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))
