III. Determination of the critical Rayleigh number

At first, we have decided to determinate the critical Rayleigh number to compare with theory.
The Rayleigh number is definied by:

It is caracteristical of the stability of the system.
We can plot the variation of the Rayleigh number depending on the wave number, the courb obtained is:

We can see that a critical number exists and its theorical value is Rac = 1708.
When the Rayleigh number is lower, there is stability because the viscosity and the heat diffusion tend to stabilize the different perturbations.
When the Rayleigh number is higher, instabilities appear.

We tried different ways to determinate the critical Rayleigh number .

 Evolution for each method
 

1. From a high Rayleigh number with steady option

We work at first with the steady option.

Our method was simple, we start a first simulation with a high Rayleigh number which means a high difference of temperature between the two walls.
After iterating, we observed the formation of rolls.
So, we have decreased the difference of temperature and have reiterated with last results as initial conditions(here, with rolls as initial conditions).
For each simulation, we determinate the Rayleigh number and plot the velocity magnitude which permits us to determinate the stability of the simulation.
We stopped the simulations when we were sure that the system was stable with a gradient of temperature as results.

So, we could plot the maximal velocity magnitude depending on Rayleigh number to determinate the critical Rayleigh number.
We obtain the following results:  critical Rayleigh number Rac1=1400

2. From low Rayleigh number with steady option

We did the same simulations but in the reverse way.
We start the first simulation with a low Rayleigh number which means a low difference of temperature between the two walls.
We observed a gradient of temperature.
So, this time, we increased the difference of temperature and we always reiterated with last results as initial conditions(here, with gradient as initial conditions)

We obtained the following results:    Rac2= 1800

3. Comparison between the two methods with steady option

As we can see, we have found two different critical Rayleigh numbers depending on the way and the different initial conditions that we started.

We can see that when we started with initial rolls and we decrease the Rayleigh number, we found a lower critical Rayleigh number than the theorical one.
That means that instability is visible longer.

Conversely, when we started with a gradient,, we found a higher Rayleigh critical number than the theorical one.
That means that the system stayed longer stable.

Theorically, we must find the same critical Rayleigh number. However, it seems that the system has a hysteresis phenomenon.

Two reasons can explain this phenomenon:

At first, that could show that Fluent could be influenced by initial conditions. Fluent would give different results with different initial conditions.

Second, the problem could be due to the fact that Fluent needs to calculate longer when the Rayleigh number is closed to the critical one. Moreover, the convergence criterion used could be not suffisant to permit Fluent to calculate exactly the solution.(If we had more time, we could change and add more precision to this criterion).

4. From high Rayleigh number with unsteady option

The  unsteady option integrates the time variable in the calcul, so, we had to iterate a long time to obtain results closed to the same ones obtained with steady option.
It is important to note that all simulations had been done with the same simulation time.

We obtained the following results:    Rac3 = 1300

5. From low Rayleigh number with unsteady option

We obtained the following results: Rac4 = 1800

6. Comparison between the two methods with unsteady option

We obtained the same results than formerly: a hysteresys phenomenon appears.

 
7. Comparison between the two options unsteady and steady.

Theorical Rayleigh critical number Rac = 1708

Recapitulative table:
 

Method	High Rayleigh number / steady option	Low Rayleigh number / steady option	High Rayleigh number / unsteady option	Low Rayleigh number / unsteady option
Rac	1400	1800	1350	1800

We can see that the results are not exactly identical.

When we start with two rolls with unsteady option, we found a critical Rayleigh number which was lower than the same simulations with steady option.
That means that the initial conditions with rolls need more time to be transformed in temperature gradient.
Actually, the results we obtained are not enough closed from the stationnary solution. We had to increase the time of simulations near the critical Rayleigh number to find better results.

To conlude, if we consider that Fluent can not give different results with different initial conditions (Which is true , in theory, for Rayleigh-Benard instability), we have shown that:

With steady option, when we are closed to the critical Rayleigh number, we have to decrease the value of the numerical residual to find good results.

With unsteady option, when we are closed to the critical Rayleigh number, we have to decrease the value of the numerical residual  and to increase the simulation time to find good results.