As many other persons, we tested the influence of the mesh, specially on the critical Rayleigh number. Our results lead to the same conclusions. Indeed, the more you increase the number of nodes, the closer to the theorical critical Rayleigh number you get. However, you always remain under the theorical number, so as you decrease the number of nodes, you decrease the numerical critical Rayleigh number.
From a mesh composed of about 500-1000 nodes, you start to have quite good results. But it is not very interesting, at least for our utilisation, to have very precise meshes. Indeed, between two meshes of 2500 nodes and 5000 nodes, the differences are very small concerning the results but of course, the more precise mesh takes much more time to converge.

As a remark, we can say that we tried to get closer and closer from the critical Rayleigh number. Of course, the calculations take much more time as it is difficult for the solver to satisfy the convergence criterium.
Here are some visualisations of simulations made close to the theorical critical Rayleigh number. We can observe how the instability starts.