Stability Of Thermal Convection In Cylindrical And Annular Containers

Thermal convection problem is not only relevant to nature phenomenon in planetary convection, in the atmosphere, in the oceans, in the earth’s mantle, but also relevant to engi-neering applications including crystal growth, nuclear systems, industrial cooling and so on. The problem has been studied by researchers for more than a century by now and is still a topic of interest to them. In this dissertation, instabilities and nonlinear evolutions of thermal convection in cylindrical and annular containers are explored by linear stability analysis and direct numerical simulation. The instability mechanisms and the processes of transition to chaos are investigated. The main contents of this thesis including:(1) For cylindrical Rayleigh-Benard convection, the stability boundaries for the axisym-metric flow are derived for Prandtl numbers from0.02to6.7, for aspect ratio A (height to radius ratio) equal to1,0.9,0.8,0.7, respectively. We found that the stability curves for A=1,0.9and0.8contain hysteresis loops. The finding for A=0.7is that very frequent changes of critical mode (azimuthal Fourier mode) of the second bifurcation occur when the Prandtl number is changed. The instability mechanism of the flow is explained by kinetic energy transfer analysis, which shows that instability is result from the radial or axial shear of base flow combining with buoyancy mechanism. The contributions of different instability factors vary with the control parameters. Direct numerical simulations of the three dimensional flow after axisymmetry-breaking bifurcation have been performed. The development of flow in nonlinear regime is diversity, including multiple flow patterns at A=1and different routes to chaos at A=0.7.(2) For sidewall heated cylindrical convection, two kinds of thermal boundary condi-tions at sidewall are considered to study the stability of axisymmetric base flow, i. e. the sidewall is partially heated with a uniform temperature and the sidewall is heated with a parabolic temperature profile. Besides, the effect of rotating horizontal wall on flow instabil-ity is considered. The stability results for both sidewall conditions are qualitatively the same. The instability mechanisms are also corresponding to the combination of inertial mechanism and buoyancy mechanism. However, the way of combination of two mechanisms is different from that in cylindrical Rayleigh-Benard convection. The main components of inertial mech-anisms are the amplification of radial disturbances by radial gradients of the basic radial flow and amplification of axial disturbances by axial gradients of the basic axial flow. When the top wall is rotating, the centrifugal force acts against the thermal convective circulation and split the main vortex into two counter rotating vortices, where the interaction leads to insta-bility. When the bottom wall is rotating, the centrifugal force intensifying the main vortex and thus stabilize the flow, the flow instability is also determined by the aspect ratio and the Prandtl number.(3) For annular Rayleigh-Benard convection, the flow instabilities are investigated for the onset of convection and the axisymmetric flow. The transitions to chaos are also investi-gated. The onset of convective instabilities are determined by the radius ratio and the radius to height ratio. The pattern is chosen by using the minimum thermal energy to supply the fluid motion. The stability of the axisymmetric flow depends on both the geometry param-eters and the Prandtl number. The stability results are qualitatively consistent with that for cylindrical Rayleigh-Benard convection. The inner wall of the annulus plays the role to in-crease the critical Rayleigh number and change the critical mode, whereas it has little effect on instability mechanism. The numerical simulation results show that the three dimensional flow is very stable for large radius ratio annulus. The development of three dimensional flow after axisymmetry-breaking is affected by the inner wall of the annulus, which plays a role to stabilize the flow.