同轴圆筒间旋转流动的吸引子及混沌仿真与控制

The Couette-Taylor flow between two concentric rotating cylinders, one of classic rotating flow, has extensive applications in lubricating theory and engineering practice. With the increasing of Reynolds number, the flow displays some behaviors of nonlinear dynamics. Accordingly, it is very important to study this problem for knowing and controlling of turbulence. This research work will study some behaviors of dynamics of the Couette-Taylor flow, and discuss chaos simulation and controlling of the Couette-Taylor flow. Using method in thin domain we will investigate the existence and regularity of strong solutions and attractor of Navier-Stokes equations for the narrow gap between two concentric rotating cylinders in periodic boundary conditions in the Z-axis and proper initial conditions, and establish the dimension estimations of the attractor；we will not only introduce the eigenfunction of the Stokes operator in the cylindrical gap region and give low-dimensional analysis method, but also present some detailed numerical result of flow phenomenon of the Couette-Taylor flow；We will derive low-dimensional Lorenz systems containing only finite Fourier modes by using the eigen spectral method, and discuss the stability of the Lorenz system, and we will present investigations of conditions of the bifurcation and chaotic behavior. We will not only discuss the existence and dimensions estimations of the attractor but also establish the globally exponentially attractive set and positive invariant set of the Lorenz system. we will investigate the general features of chaotic behavior and a route to chaos via bifurcations of Lorenz system,and discuss some relative problem of chaos controlling and simulation as well as synchronization of the Lorenz system.

同轴圆筒间Couette-Taylor流是一种典型的旋转流动,在润滑理论和工程实践等领域有广泛应用。随雷诺数的增大,这种流动表现出丰富的动力学行为，研究它对人们认识和控制湍流至关重要，本项目致力于探讨这种旋转流动的部分动力学行为及混沌仿真与控制等问题。利用薄区域方法探讨周期边界条件和适当初始条件下窄间隙圆筒间Navier-Stokes方程强解和吸引子的存在性、正则性及维数估计等问题；借助圆筒间隙区域的Stokes算子的特征函数，采用低模分析方法对Couette-Taylor流问题的流动现象进行部分数值仿真；探讨旋转流动Navier-Stokes方程特征谱展开有限模态简化后所得到类Lorenz方程组的稳定性、分歧、混沌发生的条件、吸引子的存在性和维数估计、全局指数吸引集和正向不变集、混沌行为的普适特征和通向混沌的道路等以及混沌控制与同步及仿真等相关问题。

同轴圆筒间Couette-Taylor流动是典型的旋转流动问题，随雷诺数的增大，这种旋转流动在从稳态层流发展到湍流的过程中,表现出丰富的典型的非线性动力学行为。它们所对应的无穷维非线性动力系统解的长时间行为及其数值仿真等问题是非线性科学领域的重大课题，也是国家自然科学基金重大研究计划立项“湍流结构的生成演化及作用机理”的重要问题之一。本项目探讨了同轴圆筒间Couette-Taylor流动的部分动力学行为及混沌仿真与控制等问题，获得如下研究成果：(1) 探讨了同轴圆筒间旋转流动Couette-Taylor流系统全局吸引子的存在性，给出了吸引子的Hausdorff维数上界的估计。(2) 探讨了Couette-Taylor流系统的动力学行为及其数值仿真问题，并借此解释了Couette-Taylor流试验中观察到的部分涡流的演化过程。Couette-Taylor流系统的湍流行为是由于雷诺数的增大，系统的不动点和周期轨道持续丧失稳定性而逐渐产生的，数值仿真结果反映了Couette-Taylor流湍流行为的某些特征。研究成果既有助于人们对分歧、湍流等非线性现象加深认识，同时在流体机械、航空航天、地球物理等工程实践领域有着广泛的应用。