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The effect of nonlinear drag on the rise velocity of bubbles in turbulence

Published online by Cambridge University Press:  04 August 2021

Daniel J. Ruth Open the ORCID record for Daniel J. Ruth [Opens in a new window] ,
Marlone Vernet Open the ORCID record for Marlone Vernet [Opens in a new window] ,
Stéphane Perrard Open the ORCID record for Stéphane Perrard [Opens in a new window]  and
Luc Deike Open the ORCID record for Luc Deike [Opens in a new window]
Daniel J. Ruth
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA
Marlone Vernet
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA Département de Physique, ENS, PSL Université, CNRS, 24 rue Lhomond, 75005 Paris, France
Stéphane Perrard
Affiliation:
Département de Physique, ENS, PSL Université, CNRS, 24 rue Lhomond, 75005 Paris, France
Luc Deike*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA High Meadows Environmental Institute, Princeton University, NJ 08544, USA
*
†Email address for correspondence: ldeike@princeton.edu
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Abstract

We investigate how turbulence in liquid affects the rising speed of gas bubbles within the inertial range. Experimentally, we employ stereoscopic tracking of bubbles rising through water turbulence created by the convergence of turbulent jets and characterized with particle image velocimetry performed throughout the measurement volume. We use the spatially varying, time-averaged mean water velocity field to consider the physically relevant bubble slip velocity relative to the mean flow. Over a range of bubble sizes within the inertial range, we find that the bubble mean rise velocity $\left \langle v_z \right \rangle$ decreases with the intensity of the turbulence as characterized by its root-mean-square fluctuation velocity, $u'$. Non-dimensionalized by the quiescent rise velocity $v_{q}$, the average rise speed follows $\left \langle v_z \right \rangle /v_{q}\propto 1/{\textit {Fr}}$ at high ${\textit {Fr}}$, where ${\textit {Fr}}=u'/\sqrt {dg}$ is a Froude number comparing the intensity of the turbulence to the bubble buoyancy, with $d$ the bubble diameter and $g$ the acceleration due to gravity. We complement these results by performing numerical integration of the Maxey–Riley equation for a point bubble experiencing nonlinear drag in three-dimensional, homogeneous and isotropic turbulence. These simulations reproduce the slowdown observed experimentally, and show that the mean magnitude of the slip velocity is proportional to the large-scale fluctuations of the flow velocity. Combining the numerical estimate of the slip velocity magnitude with a simple theoretical model, we show that the scaling $\left \langle v_z \right \rangle /v_{q}\propto 1/{\textit {Fr}}$ originates from a combination of the nonlinear drag and the nearly isotropic behaviour of the slip velocity at large ${\textit {Fr}}$ that drastically reduces the mean rise speed.

