  In nature and engineering applications, plunging jet acts as a key process
causing interface breaking and generating mixed-phase turbulence. In this
paper, high-resolution numerical simulation of water jet plunging into a
quiescent pool was perform to investigate the statistical property of the
mixed-phase turbulence, with special focus on the closure problem of the
Reynolds-averaged equation. We conducted phase-resolved simulations, with the
air--water interface captured using a coupled level-set and volume-of-fluid
method. Nine cases were performed to analyse the effects of the Froude number
and Reynolds number. The simulation results showed that the turbulent
statistics are insensitive to the Reynolds number under investigation, while
the Froude number influences the flow properties significantly. To investigate
the closure problem of the mean momentum equation, the turbulent kinetic energy
(TKE) and turbulent mass flux (TMF) and their transport equations were further
investigated. It was discovered that the balance relationship of the TKE budget
terms remained similar to many single-phase turbulent flows. The TMF is an
additional unclosed term in the mixed-phase turbulence. Our simulation results
showed that the production term of its transport equation was highly correlated
to the TKE. Based on this finding, a closure model of its production term was
further proposed.
