Partie 1 :Rayleigh Benard convection theory Rayleigh Benard Theory:

The Rayleigh Benard convection phenomena is an instability which basically occurs in a viscous fluid layer submitted to a vertical negative temperature gradient, in the same direction than the gravity. There is a coupling between fluid dynamics and heat transfer.Typically, the fluid in a rectangular cell si submitted to a mecanism of destabilisation, the Archimede force, and to 2 mecanisms of stabilisation, the drag and the thermic diffusion:

 Archimede force Drag Thermic diffusion time scale There always exists a mathematical solution for this motion, but the competition between the 3 forces creates a problem of stability. The temperature gradient is responsible for an instable density gradient in the fluid layer:  if this temperature gradient is strong enough, the hot fluid from the bottom of the cell will start to rise in a small plume, the Archimede force exceeding the weight of the plume. Depending on the fluid viscosity value and of the thermal diffusion coefficient, viscosity and diffusion processes tend to attenuate this phenomena. If these forces are not strong enough, the instability get amplified, and convection rolls develop. The phenomena is bidimensionnal, and get steady. The rolls have the following structure:  Critical Rayleigh number:

The Rayleigh Benard instabilty results from the competition between 3 different forces: the Archimede force, the drag, and the thermic diffusion.
The edge parameters value for Rayleigh instability occurs for a certain value of the Rayleigh number, an adimentionnalized number taking into account the convection cell geometry, and the physical parameters of the Rayleigh Benard phenomena.

 Rayleigh number For water, if the significative dimension of the cell, d , is 1 cm, then the critical Rayleigh number is 1708. It means that for a Rayleigh number smaller than this value, convection rolls won't develop, whereas for a Rayleigh number higher than 1708, convection rolls will appear in the flow. The number of rolls per cell depends on the Rayleigh number value:  according to the shape of the curve Rayleigh number versus number of rolls per cell, for each value, 2 configurations can develop. Grid:

The simulations are realized  with a structured grid under PreBFC.
The  mesh is a rectangle  of 0.02 m *0.01 m  divided in (40*20)  cells.
The domain is defined by 2 walls ( up and down boundaries ) and a symmetry condition at the inlet and outlet. This structure is bound to be changed for the cases related to the convection in the mantle, but will still be used to present the global  elements of the Rayleigh Benard rolls. Number of rolls:

In the following simulations the difference of temperature between the top and the bottom is 0.5 degrees.
Depending on the initial condition the converged case will generate 2, 3 or 4 rolls.
This convergence is obtained with a precision of 10-3, and takes about 100 iterations in stationnary solver.

There is no need to force the initial conditions in velocity to converge to 2 rolls. To obtain 3 rolls, we established an initial condition in velocity , otherwise, we would converge to the stable case of two rolls. We initialized the velocity field in the box by:

Vy = cos(pi * x * 3 / 0.02) The 4 rolls are obtained wit the following initial condition:

Vy = abs( cos( pi * x* 3 / 0.02)) 