JFM classification

Drops and Bubbles: Bubble dynamics Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 924 , 10 October 2021 , A2
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Aliseda, A. & Lasheras, J.C. 2011 Preferential concentration and rise velocity reduction of bubbles immersed in a homogeneous and isotropic turbulent flow. Phys. Fluids 23 (9), 093301.CrossRefGoogle Scholar
Allan, D.B., Caswell, T., Keim, N.C. & van der Wel, C.M. 2019 soft-matter/trackpy: Trackpy v0.4.2 (Version v0.4.2). Zenodo.Google Scholar
Barry, D.A. & Parlange, J.Y. 2018 Universal expression for the drag on a fluid sphere. PLoS ONE 13 (4), e0194907.CrossRefGoogle ScholarPubMed
Bel Fdhila, R. & Duineveld, P.C. 1996 The effect of surfactant on the rise of a spherical bubble at high Reynolds and Péclet numbers. Phys. Fluids 8 (2), 310–321.CrossRefGoogle Scholar
Bradski, G. 2000 The OpenCV library. Dr. Dobb's J. Softw. Tools 120, 122–125.Google Scholar
Byron, M. 2015 The rotation and translation of non-spherical particles in homogeneous isotropic turbulence. ProQuest Dissertations and Theses, p. 159. University of California, Berkeley.Google Scholar
Byron, M.L., Tao, Y., Houghton, I.A. & Variano, E.A. 2019 Slip velocity of large low-aspect-ratio cylinders in homogeneous isotropic turbulence. Intl J. Multiphase Flow 121, 103120.CrossRefGoogle Scholar
Calzavarini, E., Volk, R., Bourgoin, M., Lévêque, E., Pinton, J.F. & Toschi, F. 2009 Acceleration statistics of finite-sized particles in turbulent flow: the role of Faxén forces. J. Fluid Mech. 630, 179–189.CrossRefGoogle Scholar
Clift, R., Grace, J.R. & Weber, M.E. 1978 Bubbles, drops, and particles. Academic Press.Google Scholar
Deike, L., Berhanu, M. & Falcon, E. 2012 Decay of capillary wave turbulence. Phys. Rev. E 85 (6), 066311.CrossRefGoogle ScholarPubMed
Deike, L., Lenain, L. & Melville, W.K. 2017 Air entrainment by breaking waves. Geophys. Res. Lett. 44 (8), 3779–3787.CrossRefGoogle Scholar
Deike, L. & Melville, W.K. 2018 Gas transfer by breaking waves. Geophys. Res. Lett. 45 (19), 10482–10492.CrossRefGoogle Scholar
Deike, L., Melville, W.K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.CrossRefGoogle 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
Fornari, W., Picano, F., Sardina, G. & Brandt, L. 2016 Reduced particle settling speed in turbulence. J. Fluid Mech. 808, 153–167.CrossRefGoogle Scholar
Friedman, P.D. & Katz, J. 2002 Mean rise rate of droplets in isotropic turbulence. Phys. Fluids 14 (9), 3059–3073.CrossRefGoogle Scholar
Henderson, D.M. & Miles, J.W. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95–109.CrossRefGoogle Scholar
Homann, H. & Bec, J. 2010 Finite-size effects in the dynamics of neutrally buoyant particles in turbulent flow. J. Fluid Mech. 651, 81–91.CrossRefGoogle Scholar
Kawanisi, K., Nielsen, P. & Zeng, Q.-C. 1999 Settling and rising velocity of a spherical particle in homogeneous turbulence. Proc. Hydraul. Engng 43, 779–784.CrossRefGoogle Scholar
Li, Y., Perlman, E., Wan, M., Yang, Y., Meneveau, C., Burns, R., Chen, S., Szalay, A. & Eyink, G. 2008 A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9, N31.CrossRefGoogle Scholar
Loisy, A. & Naso, A. 2017 Interaction between a large buoyant bubble and turbulence. Phys. Rev. Fluids 2 (1), 014606.CrossRefGoogle Scholar
Machicoane, N., Aliseda, A., Volk, R. & Bourgoin, M. 2019 A simplified and versatile calibration method for multi-camera optical systems in 3D particle imaging. Rev. Sci. Instrum. 90 (3), 035112.CrossRefGoogle ScholarPubMed
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.CrossRefGoogle Scholar
Mathai, V., Huisman, S.G., Sun, C., Lohse, D. & Bourgoin, M. 2018 Dispersion of air bubbles in isotropic turbulence. Phys. Rev. Lett. 121 (5), 054501.CrossRefGoogle ScholarPubMed
Maxey, M.R. & Corrsin, S. 1986 Gravitational settling of aerosol particles in randomly oriented cellular flow fields. J. Atmos. Sci. 43 (11), 1112–1134.2.0.CO;2>CrossRefGoogle Scholar
Maxey, M.R. & Riley, J.J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883–889.CrossRefGoogle Scholar
Maxworthy, T., Gnann, C., Kürten, M. & Durst, F. 1996 Experiments on the rise of air bubbles in clean viscous liquids. J. Fluid Mech. 321, 421–441.CrossRefGoogle Scholar
Mazzitelli, I.M. & Lohse, D. 2004 Lagrangian statistics for fluid particles and bubbles in turbulence. New J. Phys. 6, 203.CrossRefGoogle Scholar
