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We describe the rising trajectory of bubbles in isotropic turbulence and quantify the slowdown of the mean rise velocity of bubbles with sizes within the inertial subrange. We perform direct numerical simulations of bubbles, for a wide range of turbulence intensity, bubble inertia and deformability, with systematic comparison with the corresponding quiescent case, with Reynolds number at the Taylor microscale from 38 to 77. Turbulent fluctuations randomise the rising trajectory and cause a reduction of the mean rise velocity$$\tilde {w}_b$$compared with the rise velocity in quiescent flow$$w_b$$. The decrease in mean rise velocity of bubbles$$\tilde {w}_b/w_b$$is shown to be primarily a function of the ratio of the turbulence intensity and the buoyancy forces, described by the Froude number$$Fr=u'/\sqrt {gd}$$, where$$u'$$is the root-mean-square velocity fluctuations,$$g$$is gravity and$$d$$is the bubble diameter. The bubble inertia, characterised by the ratio of inertial to viscous forces (Galileo number), and the bubble deformability, characterised by the ratio of buoyancy forces to surface tension (Bond number), modulate the rise trajectory and velocity in quiescent fluid. The slowdown of these bubbles in the inertial subrange is not due to preferential sampling, as is the case with sub-Kolmogorov bubbles. Instead, it is caused by the nonlinear drag–velocity relationship, where velocity fluctuations lead to an increased average drag. For$$Fr > 0.5$$, we confirm the scaling$$\tilde {w}_b / w_b \propto 1 / Fr$$, as proposed previously by Ruthet al.(J. Fluid Mech., vol. 924, 2021, p. A2), over a wide range of bubble inertia and deformability.
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The effect of nonlinear drag on the rise velocity of bubbles in turbulence
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.
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- Award ID(s):
- 1844932
- PAR ID:
- 10308644
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 924
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2021.556
