CFD Online Discussion Forums - Monitor of max velocity is const but keep on increasing in avg velocity.

Hello all,
So my brief case set-up:
Steady-state, Multiphase (Air and liquid) (VOF model), Realizable k-e turbulence model.
Three monitors: Mass-flow rate at outlet, Avg velocity magnitude in liquid domain, Max velocity magnitude in liquid domain

So, before jumping into directly transient simulations I am running steady-state simulations first. After 9500 iterations, what I had observed is that monitor of max velocity magnitude is almost constant but profile of volume avg velocity is keep on increasing.
Interestingly enough, even residuals are now in order of 10^(-3).
What does this result signify?

From what I understand is that it is because turbulent eddies generated within domain as simulation progresses.

In such case, what would be the right time to stop steady-state simulation ? or should I revisit the case set-up and change the turbulence model.