Determination of the critical Rayleigh number

1) Definition

Rayleigh number Ra is the balance between destabilisation forces (Archimede force) and the stabilisation ones (drag forces):

Where :

is the viscosity of the fluid
a_th is the thermal diffusivity
is thermal expansion coefficient
is the temperature difference across the plates
d is the layer thickness
g is the gravity

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 :

Ra < Rac : there is no convection, only conduction and steady rolls can't be observed. Besides, the evolution of temperature is linear. We have taken the example of Ra = 615 (=0.05 K):

Velocity vectors (m/s):

Temperature (K):

Ra > Rac : the exchange is made by conduction and convection and rolls appears. The evolution of temperature becomes non-linear. Here is presented the case with Ra = 6145 (= 0.5 K):

Velocity vector (m/s):

Temperature (K):

2) Determination of critical Rayleigh with Nusselt

This number is the ratio between the heat flux really exchanged through the fluid and the heat flux exchanged with conduction :

where :

is the real heat flux through the fluid
is the thermal conductivity
is the temperature gradient
e is the layer thickness
S is the area of one wall (e * 1m)

Two cases can be extracted :
-> the flux is exchanged only by conduction then Nu =1 and it is impossible to observe steady rolls
-> the heat flux is exchanged both by conduction and convection then Nu > 1 and steady rolls appear

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 of about1850.
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) Conclusion

In 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.