Energy cascade in a homogeneous swarm of bubbles rising in a vertical channel

We perform direct numerical simulations (DNS) of millimeter-size air bubbles rising in a vertical water channel. Our objective is to study the energy cascade in this simple configuration, which is a prototype of other more complex flows involving bubbles and droplets. Both rigid and deformable bubbles are considered using the immersed boundary method (IBM) and the volume of fluid (VoF) method. The energy cascade is quantified using the nonlinear interscale energy transfer term appearing in the Karman-Howarth-Monin (K-H-M) equation. We observe that the energy cascade is highly anisotropic in scale space with vertical turbulent eddies exhibiting strong inverse cascades. When scale-normalized, the interscale energy transfer term shows a plateau for turbulent eddies whose size is approximately 1 to 2 bubble diameters. However, the magnitude of that term is only 50–60% of the required viscous dissipation, indicating that the classic forward-cascade picture by Richardson and Kolmogorov is not satisfied despite the existence of a scale-invariant range.