Computing the critical Rayleigh:

I. Definitions.

II. Determination of critical Rayleigh.

III. Results.

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I. Definitions

1- Nusselt number .

The Nusselt number is the balance between the heat flow exchanged with convection and the the one exchanged with conduction. It represent also the balance between the temperature gradient near the wall and the average value of this one .

Nu = (real wall flux)/(average conduction flux).

In the stability domain the only way of heat transfer, if we ignore the diffusion, is conduction. So the Nusselt number is equal to one. But when the instabilities are formed convection participate in transferring the heat. This increase the Nusselt value, witch becomes greater then one. This frontier marks the apparition of rolls. By the way of these rolls there is a convection heat transfer.


2-Rayleigh number.

Rayleigh number Ra is the balance between destabilisation forces (Archiméde force) and the stabilisation ones(drag force).

Archimède force:


Drag force:


II- determination of critical Rayleigh number

There are two ways to determine the critical Rayleigh number.

2 D steady : we do a stationary simulations, and by choosing a temperature difference DT between the two plaques. And after this we use a dichotomous method with alternating successively stable state and unstable ones we can determine the Rac.

2D unsteady : in this we only one simulation, and so as to have a succession of an equilibrium states we increase slowly the temperature in relation with time until we reach the equilibrium.

In order to determine the critical Rayleigh number we can just increase progressively the difference of the temperature and display the velocity vector. The critical Rayleigh number is the one witch correspond to rolls apparition.

It's also possible to determine critical Rayleigh number with drawing Nusselt in function with Rayleigh number. It correspond to jump of Nusselt value from 1 to a greater value.

For Ra < Rac the motion is stable : all the perturbations a size R = d (with d the distance between plaques ) will be absorbed. In stationary regime temperature gradient is constant. In this case Nusselt number is equal to 1: He heat amount transferred by conduction is equal the one transferred by convection.

for Ra >Rac the motion is unstable : even if the velocity is set initially to 0, there is apparition of unsteady rolls. In this case the Nusselt number is greater then 1. In addition to conduction we have convection too. The none equilibrium between viscosity's forces and Archimed ones is the motor of this motion. If we continue to increase the Rayleigh the regime become unsteady and turbulent in 3D.


III- Results.

The critical Rayleigh have been done with 2d simulation and under the following hypothesis :

III.1 Description of the experience :

Two parallel plaques : width : infinite.

length : 2cm.

distance : 1cm.

Simulation data : number of iterations : 500

grid type : Cartesian.

grid size : 20 cells in the y direction and 40 in the x direction.


III.2 Determination of critical Rayleigh with Nusselt.

Under these conditions we compute the values of heat transfer and then the Nusselt corresponding to each of the Ra used.

We get the graph of the fig.1 .


fig.1 Nusselt in function of Rayleigh.

With this graph we can determine the critical Rayleigh number, it correspond to jump of Nusselt value from 1 to a greater value. The Rayleigh critique here is approximately equal to Rc=1763. To perform this result we should change the grid.  


III.3 Determination of critical Rayleigh with Vxmax.


fig.2 maximum of velocity in function of Rayleigh.


We see that maximum of velocity remains too small at the beginning and then it grows up very quickly. This jump is sign of changing the regime. So the critical Rayleigh would be between 1700 et 1800. But this don't give a good estimation of the critical value of the Rayleigh number. We can perform this result by modifying the simulation properties : grid, iterations number, relaxation parameter...etc.