakagawa, K., Zelinskaya, A. & Joseph, G. 2010b Stokes’ cradle: normal three-body collisions between wetted particles. J. Fluid Mech. 650, 479.CrossRefGoogle Scholar
19. Donahue, C., Hrenya, C., Zelinskaya, A. & Nakagawa, K. J. 2008 Newton’s cradle undone: experiments and collision models for the normal collision of three solid spheres. Phys. Fluids 20, 113301.CrossRefGoogle Scholar
20. Eames, I. & Dalziel, S. 2000 Dust resuspension by the flow around an impacting sphere. J. Fluid Mech. 403, 305–328.CrossRefGoogle Scholar
21. Feng, Z. & Michaelides, E. 2005 Proteus: a direct forcing method in the simulations of particulate flows. J. Comput. Phys. 202, 20–51.CrossRefGoogle Scholar
22. Feng, Z., Michaelides, E. & Mao, S. 2010 A three-dimensional resolved discrete particle method for studying particle–wall collision in a viscous fluid. Trans. ASME: J. Fluids Engng 132, 091302.Google Scholar
23. Foerster, S., Louge, M., Chang, H. & Allia, K. 1994 Measurements of the collision properties of small spheres. Phys. Fluids 6, 1108–1115.Google Scholar
24. Gay, D. M. 1977 Some convergence properties of Broyden’s method. Working Paper 175, National Bureau of Economic Research.CrossRefGoogle Scholar
25. Glowinski, R., Pan, T.-W., Hesla, T. & Joseph, D. 1999 A distributed Lagrange multiplier/fictitious domain method for particulate flows. Intl J. Multiphase Flow 25, 755–794.CrossRefGoogle Scholar
26. Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D. & Priaux, J. 2001 A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169, 363–426.Google Scholar
27. Goldman, A., Cox, R. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall – Part I: motion through a quiescent fluid. Chem. Engng Sci. 22, 637–651.CrossRefGoogle Scholar
28. Gondret, P., Hallouin, E., Lance, M. & Petit, L. 1999 Experiments on the motion of a solid sphere toward a wall: from viscous dissipation to elastohydrodynamic bouncing. Phys. Fluids 11, 2803–2805.CrossRefGoogle Scholar
29. Gondret, P., Lance, M. & Petit, L. 2002 Bouncing motion of spherical particles in fluids. Phys. Fluids 14, 643–652.CrossRefGoogle Scholar
30. Haff, P. & Werner, B. 1986 Computer simulation of the mechanical sorting of grains. Powder Technol. 48, 239–245.CrossRefGoogle Scholar
31. Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182–2189.CrossRefGoogle Scholar
32. Hertz, H. 1882 Über die Berührung fester elastischer Körper. J. Reine Angew. Math. 92, 156–171.Google Scholar
33. Hoomans, B. P. B., Kuipers, J. A. M., Briels, W. J. & van Swaaij, W. P. M. 1996 Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: a hard-sphere approach. Chem. Engng Sci. 51, 99–108.CrossRefGoogle Scholar
34. Hu, H. H. 1996 Direct simulation of flows of solid–liquid mixtures. Intl J. Multiphase Flow 22, 335–352.CrossRefGoogle Scholar
35. Hunter, S. 1957 Energy absorbed by elastic waves during impact. J. Mech. Phys. Solids 5, 162–171.CrossRefGoogle Scholar
36. Iaccarino, G. & Verzicco, R. 2003 Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56, 331–347.CrossRefGoogle Scholar
37. Joseph, G. 2003 Collisional dynamics of macroscopic particles in a viscous fluid. PhD thesis, California Institute of Technology, Pasadena, California.Google Scholar
38. Joseph, G. & Hunt, M. 2004 Oblique particle wall collisions in a liquid. J. Fluid Mech. 510, 71–93.CrossRefGoogle Scholar
39. Joseph, G., Zenit, R., Hunt, M. & Rosenwinkel, A. 2001 Particle wall collisions in a viscous fluid. J. Fluid Mech. 433, 329–346.Google Scholar
40. Kempe, T. & Fröhlich, J. 2012 An improved immersed boundary method with direct forcing for the simulation of particle laden flows. J. Comput. Phys. 231, 3663–3684.CrossRefGoogle Scholar
41. Kharaz, A. H., Gorham, D. A. & Salman, A. D. 1999 Accurate measurement of particle impact parameters. Meas. Sci. Technol. 10, 31–35.CrossRefGoogle Scholar
42. Kharaz, A. H., Gorham, D. A. & Salman, A. D. 2001 An experimental study of the elastic rebound of spheres. Powder Technol. 120, 281–291.Google Scholar
43. Kruggel-Emden, H., Simsek, E., Rickelt, S., Wirtz, S. & Scherer, V. 2007 Review and extension of normal force models for the discrete element method. Powder Technol. 171, 157–173.CrossRefGoogle Scholar
