Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-18T14:20:22.340Z Has data issue: false hasContentIssue false
  • English
  • Français

On the two-way interaction in two-dimensional particle-laden flows: the accumulation of particles and flow modification

Published online by Cambridge University Press:  26 April 2006

O. A. Druzhinin
O. A. Druzhinin
Affiliation:
Applied Physics Institute, Russian Academy of Sciences, 603600 Nizhni Novgorod, Russia
Get access Rights & Permissions [Opens in a new window]

Abstract

The evolution of two-dimensional regular flows laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle-particle interactions. Flows with large Reynolds and Froude numbers are considered, when effects related to viscous dissipation and gravity are negligible. A Cauchy problem is solved for an initially uniform distribution of particles with Stokes (St) and Reynolds (Rep) numbers of order unity in several types of flows representing steady solutions of the two-dimensional Euler equations. We consider flows in the vicinity of the hyperbolic stagnation point (with a uniform strain and zero vorticity) and the elliptic stagnation point (where vorticity is uniform), a circular vortex (with vorticity depending on the radius) and Stuart vortex flow. Analytical solutions are obtained, for the case of sufficiently small St, describing the accumulation of particles and corresponding modification of the fluid flow. Solutions derived show that the concentration of particles, although remaining uniform, decreases at the elliptic stagnation point and grows at the hyperbolic point. Owing to the coupling between the particulate and fluid dynamics, the flow vorticity is reduced at the elliptic point, while flow strain rate is enhanced at the hyperbolic point. Solutions obtained for the circular vortex show that the accumulation of particles proceeds in the form of a travelling wave. The concentration grows locally, forming the crest of the wave which propagates away from the vortex centre. Owing to the influence of the particulate on the carrier flow, the vorticity is reduced in the vortex centre. At the location of the crest the gradient of the flow grows owing to the drag forces between the fluid and particles and a vorticity peak is generated. Analytical solutions are also obtained for a chain of particle-laden Stuart vortices. Owing to the coupling effects, the concentration is diminished and the vorticity is reduced at the centres of the vortices. A sheet of increased concentration and vorticity is formed extending from the braid region to the periphery of the vortices, and the flow strain in the braid region is enhanced. Results of numerical simulations performed for St = 0.5 show good agreement with analytical solutions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 297 , 25 August 1995 , pp. 49 - 76
Copyright
© 1995 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.)

