Results of the simulations

__1)
Initial conditions with a temperature gradient__

1.1) A nul temperature gradient

1.2) A linear temperature gradient

1.3) A hyperbolic temperature gradient1.3.1) Temperature gradient concentrate at the top1.3.2) Temperature gradient concentrate at the bottom

2.1) I.C. adapted for one roll2.1.1) Results2.1.2) Modification of the precedent case2.2) Two rolls

__1)
Initial conditions with a temperature gradient__

1.1 A nul 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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1.2) A linear temperature gradient

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).

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1.3) A hyperbolic temperature gradient

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

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__

initial conditions2.1) I.C. adapted for one roll

2.1.1) Results

steady

unsteady

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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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.2.1.2) Modification of the precedent case

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

steady

unsteady

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

steady

unsteady

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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initial conditions2.4 Four rolls

steady

unsteady

With this last case, we understand the power
of the initialisation on the temperature. Indeed, the initialisation with
the velocity doesn't converge to four rolls and with the temperature, it
does.

So, in the reality, it seems that to play
on the temperature to control the movement is more effective than to play
with the velocity. For example, in the industry, if you want to impose
a certain kind of movement, install radiators instead of ventilators.

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