Dear all, I find difficulty in the resolution of the Navier stokes equations at high Rayleigh number (more than 1e6). The physical problem is about natural convection in rectangular shaped enclosure. I use a control volume based finite element method for obtaining the algebraic discretisation equations. I would like to know witch is the appropriate dimensionelless grouped to be the better, and how can trait this kind of problems. You can answer in French if you want

Les difficultés ne sont aucunement liées à la combinaison de nombres sans dimension qui est utilisée. Pour un tel nombre de Rayleigh, il extrèmement difficile d'obtenir une solution avec une résolution couplée du système. En effet, la solution pour la vélocité est intimement liée à celle de la température. Il est préférable d'utiliser de la relaxation: découplé le système et un faux pas de temps (résoudre l'écoulement en transitoire bien qu'il soit permanent).

Vous pouvez tenter une résolution avec ces paramètres: cavitée unitaire, maillage 40x40, dt=0.001, rho=1, mu=0.71, k=1, cp=1, g= -10, Beta =71000 (Ra=1.e+6, Pr=0.71). Dans mon cas, je n'ai aucune difficulté à résoudre ce problème avec ces paramètres.

1) Quelles sont vos fonctions d'interpolation? 2) Quelle type de discrétisation du terme convectif utilisez-vous?

Bonne chance.

I don't read French, and so have no idea what good advice Sebatian may have provided.

Recall that there are two important dimensionless groups in 'standard' natural convection: the Rayleigh number and the Prandtl number. What are you using? You can have hi-Rayleigh number convection at very low Reynolds number for example in some applications in earth sciences.

Also be aware that as Rayleigh number increases, there may be no steady solution as boundary-layer instabilities create time choatic flow even at low Reynolds number.

Perhaps provide more information please. How are you dealing with the pressure in this enclosure flow?

Dear George, What Mr Sebastien said is: From him the difficulty is not caused bye the dimensionless parameters and is not easy to obtain solution with coupled equations (velocity and temperature) and relaxation should be taken. He suggested that the problem must be considered as stationary with constant time step. He gave some parameters that he has used for this problem at Ra=1e6 and he asked me witch interpolation function and type oh the discretization of the advection term I have used.

To obtain the non dimensionel form of the conservation eqautions , I have used the Rayleigh and the Prandtl parameters in the same time, but not the Reynoldes. ( I think that the Reynolds number is applicable in mixed convection). In the literature we can find resultes at Ra=1e7 without instability??. I think that the problem is of numerical order. In the coupling velocity-pressure I use the SIMPLER algorithm.

I once used the simpler algorithm to solve this kind of flow. In order to get a solution, I had to use a small time step (dt < 0.001). The relaxation was not enough.

By the way, when you use the simpler algorithm to solve an incompresible, the equations are un-coupled (mass and momentum are not solve at the same time)

The parameters I gave you are for Pr=0.71 and Ra=10e6

(Sorry, If I have interfered in George thread....)