FORMAION OF PATTERN,TRANSITION,CHAOS AND TURBULENCE IN RAYLEIGH-BENARD CONVECTION

通过对非线性平衡解的数值计算和稳定性分析，研究了Rayleigh-Benard对流中斑图的形成、转变、混沌和湍流特性的机理。本课题包括以下四个方面的研究。研究了无限大两水平面间热对流中花纹的形成机理。利用弱非线性稳定性理论推导了一个振幅方程，并对其系数进行了数值计算。确定了六角形或卷曲花样开始出现的条件。研究了有限体积容器内对流的起始和稳定性。结果表明，旋涡的数量随船舶横向范围的不同而变化。通过数值模拟研究了热对流的稳定性和转换性。结果表明，随着底部温度的升高，首先出现大尺度的圆周运动，但伴随两个圆涡的热对流取代了大尺度的圆周运动。当容器放置在与水平面成一定角度时，即使顶部和底部的温差很小，也总是会出现热对流。对流的发生是由于干草分叉的不完善造成的。通过数值计算得到了热对流的平衡解，并阐明了其稳定性。

The mechanism of the pattern formation, transition, chaos and properties of turbulence in Rayleigh-Benard convection was investigated by numerical calculation of the nonlinear equilibrium solution and by the stability analysis. The present project consists of the following four researches.1. The mechanism of the pattern formation in the thermal convection between two hoizontal planes with infinite extent is investigated. An amplitude equation is derived by the weakly nonlinear stability theory and its coefficients are evaluated numerically. The conditions for the onset of hexagonal or roll-like patterns are determined.2. The onset and stability of the convection in a vessel of finite volume is investigated. It is shown that the number of vortices varies depending on the lateral extent of the vessel.3. The stability and the transition of the thermal convection is investigated by numerical simulation. It is found that a large scale circular motion appears as a result of the thermal instability first, but thermal convection with two circular vorties take the place of the large scale circular motion when the temperature of the bottom is raised.4. When the vessel is placed at an angle from the horizontal plan, the thermal convection always appears even if the difference of the temperatures at the top and bottom is very small. The occurrence of the convection is found due to the imperfect pitchfork bifurcation. The equilibrium solution for the thermal convection was obtained numerically and its stability was clarified.