Scale-by-scale energy transfers in bubbly flows

Buoyancy-driven bubbly flows naturally have spatially dependent density fields, which allow for multiple definitions of the scale-dependent (or filtered) energy. A priori , it is not obvious which of these provide the most physically apt scale-by-scale budget. In the present study, we compare two such definitions, based on (i) filtered momentum and filtered velocity (Pandey et al., J. Fluid Mech ., 2020, vol. 884, p. R6), and (ii) Favre-filtered energy (Aluie, Phys. D: Nonlinear Phenom. , 2013, vol. 247, pp. 54–65; Pandey et al., Phys. Rev. Lett. , 2023, vol. 131, p. 114002). We also derive a Kármán–Howarth–Monin relation using the momentum–velocity correlation function and contrast it with the scale-by-scale energy budget obtained in (i). We find that, for the volume fraction and Atwood number explored, irrespective of the definition, energy transfers due to the advective nonlinearity and surface tension are identical. However, discrepancies arise for the buoyancy and pressure contributions. We show that the Favre-filtered definition is the more appropriate choice, within which buoyancy injects energy, pressure transfers energy to large scales and both advective nonlinearity and surface tension transfer energy downscales where it is dissipated by viscosity.