Hostname: page-component-68945f75b7-72kh6 Total loading time: 0 Render date: 2024-08-06T00:45:28.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Drag and lift forces on clean spherical and ellipsoidal bubbles in a solid-body rotating flow

Published online by Cambridge University Press:  19 July 2011

MARIE RASTELLO ,
JEAN-LOUIS MARIÉ  and
MICHEL LANCE
MARIE RASTELLO*
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, Ecole Centrale de Lyon, CNRS, Université Claude Bernard Lyon 1, INSA Lyon, 36 Avenue Guy de Collongue, 69134 Ecully CEDEX, France
JEAN-LOUIS MARIÉ
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, Ecole Centrale de Lyon, CNRS, Université Claude Bernard Lyon 1, INSA Lyon, 36 Avenue Guy de Collongue, 69134 Ecully CEDEX, France
MICHEL LANCE
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, Ecole Centrale de Lyon, CNRS, Université Claude Bernard Lyon 1, INSA Lyon, 36 Avenue Guy de Collongue, 69134 Ecully CEDEX, France
*
†Email address for correspondence: marie.rastello@ec-lyon.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

A single bubble is placed in a solid-body rotating flow of silicon oil. From the measurement of its equilibrium position, lift and drag forces are determined. Five different silicon oils have been used, providing five different viscosities and Morton numbers. Experiments have been performed over a wide range of bubble Reynolds numbers (0.7 ≤ Re ≤ 380), Rossby numbers (0.58 ≤ Ro ≤ 26) and bubble aspect ratios (1 ≤ χ ≤ 3). For spherical bubbles, the drag coefficient at the first order is the same as that of clean spherical bubbles in a uniform flow. It noticeably increases with the local shear S = Ro−1, following a Ro−5/2 power law. The lift coefficient tends to 0.5 for large Re numbers and rapidly decreases as Re tends to zero, in agreement with existing simulations. It becomes hardly measurable for Re approaching unity. When bubbles start to shrink with Re numbers decreasing slowly, drag and lift coefficients instantaneously follow their stationary curves versus Re. In the standard Eötvös–Reynolds diagram, the transitions from spherical to deformed shapes slightly differ from the uniform flow case, with asymmetric shapes appearing. The aspect ratio χ for deformed bubbles increases with the Weber number following a law which lies in between the two expressions derived from the potential flow theory by Moore (J. Fluid Mech., vol. 6, 1959, pp. 113–130) and Moore (J. Fluid Mech., vol. 23, 1965, pp. 749–766) at low- and moderate We, and the bubble orients with an angle between its minor axis and the direction of the flow that increases for low Ro. The drag coefficient increases with χ, to an extent which is well predicted by the Moore (1965) drag law at high Re and Ro. The lift coefficient is a function of both χ and Re. It increases linearly with (χ − 1) at high Re, in line with the inviscid theory, while in the intermediate range of Reynolds numbers, a decrease of lift with aspect ratio is observed. However, the deformation is not sufficient for a reversal of lift to occur.

