- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability

José Carlos Cano-Lozano and Carlos Martínez-Bazán

Jacques Magnaudet

Joël Tchoufag

  • Área de Mecánica de Fluidos, Departamento de Ingeniería Mecánica y Minera, Universidad de Jaén, Campus de las Lagunillas, 23071 Jaén, Spain

  • Institut de Mécanique des Fluides de Toulouse, Université de Toulouse, CNRS, INPT, UPS, 31062 Toulouse, France

  • Institut de Mécanique des Fluides de Toulouse, Université de Toulouse, CNRS, INPT, UPS, 31062 Toulouse, France and Department of Mechanical Engineering, University of California, Berkeley, Berkeley, California 94720, USA

PDF

Phys. Rev. Fluids 1, 053604 – Published 27 September, 2016

DOI: https://doi.org/10.1103/PhysRevFluids.1.053604

Abstract

We report on a series of results provided by three-dimensional numerical simulations of nearly spheroidal bubbles freely rising and deforming in a still liquid in the regime close to the transition to path instability. These results improve upon those of recent computational studies [Cano-Lozano et al., Int. J. Multiphase Flow 51, 11 (2013); Phys. Fluids 28, 014102 (2016)] in which the neutral curve associated with this transition was obtained by considering realistic but frozen bubble shapes. Depending on the dimensionless parameters that characterize the system, various paths geometries are observed by letting an initially spherical bubble starting from rest rise under the effect of buoyancy and adjust its shape to the surrounding flow. These include the well-documented rectilinear axisymmetric, planar zigzagging, and spiraling (or helical) regimes. A flattened spiraling regime that most often eventually turns into either a planar zigzagging or a helical regime is also frequently observed. Finally, a chaotic regime in which the bubble experiences small horizontal displacements (typically one order of magnitude smaller than in the other regimes) is found to take place in a region of the parameter space where no standing eddy exists at the back of the bubble. The discovery of this regime provides evidence that path instability does not always result from a wake instability as previously believed. In each regime, we examine the characteristics of the path, bubble shape, and vortical structure in the wake, as well as their couplings. In particular, we observe that, depending on the fluctuations of the rise velocity, two different vortex shedding modes exist in the zigzagging regime, confirming earlier findings with falling spheres. The simulations also reveal that significant bubble deformations may take place along zigzagging or spiraling paths and that, under certain circumstances, they dramatically alter the wake structure. The instability thresholds that can be inferred from the computations compare favorably with experimental data provided by various sets of recent experiments guaranteeing that the bubble surface is free of surfactants.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (73)

