The turbulent kinetic energy budget in a bubble plume

Abstract

We present the turbulent kinetic energy (t.k.e.) budget of a dilute bubble plume in its asymptotic state. The budget is derived from an experimental dataset of bubble plumes formed inside an unstratified water tank. The experiments cover both the adjustment phase and asymptotic state of the plume. The diameters $d$ of air bubbles are in the range 1–4 mm and the air void fraction $\unicode[STIX]{x1D6FC}_{g}$ is between 0.7 % and 1.8 %. We measured the three components of the instantaneous liquid velocity vector with a profiling acoustic Doppler velocimeter. From the experiments, we found the following inside the heterogeneous bubble core of the plume: (i) the probability density functions of the standardized liquid fluctuations are very similar to those of homogeneous bubble swarms rising with and without background liquid turbulence; (ii) the characteristic temporal frequency $f_{cwi}$ at which bubbles inject t.k.e. into the liquid agrees with the prediction $f_{cwi}=0.14u_{s}/d$ observed and theoretically derived for homogeneous bubble swarms ($u_{s}$ is the bubble slip velocity); (iii) the liquid turbulence is anisotropic with the ratio of turbulence intensities between the vertical and horizontal components in the range 1.9–2.1; (iv) the t.k.e. production by air bubbles is much larger than that by liquid mean shear; and (v) an increasing fraction of the available work done by bubbles is deposited into liquid turbulence as one moves away from the plume centreline. Together with the existing knowledge of homogeneous bubble swarms, our results of the heterogeneous bubble plume support the view that millimetre-sized bubbles create specific patterns of liquid fluctuations that are insensitive to flow conditions and can therefore be possibly modelled by a universal form.