Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T16:05:17.842Z Has data issue: false hasContentIssue false
  • English
  • Français

The effective diffusivity of ordered and freely evolving bubbly suspensions

Published online by Cambridge University Press:  09 February 2018

Aurore Loisy Open the ORCID record for Aurore Loisy [Opens in a new window] ,
Aurore Naso Open the ORCID record for Aurore Naso [Opens in a new window]  and
Peter D. M. Spelt
Aurore Loisy
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS, Université Claude Bernard Lyon 1, École Centrale de Lyon, INSA de Lyon, 36 avenue Guy de Collongue, 69134 Écully CEDEX, France
Aurore Naso
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS, Université Claude Bernard Lyon 1, École Centrale de Lyon, INSA de Lyon, 36 avenue Guy de Collongue, 69134 Écully CEDEX, France
Peter D. M. Spelt
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS, Université Claude Bernard Lyon 1, École Centrale de Lyon, INSA de Lyon, 36 avenue Guy de Collongue, 69134 Écully CEDEX, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the dispersion of a passive scalar such as the concentration of a chemical species, or temperature, in homogeneous bubbly suspensions, by determining an effective diffusivity tensor. Defining the longitudinal and transverse components of this tensor with respect to the direction of averaged bubble rise velocity in a zero mixture velocity frame of reference, we focus on the convective contribution thereof, this being expected to be dominant in commonly encountered bubbly flows. We first extend the theory of Koch et al. (J. Fluid Mech., vol. 200, 1989, pp. 173–188) (which is for dispersion in fixed beds of solid particles under Stokes flow) to account for weak inertial effects in the case of ordered suspensions. In the limits of low and of high Péclet number, including the inertial effect of the flow does not affect the scaling of the effective diffusivity with respect to the Péclet number. These results are confirmed by direct numerical simulations performed in different flow regimes, for spherical or very deformed bubbles and from vanishingly small to moderate values of the Reynolds number. Scalar transport in arrays of freely rising bubbles is considered by us subsequently, using numerical simulations. In this case, the dispersion is found to be convectively enhanced at low Péclet number, like in ordered arrays. At high Péclet number, the Taylor dispersion scaling obtained for ordered configurations is replaced by one characterizing a purely mechanical dispersion, as in random media, even if the level of disorder is very low.

JFM classification

Multiphase and Particle-laden Flows: Gas/liquid flow Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 840 , 10 April 2018 , pp. 215 - 237
Check for updates
Copyright
© 2018 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

