Progress in the second-moment closure for bubbly flow based on direct numerical simulation data

Data from direct numerical simulations (DNS) of disperse bubbly flow in an upward vertical channel are used to develop a new second-moment closure for bubble-induced turbulence (BIT) in the Euler-Euler framework. The closure is an extension of a BIT model originally proposed by Ma et al. (Phys. Rev. Fluids, vol. 2, 2017, 034301) for two-equation eddy-viscosity models and focuses on the core region of the channel, where the interfacial term and dissipation term are in balance. Particular attention in this study is given to the treatment of the pressure-strain term for bubbly flows and the form of the interfacial term to account for BIT. For the latter, the concept of an effective BIT source is proposed, which leads to a significant simplification of the modelling work for both the pressure-strain correlation and the interfacial term itself. The anisotropy of bubbly flow is analysed with the aid of the anisotropy-invariant map obtained from the DNS data, and a parameter governing this issue is established. The complete second-moment closure is tested against reference data for different bubbly channel flows and a case of a bubble column. The agreement achieved with the DNS data is very good and the performance of the new model is better than obtained with the standard procedure. Furthermore, the model is shown to be robust and to fulfil the requirements of realizability.