Numerical study of natural convection stability

Abstract

The analytical theory of natural convection stability, founded in the middle of the 20th c., is practically applicable only to the analysis of simple model objects: liquid between solid planes, in cavities of spherical and cubic shape, in channels of round, rectangular and annular sections, among others. The analysis of modern technical systems requires the use of numerical methods, which are currently the most powerful methods of mathematical analysis. Yet, analytical methods are necessary for testing calculation codes and verifying the results obtained using numerical methods. This article presents a series of numerical experiments aimed at investigating the stability of stationary one- and two-vortex convection regimes, establishing bifurcation regions in which transitions between these regimes occur, and the relationship of these transitions with changes in the energy parameters of convective flows. Numerical simulation was performed in square cells on a 21 × 21 grid using the control volume method and the SIMPLER algorithm. In place of a liquid, water was taken in the temperature range of 20–50 °C, and Grashof numbers from 100 to 22,400. The dependence of the Prandtl number on temperature was considered in the simulation. The results have established 4 bifurcation regions in which the established type of convective flows loses stability and changes to another form: 313.6 < Gr < 396.8; 3135.8 < Gr < 3527.3; 10913.3 < Gr < 13307.2; Gr > 22406.0. Four critical Rayleigh numbers corresponding to these regions have been found, which, considering the dependence of the number Pr of the temperature is equal to: Racr1 = 1,790.7; Racr2 = 14,738.3; Racr3 = 45,835.9; Racr4 = 79,317.2. All these bifurcation regions are associated with transitions of the potential energy of the liquid into kinetic energy and vice versa. The limit of the Boussinesq approximation applicability corresponds to the values Gr ≈ 13,307.2, or Ra ≈ 55,890.2. The comparison of the values of the critical Rayleigh numbers obtained in numerical experiments with the numbers of the analytical theory shows a very good coincidence of the first two critical numbers Racr1 and Racr2. The Racr3 values coincide in order of magnitude, and Racr4 differ almost twofold, which is explained by the Racr4 value going beyond the scope of the Boussinesq approximation, and, strictly speaking, has no physical meaning.