Aboulhasanzadeh, B. & Tryggvason, G. 2014 Effect of bubble interactions on mass transfer in bubbly flow. Intl J. Heat Mass Transfer 79, 390–396.Google Scholar
Acrivos, A., Hinch, E. J. & Jeffrey, D. J. 1980 Heat transfer to a slowly moving fluid from a dilute fixed bed of heated spheres. J. Fluid Mech. 101, 403–421.CrossRefGoogle Scholar
Alméras, E., Risso, F., Roig, V., Cazin, S., Plais, C. & Augier, F. 2015 Mixing by bubble-induced turbulence. J. Fluid Mech. 776, 458–474.Google Scholar
Batchelor, G. K. 1974 Transport properties of two-phase materials with random structure. Annu. Rev. Fluid Mech. 6, 227–255.Google Scholar
Batchelor, G. K. & O’Brien, R. W. 1977 Thermal or electrical conduction through a granular material. Proc. R. Soc. Lond. A 355, 313–333.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.Google Scholar
Brenner, H. 1980 Dispersion resulting from flow through spatially periodic porous media. Phil. Trans. R. Soc. Lond. A 297 (1430), 81–133.Google Scholar
Brenner, H. & Adler, P. M. 1982 Dispersion resulting from flow through spatially periodic porous media. Part II. Surface and intraparticle transport. Phil. Trans. R. Soc. Lond. A 307 (1498), 149–200.Google Scholar
Brenner, H. & Cox, R. G. 1963 The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers. J. Fluid Mech. 17 (4), 561–595.Google Scholar
Bunner, B. & Tryggvason, G. 2002 Dynamics of homogeneous bubbly flows. Part 1. Rise velocity and microstructure of the bubbles. J. Fluid Mech. 466, 17–52.Google Scholar
Chorin, A 1968 Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762.Google Scholar
Colombet, D., Legendre, D., Cockx, A., Guiraud, P., Risso, F., Daniel, C. & Galinat, S. 2011 Experimental study of mass transfer in a dense bubble swarm. Chem. Engng Sci. 66 (14), 3432–3440.CrossRefGoogle Scholar
Colombet, D., Legendre, D., Risso, F., Cockx, A. & Guiraud, P. 2015 Dynamics and mass transfer of rising bubbles in a homogenous swarm at large gas volume fraction. J. Fluid Mech. 763, 254–285.CrossRefGoogle Scholar
Deckwer, W.-D. 1992 Bubble Column Reactors. Wiley.Google Scholar
Hadamard, J. 1911 Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquide visqueux. Comptes Rend. l’Acad. Sci. 152 (25), 1735–1738.Google Scholar
Harfield, N. 1999 Conductivity calculation for a two-phase composite with spheroidal inclusions. J. Phys. D 32 (10), 1104–1113.CrossRefGoogle Scholar
Hewitt, G. F., Shires, G. L. & Bott, T. R. 1994 Process Heat Transfer. CRC Press.Google Scholar
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83 (4), 695–720.CrossRefGoogle Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. A 335 (1602), 355–367.Google Scholar
Koch, D. L. & Brady, J. F. 1985 Dispersion in fixed beds. J. Fluid Mech. 154, 399–427.Google Scholar
Koch, D. L. & Brady, J. F. 1987 The symmetry properties of the effective diffusivity tensor in anisotropic porous media. Phys. Fluids 30 (3), 642–650.Google Scholar
Koch, D. L., Cox, R. G., Brenner, H. & Brady, J. F. 1989 The effect of order on dispersion in porous media. J. Fluid Mech. 200, 173–188.Google Scholar
Kushch, V. I. 1997 Conductivity of a periodic particle composite with transversely isotropic phases. Proc. R. Soc. Lond. A 453 (1956), 65–76.Google Scholar
Loisy, A.2016 Direct numerical simulation of bubbly flows: coupling with scalar transport and turbulence. PhD thesis, Université de Lyon.Google Scholar
Loisy, A., Naso, A. & Spelt, P. D. M. 2017 Buoyancy-driven bubbly flows: ordered and free rise at small and intermediate volume fraction. J. Fluid Mech. 816, 94–141.Google Scholar
Mareuge, I. & Lance, M. 1995 Bubble induced dispersion of a passive scalar in bubbly flows. In Proceedings of the 2nd International Conference on Multiphase Flow, pp. PT1–3–8.Google Scholar
Maxwell, J. C. 1873 A Treatise on Electricity and Magnetism. Clarendon Press.Google Scholar
Rayleigh, R. S. 1892 LVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium. Phil. Mag. 5 34 (211), 481–502.Google Scholar
Russo, G. & Smereka, P. 2000 A remark on computing distance functions. J. Comput. Phys. 163 (1), 51–67.Google Scholar
Rybczynski, W. 1911 Über die fortschreitende Bewegung einer flüssigen Kugel in einen zähen Medium. Bull. Intl l’Acad. Sci. Cracovie A 1, 40–46.Google Scholar
Sabelnikov, V., Ovsyannikov, A. Y. & Gorokhovski, M. 2014 Modified level set equation and its numerical assessment. J. Comput. Phys. 278, 1–30.CrossRefGoogle Scholar
Sangani, A. S. & Acrivos, A. 1983 The effective conductivity of a periodic array of spheres. Proc. R. Soc. Lond. A 386 (1791), 263–275.Google Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1), 146–159.CrossRefGoogle Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. s2‐20 (1), 196–212.Google Scholar