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 gradient

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

2.3) Three rolls

2.4) Four 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:
 

velocity
 

In the unsteady case, we get the results following:
 

Click on the image to animate it
 

   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.
 

velocity
 

 In the instady case, we obtained the same results:
 

click to animate it
 

   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
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


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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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
 


 

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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2.4 Four rolls
initial conditions
 


 

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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