  1. C. D. Ohl, A. Tijink, and A. Prosperetti, The added mass of an expanding bubble, J. Fluid Mech. 482, 271 (2003).
  2. W. L. Haberman and R. K. Morton, David Taylor Model Basin Report No. 802, 1953 (unpublished).
  3. R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops and Particles (Academic, New York, 1978).
  4. A. W. G. De Vries, Path and wake of a rising bubble, Ph.D. thesis, University of Twente, 2001.
  5. C. H. J. Veldhuis, Leonardo's paradox: Path and shape instabilities of particles and bubbles. Ph.D. thesis, University of Twente, 2007.
  6. P. Ern, F. Risso, D. Fabre, and J. Magnaudet, Wake-induced oscillatory paths of bodies freely rising or falling in fluids, Annu. Rev. Fluid Mech. 44, 97 (2012).
  7. P. G. Saffman, On the rise of small air bubbles in water, J. Fluid Mech. 1, 249 (1956).
  8. T. Maxworthy, C. Gnann, M. Kürten, and F. Durst, Experiments on the rise of air bubbles in clean viscous liquids, J. Fluid Mech. 321, 421 (1996).
  9. R. Zenit and J. Magnaudet, Path instability of rising spheroidal air bubbles: A shape-controlled process, Phys. Fluids 20, 061702 (2008).
  10. R. Zenit and J. Magnaudet, Measurements of the streamwise vorticity in the wake of an oscillating bubble, Int. J. Multiphase Flow 35, 195 (2009).
  11. A. Sato, Deformation, wake and collision of rising bubbles, Ph.D. thesis, Kyushu University, 2009.
  12. R. A. Hartunian and W. R. Sears, On the instability of small gas bubbles moving uniformly in various liquids, J. Fluid Mech. 3, 27 (1957).
  13. H. Tsuge and S. I. Hibino, The onset conditions of oscillatory motion of single gas bubbles rising in various liquids, J. Chem. Eng. Jpn. 10, 66 (1977).
  14. P. C. Duineveld, Rise velocity and shape of bubbles in pure water at high Reynolds number, J. Fluid Mech. 292, 325 (1995).
  15. T. Sanada, K. Sugiara, M. Shirota, and M. Watanabe, Motion and drag of a single bubble in super-purified water, Fluid Dyn. Res. 40, 534 (2008).
  16. J. Magnaudet and I. Eames, The motion of high-Reynolds number bubbles in inhomogeneous flows, Annu. Rev. Fluid Mech. 32, 659 (2000).
  17. Y. Tagawa, S. Takagi, and Y. Matsumoto, Surfactant effect on path instability of a rising bubble, J. Fluid Mech. 738, 124 (2014).
  18. J. Magnaudet and G. Mougin, Wake instability of a fixed spheroidal bubble, J. Fluid Mech. 572, 311 (2007).
  19. B. Yang and A. Prosperetti, Linear stability of the flow past a spheroidal bubble, J. Fluid Mech. 582, 53 (2007).
  20. J. Tchoufag, J. Magnaudet, and D. Fabre, Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble, Phys. Fluids 25, 054108 (2013).
  21. G. Mougin and J. Magnaudet, Path Instability of a Rising Bubble, Phys. Rev. Lett. 88, 014502 (2001).
  22. J. Tchoufag, D. Fabre, and J. Magnaudet, Weakly Nonlinear Model with Exact Coefficients for the Fluttering and Spiraling Motion of Buoyancy-Driven Bodies, Phys. Rev. Lett. 115, 114501 (2015).
  23. J. C. Cano-Lozano, P. Bohorquez, and C. Martínez-Bazán, Wake instability of a fixed axisymmetric bubble of realistic shape, Int. J. Multiphase Flow 51, 11 (2013).
  24. J. C. Cano-Lozano, J. Tchoufag, J. Magnaudet, and C. Martínez-Bazán, A global stability approach to wake and path instabilities of nearly oblate spheroidal rising bubbles, Phys. Fluids 28, 014102 (2016).
  25. G. Mougin and J. Magnaudet, Wake-induced forces and torques on a zigzagging/spiralling bubble, J. Fluid Mech. 567, 185 (2006).
  26. A. W. G. De Vries, A. Biesheuvel, and L. Van Wijngaarden, Notes on the path and wake of a gas bubble rising in pure water, Int. J. Multiphase Flow 28, 1823 (2002).
  27. D. Gaudlitz and N. A. Adams, Numerical investigation of rising bubble wake and shape variations, Phys. Fluids 21, 122102 (2009).
  28. J. Zhang and M.-J. Ni, Direct simulation of single bubble motion under vertical magnetic field: Paths and wakes, Phys. Fluids 26, 102102 (2014).
  29. M. K. Tripathi, K. C. Sahu, and R. Govindarajan, Dynamics of an initially spherical bubble rising in quiescent liquid, Nat. Commun. 6, 6268 (2015).
  30. S. Popinet, Gerris: A tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. Comput. Phys. 190, 572 (2003).
  31. J. C. Cano-Lozano, R. Bolaños-Jiménez, C. Gutiérrez-Montes, and C. Martínez-Bazán, The use of volume of fluid technique to analyze multiphase flows: Specific case of bubble rising in still liquids, Appl. Math. Model. 39, 3290 (2015).
  32. S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys. 228, 5838 (2009).
  33. Sir H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1932).
  34. A. Prosperetti, Normal mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid, J. Méc. 19, 149 (1980).
  35. D. I. Meiron, On the stability of gas bubbles rising in an inviscid fluid, J. Fluid Mech. 198, 101 (1989).
  36. B. Lalanne, Simulation numérique directe de la déformation, des oscillations et de la rupture d'une bulle en ascension dans un écoulement instationnaire, Ph.D. thesis, Institut National Polytechnique de Toulouse, 2012.
  37. B. Lalanne, S. Tanguy, and F. Risso, Effect of rising motion on the damped shape oscillations of drops and bubbles, Phys. Fluids 25, 112107 (2013).
