According to us, the first conclusion that we could learn from this hands-on is that it is not obvious to determine precisely a critical number (i.e. associated to a transition of the flow) with an industrial code: there are lots of models (numerical schemes, physical models, ...), and there are also several techniques to get a value from the same set of datas.

What we have retained from all these computations may be summed up this way:

The mesh influence is very important. Typically, when the number N of cells in the mesh increases, the result (velocity, temperature,...) at a given point tends to a limit value, but the number of iterations required for the achievement of the convergence increase like N too. Moreover, the computation time incresases like (Number of cell)²x(Number of iterations).
Finally, the CPU time to observe the convergence on a N-cells mesh increases like N³, for a given e=(Ra-Ra_c)/Ra_c.
This is a very fast evolution law and it seems to us very difficult to achieve an accurate determination of Ra_c, because the number of iterations required to converge increases a lot when e tends to zero.

We can sum this up into a single formula:

T_CPU ~ N³/e.

Concerning the different methods we proposed, they all have different advantages and drawbacks. We recall them to the mind of the reader below.

• The one based on the analysis of the convergence history is very fast. Though, it is not very precise and should only be used as a way to detect if the motion is clearly super-critical or might be trans- or sub-critical.
• The one based on the number of iterations needed to achieve convergence is based on the previous method and so, is not very precise too. However, by plotting this number of iteration versus the Rayleigh number, it is more easy to determine the asymptot and so the critical value of Ra. In reality, this method can only give us a maximum value for Ra_c.
• The one based on the evaluation of velocity or temperature at a point after convergence is clearly better. It is much more accurate but the major drawback is that it is very time-expensive. Moreover, for high Prandtl number, the variations of the temperature are more easy to observe than the velocity variations, and vice and versa for a low Prandtl number.
• The last one, based on the observation of the evolutions of T and V at a point seems to be the best one, according to us. Actually, one of the previous method drawback was also that when coming closer and closer to Ra_c, after a given number of iteration, it was not easy to decide if the case was really sub-critical or if it was a very slowly evoluting super-critical case. This method allowed us to decide if the case was sub or super critical, by observing the evolution of the results during the convergence process.

Finally, the best method to obtain an accurate value of Ra_c in a short amount of time may be to use this last method on a small mesh, and then to interpolate the "real" result by scaling the values we obtained thanks to a mesh-correction factor, which could be found with a single computation on the coarse grid.