References

Batchelor, G. K. 1972 Sedimentation in a dilute suspension of spheres. J. Fluid Mech. 52, 245–268.Google Scholar
Chung, J. N. & Troutt, T. R. 1988 Simulations of particle dispersion in an axisymmetric jet. J. Fluid Mech. 186, 199–222.Google Scholar
Clift, H., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.
Crow, C. T., Gore, R. & Troutt, T. R. 1985 Particle dispersion by coherent structures in free shear flows. Particulate Sci. Tech. 3, 149–158.Google Scholar
Druzhinin, O. A. 1994 Concentration waves and flow modification in a particle-laden circular vortex. Phys. Fluids 6, 3276–3284.Google Scholar
Druzhinin, O. A. & Ostrovsky, L. A. 1994 The influence of Basset force on particle dynamics in two-dimensional flows. Proc. NATO ARW Chaotic Advection, Tracer Dynamics and Turbulent Dispersion (May 1993). Physica D76, 34-43.Google Scholar
Elghobashi, S. E. & Abou-Arab, T. W. 1983 A two-equation turbulence model for two-phase flows. Phys. Fluids 26, 931.Google Scholar
Elghobashi, S. E. & Truesdell, G. C. 1993 On the two-way interaction between homogeneous turbulence and dispersed solid particles. I. Turbulence modification. Phys. Fluids A 5, 1790–1802.Google Scholar
Fletcher, C. A. J. 1990 Computational Techniques for Fluid Dynamics, Vols. 1 and 2. Springer.
Gantáan-Calvo, A. M. & Lasheras, J. C. 1991 The dynamics and mixing of small spherical particles in a plane free shear layer. Phys. Fluids A 3, 1207–1217.Google Scholar
Lamb, H. 1975 Hydrodynamics, 7th edn. Cambridge University Press.
Landau, L. D. & Lifshitz, E. M. 1987 Hydrodynamics. Pergamon.
Lazaro, B. J. & Lasheras, J. C. 1992a Particle dispersion in the developing free shear layer. Part 1. Unforced flow. J. Fluid Mech. 235, 143–178.Google Scholar
Lazaro, B. J. & Lasheras, J. C. 1992b Particle dispersion in the developing free shear layer. Part 2. Forced flow. J. Fluid Mech. 235, 179–221.Google Scholar
Longmire, E. K. & Eaton, J. K. 1992 Structure of a particle-laden round jet. J. Fluid Mech. 236, 217–257.Google Scholar
Manton, M. J. 1974 On the motion of a small particle in the atmosphere. Boundary-Layer Met. 6, 487–504.Google Scholar
Martin, J. E. & Meiburg, E. 1994 The accumulation and dispersion in forced two-dimensional mixing layers. I. The fundamental and subharmonic cases. Phys. Fluids 6, 1116–1132.Google Scholar
Maxey, M. R. 1990 On the advection of spherical and non-spherical particles in a non-uniform flow. Phil. Trans. R. Soc. Lond. A 333, 289–307.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for small rigid sphere in a nonuniform flow. Phys. Fluids. 26, 883–889.Google Scholar
Nielsen, P. 1984 On the motion of suspended sand particles. J. Geophys. Res. 89, 616–626.Google Scholar
Nigmatulin, R. I. 1987 Dynamics of Multiphase Flows, Vols. 1 and 2. Moscow: Nauka (in Russian).
Ruetsch, G. R. & Maxey, M. R. 1991 Small-scale features of vorticity and passive scalar fields in homogeneous isotropic turbulence. Phys. Fluids A 3, 1587–1597.Google Scholar
Ruetsch, G. R. & Maxey, M. R. 1992 The evolution of small-scale structures in homogeneous isotropic turbulence. Phys. Fluids A 4, 2747–2760.Google Scholar
Schreck, S. & Kleis, S. 1993 Modification of grid-generated turbulence by solid particles. J. Fluid Mech. 249, 665–688.Google Scholar
She, Z.-S., Jackson, E. & Orszag, S. A. 1990 Intermittent vortex structures in homogeneous turbulence. Nature 344, 226.Google Scholar
Shraiber, A. A., Gavin, L. B., Naumov, V. A. & Yatsenko, V. P. 1988 Turbulent Flows in Gas Suspension. Hemisphere.
Soo, S. L. 1967 Fluid Dynamics of Multiphase Systems. Waltham, MA: Blaisdell.
Squires, K. D. & Eaton, J. K. 1991 Preferential concentration of particles by turbulence. Phys. Fluids A. 3, 1169-1178.Google Scholar
Tang, L., Wen, E., Yang, T., Crowe, C. T., Chung, J. N. & Troutt, T. R. 1992 Self-organizing particle dispersion mechanism in a plane wake. Phys. Fluids A 4, 2244–2251.Google Scholar
Tio, K.-K., Gantáan-Calvo, A. M. & Lasheras, J. C. 1993 a The dynamics of small heavy rigid spherical particles in a periodic Stuart vortex flow. Phys. Fluids A 5, 1679–1693.Google Scholar
Tio, K.-K., Lintáan, A., Lasheras, J. C. & Gantáan-Calvo, A. M. 1993b On the dynamics of buoyant and heavy particles in a periodic Stuart vortex flow. J. Fluid Mech. 254, 671–699.Google Scholar
Wang, L.-P. & Maxey, M. R. 1993 Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 27–68.Google Scholar