An inverse energy cascade in two-dimensional low Reynolds number bubbly flows

Two direct numerical simulations of several buoyant bubbles in a two-dimensional periodic domain are presented. The average rise Reynolds number of the bubbles is close to 2, and surface tension is high, resulting in small bubble deformation. The void fraction is relatively high, and the bubbles interact strongly. Simulations of the motion of both 144 and 324 bubbles show the formation of flow structures much larger than the bubble size, and a continuous increase in the energy of the low-wavenumber velocity modes. Plots of the energy spectrum show a range of wavenumbers with an approximately -5/3 slope. This suggests that a part of the work done by the buoyant bubbles is not dissipated, but instead increases the energy of flow structures much larger than the bubbles. This phenomenon, which is also seen in numerical simulation of forced two-dimensional turbulence, prevents the appearance of a statistically steady-state motion that is independent of the size of the computational domain.