Stability Of Thermal Convection In Annular And Cylindrical Containers

Thermal convection plays an important role in geophysics,astrophysics and industrial applications,such as the convection in the earth's mantle,in the ocean and in the sun,moreover,in the growth of crystal,the thermal convective instability has a great influence on the quality of the crystal.By means of linear stability analysis and direct numerical simulation,the thermal instability and nonlinear evolution of the top-bottom symmetry breaking system and the thermal convective instability in annular containers under rotating magnetic fields are studied.The research results are as follows:The onset instability of cylindrical Rayleigh-Bénard convection is studied by means of linear stability analysis with height radius ratio A ranges from 0.5 to 1.5,and density inversion parameter ?m ranges from 0 to 0.5.The results show that the critical Rayleigh number and critical mode of the first instable convection are affected by the high radius ratio A,the density inversion parameter.And the critical modes for the onset convection under the same configuration may be different.Moreover,in the range of parameters studied in this paper,the influence of density inversion parameters on the critical Rayleigh number of the convection onset is the same.The method of linear stability analysis and direct numerical simulation are used to study the bifurcation process of convection in the parameter range of A=1.0 and 0<?m<0.5.It is found that,because of the top-bottom symmetry breaking,the heat conduction solution with the density inversion parameter changing from 0 to 0.5 is bifurcated through the trans-critical bifurcation to the axisymmetric solution,and is different from the pitchfork bifurcation within OB approximation at A-1.Two different axisymmetric solutions of the central hot fluid upward(a0)and cold fluid downward(b0)are observed.The hysteresis of the a0 solution is found,but no hysteresis is found in the b0 solution.The pattern evolution of a0 solution and b0 solution are studied.It is found that the b0 solution can still be stable beyond the secondary bifurcation.Direct numerical simulation of three-dimensional convections are conducted,and the existence of many stable three-dimensional flow and periodic convection are observed.The direct numerical simulation method is used to study the influence of the magnetic Taylor number and the Prandtl number on the basic flow of heat convection in annular Rayleigh-Bénard convection under rotating magnetic field.It is found that the basic flow is greatly infected by the magnetic Taylor number and the Prandtl number.The influence of Prandtl number and magnetic Taylor number on the instabilities of the base flow is studied by using the linear stability analysis method at the height radius ratio A of 1,0.8 and 0.65.The critical Rayleigh number and critical frequency of the instability increase with the increase of the magnetic Taylor number,but the effect of the critical Rayleigh number on the height radius ratio of 0.8 and 0.65 is not the same at A=1.0.The critical Rayleigh number of the working fluid with larger Prandtl number is more sensitive to the increase of the magnetic Taylor number,and a steady axisymmetric solution beyond the primary bifurcation is observed.