RANS
The first one is the dispersed phase NS equation, I used the CLSVOF method to capture the interface .The second one is the continuous phase NS equation, which requires lots of empirical coefficients to represent the drag force, virtual force, lift force due to the absence of surface tension term.
Two Phases
In the research, we deal with the dispersed phase and the continuous phase separately to compensate for the inaccuracy caused by the limited resolution which cannot capture those extreme small bubbles. Therefore, we introduce the population balance equation. We use the CLSVOF method to capture those relative bigger bubbles and population balance equation will govern those small bubbles. This method can deal with the all size bubble without requiring much more calculation resource by increasing the resolution.
Drag coefficient
According to the Navier-Stokes equation for continuous phase, we can get the terminal bubble rising speed based on the drag coefficient in different water purity and compare them to the benchmark.