Critical Rayleigh

Velocity study

Unsteady case


Critical Rayleigh

In this paragraph, we are going to study, in a first point, the temperature for which the steady rolls appear. In a second part, we'íll study the meshing influence on the computation.

Determination of Rac

Rac is defined as the number for which the steady rolls appear. So to determined it, we are going to simulate some cases for different temperature and calculate the heat flow between the two walls. Indeed, we are going to use the Nusselt number:


This number is the ratio of two heat flux: the actual and those exchanged as if the exchange was only made by conduction. Then we have:

    • Nu=1, if the flux is exchanged only by conduction

    • Nu>1, if the flux is exchange by conduction and convection

In our case:

    • If any rolls appear, we have only conduction, then Nu=1

    • If the rolls appear, the exchange is made by conduction and convection, so Nu>1

So, to determine Rac, we just have to plot Ra, in function of Nu. This have been made for a grid containing 800 cells (20x40):

We find: Rac=1743

To check our calculations, we plot the contour of temperature at Ra=1763 and Ra=1723.

So, our computations seem to be correct:

    • at Ra=1723, the temperature is linear

    • at Ra=1763, the temperature isn't linear. It's due to the formation of rolls.

Experiment show us that the critical Rayleigh is about 1708. Our computations gave us 1743. So our results are quite good, they are corresponding with the actual Rac ( at about 2%).

Meshing influence

Now, we are trying to determine the influence of the grid on Rac. We have realised new computations on a 10x5 grid. A new critical Rayleigh Rac has been calculated by using the same method:

This time, we find Rac=1565

We check this result by plotting the contour of temperature at Ra=1603 and Ra=1522:



For the same physic case, we didnít found the case result: the grid has a heavy influence on the computations. Then, to find the actual Rac, a investigation would be to lead the simulation on grids more and more fine, until Rac donít depend on the mesh.