Three-dimensional instabilities of natural convection flow in a vertical cylinder with partially heated sidewall

The three-dimensional axisymmetry-breaking instability of an axisymmetric convective flow in a vertical cylinder with a partially heated sidewall is studied numerically. The central part of the sidewall is maintained at constant temperature, while its upper and lower parts are thermally insulated. The dependence of the critical Grashof number on the cylinder aspect ratio (A = height/radius) is obtained for a fixed value of the Prandtl number Pr=0.021, and fixed length of the heated central region, equal to the cylinder radius. Three different modes of the most dangerous three-dimensional perturbations, which replace each other with the variation of the aspect ratio, are found. Comparison with experiment shows a good agreement at the aspect ratio A = 8 and 12, while at A = 4 a significant disagreement is observed. Possible reasons for this disagreement are discussed. At A = 4, the dependence of the critical Grashof number on the Prandtl number is studied in the range 0<Pr<0.05, to rule out the possibility that the disagreement is due to uncertainty in values of fluid properties. The similarities and differences of instabilities in the cylindrical and rectangular geometries are examined. The computations are carried out using two independent numerical approaches, which cross-validate each other.