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Energy spectra in turbulent bubbly flows

We conduct experiments in a turbulent bubbly flow to study the nature of the transition between the classical $-$5/3 energy spectrum scaling for a single-phase turbulent flow and the $-$3 scaling for a swarm of bubbles rising in a quiescent liquid and of bubble-dominated turbulence. The bubblance parameter, which measures the ratio of the bubble-induced kinetic energy to the kinetic energy induced by the turbulent liquid fluctuations before bubble injection, is often used to characterise the bubbly flow. We vary the bubblance parameter from $b = \infty$ (pseudo-turbulence) to $b = 0$ (single-phase flow) over 2-3 orders of magnitude ($0.01 - 5$) to study its effect on the turbulent energy spectrum and liquid velocity fluctuations. The probability density functions (PDFs) of the liquid velocity fluctuations show deviations from the Gaussian profile for $b > 0$, i.e. when bubbles are present in the system. The PDFs are asymmetric with higher probability in the positive tails. The energy spectra are found to follow the $-$3 scaling at length scales smaller than the size of the bubbles for bubbly flows. This $-$3 spectrum scaling holds not only in the well-established case of pseudo-turbulence, but surprisingly in all cases where bubbles are present in the system ($b > 0$). Therefore, it is a generic feature of turbulent bubbly flows, and the bubblance parameter is probably not a suitable parameter to characterise the energy spectrum in bubbly turbulent flows. The physical reason is that the energy input by the bubbles passes over only to higher wave numbers, and the energy production due to the bubbles can be directly balanced by the viscous dissipation in the bubble wakes as suggested by Lance $\&$ Bataille (1991). In addition, we provide an alternative explanation by balancing the energy production of the bubbles with viscous dissipation in the Fourier space.