Mei, R. 1994 Effect of turbulence on the particle settling velocity in the nonlinear drag range. Intl J. Multiphase Flow 20 (2), 273–284.CrossRefGoogle Scholar
Mercado, J.M., Prakash, V.N., Tagawa, Y., Sun, C. & Lohse, D. 2012 Lagrangian statistics of light particles in turbulence. Phys. Fluids 24 (5), 055106.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2001 Path instability of a rising bubble. Phys. Rev. Lett. 88 (1), 014502.CrossRefGoogle ScholarPubMed
Nielsen, P. 1992 Effects of turbulence on the settling or rise velocity of isolated suspended particles. In 11th Australasian Fluid Mechanics Conference, University of Tasmania, Hobart, Australia, 14-18 December 1992, paper 2C-2.Google Scholar
Nielsen, P. 2007 Mean and variance of the velocity of solid particles in turbulence. ERCOFTAC Ser. 11, 385–391.CrossRefGoogle Scholar
Park, S.H., Park, C., Lee, J.Y. & Lee, B. 2017 A simple parameterization for the rising velocity of bubbles in a liquid pool. Nucl. Engng Technol. 49 (4), 692–699.CrossRefGoogle Scholar
Perlman, E., Burns, R., Li, Y. & Meneveau, C. 2007 Data exploration of turbulence simulations using a database cluster. In Proceedings of the 2007 ACM/IEEE Conference on Supercomputing, SC’07, paper 23.Google Scholar
Poorte, R.E.G. & Biesheuvel, A. 2002 Experiments on the motion of gas bubbles in turbulence generated by an active grid. J. Fluid Mech. 461, 127–154.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Prakash, V.N., Tagawa, Y., Calzavarini, E., Mercado, J.M., Toschi, F., Lohse, D. & Sun, C. 2012 How gravity and size affect the acceleration statistics of bubbles in turbulence. New J. Phys. 14 (1991), 105017.CrossRefGoogle Scholar
Reichardt, T., Tryggvason, G. & Sommerfeld, M. 2017 Effect of velocity fluctuations on the rise of buoyant bubbles. Comput. Fluids 150, 8–30.CrossRefGoogle Scholar
Rensen, J., Luther, S. & Lohse, D. 2005 The effect of bubbles on developed turbulence. J. Fluid Mech. 538, 153–187.CrossRefGoogle Scholar
Risso, F. 2018 Agitation, mixing, and transfers induced by bubbles. Annu. Rev. Fluid Mech. 50 (1), 25–48.CrossRefGoogle Scholar
Salibindla, A.K.R., Masuk, A.U.M., Tan, S. & Ni, R. 2020 Lift and drag coefficients of deformable bubbles in intense turbulence determined from bubble rise velocity. J. Fluid Mech. 912, A50.CrossRefGoogle Scholar
Snyder, M.R., Knio, O.M., Katz, J. & Le Maître, O.P. 2007 Statistical analysis of small bubble dynamics in isotropic turbulence. Phys. Fluids 19 (6), 065108.CrossRefGoogle Scholar
Spelt, P.D.M. & Biesheuvel, A. 1997 On the motion of gas bubbles in homogeneous isotropic turbulence. J. Fluid Mech. 336 (1–4), 221–244.CrossRefGoogle Scholar
Sreenivasan, K.R. 1998 An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids 10 (2), 528–529.CrossRefGoogle Scholar
Stöhr, M., Schanze, J. & Khalili, A. 2009 Visualization of gas-liquid mass transfer and wake structure of rising bubbles using pH-sensitive PLIF. Exp. Fluids 47 (1), 135–143.CrossRefGoogle Scholar
Taghizadeh-Popp, M., et al. 2020 SciServer: a science platform for astronomy and beyond. Astron. Comput. 33, 100412.CrossRefGoogle Scholar
Thielicke, W. & Stamhuis, E.J. 2014 PIVlab – towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB. J. Open Res. Softw. 2 (1), e30.CrossRefGoogle Scholar
Thorpe, S.A. 1982 On the clouds of bubbles formed by breaking wind-waves in deep water, and their role in air – sea gas transfer. Phil. Trans. R. Soc. Lond. A 304 (1483), 155–210.Google Scholar
Tomiyama, A., Tamai, H., Zun, I. & Hosokawa, S. 2002 Transverse migration of single bubbles in simple shear flows. Chem. Engng Sci. 57 (11), 1849–1858.CrossRefGoogle Scholar
Van Dorn, W.G. 1966 Boundary dissipation of oscillatory waves. J. Fluid Mech. 24 (4), 769–779.CrossRefGoogle Scholar
Variano, E.A. & Cowen, E.A. 2008 A random-jet-stirred turbulence tank. J. Fluid Mech. 604, 1–32.CrossRefGoogle Scholar
Vejražka, J., Zedníková, M. & Stanovský, P. 2018 Experiments on breakup of bubbles in a turbulent flow. AIChE J. 64 (2), 740–757.CrossRefGoogle Scholar
Voth, G.A., La Porta, A., Crawford, A.M., Alexander, J. & Bodenschatz, E. 2002 Measurement of particle accelerations in fully developed turbulence. J. Fluid Mech. 469, 121–160.CrossRefGoogle Scholar
Wang, L.P. & Maxey, M.R. 1993 Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256 (3), 27–68.CrossRefGoogle Scholar