Karaa Samir et Talbi Samia

Rayleigh-Bénard instabilies



1. Introduction

2. The Critical Rayleigh number

3. Comparison de two turbulence models

4. The three dimensionnel case

5. Conclusion

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1. Introduction
The problem concerns a fluid contained between two parallel horizontal plates at different temperautures.

When the upper plan temperature is higher than the lower's, we find that the system is stable. If a given particule goes up, it will be in a denser place. Exposed to Archimede force, it goes down again, back to its inital position. Thus, the domain stays stratified in terms of temperature.

Temperature for Ra =1606

On the opposite, if the upper tempertature is smaller than the lower's, we notice that while the difference of temperatures is low enough, the system remains stable. When it becomes greater than a certain value, we call critical difference of temperature, periodic vortex mouvements take place within the fluid, pointing out that the flow mode is instable. This is Rayleigh-Benard instability.

Temperature for Ra =4015

The Rayleigh number

Ra = alpha * g * dT * d3 / ( nu DTh )

is a non dimensional number involving the difference of temperature dT. This parameter is basically the one that will enable us to describe our convective state.

alpha : volumic dilatation coefficient

g : graviotational acceleration

d : the separating distance between the two plans

nu : fluide cinematical viscosity

DTh : fluid's thermical diffusivity.

We theoritically determine that Rayleigh number from which we start noticing fluid's instabilities is Rac = 1708

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2. The critical Rayleigh number

As we found the theoritical critical Ra as being 1708, we launched simulations increasing Rayleigh number from 1615.

We limit the number of iterations to 20 000. We find that the critical Ra is located between 1664 and 1688. This value is pretty close to theoretical number, but we can not be significantely more precise. Moreover, this value is initial conditions dependant.

For furhter calculations, Rac has been taken as its theoretical value.

As there are two softwares that can be used, wethought about a way of comparing their results. Thus, we superpose for each plot PHOENICS and FLUENT curves and interpret the outoputs.

Exploitation of PHOENICS

The first plot represents Phoenics results.

Because of Phoenics capabilities, it was more practical to directly get the maximum velocity magnitude and not the detail of both x-velocity and y-velocity.

As predicted, we notice that after a certaint value of Ra, velocity suddenly gets bigger. We can approximately determine that the critical Rayleigh number is located between 1400 and 1600.

Exploitation of FLUENT

We present here different curves provided by Fluent.

This software enables us to get after each simulation maximum velocities in the two spatial directions. Thus, we plot on the same graphic maximum x-velovity and maximum y-velocity. We present them as a function of Ra.

Here again, we clearly see an explosion of velocity value when the Ra gets greater than a certain value. The critical Rayleigh number we graphically extract is here around 1750.

Comparison of softwares results

Several notes deserve to be mentionned :

The equation of the cases are: square ( V ) = K * eps


The curves equation can be read as a linear relation between square velocity and Ra, when Ra > Rac. This is pretty true in both softwares simulations: we get a line with a positive pente.

Phoenics provides Rac = 1500, while Fluent gives a vlaue around 1750. The first value does not exactly correspond to theoretical result, the second one is much closer to the 1708 we theoretically expect.

In terms of precision, Fluent seems to give better results than Phoenics. This is probably due to a wider exploitation of Fluent, thanks to serveral statistics and data it offers to compute. Getting the same results may be doable with Phoenics, but it will take a few lines to program. The commands are, no the opposite, ready to use in Fluent.

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3. Comparison of two turbulence models

High Rayleigh number turbulent is simulated to determine the effects of Rayleigh number on heat transfer and flow characteristics. Changing Ra from 0.5 10^7 to 8 10^7, we computed the total heat transfer rate between the two walls. We have tested two turbulence models:

We discover a linear relation between the Rayleigh number and the total heat transfer rate. Besides, the results given by the models are quite the same.

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4. The three dimensional case

Here is a list of three dimensional views of the temperature for small Rayleigh numbers. In the first figure, isothermal boundary conditions are used at both top and bottom bounding planes . Periodic boundary conditions are used along the horizontal directions x and z . In the second case, we add parallel walls to the xy-plane.

The first case:

In the second figure, the convection rolls are parallel to the x-axis. This agrees with the two dimensional case.

The second case:

Unlike the preceding case, it is difficult here to know the direction of the convection rolls .

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5. Conclusion

This BES enabled us to use PHOENICS and FLUENT softwares. We exploited their specific capabilities, and compared their results.

Phoenics gives a unique way of imposing intial conditions (sinusoidal form). However, Fluent stayed the main tool for further analysis.

We studied the influence of choosing a turbulence model and found out that the results are identical. Then we moved to the 3D modelization. Unlike the 2D case, this one gives a wide view of the occuring phenomenon: we see how the rolls move. This also shows how difficult it is to study the problem three dimensionnally. In fact, rolls behavior is totally unpredictible.

Physically speaking, the BES was an opportunity to have a first approach of Rayleigh-Benard theory. The phenomenon is very complex. Our study involved only the stationnary case. But there are a lot of other interesting cases to experience and analyze, le instationnary regime for example.

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