If the plotted residuals are found to diverge or fluctuate significantly, in many cases you will find that the situation can be improved by reducing the false time steps for the velocity components. The recommended procedure is to continuously half the time steps until a more stable solution is found.
In some cases a solution can be very difficult, if not impossible, to achieve using the standard k-e turbulence model. Particularly difficult problems involve very low velocity buoyancy-driven flows where diffusion is dominated by convection and this can cause velocities to fluctuate around a mean value. In these cases, reducing false time steps may not improve the situation. In order to force a solution, you may consider introducing additional boundary conditions such as occupants, which can change the flow sufficiently to obtain a solution without the problem necessarily becoming unrealistic.
Alternatively, you may consider using the constant effective viscosity turbulence model with a relatively high turbulent viscosity or turbulent viscosity multiplier but bear in mind that this will introduce an artificially high level of diffusion into the flow and also, the constant effective viscosity model is incapable of modelling the transport of turbulence.
If the residuals have dropped to very low levels and the monitored variables are fairly constant (they don't have to be absolutely constant), then you can be confident that you have a good solution even though the lack of convergence message is displayed. The message is always displayed even if the residuals are very slightly above the termination values. These are conservative values and don't need to be strictly adhered to.
You should only need to reduce the velocity false time steps if you experience a very unstable solution which exhibits diverging residuals or high amplitude oscillations. The time steps have a 'damping' effect on the solution which can increase stability but increases the overall simulation time. Generally, if you experience a solution that doesn't converge easily where the residuals appear to remain constant, it's better to re-structure the model definition by removing unnecessary obstructions and aligning any repeating component/assemblies with each other in order to reduce the number of grid lines.