Linear stability analysis of Rayleigh–Bénard convection for cold water near its density maximum in a cylindrical container

Abstract

In this paper, the onset of Rayleigh–Bénard convection of cold water near its density maximum in a vertical cylindrical container and the stability of the steady axisymmetric flow were studied by using linear stability analysis. The results show that the critical Rayleigh number for the onset of convection increases with increasing the density inversion parameter Tm (Tm=(Фmax-Фc)/(Фh-Фc), where Фmax is the temperature at the maximum density, and Фh and Фc are the temperature at the bottom wall and top wall, respectively), and some new flow patterns not found in the Rayleigh–Bénard convection of common fluids are presented.Within a certain range of density inversion parameters, the onset of convection is steady and axisymmetric. The steady axisymmetric basic flow is found to depend on the initial condition, and two distinctly different basic flow are obtained at identical parameters, due to the top-bottom symmetry breaking. For the upward basic flow, four different flow patterns are observed after the axisymmetry-breaking instability. In addition, a hysteresis phenomenon of the flow pattern transition is found when Tm=0.57. For the downward basic flow, six flow patterns are observed after the axisymmetry-breaking instability. In particular, it is found that there still exists stable axisymmetric flow beyond the second bifurcation in certain ranges of Rayleigh numbers. The energy analysis shows that both the primary instability and the axisymmetry-breaking instability are caused by the thermal-buoyancy mechanism.