Research On The Stability Of Rayleigh-Bénard Convection Of Cold Water Near Its Density Maximum In Typical Confined Enclosures

Thermal convection widely exists in natural phenomena,such as atmospheric circulation,ocean circulation,mantle convection.Moreover,thermal convection plays a significant role in the industry,such as in the process of crystal growth,cooling of the nuclear reactor and the energy storage device.In recent one hundred years,it has become a classic research topic for researchers.Up to now,many works about Rayleigh-Bénard convection have been carried out on common fluid that the density is regarded as a linear function of temperature.Research of thermal convection and the flow behaviour in the fluid with the density maximum has caught attention on natural convection.However,stability study on the Rayleigh-Bénard convection of the fluid with density maximum in a confined enclosure is lacked.In this thesis,a systematic study on Rayleigh-Bénard convection of cold water near its density maximum in a cylindrical container,annular container and rectangular cavity with different boundary conditions is performed by both linear stability analysis and direct numerical simulation.The critical Rayleigh number and flow pattern at the onset of convection for cold water near its density maximum are obtained for different density inversion parameters,boundary conditions and geometric configurations.The effect of the major parameters on the stability of Rayleigh-Bénard convection of cold water in cylindrical,annular and rectangular cavities is analyzed.Firstly,it is the first time to use the method of linear stability analysis to study the Rayleigh-Bénard convection of cold water with density maximum in a cylindrical container.Stability curves at different conditions are obtained.The effect of the aspect ratio and density inversion parameter on the critical Rayleigh number and critical wave number is discussed.The patterns of velocity and temperature perturbations near the critical value are displayed.The influence of the thermal boundary conditions on the critical conditions is discussed by adopting perfect insulating and conducting sidewalls.The effect of the density inversion parameters of cold water with density maximum on the stability of Rayleigh-Bénard convection is analyzed by comparing the results of cold water with those of common fluid.When the density inversion parameter is relatively small,the critical wave number at the onset of convection for different aspect ratios is mostly m_cri=0 or 1,which is similar to that in common fluid.When the density inversion parameter is large,the stability is enhanced and the critical wave number is increasing with the aspect ratio.Moreover,three-dimensional numerical simulation is also carried out to get the critical Rayleigh number at the onset of convection and the steady flow patterns.Results show that the critical Rayleigh number gradually decreases with the increase of the aspect ratio and the reductions rate is higher at smaller aspect ratios.Density inversion parameter has a strong effect on the critical Rayleigh number and the critical flow patterns at the onset of convection.The flow pattern at a small density inversion parameter is axisymmetric one-torus or two-torus pattern and the flow can reach to the top wall of the container.With the increase of the density inversion parameter,critical Rayleigh number gets dramatically increased.Furthermore,the flow pattern in cold water shows an obvious difference with that in common fluid.When the density inversion parameter is relatively larger,the area with strong intensity of flow gradually approaches to the bottom wall of the container.The flow pattern appears as multiple-cell pattern which can be found in the lower fluid layer and a motionless fluid layer near the top wall of the container is observed,which is not found in common fluid.When compared the results to those for perfect insulating sidewalls,conducting boundary condition enhances the stability of the system.The critical Rayleigh number is larger and the flow patterns at the onset of convection are quite different.Secondly,take the cold water with density maximum in confined annular containers as a whole system,research works are performed by linear stability analysis,aiming to understand the dependence of the critical Rayleigh number at the threshold and the critical flow patterns on the aspect ratio,radius ratio and the density inversion parameter.On the other hand,the critical conditions are also determined by three-dimensional numerical simulation.Comparisons of the results are made.Results indicate that the critical conditions at the onset of convection in annular containers are dependent on the aspect ratio,radius ratio and the density inversion parameter.With the increase of the aspect ratio,the restriction of the sidewall on the flow is more obvious,thus the critical Rayleigh number at the threshold is decreased gradually but the critical wave number is increasing.With the increase of the radius ratio,both the critical Rayleigh number at the onset of convection and the critical wave number is continuously increasing.The density inversion parameter has a decisive influence on the critical conditions at the onset of convection.When the density inversion parameter is small,the critical wave number for cold water is the same as that for common fluid.However,when the density inversion parameter is larger the critical Rayleigh number and the critical wave number at the onset of convection dramatically increase,which are larger than that in common fluid.Most importantly,it is found that a motionless fluid layer appears near the top wall of the container,which is similar to that in cylindrical container.The larger the density inversion parameter is,the thicker the motionless fluid layer is.Compared the results of the two containers,it seems that the inner wall of the annular container is to enhance the stability of the system,increasing the critical Rayleigh number and the critical wave number.Lastly,the critical conditions at the onset of convection for cold water near its density maximum in a rectangular cavity are determined by linear stability analysis.Results show that with the increase of the aspect ratio,the critical Rayleigh number is decreasing.Density inversion parameter plays a significant effect on the critical Rayleigh number and the critical wave number.With the increase of the density inversion parameter,the critical Rayleigh number at the threshold is dramatically increased,which is similar to the results of the other two containers.