The problem is defined for a Rayleigh number Ra= 4800, which is 2.8 times the critical Rayleigh number. The results obtained for the different variables are sumed up in the following table.
Fluid 1 Fluid 2 Fluid 3 Fluid 4 Fluid 5 Pr 7 0.74 0.02 13 460 100 Number of rolls 3 3 2 2 2 Vx calculated
Tmax max 0.147 0.154 0.0875 0.182 0.182The number of rolls
With the Rayleigh number of the problem 2 or 3 rolls can appear. But it seems that the Prandtl number does not influence rolls since two of them only appear for very high and very low Prandtl numbers.Velocity
The velocities involved in the chosen geometry are quiet slight but very different comparing each other( between 1e-6 and 1e-3).
The results obtained with Fluent are quiet good and it can be remarked that they are better and better as soon as the Prandtl number is big: the results are really accurate for Pr>100 while they seems less good for fluid 3 ( The theoretical result is 4 times the calculated one). Anyway the order of results approximately remains reasonable.
For example, click here to see velocity profiles for fluid2
In a quantitative way the velocities are less important when fluids are viscous and that is very logical. As the Prandtl number stands for the ratio of the viscous stresses and the heat transfert power, it normal to have low velocities for fluid 4 (that have the same behaviour as oil with regards to thermal instabilities) and higher ones for fluid 3 ( that have a behaviour of mercury).
It also seems that results given by Fluent are less isotropic than the real ones.So theoretically Vx max/Vy max = 0.987
And it is not what is encountered in all the calculated cases.
The temperature fields are quiet different because they are functions of the velocities. For example the temperature field of fluid 3 is nearly the contrary of the one of the fluid 4 because the sense of roation of rolls are not the same. In fact it happens a stream like what can be encountered in oceans. The velocity field drives the temperatures that it meets in its sense.
The function Tmax that is defined in a previous page (click here to go to the definition) gives some interesting informations that are not perceptible looking at the temperature field.