Hostname: page-component-7d684dbfc8-mqbnt Total loading time: 0 Render date: 2023-09-24T05:11:09.552Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Exact regularized point particle method for multiphase flows in the two-way coupling regime

Published online by Cambridge University Press:  27 May 2015

P. Gualtieri*
Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
F. Picano
Department of Industrial Engineering, University of Padova, Via Venezia 1, 35131 Padua, Italy
G. Sardina
Department of Meteorology and SeRC, Stockholm University, Stockholm, Sweden
C. M. Casciola
Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
Email address for correspondence:


Particulate flows have mainly been studied under the simplifying assumption of a one-way coupling regime where the disperse phase does not modify the carrier fluid. A more complete view of multiphase flows can be gained calling into play two-way coupling effects, i.e. by accounting for the inter-phase momentum exchange, which is certainly relevant at increasing mass loading. In this paper we present a new methodology rigorously designed to capture the inter-phase momentum exchange for particles smaller than the smallest hydrodynamical scale, e.g. the Kolmogorov scale in a turbulent flow. The momentum coupling mechanism exploits the unsteady Stokes flow around a small rigid sphere, where the transient disturbance produced by each particle is evaluated in a closed form. The particles are described as lumped point masses, which would lead to the appearance of singularities. A rigorous regularization procedure is conceived to extract the physically relevant interactions between the particles and the fluid which avoids any ‘ad hoc’ assumption. The approach is suited for high-efficiency implementation on massively parallel machines since the transient disturbance produced by the particles is strongly localized in space. We will show that hundreds of thousands of particles can be handled at an affordable computational cost, as demonstrated by a preliminary application to a particle-laden turbulent shear flow.

© 2015 Cambridge University Press 

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.)


