Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-25T06:55:53.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Physical interpretation of probability density functions of bubble-induced agitation

Published online by Cambridge University Press:  09 November 2016

Frédéric Risso
Frédéric Risso*
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), CNRS-INPT-UPS, Université de Toulouse, 31400 Toulouse, France
*
†Email address for correspondence: frisso@imfr.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

A stochastic model is presented for the probability density function (p.d.f.) of the liquid velocity fluctuations generated by high-Reynolds-number rising bubbles. It considers three elementary sources of fluctuations: the potential flow disturbance around each bubble; the average bubble wakes, which are assumed to decay exponentially; and the turbulent agitation resulting from the flow instability, which is assumed to be isotropic, homogeneously distributed all over the flow and statistically independent of the two others. The model reproduces well and explains the characteristics of the experimental p.d.f.s: exponential tails, asymmetry of vertical fluctuations and evolution with the gas volume fraction. The model involves two a priori unknown parameters: the volume of the wake and the velocity scale of the turbulent agitation. Because some parts of the probability functions depend only on a single contribution, these two parameters can be uniquely and independently determined from experimental p.d.f.s. This defines an objective method to separate the various kinds of fluctuations and allows one to determine the contribution of each of them to the total agitation.

JFM classification

Multiphase and Particle-laden Flows: Gas/liquid flow Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 809 , 25 December 2016 , pp. 240 - 263
Check for updates
Copyright
© 2016 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

Abbas, M., Climent, E., Simonin, O. & Maxey, M. R. 2006 Dynamics of bidisperse suspensions under Stokes flows: linear shear flow and sedimentation. Phys. Fluids 18, 121504.Google Scholar
Amoura, Z.2008 Etude hydrodynamique de l’écoulement traversant un réseau aléatoire de sphères fixes. PhD thesis, Institut National Polytechnique de Toulouse.Google Scholar
Bagchi, P. & Balachandar, S. 2004 Response of the wake of an isolated particle to an isotropic turbulent flow. J. Fluid Mech. 518, 95–123.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Biesheuvel, A. & Van Wijngaarden, L. 1984 Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid Mech. 148, 301–318.Google Scholar
Bush, J. W. M. & Eames, I. 1998 Fluid displacement by high Reynolds number bubble motion in a thin gap. Intl J. Multiphase Flow 24 (3), 411–430.CrossRefGoogle Scholar
Cartellier, A., Andreotti, M. & Sechet, P. 2009 Induced agitation in homogeneous bubbly flows at moderate particle Reynolds number. Phys. Rev. E 80, 065301.Google Scholar
Drazer, G., Koplik, J., Khusid, B. & Acrivos, A. 2002 Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions. J. Fluid Mech. 460, 307–335.Google Scholar
Ellingsen, K. & Risso, F. 2001 On the rise of an ellipsoidal bubble in water: oscillatory paths and liquid-induced velocity. J. Fluid Mech. 440, 235–268.Google Scholar
Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97–121.CrossRefGoogle Scholar
Hunt, J. C. R. & Eames, I. 2002 The disappearance of laminar and turbulent wakes in complex flows. J. Fluid Mech. 457, 111–132.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lance, M. & Bataille, J. 1991 Turbulence in the liquid phase of a uniform bubbly air–water flow. J. Fluid Mech. 222, 95–118.CrossRefGoogle Scholar
Legendre, D., Merle, A. & Magnaudet, J. 2006 Wake of a spherical bubble or a solid sphere set fixed in a turbulent environment. Phys. Fluids 18, 048102.Google Scholar
Martínez Mercado, J., Chehata Gómez, D., Van Gils, D., Sun, C. & Lohse, D. 2010 On bubble clustering and energy spectra in pseudo-turbulence. J. Fluid Mech. 650, 287–306.Google Scholar
Martínez Mercado, J., Palacios-Morales, C. A. & Zenit, R. 2007 Measurement of pseudoturbulence intensity in monodispersed bubbly liquids for 10 < Re < 500. Phys. Fluids 19, 103302.CrossRefGoogle Scholar
Mendez-Diaz, S., Serrano-Garcia, J. C., Zenit, R. & Hernández-Cordero, J. A. 2013 Power spectral distributions of pseudo-turbulent bubbly flows. Phys. Fluids 25, 043303.Google Scholar
Parthasarathy, R. N. & Faeth, G. M. 1990 Turbulence modulation in homogeneous dilute particle-laden flows. J. Fluid Mech. 220, 485–514.Google Scholar
Prakash, V. N., Martínez Mercado, J., van Wijngaarden, L., Mancilla, E., Tagawa, Y., Lohse, D. & Sun, C. 2016 Energy spectra in turbulent bubbly flows. J. Fluid Mech. 791, 174–190.Google Scholar
Rensen, J., Luther, S. & Lohse, D. 2005 The effect of bubbles on developed turbulence. J. Fluid Mech. 538, 153–187.Google Scholar
Riboux, G., Legendre, D. & Risso, F. 2013 A model of bubble-induced turbulence based on large-scale wake interactions. J. Fluid Mech. 719, 362–387.Google Scholar
Riboux, G., Risso, F. & Legendre, D. 2010 Experimental characterization of the agitation generated by bubbles rising at high Reynolds number. J. Fluid Mech. 643, 509–539.Google Scholar
Rice, S. O. 1944 Mathematical analysis of random noise. Bell System Tech. J. 23, 282–332.Google Scholar
Risso, F. 2011 Theoretical model for k -3 spectra in dispersed multiphase flow. Phys. Fluids 23, 011701.Google Scholar
Risso, F. & Ellingsen, K. 2002 Velocity fluctuations in a homogeneous dilute dispersion of high-Reynolds-number rising bubbles. J. Fluid Mech. 453, 395–410.Google Scholar
Risso, F., Roig, V., Amoura, Z., Riboux, G. & Billet, A.-M. 2008 Wake attenuation in large Reynolds number dispersed two-phase flows. Phil. Trans. R. Soc. Lond. A 366, 2177–2190.Google Scholar
Roghair, I., Martínez Mercado, J., van Sint Annaland, M., Kuipers, H., Sun, C. & Lohse, D. 2011 Energy spectra and bubble velocity distributions in pseudo-turbulence: numerical simulations versus experiments. Intl J. Multiphase Flow 37, 1093–1098.Google Scholar
Tennekes, K. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Google Scholar
Wu, J. S. & Faeth, G. M. 1994 Sphere wakes at moderate Reynolds numbers in a turbulent environment. AIAA J. 32, 535–541.Google Scholar
Wu, J. S. & Faeth, G. M. 1995 Effect of ambient turbulence intensity on sphere wakes at intermediate Reynolds numbers. AIAA J. 33, 171–173.CrossRefGoogle Scholar
Zenit, R., Koch, D. L. & Sangani, A. S. 2001 Measurements of the average properties of a suspension of bubbles rising in a vertical channel. J. Fluid Mech. 429, 307–342.Google Scholar