Critical Rayleigh number

I . Definition

We have seen in a previous part that the Rayleigh-Benard instability is directed by three main mecanisms :

 Archimede's force Thermic diffusion Drag force

This competition of forces is parameterized by the Rayleigh number, which is the temperature difference, but appropriately normalized to take into account the geometry of the convection cell and the physical properties of the fluid.

If the Rayleigh number is greater than 1708, then convection occurs. If it is below this value called the critical Rayleigh number Rac, there is no convective flow.
If nothing else is specified, d ( distance between the two walls ) will always be 1 cm.

II . Determination - Grid influence

As the Rayleigh number is function of the geometry of the convection cell, the determination of the numeric Rac has to be done with different kind of mesh.

I . Structured grid
The grids used are such as the followings :

The following table summarises the differents results :

 Size ( nodes ) Rac 5 * 5 1390 10 * 10 1560 20 * 20 1641 40 * 40 1685 80 * 80 1711

We can see that the value of Rac decreases as well as the number of nodes. This phenomena can be explained by the fact that the flux at the wall is determined with the temperature at the wall and the one at the middle of the first mesh. So the estimation of the turbulent diffusion is reduced to the calcul of a difference of temperature. But the temperature law is near the wall , as well as the velocity field, parabolic. Thus the diminution falses the evaluation to the gradients of temperature and makes the Rac decreases.

We could do this study with a 3D grid to see if an other dimension influenced the computations such as :

II . Unstructured grid

The mesh used is :

Here the numerics control PRESTO! can cont de used . The results are for a Rayleigh number about 2214 :

As the velocity vectors near the walls can go through it , we gave up this option after not having found an appropriate numerics control.