Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T10:50:00.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Simulations of Particle Sedimentation Using the Immersed Boundary Method

Part of: Two-phase and multiphase flows Basic methods in fluid mechanics Coupling of solid mechanics with other effects Incompressible viscous fluids

Published online by Cambridge University Press:  30 July 2015

Sudeshna Ghosh  and
John M. Stockie
Sudeshna Ghosh
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada
John M. Stockie*
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada
*
*Corresponding author. Email addresses: sud1800@yahoo.co.in (S. Ghosh), jstockie@sfu.ca (J. M. Stockie)
Get access

Abstract

We study the settling of solid particles in a viscous incompressible fluid contained within a two-dimensional channel, where the mass density of the particles is greater than that of the fluid. The fluid-structure interaction problem is simulated numerically using the immersed boundary method, where the added mass is incorporated using a Boussinesq approximation. Simulations are performed with a single circular particle, and also with two particles in various initial configurations. The terminal particle settling velocity and drag coefficient correspond closely with other theoretical, experimental and numerical results, and the particle trajectories reproduce the expected behavior qualitatively. In particular, simulations of a pair of interacting particles similar drafting-kissing-tumbling dynamics to that observed in other experimental and numerical studies.

Keywords

Immersed boundary methodparticle suspensionsedimentationsettling velocityfluid-structure interaction

MSC classification

Secondary: 45K05: Integro-partial differential equations 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76D05: Navier-Stokes equations 76M20: Finite difference methods 76T20: Suspensions
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 380 - 416
Copyright
Copyright © Global-Science Press 2015 

