As we decided to study the influence of the mesh on differents aspects of the solution,  we open Fluent in its double precision version (version>2ddp) in order to avoid errors due to the lack of precision of the code. This way, it is easier to compare the effects of the different meshes or schemes which are predominant over precision effects in the double precision mode.

In all the simulations, the option PRESTO! was used . Indeed, it is specified that for flows with high Rayleigh number natural convection, the pressure interpolation scheme  should be the PRESTO! scheme.

The  fluid we used for the numerical simulations is water. In order to take into account the buoyancy effects, which are of course necessary for this study as  they directly cause the Rayleigh-Benard instabilities, we took the Boussinesq approximation in the definition of the density. The Boussinesq approximation is employed to simplify the equations. Indeed, when the density varies by thermal dilatation, there are some situations in which you can only take into account this variation in the term concerning the forces of gravity and consider denisty as a constant everywhere else. It's the origin of Boussinesq approximation, which is in fact a simplification of the global model for viscous fluid with density contrast (which leads to the forming and the development of the convection phenomenon) but the complete model of the compressible fluid is not solved. The density depends on temperature in a polynomial form as follows:

In Fluent, the different parameters taken for the choice of the material are:

In order to study the influence of the mesh, we decided to  keep on studying always the same physical situation. The geometry of the problem is a rectangular box of 2cm lenght and 1cm wide.

As we always keep the same geometry and the same fluid characteristics, the only parameter we changed in order to vary the Rayleigh number is the temperature difference between the top and the bottom of the box. We can express the Rayleigh number as a direct function of the temperature difference as follows: