Determination of the critical Rayleigh number

    The instabilities of Rayleigh-Benard are only present when the difference of temperature between the plates is sufficient. However, it is difficult to define the critical temperature difference from a visual point of view. For example, we cannot say that the instabilities are only present when we see rolls. We use the Nusselt number (Nu), that is the ratio between the total heat flux and the diffusion heat flux.

When the Nu is equal to 1, it means that all the heat transfer is made by the diffusion.

- The diffusion term in fluids is equivalent to the conduction term with solids and can easily be calculated with the Fourrier law:

where Phi is the surfacic heat transfer and lambda the thermal conductivity. The flux is reduced in our case to:
 

     as the thermal gradient is constant

As soon as there is a movement of the fluid particles at the macroscopic level, it creates a heat flux due to convection. As a consequence, Nu become greater than one.

- We determine the value of the total heat flux using one property of Fluent.
 

       Report => Fluxes

A window appears and we can choose to compute the Total heat transfer rate at a Wall, and the result is given in W.
 

Using both calculation formerly defined, we can plot the value of the Nusselt number depending on the Rayleigh number (equivalent to the difference of temperature) for several configuration.
 


    We see on the graph above that the critical Rayleigh number is approximately 2000. This is the result given by Fluent. The theoretical critical Ra is 1708.
The difference can be explained by the sensibility of the results to several conditions, such as the initial conditions. In our case, the initial conditions are a constant temperature in the domain, equal to the average between the walls temperatures, and a velocity equal to zero.

Furthermore, there is an approximation inherent to the solving. We can have an example of it when we compare the heat fluxes on each walls. They are supposed to be equal, as there is a symmetric condition on the vertical boundaries, but the computation given by Fluent shows some differences.