1) DefinitionRayleigh number Ra is the balance between destabilisation forces (Archimede force) and the stabilisation ones (drag forces):
It is known that when the Rayleigh number Ra exceeds a certain critical value Rac = 1708, stationary contrarotatifs rolls occurs.
Then we have two cases :
2) Determination of critical Rayleigh with NusseltThis number is the ratio between the heat flux really exchanged through the fluid and the heat flux exchanged with conduction :
The jump of Nusselt number value from 1 to a greater value
corresponds to the critical value of Rayleigh number. So, to determinate
the critical number Rac, we have to draw for many Rayleigh numbers (i.e many values of the temperature gradient), the variation
of Nusselt number.
We get the following graph : (go to the table of values)
Variation of Nusselt number with Rayleigh number
At first, we notice, as it seems logical, that the heat
flux increase with the convection. We obtain a critical Rayleigh number
Our estimation could be better by performing the parameters of the numerical simulation such as the grid, mesh, the number of iterations, the relaxation parameter...
3) Determination of critical Rayleigh with Vmax
There is an another way to obtain the critical rayleigh number by drawing the graph of Vmax in fonction of the rayleigh number (we calculate the Vmax value for a given value of the temperature gradient). We get the following results :
Variation of Vmax with Rayleigh number
As expected, there is almost no variation of Vmax at the
beginning and then it grows up very quickly, which corresponds to the appearance
of rolls. The critical rayleigh number corresponds to the take off of Vmax
and it is approximatively egual to1800. Moreover, we think it is possible
to perform the accuracy of this number for the same reasons as previously.
4) ConclusionIn two different ways, we found a critical Rayleigh equal to : Rac ~ 1800-1850. Since the theoretical value is 1708 and also since this value is already an approximation, the results we got are quite good.