44. Kruggel-Emden, H., Wirtz, S. & Scherer, V. 2008 A study on tangential force laws applicable to the discrete element method (DEM) for materials with viscoelastic or plastic behaviour. Chem. Engng Sci. 63, 1523–1541.CrossRefGoogle Scholar
45. Ladd, A. 1997 Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluids 9, 491–499.Google Scholar
46. Lain, S., Sommerfeld, M. & Kussin, J. 2002 Experimental studies and modelling of four-way coupling in particle-laden horizontal channel flow. Intl J. Heat Fluid Flow 23, 647–656.Google Scholar
47. Legendre, D., Zenit, R., Daniel, C. & Guiraud, P. 2006 A note on the modelling of the bouncing of spherical drops or solid spheres on a wall in viscous fluid. Chem. Engng Sci. 61, 3543–3549.Google Scholar
48. Leopardi, P. 2006 A partition of the unit sphere into regions of equal area and small diameter. Electron. Trans. Numer. Anal. 25, 309–327.Google Scholar
49. Leweke, T., Thompson, M. & Hourigan, K. 2004 Vortex dynamics associated with the collision of a sphere with a wall. Phys. Fluids 16, L74–L77.CrossRefGoogle Scholar
50. Leweke, T., Thompson, M. & Hourigan, K. 2006 Instability of the flow around an impacting sphere. J. Fluids Struct. 22, 961–971.CrossRefGoogle Scholar
51. Lorenz, A., Tuozzolo, C. & Louge, M. Y. 1997 Measurements of impact properties of small, nearly spherical particles. Exp. Mech. 37, 292–298.CrossRefGoogle Scholar
52. Lun, C. & Savage, S. 1987 A simple kinetic theory for granular flow of rough, inelastic, spherical particles. J. Appl. Mech. 54, 47–53.CrossRefGoogle Scholar
53. Maw, N., Barber, J. & Fawcett, J. 1976 The oblique impact of elastic spheres. Wear 38, 101–114.Google Scholar
54. Maw, N., Barber, J. & Fawcett, J. 1977 The rebound of elastic bodies in oblique contact. Mech. Res. Comm. 4, 17–22.CrossRefGoogle Scholar
55. McLaughlin, M. 1968 An experimental study of particle–wall collision relating to flow of solid particles in fluid. Master’s thesis, California Institute of Technology, Pasadena, California.Google Scholar
56. Mindlin, R. 1949 Compliance of elastic bodies in contact. J. Appl. Mech. 16, 259–268.CrossRefGoogle Scholar
57. Mindlin, R. & Deresiewicz, H. 1953 Elastic spheres in contact under varying oblique forces. J. Appl. Mech. 20, 327–344.CrossRefGoogle Scholar
58. Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261.CrossRefGoogle Scholar
59. Mohd-Yusof, J. 1997 Combined immersed boundary/B-Spline method for simulations of flows in complex geometries. Center for Turbulence Research. Annual Research Briefs. NASA Ames/Stanford University, pp. 317–327.Google Scholar
60. Nguyen, N. & Ladd, A. 2002 Lubrication corrections for lattice-Boltzmann simulations of particle suspensions. Phys. Rev. E 66, 046708.Google Scholar
61. Pan, T. & Glowinski, R. 2002 Direct simulation of the motion of neutrally buoyant circular cylinders in plane poiseuille flow. J. Comput. Phys. 181, 260–279.Google Scholar
62. Papista, E., Dimitrakis, D. & Yiantsios, S. G. 2011 Direct numerical simulation of incipient sediment motion and hydraulic conveying. Ind. Engng Chem. Res. 50, 630–638.Google Scholar
63. Peskin, C. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252.Google Scholar
64. Pianet, G., Ten Cate, A., Derksen, J. & Arquis, E. 2007 Assessment of the 1-fluid method for DNS of particulate flows: sedimentation of a single sphere at moderate to high Reynolds numbers. Comput. Fluids 36, 359–375.Google Scholar
65. Popov, V. & Psakhie, S. 2007 Numerical simulation methods in tribology. Tribol. Intl 40, 916–923.CrossRefGoogle Scholar
66. Pöschel, T. & Schwager, T. 2005 Computational Granular Dynamics: Models and Algorithms. Springer.Google Scholar
67. Prokunin, A. & Williams, M. 1996 Spherical particle sedimentation along an inclined plane at high Reynolds numbers. Fluid Dyn. 31, 567–572.Google Scholar
68. Prokunin, A. N. 1998 Spherical particle sedimentation along an inclined plane at small Reynolds numbers. Fluid Dyn. 23, 573–579.Google Scholar
69. Prosperetti, A. & Tryggvason, G. 2007 Computational Methods for Multiphase Flow. Cambridge University Press.Google Scholar
70. Reed, J. 1985 Energy losses due to elastic wave propagation during an elastic impact. J. Phys. D: Appl. Phys. 18, 2329–2337.CrossRefGoogle Scholar