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]Aidun, C. K. and Ding, E.-J.. Dynamics of particle sedimentation in a vertical channel: Period-doubling bifurcation and chaotic state. Phys. Fluids, 15(6):1612–1621, 2003.Google Scholar
[2]Alpkvist, E. and Klapper, I.. Description of mechanical response including detachment using a novel particle model of biofilm/flow interaction. Water Sci. Tech., 55:265–273, 2007.CrossRefGoogle ScholarPubMed
[3]Batchelor, G. K.. An Introduction to Fluid Dynamics. Cambridge University Press, 1967.Google Scholar
[4]Richou, A. Ben, Ambari, A., Lebey, M., and Naciri, J. K.. Drag force on a circular cylinder midway between two parallel plates at Re ≪ 1. Part 2: moving uniformly (numerical and experimental). Chem. Eng. Sci., 60(10):2535–2543, 2005.Google Scholar
[5]Bhalla, A. P. S., Bale, R., Griffith, B. E., and Patankar, N. A.. A unified mathematical framework and an adaptive numerical method for fluid-structure interaction with rigid, deforming, and elastic bodies. J. Comput. Phys., 250:446–476, 2013.Google Scholar
[6]Boffi, D., Gastaldi, L., and Heltai, L.. On the CFL condition for the finite element immersed boundary method. Comput. Struct., 85:775–783, 2007.CrossRefGoogle Scholar
[7]Brenner, H.. Hydrodynamic resistance of particles at small Reynolds number. Adv. Chem. Eng., 6:287–438, 1966.CrossRefGoogle Scholar
[8]Breugem, W.-P.. A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows. J. Comput. Phys., 231(13):4469–4498, 2012.Google Scholar
[9]Bringley, T. T.. Analysis of the Immersed Boundary Method for Stokes Flow. PhD thesis, Department of Mathematics, New York University, May 2008.Google Scholar
[10]Bürger, R. and Wendland, W. L.. Sedimentation and suspension flows: Historical perspective and some recent developments. J. Eng. Math., 41:101–116, 2001.Google Scholar
[11]Champmartin, S. and Ambari, A.. Kinematics of a symmetrically confined cylindrical particle in a “Stokes-type” regime. Phys. Fluids, 19:073303, 2007.Google Scholar
[12]Cheng, Y., Zhu, L., and Zhang, C.. Numerical study of stability and accuracy of the immersed boundary method coupled to the lattice Boltzmann BGK model. Commun. Comput. Phys., 16(1):136–168, 2014.Google Scholar
[13]Chhabra, R. P., Agarwal, S., and Chaudhary, K.. A note on wall effect on the terminal falling velocity of a sphere in quiescent Newtonian media in cylindrical tubes. Powder Tech., 129:53–58, 2003.CrossRefGoogle Scholar
[14]Davis, R. H. and Acrivos, A.. Sedimentation of noncolloidal particles at low Reynolds numbers. Annu. Rev. Fluid Mech., 17:91–118, 1985.Google Scholar
[15]Felice, R. Di. The sedimentation velocity of dilute suspensions of nearly monosized spheres. Int. J. Multiphase Flow, 25:559–574, 1999.Google Scholar
[16]Dupuis, A., Chatelain, P., and Koumoutsakos, P.. An immersed boundary-lattice-Boltzmann method for the simulation of the flow past an impulsively started cylinder. J. Comput. Phys., 227(9):4486–4498, 2008.CrossRefGoogle Scholar
[17]Faxén, O. H.. Forces exerted on a rigid cylinder in a viscous fluid between two parallel fixed planes. Proc. Roy. Swed. Acad. Sci., 187:1–13, 1946.Google Scholar
[18]Feng, J., Hu, H. H., and Joseph, D. D.. Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech., 261:95–134, 1994.CrossRefGoogle Scholar
[19]Fortes, A. F., Joseph, D. D., and Lundgren, T. S.. Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech., 177:467–483, 1987.CrossRefGoogle Scholar
[20]Ghosh, S.. The immersed boundary method for simulating gravitational settling and fluid shear-induced deformation of elastic structures. PhD thesis, Department of Mathematics, Simon Fraser University, Burnaby, Canada, Mar. 2013. Available at http://summit.sfu.ca/item/12775.Google Scholar
[21]Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D., and Périaux, J.. 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(2):363–426, 2001.Google Scholar
[22]Griffith, B. E., Hornung, R. D., McQueen, D. M., and Peskin, C. S.. An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys., 223(1):10–49, 2007.CrossRefGoogle Scholar
[23]Haeri, S. and Shrimpton, J. S.. On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows. Int. J. Multiphase Flow, 40:38–55, 2012.Google Scholar
[24]Happel, J. and Brenner, H.. Low Reynolds number hydrodynamics, with special applications to particulate media. Mechanics of fluids and transport processes. M. Nijhoff Publishers, 1983.Google Scholar
