Thermoconvective instabilities in a vertical cylinder of water with maximum density effects part 2. Theory: Onset of convection

The onset of convection in a vertical cylinder of water with regard to the density anomaly near 4°C is studied using linear stability analysis. Since the maximum density is located at the top of the cylinder, making the whole height of the cell gravitationally unstable, the Boussinesq equations are solved by the Galerkin method. Also for the special case of axisymmetrical flow pattern, direct numerical procedure and analytical method have been used. Special emphasis is given to the large aspect ratio limit (height of cylinder much larger than radius) for which the combined influence of the sidewalls and the non linear density gradient gives rise to a non symmetrical axial velocity profile. The maximum of the velocity amplitude is shifted towards the bottom of the cylinder as the aspect ratio increases and consequently the critical Rayleigh number is lower than when a linear density gradient is assumed.