Results of the simulations

1.3.1) Temperature gradient concentrate at the top
1.3.2) Temperature gradient concentrate at the bottom
2) Initial conditions adapted for different numbers of rolls
2.1) I.C. adapted for one roll
2.1.1) Results
2.1.2) Modification of the precedent case
2.2) Two rolls

1) Initial conditions with a temperature gradient

For the first simulation, the initial condition were very simple: a constant temperature and an initial velocity nul.
We get the results following in the steady case:

In the unsteady case, we get the results following:

It seems that the convergence into two rolls is the most stable because there is no initial condition that could favorise a particular convergence.
The initial temperature is not very physic, so we impose different temperature gradients in the next simulations.

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We impose a linear gradient of temperature and no condition for the velocity.
We obtained the results following  in a steady case.

In the instady case, we obtained the same results:

We remarked that the solution take a lot of time to converge to two rolls.
First of all, the residus converge to a solution with a linear temperature gradient and a nul velocity.
So, our initial conditions seem to be stable.
But, after a few moment, the water starts moving and the system converge to another solution which is more stable (two rolls). Back to the top of the page

1.3.1) Temperature gradient concentrate at the top
In the steady case, we obtained the results:

In the unsteady case, we obtained the results:

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1.3.2) Temperature gradient concentrate at the bottom

In the steady case, we obtained the results:

In the unsteady case, we obtained the results:

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2) Initial conditions adapted for different numbers of rolls

2.1) I.C. adapted for one roll
initial conditions

2.1.1) Results

We remark that the solution converge first to a one roll solution but after 500 s, the convergence changes of direction. The roll lose some energy and two rolls appear. The problem is that the two rolls don't mouve in the same way. This situation is completly unstable,  and a roll appears to make the junction between the two others. So the system converge to three rolls.
We can say that in this situation, we are at the limit of the convergence to one roll. Indeed, it depends on the criters of convergence because for a criter of 1e-3 for the velocity, it converges to one roll but for a criter of 1e-4, it converges to three rolls as it is described.
So, we will try in the next part to modify the boundary conditions in order to get a convergence to one roll with a convergence of 1e-3.

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2.1.2) Modification of the precedent case
In this part, we keep the same initial conditions but we modify the boundary conditions: In the hoter wall, we warm up the area where we patched the hoter temperature. And in the colder wall, we reduce the temperature where we patched the colder temperature. The difference of temperature we impose compare to the initial temperature is 0.06 K. So,  physically, a such difference of temperature schouldn't affect the results. Off course, the Rayleigh number is not modifiate.
The results we get are the next one.

The system converge very quickly to one roll. Even with a strict criter of convergence, there is no problem. This case prouve that if the system converges first to a solution and then to an other solution, we can't conclude for a real case. A difference of temperature of 0.06 K, represent nothing physically, and yet, the results are different for the two simulations.

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2.2 Two rolls

initial conditions

As we saw, this case is very stable. So, with favorable initial conditions, the convergence is not a problem.

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2.3 Three rolls

initial conditions

The initial temperatures allowe the system to converge to three rolls. Indeed, we allready saw that three rolls were a possible convergence. It prouves that the initiation with temperature is a good way to control the solution you want.

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2.4 Four rolls
initial conditions