Deciding that a solution is finished takes more experience than starting one. The most common assumption is this: the residuals fell below some threshold, so the job is done. On its own that tells you nothing [1] [2].
What a residual actually measures
A residual measures how far the discretised conservation equations are from being satisfied. It is a scaled quantity, and the scaling is taken from the initial condition. Start from a poor initial guess and the residuals will "fall" quickly — that fall says nothing about the accuracy of the solution.
Start the same problem from two different initial conditions: the residual curves look completely different and the solution is the same. That is how weakly the usual criterion relates to what you actually want to measure.
The three criteria
1. Residuals must have fallen and levelled off. The behaviour matters more than the value: if they are still falling the solution has not settled; if they oscillate the physics may be time-dependent.
2. The monitored quantities must be steady. Record the quantities that will drive the decision — head, drag force, outlet temperature, pressure drop — throughout the iterations. If the residuals are flat but these quantities are still drifting, the solution has not converged.
3. Mass and energy balances must close. The difference between what enters and what leaves should be below one part in a thousand of the inlet flow, and the same for energy. If the balance does not close, every remaining number is suspect.
Oscillation is not always a defect
If a monitored quantity oscillates with a regular amplitude, that may be the physics rather than a numerical fault: vortex shedding, blade passing, sloshing. Insisting on a steady-state solver here is a mistake; the problem is time-dependent and should be solved that way.
The practical way to tell them apart: refine the mesh and lower the relaxation factors. If the amplitude shrinks it is numerical; if it does not, it is physical.
What belongs in the report
Residual curves alone are not evidence. Include the iteration history of the monitored quantities, the mass and energy balance table, and a statement of what the convergence criterion was.
References
- Ferziger, J. H., Perić, M. & Street, R. L. (2020). Computational Methods for Fluid Dynamics, 4th ed. Springer. DOI: 10.1007/978-3-319-99693-6
- Celik, I. B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H. & Raad, P. E. (2008). Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. Journal of Fluids Engineering, 130(7), 078001. DOI: 10.1115/1.2960953