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 (vz) decreases with the intensity of the turbulence as characterized by its root-mean-square fluctuation velocity, u'. Non-dimensionalized by the quiescent rise velocity vq, the average rise speed follows {vz) /vq ∝ 1/Fr at high Fr, where Fr = u'/√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 (vz) /vq ∝ 1/Fr originates from a combination of the nonlinear drag and the nearly isotropic behaviour of the slip velocity at large Fr that drastically reduces the mean rise speed.