[25]Hernández-Ortiz, J. P.. Boundary Integral Equations for Viscous Flows — non-Newtonian Behavior and Solid Inclusions. PhD thesis, Department of Mechanical Engineering, University of Wisconsin-Madison, 2004.Google Scholar
[26]Hopkins, M. M. and Fauci, L. J.. A computational model of the collective fluid dynamics of motile microorganisms. J. Fluid Mech., 455:149–174, 2002.Google Scholar
[27]Hu, H. H., Joseph, D. D., and Fortes, A. F.. Experiments and direct simulation of fluid particle motions. Int. Vid. J. Eng. Res., 2:17–24, 1997.Google Scholar
[28]Jayaweera, K. O. L. F. and Mason, B. J.. The behaviour of freely falling cylinders and cones in a viscous fluid. J. Fluid Mech., 22(4):709–720, 1965.Google Scholar
[29]Joseph, D. D., Fortes, A., Lundgren, T. S., and Singh, P.. Nonlinear mechanics of fluidization of spheres, cylinders and disks in water. In Advances in Multiphase Flow and Related Problems, pages 101–122. SIAM, Philadelphia, PA, 1987.Google Scholar
[30]Kim, Y. and Peskin, C. S.. Penalty immersed boundary method for an elastic boundary with mass. Phys. Fluids, 19(5):053103, 2007.CrossRefGoogle Scholar
[31]Ladd, A. J. C.. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part II. Numerical results. J. Fluid Mech., 271:311–339, 1994.Google Scholar
[32]Lai, M.-C.. Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method. PhD thesis, New York University, Sept. 1998.Google Scholar
[33]Lai, M.-C. and Peskin, C. S.. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys., 160(2):705–719, 2000.Google Scholar
[34]Mittal, R. and Iaccarino, G.. Immersed boundary methods. Annu. Rev. Fluid Mech., 37:239–261, 2005.Google Scholar
[35]Münster, R., Mierka, O., and Turek, S.. Finite element-fictitious boundary methods (FEM-FBM) for 3D particulate flow. Int. J. Numer. Meth. Fluids, 69(2):294–313, 2012.Google Scholar
[36]Persson, P. O. and Strang, G.. A simple mesh generator in MATLAB. SIAM Review, 46(2):329–345, 2004.Google Scholar
[37]Peskin, C. S.. The immersed boundary method. Acta Numer., 11:1–39, 2002.Google Scholar
[38]Phan-Thien, N. and Fan, X.-J.. Viscoelastic mobility problem using a boundary element method. J. Non-Newton. Fluid Mech., 105(2–3):131–152, 2002.Google Scholar
[39]Pianet, G. and Arquis, E.. Simulation of particles in fluid: a two-dimensional benchmark for a cylinder settling in a wall-bounded box. Euro. J. Mech. B Fluids, 27:309–321, 2008.Google Scholar
[40]Richardson, J. F. and Zaki, W. N.. Sedimentation and fluidisation: Part I. Trans. Inst. Chem. Eng., 32:35–53, 1954.Google Scholar
[41]Stockie, J. M.. Analysis and Computation of Immersed Boundaries, with Application to Pulp Fibres. PhD thesis, Institute of Applied Mathematics, University of British Columbia, Vancouver, Canada, 1997. Available at http://circle.ubc.ca/handle/2429/7346.Google Scholar
[42]Stockie, J. M.. Modelling and simulation of porous immersed boundaries. Comput. Struct., 87(11–12):701–709, 2009.Google Scholar
[43]Stokes, G. G.. Section IV. Determination of the motion of a fluid about a sphere which moves uniformly with a small velocity. In Mathematical and Physical Papers, volume III of The Sources of Science, No. 33, pages 55–67. Cambridge University Press, Teddington, UK, second edition, 1966. First published in 1901.Google Scholar
[44]Sucker, D. and Brauer, H.. Fluiddynamik bei quer angeströmten Zylindern. Heat Mass Transfer, 8(3):149–158, 1975.Google Scholar
[45]Takaisi, Y.. The drag on a circular cylinder moving with low speeds in a viscous liquid between two parallel walls. J. Phys. Soc. Japan, 10:685–693, 1955.Google Scholar
[46]Uhlmann, M.. An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys., 209:448–476, 2005.Google Scholar
[47]Vasseur, P. and Cox, R. G.. The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech., 80(3):561–591, 1977.CrossRefGoogle Scholar
[48]Wang, J. and Layton, A.. Numerical simulations of fiber sedimentation in Navier-Stokes flow. Commun. Comput. Phys., 5(1):61–83, 2009.Google Scholar
[49]Wang, Z., Fan, J., and Luo, K.. Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles. Int. J. Multiphase Flow, 34:283–302, 2008.Google Scholar
[50]White, C. M.. The drag of cylinders in fluids at slow speeds. Proc. Roy. Soc. A, 186:472–479, 1946.Google Scholar
[51]Wiens, J. K. and Stockie, J. M.. An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver. J. Comput. Phys., 281:917–941, 2015.Google Scholar
[52]Zhu, L. and Peskin, C. S.. Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. J. Comput. Phys., 179(2):452–468, 2002.Google Scholar