Ayala, O., Grabowski, W. W. & Wang, L.-P. 2007 A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles. J. Comput. Phys. 225 (1), 5173.CrossRefGoogle Scholar
Balachandar, S. 2009 A scaling analysis for point-particle approaches to turbulent multiphase flows. Intl J. Multiphase Flow 35 (9), 801810.CrossRefGoogle Scholar
Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111133.CrossRefGoogle Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245268.CrossRefGoogle Scholar
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74, 129.CrossRefGoogle Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. & Toschi, F. 2010 Intermittency in the velocity distribution of heavy particles in turbulence. J. Fluid Mech. 646, 527536.CrossRefGoogle Scholar
Boivin, M., Simonin, O. & Squires, K. D. 1998 Direct numerical simulation of turbulence modulation by particles in isotropic turbulence. J. Fluid Mech. 375, 235263.CrossRefGoogle Scholar
Burton, T. M. & Eaton, J. K. 2005 Fully resolved simulations of a particle–turbulence interaction. J. Fluid Mech. 545, 67111.CrossRefGoogle Scholar
Cate, A. T., Derksen, J. J., Portela, L. M. & Akken, H. E. A. V. D. 2004 Fully resolved simulations of colliding monodisperse spheres in forced isotropic turbulence. J. Fluid Mech. 519, 233271.CrossRefGoogle Scholar
Climent, E. & Magnaudet, J. 2006 Dynamics of a two-dimensional upflowing mixing layer seeded with bubbles: bubble dispersion and effect of two-way coupling. Phys. Fluids 18 (10), 103304.CrossRefGoogle Scholar
Crowe, C. T., Sharma, M. P. & Stock, D. E. 1977 The particle-source in cell method for gas droplet flow. J. Fluid Eng. 99, 325332.CrossRefGoogle Scholar
Dance, S. L. & Maxey, M. R. 2003 Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow. J. Comput. Phys. 189, 212238.CrossRefGoogle Scholar
Eckhardt, B. & Buehrle, J. 2008 Time-dependent effects in high viscosity fluid dynamics. Eur. Phys. J. 157, 135148.Google Scholar
Elgobashi, S. 2006 An updated classification map of particle-laden turbulent flows. In IUTAM Symposium on Computational Approaches to Multiphase Flow, pp. 310. Springer.CrossRefGoogle Scholar
Gao, H., Li, H. & Wang, L.-P. 2013 Lattice Boltzmann simulation of turbulent flow laden with finite-size particles. Comput. Math. Appl. 65 (2), 194210.CrossRefGoogle Scholar
Gatignol, R. 1983 The Faxén formulas for a rigid particle in an unsteady non-uniform Stokes-flow. J. Méc. Théor. Appl. 2 (2), 143160.Google Scholar
Gualtieri, P., Casciola, C. M., Benzi, R., Amati, G. & Piva, R. 2002 Scaling laws and intermittency in homogeneous shear flow. Phys. Fluids 14 (2), 583596.CrossRefGoogle Scholar
Gualtieri, P., Casciola, C. M., Benzi, R. & Piva, R. 2007 Preservation of statistical properties in large eddy simulation of shear turbulence. J. Fluid Mech. 592, 471494.CrossRefGoogle Scholar
Gualtieri, P., Picano, F. & Casciola, C. M. 2009 Anisotropic clustering of inertial particles in homogeneous shear flow. J. Fluid Mech. 629, 2539.CrossRefGoogle Scholar
Gualtieri, P., Picano, F., Sardina, G. & Casciola, C. M. 2013 Clustering and turbulence modulation in particle-laden shear flow. J. Fluid Mech. 715, 134162.CrossRefGoogle Scholar
Hasimoto, H. 1959 On the periodic fundamental solutions of Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5, 317328.CrossRefGoogle Scholar
Homann, H. & Bec, J. 2010 Finite-size effects in the dynamics of neutrally buoyant particles in turbulent flow. J. Fluid Mech. 651 (1), 8191.CrossRefGoogle Scholar
Jenny, P., Roekaerts, D. & Beishuizen, N. 2012 Modeling of turbulent dilute spray combustion. Prog. Energy Combust. Sci. 38 (6), 846887.CrossRefGoogle Scholar
Kim, S. & Karilla, S. J. 2005 Microhydrodynamics. Dover.Google Scholar
Lamb, H. 1993 Hydrodynamics. Cambridge University Press.Google Scholar
Liu, D., Keaveny, E. E., Maxey, M. R. & Karniadakis, G. E. 2009 Force-coupling method for flows with ellipsoidal particles. J. Comput. Phys. 228 (10), 35593581.CrossRefGoogle Scholar
Lomholt, S. & Maxey, M. R. 2003 Force-coupling method for particulate two phase flow: Stokes flow. J. Comput. Phys. 184, 381405.CrossRefGoogle Scholar
Lucci, F., Ferrante, A. & Elghobashi, S. 2010 Modulation of isotropic turbulence by particles of Taylor length-scale size. J. Fluid Mech. 650 (1), 555.CrossRefGoogle Scholar
Luo, X., Maxey, M. R. & Karniadakis, G. E. 2009 Smoothed profile method for particulate flows: error analysis and simulations. J. Comput. Phys. 228 (5), 17501769.CrossRefGoogle Scholar
Marchioli, C. & Soldati, A. 2002 Mechanisms for particle transfer and segregation in a turbulent boundary layer. J. Fluid. Mech. 468, 283315.CrossRefGoogle Scholar
Maxey, M. R. & Patel, B. K. 2001 Localized force representations for particles sedimenting in 1341 Stokes flow. Intl J. Multiphase Flow 27, 16031626.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883889.CrossRefGoogle Scholar
Meneguz, E. & Reeks, M. W. 2011 Statistical properties of particle segregation in homogeneous isotropic turbulence. J. Fluid Mech. 686, 338351.CrossRefGoogle Scholar
Monchaux, R., Bourgoin, M. & Cartellier, A. 2010 Preferential concentration of heavy particles: a Voronoi analysis. Phys. Fluids 22 (10), 103304.CrossRefGoogle Scholar
Naso, A. & Prosperetti, A. 2010 The interaction between solid particle and turbulent flow. New J. Phys. 12, 120.CrossRefGoogle Scholar
Pan, Y. & Banerjee, S. 2001 Numerical simulation of particle interaction with wall turbulence. Phys. Fluids 8 (10), 27332755.CrossRefGoogle Scholar
Pasquetti, R., Bwemba, R. & Cousin, L. 2008 A pseudo-penalization method for high Reynolds number unsteady flows. Appl. Numer. Maths 58 (7), 946954.CrossRefGoogle Scholar
Pawlowski, L. 2008 The Science and Engineering of Thermal Spray Coatings. Wiley.CrossRefGoogle Scholar
Picano, F., Sardina, G. & Casciola, C. M. 2009 Spatial development of particle-laden turbulent pipe flow. Phys. Fluids 21, 093305.CrossRefGoogle Scholar
Pignatel, F., Nicolas, M. & Guazzelli, E. 2011 A falling cloud of particles at a small but finite Reynolds number. J. Fluid Mech. 671, 3451.CrossRefGoogle Scholar
Post, S. L. & Abraham, J. 2002 Modeling the outcome of drop–drop collisions in diesel sprays. Intl J. Multiphase Flow 28 (6), 9971019.CrossRefGoogle Scholar
Pumir, A. 1996 Turbulence in homogeneous shear flow. Phys. Fluids 8 (11), 31123127.CrossRefGoogle Scholar
Reade, W. C. & Collins, L. R. 2000 Effect of preferential concentration on turbulent collision rates. Phys. Fluids 12 (10), 25302540.CrossRefGoogle Scholar
Rogallo, R. S.1981 Numerical experiments in homogeneous turbulence. NASA, TM 81315.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sangani, A. S. & Acrivos, A. 1982 Slow flow through a periodic array of spheres. Intl J. Multiphase Flow 4, 343360.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381411.CrossRefGoogle Scholar
Subramanian, G. & Koch, D. L. 2008 Evolution of clusters of sedimenting low-Reynolds-number particles with Oseen interactions. J. Fluid Mech. 603 (1), 63100.CrossRefGoogle Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41, 375404.CrossRefGoogle Scholar
Wang, L.-P., Ayala, O. & Grabowski, W. W. 2005a Improved formulations of the superposition method. J. Atmos. Sci. 62 (4), 12551266.CrossRefGoogle Scholar
Wang, L.-P., Ayala, O., Kasprzak, S. E. & Grabowski, W. W. 2005b Theoretical formulation of collision rate and collision efficiency of hydrodynamically interacting cloud droplets in turbulent atmosphere. J. Atmos. Sci. 62 (7), 24332450.CrossRefGoogle Scholar
Yakhot, V. 2003 A simple model for self-sustained oscillations in homogeneous shear flow. Phys. Fluids 15 (2), L17L20.CrossRefGoogle Scholar
Yeo, K., Dong, S., Climent, E. & Maxey, M. R. 2010 Modulation of homogeneous turbulence seeded with finite size bubbles or particles. Intl J. Multiphase Flow 36 (3), 221233.CrossRefGoogle Scholar
Zapryanov, Z. & Tabakova, S. 1998 Dynamics of Bubbles, Drops and Rigid Particles, Vol. 50. Springer.Google Scholar
Zhang, Z. & Prosperetti, A. 2005 A second order method for three dimensional particle simulation. J. Comput. Phys. 210, 292324.CrossRefGoogle Scholar