Three-Dimensional Instabilities of Natural Convection Flow in a Vertical Cylinder With Partially Heated Sidewall

Abstract

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.

Three-dimensional numerical study of natural convection in vertical cylinders partially heated from the side

“…As has been shown by Rubinov et al, 11 the flows which will bifurcate from these toroidal flows will break the axisymmetry property. More precisely, they have shown that, depending on the aspect ratio of the cavity, different perturbation modes ͑defined by their azimuthal wave number k͒ are associated with this axisymmetry breaking.…”

“…The cylinder aspect ratio ͑A = height/ radius͒ ranges from 2 to 8, whereas a fixed Prandtl number, Pr= 0.021, is considered as well as a fixed length of heated zone, equal to the cylinder radius. Our study is a continuation of the work of Rubinov et al: 11 by linear stability analysis, they have determined the thresholds at which the three-dimensional axisymmetry breaking of the flow occurs; by three-dimensional numerical simulation, we will study the three-dimensional patterns appearing beyond these thresholds and their further transitions. A finite volume approach based on the multigrid SIMPLE scheme is used for the computation of steady states, while an improved fractionstep finite-difference method is used for the oscillatory cases.…”

“…The twodimensional work of Erenburg et al 10 has revealed complicated bifurcation diagrams and the existence of multiple solutions. Finally, the recently published paper of Rubinov et al 11 carefully studied the transition leading to the threedimensional axisymmetry breaking of the flow. They presented the dependence of the critical Grashof number on the aspect ratio and they showed that three different modes are the most dangerous perturbations and that they replace each other with the variation of the aspect ratio.…”

Three‐dimensional instability of axisymmetric flows: solution of benchmark problems by a low‐order finite volume method

“…Details on this problem can be found in [4], where it was studied by the global Galerkin method. The results of [4] were then used in [9,11] as a benchmark data. Here, we consider one particular case defined by A = 2 and Pr = 0.03.…”

Numerical analysis of the unsteady character of “spoke pattern” in Bridgman top seeding geometry

“…The three dimensional axisymmetry-breaking instability of natural convection in a vertical Bridgman growth configuration with different thermal boundary conditions was studied by Gelfgat's group [6,7]. It was shown that the first instability of the considered flow is always threedimensional.…”