Dear Violeta & Sébastien,
we deal at BAW a lot with quasi-stationary flows in river stretches (BCs like inflow flux and outflow water level kept constant), i.e flows in which the overall picture and/or some variables reach an almost stationary state, but here or there oscillations (e.g. eddies) may occur.
I have written once a module (I think it was called, ehm, variat) to be called in the Fortran file which computed the norm of changes in required variables as a sum over all nodes (or maximum change...?) in the last n steps of the computation in order to have a statistically valid measure if the (quasi-)stationary state has been reached or not. (Other usual criteria are fluxes over given cuts along the river.)
Unfortunately I cannot find the code anymore, but maybe someone at BAW still has it. It would be easy to reprogram it to get sound stopping criteria.
Best regards,
jaj