Determination of critical Rayleigh number

The fundamental parameter describing the convective state of fluid is the Rayleigh number, it balances the drag force that works against convection and the Archimede force the works for the convection. If this number exceeds a critical value the convection sets in. It is defined by the following expression:

In this expression we can notice that the variation of this number is only function of the temperature because the other variables only depends of the fluid. When the Rayleigh-Benard's instability develops the heat transfer increases suddenly because the convection sets in. At this moment, the derivative of the velocity has an infinite value and we can determine the .
We can see below the velocity variation with the Rayleigh number.

We tried to study the critical Rayleigh number with the refined mesh but the convergence was much more difficult and the calculation time was really too long. Furthermore the appearance of the convection rolls seems to occur later than with the "rough" grid. For example, the convection has not yet started at a Rayleigh of 2000 with the 50x100 mesh, while convection starts at 1900 with the 20x40 mesh.
The next chart presents the results obtained using a first order upwind scheme for solving the energy and momentum equations, while for the previous one we used a "QUICK" scheme. We think that the upwind scheme introduces too much numerical viscosity hence the calculus converges towards a non physic solution.

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