  38. M. Horowitz and C. H. K. Williamson, The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres, J. Fluid Mech. 651, 251 (2010).
  39. L. G. Leal, Vorticity transport and wake structure for bluff-bodies at finite Reynolds number, Phys. Fluids A 1, 124 (1989).
  40. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1967).
  41. A. Blanco and J. Magnaudet, The structure of the axisymmetric high-Reynolds number flow around an ellipsoidal bubble of fixed shape, Phys. Fluids 7, 1265 (1995).
  42. P. C. Fernandes, P. Ern, F. Risso, and J. Magnaudet, On the zigzag dynamics of freely moving axisymmetric bodies, Phys. Fluids 17, 098107 (2005).
  43. C. Brücker, Structure and dynamics of the wake of bubbles and its relevance for bubble interaction, Phys. Fluids 11, 1781 (1999).
  44. J. Jeong and F. Hussain, On the identification of a vortex, J. Fluid Mech. 285, 69 (1995).
  45. T. A. Johnson and V. C. Patel, Flow past a sphere up to a Reynolds number of 300, J. Fluid Mech. 378, 19 (1999).
  46. P. Ploumhans, G. S. Winckelmans, J. K. Salmon, A. Leonard, and M. S. Warren. Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at Re = 300, 500, and 1000, J. Comput. Phys. 178, 427 (2002).
  47. J. Yang, G. Wu, W. Zhong, and M. Liu, Numerical study on bifurcations in the wake of a circular disk, Int. J. Comput. Fluid Dyn. 28, 187 (2014).
  48. A. R. Shenoy and C. Kleinstreuer, Flow over a thin circular disk at low to moderate Reynolds numbers, J. Fluid Mech. 605, 253 (2008).
  49. J. I. Jiménez-González, E. Sanmiguel-Rojas, and C. Martínez-Bazán, Laminar flow past a spinning bullet-shaped body at moderate angular velocities, J. Fluids Struct. 43, 200 (2013).
  50. J. I. Jiménez-González, A. Sevilla, E. Sanmiguel-Rojas, and C. Martínez-Bazán, Global stability analysis of the axisymmetric wake past a spinning bullet-shaped body, J. Fluid Mech. 748, 302 (2014).
  51. C. Veldhuis, A. Biesheuvel, and L. Van Wijngaarden, Shape oscillations on bubbles rising in clean and in tap water, Phys. Fluids 20, 040705 (2008).
  52. T. Sanada, M. Shirota, and M. Watanabe, Bubble wake visualization by using photochromic dye, Chem. Eng. Sci. 62, 7264 (2007).
  53. T. Saito and M. Toriu, Effects of a bubble and the surrounding liquid motions on the instantaneous mass transfer across the gas-liquid interface, Chem. Eng. J. 265, 164 (2015).
  54. S. C. Crow, Stability theory for a pair of trailing vortices, AIAA J. 8, 2172 (1970).
  55. J. T. Lindt, On the periodic nature of the drag of a rising bubble, Chem. Eng. Sci. 27, 1775 (1972).
  56. K. Lunde and R. J. Perkins, Shape oscillations of rising bubbles, Appl. Sci. Res. 58, 387 (1998).
  57. D. Fabre, F. Auguste, and J. Magnaudet, Bifurcations and symmetry breaking in the wake of axisymmetric bodies, Phys. Fluids 20, 051702 (2008).
  58. F. Auguste, D. Fabre, and J. Magnaudet, Bifurcations in the wake of a thick circular disk, Theor. Comput. Fluid Dyn. 24, 305 (2010).
  59. M. Jenny, J. Dusek, and G. Bouchet, Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid, J. Fluid Mech. 508, 201 (2004).
  60. M. Chrust, G. Bouchet, and J. Dušek, Parametric study of the transition in the wake of oblate spheroids and flat cylinders, J. Fluid Mech. 665, 199 (2010).
  61. W. Zhou and J. Dušek, Chaotic states and order in the chaos of the paths of freely falling and ascending spheres, Int. J. Multiphase Flow 75, 205 (2015).
  62. K. Lunde and R. J. Perkins, ASME Fluids Engineering Division Summer Meeting (ASME, New York, 1997), paper FEDSM97-3530.
  63. F. Giannetti and P. Luchini, Structural sensitivity of the first instability of the cylinder wake, J. Fluid Mech. 581, 167 (2007).
  64. F. Auguste, J. Magnaudet, and D. Fabre, Falling styles of disks, J. Fluid Mech. 719, 388 (2013).
  65. J. Tchoufag, D. Fabre, and J. Magnaudet, Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders, J. Fluid Mech. 740, 278 (2014).
  66. D. Fabre, P. Assemat, and J. Magnaudet, A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid, J. Fluids Struct. 27, 758 (2011).
  67. M. Chrust, G. Bouchet, and J. Dusek, Effect of solid body degrees of freedom on the path instabilities of freely falling or rising flat cylinders, J. Fluids Struct. 47, 55 (2014).
  68. D. W. Moore, The velocity of rise of distorted gas bubbles in a liquid of small viscosity, J. Fluid Mech. 23, 749 (1965).
  69. N. M. Aybers and A. Tapucu, The motion of gas bubbles rising through stagnant liquid, Wärme-Stoffübertragung 2, 118 (1969).
  70. W. L. Shew, S. Poncet, and J.-F. Pinton, Force measurements on rising bubbles, J. Fluid Mech. 569, 51 (2006).
  71. M. Chrust, G. Bouchet, and J. Dusek, Numerical simulation of the dynamics of freely falling discs, Phys. Fluids 25, 044102 (2013).
  72. D. Fabre, J. Tchoufag, and J. Magnaudet, The steady oblique path of buoyancy-driven disks and spheres, J. Fluid Mech. 707, 24 (2012).
  73. P. C. Fernandes, F. Risso, P. Ern, and J. Magnaudet, Oscillatory motion and wake instability of freely rising axisymmetric bodies, J. Fluid Mech. 573, 479 (2007).

Outline

Information

Phys. Rev. Fluids 1, 053604– Published 27 September, 2016
  • Received 27 May 2016

©2016 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation