Ansys Employee

The monitor isn't a residual. It's a report from Fluent of a value at every iteration/time step. When it's level it's a good sign that the model isn't changing at that location. However, as you're monitoring an average you may be masking localised changes in value. 

Residuals are the solver reports and are a function of how much the overall solution is changing. 

Convergence is typically OK when monitors stop changing, residuals drop by the required number of orders of magnitude and fluxes balance. For a transient run the extra requirement is that previous time steps have converged otherwise you risk a model that's wrong but returning to "correct" over time, so at some point it will be OK. If you check the AIS courses (Learning on here) there will be something covering this in the Fluids/CFD modules.