Some notes on eddy viscosity in wall-bounded turbulent bubbly flows

Recently, based on data from Direct Numerical Simulations (DNS), Ma et al. (Phys. Rev. Fluids 2, 034301, 2017) proposed a model for closing the bubble-induced turbulence (BIT) in a typical Euler-Euler two-equation model, which appears to yield improved performance for predicting $k$ and $\varepsilon$ over the previous models. The present study departures from this BIT model and purpose to use the same DNS data to investigate the behavior of the $C_\mu$ constant and standard eddy viscosity definition. It can be shown that $C_\mu$ constant computed using the DNS database has a very different behavior than that in single-phase flow. Checking closely, the deficiency originates from the description of the standard eddy viscosity that is intrinsic to this general hierarchy of Euler-Euler $k-\varepsilon$ type model, hence, cannot be overcome by a more complex correction function for $C_\mu$. Departing from this point, a modification to the definition of the eddy viscosity in bubbly flows is derived for the Euler-Euler two-equation models. The new expression is based on the bubble length-scale and its corresponding velocity scale. We focus on the intermediate region -- a region extended from the core region, where bubble-induced production and dissipation are nearly in balance, and find that the modified model can lead to significantly improved predictions for the mean liquid, when compared with DNS data.