Number of iterations for convergence.
 

The purpose of this section is to show that we can determine Rac just by looking at the convergence history.

We noticed that, when we made computations for high Rayleigh numbers, these computations were faster than when we were close from Rac.
We decided to check this assumption and to see if it was possible to determine the critical Rayleigh number from this observation.

We have made several computations on the same mesh and for different Rayleigh number, and just by looking at the convergence history, we tried to estimate the number of iteration that was recquired for convergence.
As it has been explained earlier, this convergence history can always be decomposed in four steps.
The main difficuly was to decide when convergence was reached.

In fact, as you can see here, when Ra is going closer and closer to Rac, these four steps are not easy to distinguish because we continuously go from a super-critical to a sub-critical behaviour.
 

Nevertheless, as long as it was possible to clearly determine the number of iterations recquired for convergence, we tried to estimate this number and we gather our results in the plots above.
 

We can clearly see that when Ra is decreasing, this number of iterations increases very rapidly.
We can then interpolate an asymptot to this curve and determine this way a value for Rac.

This method gives us roughly Rac~1560.
 

Conclusion:

The main advantage of this method is that it is very visual and very fast.
In fact, it's the first thing to do when making a computation: looking at the convergence history to check if we are in the good zone (sub or super critical).
Just in a glance, you can determine if your run has a chance to be good or if it is totally wrong.

On the other hand, it is not very precise because it is not easy to clearly determine the limit between the two convergence behaviour.
Also, it is not really obvious to draw precisely the asymptot of the above graph and we can globally say that this method suffers from a big lack of  precision.