Principle of RayleighBenard instability
Mathematical theory : determination of the critical
Rayleigh number
Manifestations of RayleighBenard instability
Principle of RayleighBenard instability
This problem of RayleighBenard instability is a problem of convection inside a fluid.
This instability occurs inside a fluid confined between two infinite horizontal planes at different temperatures and .

If then the system is stable and stays stratified in temperature. But, if and if a perturbation is introduced in the system, then for a critical value of the difference of temperature () it can appear movements inside the fluid : the system is unstable and movements are organized in periodic contrarotative rolls. These rolls, or also called cells of RayleighBenard, appear when there is a coupling between the dynamic field and the thermal field.
Rolls of RayleighBenard
The principle of the instability is simple. Consider a drop of fluid near the lower plane. The drop is heated so its density decreases and it goes up inside the fluid due to the Archimede force. When the drop reaches the upper plane, more cold, it is cooled so its density increases and it can goes down inside the fluid. Rolls of RayleighBenard instability are caused by this mechanism.
The Archimede force must surpass the viscous drag force and the heat diffusion to allow convection inside the fluid. So there is a notion of threshold.
Effectively, this instability occurs following the
value of the Rayleigh number which is:
Enthalpie field for Ra=1468  Enthalpie field for Ra=4405 
The critical value of the Rayleigh number for apparition
of RayleighBenard instability is 1707. For this critical value we can
observe 2 rolls inside the fluid. This state of the fluid with two rolls
is the most stable. But if initial conditions are well chosen, if Rayleigh
number increases, number of rolls inside the fluid increases too.



To study this problem, we have to solve classic equations of continuity, motion and heat conduction.
For this problem, Boussinesq approximation is used and we have following equations for incompressible fluid:
The velocity limit conditions are for z=0 and z=d. For temperature we can take constant temperatures or constant gradients of temperature for z=0 and z=d.
The equations become, in thermal adimensionnal form (, and ), and for the 2D case:
Velocity limit conditions become :
for z=0 and z=1.
Determination of the critical Rayleigh number :
The fonction is introduced : and . The problem become after linearisation :
If we look for solutions with the following form, the system of equations gives :
Limit conditions for z=0 and z=1 are :
By solving this sytem we can find the value of critical Rayleigh number et critical value of k :
Graphic of stability
Note:
We have studied the case where there are two planes under and above the fluid. Then, velocity limit conditions are two rigid conditions : for z=0 and z=d.
Two others cases exist following the type of boundary. Fluid can be placed on a heated plane but its upper face can be a free surface.
Case where there is one free surface
In this case we have a rigid condition for the lower
boundary
( for z=0 ) and a free condition
for the upper boundary (
and w=0 for z=d) . So critical values to see instability change :
The last case is the case where the fluid is confined between two free surfaces : and w=0 for z=0 and z=d .
Case where there is two free surfaces
The critical values are now :
RayleighBenard instability visualization  3D RayleighBenard instability 
This phenomenon of RayleighBenard instability is
very important in the industrial domain. Examples are numerous : cooling
of nuclear plants, heating of building, exchanges between atmosphere and
oceans..... But manifestations of RayleighBenard instability are not always
visible to our eyes that's why we have not a lot of pictures of it.