Theorical Presentation












Rayleigh Benard instability is one of the most famous thermal instability. Innumerable number of publications shows great interest of scientific community. If this phenomenon take part in elaboration of recent theories as such dynamical system, the fact remains that the first study dates back beginning of the century. Indeed, Benard suggested experimental study :
 
 












This study showed clearly the existence of a critical threshold between two mode of heat transferts:

- a mode of pure thermal conduction:

So, it represents the solution of heat equation, consequently the profile of temperature is linear ( in this case, heat transfert is
consequence of molecular phenomenon )
 
 

( temperature )









- a mode of convection  heat transfert :

This mode is characterized by forming of stationnaries contra-rotative rolls ( here, heat transfert is macroscopic phenomenon)
As difference of temperature increase,   there is, first, forming secondaries instabilities which are characterized by
 oscillating motions of contra-rotative rolls and finally, as T1 -T2 becomes bigger and bigger, more and more "unstructured' behaviour  appear ( until appearence of turbulence ).
 
 

( temperature )












Lord Rayleigh established by using theorical analysis concordance with Benard 's experience. More particularly, he showed existence of Ra the characteristic number of this instability :
 
 








for which :

                - if Ra < Rac, heat tranfert 's mode is purely conductive

                    - if Ra > Rac, convection is established

where Rac is the critical Rayleigh number. Rayleigh showed Rac = 1709.
 

The Modelisation of Rayleigh Benard relies on Navier Stokes system with Boussinesq assumption : in other words, density is supposed constant except that it feignes only quantities of motion by means of only buyoncy forces (generator of the instability). Thus, variation of density destabilizes fluid while drag force and thermal force stabilize it. Ra repesents ratio between these two quantities.
 

A study of linear stability allows to obtain curve of stabilty representing Ra in function of wave number
 
 










For Ra < 1708, all perturbations are absorbed

For Ra > 1708,  a certain number of perturbations grows up.

This curve will be very useful in searching of initials conditions  ( see chapter III )
 

Finally, we give a figure showing differents states of th instability :
 
 


 
 

with






In order to study stationnary rolls, our study will be limited to epsilon < 30.
 
 











 

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