JFM classification

Drops and Bubbles: Bubble dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 434 - 459
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Adoua, R., Legendre, D. & Magnaudet, J. 2009 Reversal of the lift force on an oblate bubble in a weakly viscous shear flow. J. Fluid Mech. 628, 23–41.CrossRefGoogle Scholar
Adoua, R. 2007 Hydrodynamique d'une bulle déformée dans un écoulement cisaillé. PhD thesis, Institut National Polytechnique de Toulouse.Google Scholar
Auton, T. R. 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199–218.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2002 Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate Re. Phys. Fluids 14 (8), 2719–2737.CrossRefGoogle Scholar
Blanco, A. & Magnaudet, J. 1995 The structure of the axisymmetric high-Reynolds number flow around an ellipsoidal bubble of fixed shape. Phys. Fluids 7, 1265–1274.CrossRefGoogle Scholar
Bluemink, J. J., Lohse, D., Prosperetti, A. & Van Wijngaarden, L. 2008 A sphere in a uniformly rotating or shearing flow. J. Fluid Mech. 600, 201–233.CrossRefGoogle Scholar
Bluemink, J. J., Lohse, D., Prosperetti, A. & Van Wijngaarden, L. 2010 Drag and lift forces on particles in a rotating flow. J. Fluid Mech. 643, 1–31.CrossRefGoogle Scholar
Breach, D. R. 1961 Slow flow past ellipsoids of revolution. J. Fluid Mech. 10, 303–314.CrossRefGoogle Scholar
Brenner, H. 1961 The Oseen resistance of a particle of arbitrary shape. J. Fluid Mech. 11, 604–610.CrossRefGoogle 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, 561–595.CrossRefGoogle Scholar
Candelier, F., Angilella, J.-R. & Souhar, M. 2004 On the effect of the Boussinesq–Basset force on the radial migration of a Stokes particle in a vortex. Phys. Fluids 16 (5), 1765–1776.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.Google Scholar
Duineveld, P. C. 1995 The rise velocity and shape of bubbles in pure water at high Reynolds number. J. Fluid Mech. 292, 325–332.CrossRefGoogle Scholar
Ervin, E. A. & Tryggvason, G. 1997 The rise of bubbles in a vertical shear flow. J. Fluid Engng 119, 443–449.CrossRefGoogle Scholar
Gotoh, T. 1990 Brownian motion in a rotating flow. J. Stat. Phys. 59, 371–402.CrossRefGoogle Scholar
Grace, J. R., Wairegi, T. & Nguyen, T. H. 1976 Shapes and velocities of single drops and bubbles moving freely through immiscible liquids. Trans. Inst. Chem. Engrs 54, 167–173.Google Scholar
Harper, E. Y. & Chang, I. D. 1968 Maximum dissipation resulting from lift in a slow viscous shear flow. J. Fluid Mech. 33, 209–225.CrossRefGoogle Scholar
Kariyasaki, A. 1987 Behavior of a single gas bubble in a liquid flow with a linear velocity profile. In Proc. ASME–JSME Thermal Engng Joint Conf., Honolulu, Hawaii, pp. 261–267.Google Scholar
Lamb, H. 1934 Hydrodynamics, 6th edn. Dover.Google Scholar
Legendre, D. & Magnaudet, J. 1998 The lift force on a spherical bubble in a viscous linear shear flow. J. Fluid Mech. 368, 81–126.CrossRefGoogle Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice Hall.Google Scholar
Loth, E. 2008 Quasi-steady shape and drag of deformable bubbles and drops. Intl J. Multiphase Flow 34 (6), 523–546.CrossRefGoogle Scholar
Lunde, K. & Perkins, R. J. 1998 Shape oscillations of rising bubbles. Appl. Sci. Res. 58, 387–408.CrossRefGoogle Scholar
Magnaudet, J. & Eames, I. 2000 Dynamics of high re bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.CrossRefGoogle Scholar
Magnaudet, J. & Legendre, D. 1998 Some aspects of the lift force on a spherical bubble. Appl. Sci. Res. 58, 441–461.CrossRefGoogle Scholar
Mei, R., Klausner, J. & Lawrence, C. 1994 A note on the history force on a spherical bubble at finite Reynolds number. Phys. Fluids 6, 418–420.CrossRefGoogle Scholar
Moore, D. W. 1959 The rise of a gas bubble in a viscous liquid. J. Fluid Mech. 6, 113–130.CrossRefGoogle Scholar
Moore, D. W. 1965 The velocity rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.CrossRefGoogle Scholar
Naciri, A. 1992 Contribution à l'étude des forces exercées par un liquide sur une bulle de gaz: portance, masse ajoutée et interactions hydrodynamiques. PhD thesis, Ecole Centrale de Lyon.Google Scholar
van Nierop, E. A., Luther, S., Bluemink, J. J., Magnaudet, J., Prosperetti, A. & Lohse, D. 2007 Drag and lift forces on bubbles in a rotating flow. J. Fluid Mech. 571, 439–454.CrossRefGoogle Scholar
Rastello, M., Marié, J.-L., Grosjean, N. & Lance, M. 2007 Study of bubble's equilibrium in a rotating flow. In 6th Intl Conf. Multiphase Flows, Leipzig, 9–13 July.Google Scholar
Rastello, M., Marié, J. L., Grosjean, N. & Lance, M. 2009 Drag and lift forces on interface-contaminated bubbles spinning in a rotating flow. J. Fluid Mech. 624, 159–178.CrossRefGoogle Scholar
Saffman, P. G. 1965 The lift force on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385–400.CrossRefGoogle Scholar
Serizawa, A., Kataoka, I. & Michiyoshi, I. 1975 Turbulence structure of air–water bubbly flow II. Intl J. Multiphase Flow 2, 235–246.CrossRefGoogle Scholar
Takemura, F., Magnaudet, J. & Dimitrakopoulos, P. 2009 Migration and deformation of bubbles rising in a wall-bounded shear flow at finite Reynolds number. J. Fluid Mech. 634, 463–486.CrossRefGoogle Scholar
Takemura, F., Takagi, S., Magnaudet, J. & Matsumoto, Y. 2002 Drag and lift forces on a bubble rising near a vertical wall in a viscous liquid. J. Fluid Mech. 461, 277–300.CrossRefGoogle Scholar
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number. J. Fluid Mech. 18, 466–476.CrossRefGoogle Scholar
Teague, M. 1980 Image analysis via the general theory of moments. J. Opt. Soc. Am. 70 (8), 920–930.CrossRefGoogle Scholar
Tomiyama, A., Tamai, H., Zun, I. & Hosokawa, S. 2002 Transverse migration of single bubbles in simple shear flows. Chem. Engng Sci. 57, 1849–1859.CrossRefGoogle Scholar
Veldhuis, C., Biesheuvel, A. & Van Wijngaarden, L. 2008 Shape oscillations on bubbles rising in clean and tap water. Phys. Fluids 4 (20), 040705.CrossRefGoogle Scholar
de Vries, A. W. G., Biesheuvel, A. & van Wijngaarden, L. 2002 a Notes on the path and wake of a gas bubble rising in pure water. Intl J. Multiphase Flow 28, 1823–1835.CrossRefGoogle Scholar
de Vries, J., Luther, S. & Lohse, D. 2002 b Induced bubble shape oscillations and their impact on the rise velocity. Eur. Phys. J. B 29 (3), 503–509.CrossRefGoogle Scholar
Wakaba, L. & Balachandar, S. 2005 History force on a sphere in a weak linear shear flow. Intl J. Multiphase Flow 31 (9), 996–1014.CrossRefGoogle Scholar
Zenit, R. & Magnaudet, J. 2008 Path instability of rising spheroidal air bubbles: a shape-controlled process. Phys. Fluids 20, 061702.CrossRefGoogle Scholar