71. Roma, A. M., Peskin, C. S. & Berger, M. J. 1999 An adaptive version of the immersed boundary method. J. Comput. Phys. 153, 509–534.Google Scholar
72. Schiller, L. & Naumann, Z. 1933 Über die grundlegenden berechnungen bei der schwerkraftaufbereitung. Z. Ver. Dtsch. Ing. 318–338.Google Scholar
73. Schwarzer, S. 1995 Sedimentation and flow through porous media: simulating dynamically coupled discrete and continuum phases. Phys. Rev. E 52, 6461–6475.Google Scholar
74. Stevens, A. & Hrenya, C. 2005 Comparison of soft-sphere models to measurements of collision properties during normal impacts. Powder Technol. 154, 99–109.CrossRefGoogle Scholar
75. Sundaram, S. & Collins, L. 1999 A numerical study of the modulation of isotropic turbulence by suspended particles. J. Fluid Mech. 379, 105–143.Google Scholar
76. Ten Cate, A., Derksen, J. J., Portela, L. M. & van den Akker, H. E. A. 2004 Fully resolved simulations of colliding monodisperse spheres in forced isotropic turbulence. J. Fluid Mech. 519, 233–271.Google Scholar
77. Ten Cate, A., Nieuwstad, C., Derksen, J. & den Akker, H. V. 2002 Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity. Phys. Fluids 14, 4012–4025.Google Scholar
78. Thompson, M., Leweke, T. & Hourigan, K. 2007 Sphere-wall collisions: vortex dynamics and stability. J. Fluid Mech. 575, 121–148.Google Scholar
79. Timoshenko, S. & Goodier, J. 1970 Theory of Elasticity. McGraw-Hill.Google Scholar
80. Tryggvason, G., Thomas, S., Lu, J., Aboulhasanzadeh, B. & Tsengue, V. 2010 Multiscale computation of multiphase flows. In 7th International Conference on Multiphase Flows, Tampa, Florida, USA.CrossRefGoogle Scholar
81. Uhlmann, M. 2005 An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209, 448–476.CrossRefGoogle Scholar
82. Uhlmann, M. 2008 Interface-resolved direct numerical simulation of vertical particulate channel flow in the turbulent regime. Phys. Fluids 20, 053305.Google Scholar
83. Uzgoren, E., Singh, R., Sim, J. & Shyy, W. 2007 Computational modelling for multiphase flows with spacecraft application. Prog. Aerosp. Sci. 43, 138–192.Google Scholar
84. Vanella, M. & Balaras, E. 2009 A moving-least-squares reconstruction for embedded-boundary formulations. J. Comput. Phys. 228, 6617–6628.Google Scholar
85. Veeramani, C., Minev, P. & Nandakumar, K. 2009 Collision modelling between two non-Brownian particles in multiphase flow. Intl J. Therm. Sci. 48, 226–233.CrossRefGoogle Scholar
86. Vowinckel, B., Kempe, T. & Fröhlich, 2011 Impact of collision models on particle transport in open channel flow. In 7th International Symposium on Turbulence and Shear Flow Phenomena. Ottawa, Canada.Google Scholar
87. Vu-Quoc, L., Lesburg, L. & Zhang, X. 2004 An accurate tangential force–displacement model for granular-flow simulations: contacting spheres with plastic deformation, force-driven formulation. J. Comput. Phys. 196, 298–326.Google Scholar
88. Walton, O. & Braun, R. 1986 Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks. J. Rheol. 30, 949–980.CrossRefGoogle Scholar
89. Walton, O. R. 1993 Particulate two-phase flow. In Numerical Simulation of Inelastic, Frictional Particle–particle Interactions, pp. 884–911. Butterworth-Heinemann.Google Scholar
90. Wan, D. & Turek, S. 2006 Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method. Intl J. Numer. Meth. Fluids 51, 531–566.Google Scholar
91. Xu, B. & Yu, A. 1997 Numerical simulation of the gas–solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics. Chem. Engng Sci. 52, 2785–2809.Google Scholar
92. Yang, C., York, D. & Broeckx, W. 2008 Numerical simulation of sedimentation of microparticles using the discrete particle method. Particuology 6, 38–49.Google Scholar
93. Yang, F. & Hunt, M. 2006 Dynamics of particle–particle collisions in a viscous liquid. Phys. Fluids 18, 121506.Google Scholar
94. Yang, L., Seddon, J., Mullin, T., Del Pino, C. & Ashmore, J. 2006 The motion of a rough particle in a Stokes flow adjacent to a boundary. J. Fluid Mech. 557, 337–346.Google Scholar
95. Zenit, R. & Hunt, M. 1999 Mechanics of immersed particle collisions. Trans. ASME: J. Fluids Engng 121, 179–184.Google Scholar
96. Zhang, J., Fan, L., Zhu, C., Pfeffer, R. & Qi, D. 1999 Dynamic behaviour of collision of elastic spheres in viscous fluids. Powder Technol. 106, 98–109.Google Scholar