Evaluation of T and V after convergence.


At first we fix a convergence criterion e.g. 1.e-3. The main goal is to mesure the temperature and velocity at a location which caracterizes the regime. We choose the center of the box because this location shows the greatest variations for V or T between under-critical and over-critical regimes.
However the aspect of residuals curves seems to be regular in each regime :
Typical shape of residuals curves for under-critical regime Typical shape of residuals curves for over-critical regime


We noticed for under-critical regime the no convergence of the computation even after extended run.
Hence we decide to take into acount the mean value of T and V on the set of iterations for which the residuals are levelled.

1) Evaluation of V.

Here we use the value (V) of the vertical component of the velecity at the box center point to caracterize the critical Rayleigh number. The evolution of V vs Ra number is drawn below. We use the tangent at the inflexion point of the experimental curve to extrapolate the curve to V = 0. This gives an estimated value of the critical Ra.


We can plot the Velocity vs Ra number :

The evaluation of the critical Rayleigh number Rac is made graphically using the inflexion point of the curve. The discrete set of points does not permit us to determine the slope at the inflexion point. Then we use the two  lines defined by the neighbouring segments around the discrete inflexion point.


We find 1572 < Rac < 1578 . This low value compared to the admitted value of 1708 is due to the poor mesh refinement whose effect has been describe in a previous report (Cayrol & Perchat):


2) Profile of T .

We defined a dimensionless temperature to be able to compare measurement of T :


The Rac is defined when Tadim=0.5 at the box center because it correspond to the value for linear temperature gradient.
We notice the flatness ot temperature profile when dT tends to critical value.


We find Rac =1565 for the 05x10 mesh.

3) Conclusion.

The first remark is that we find different Rac for the two methods. But the difference is less than 1% between the two values.

Determination of Rac with V.
We can see that the method is not very accurate.
The method tends to under-estimate the Rac because the extrapolation method is not sufficiently accurate.
Determination of Rac with Tadim.
This method seems to be more reliable. We seek the value of Tadim wich may tend to 0.5 for critical and under-critical Ra.
Nevertheless this yields to a slightly lower value of Ra than using V.