Chaos Structure of the Flows in a Curved Duct of Rectangular Cross-section

我们研究了长径比为8的矩形截面弯曲管道内的流动，特别是对Hopf分叉上方的周期和准周期流动进行了数值计算。将符号动力学应用于包含一系列流型的多周期流动。结果表明，没有一种流型具有稳定的平衡状态，但由每种流型组成的流型是稳定的。存在间歇性非对称流动的Dean数。相应的吸引子表明，吸引子的分数维近于二维。发现了吸引子沿轨道的伸展和折叠，从中可以看出邵氏吸引子似乎是奇怪的吸引子。研究结果发表在《日本机械工程师学会会刊》上。在此基础上，用同步置换代替逐次置换的方法对相同的流动进行了数值模拟。预期后一种方法可以尽可能地减少不对称干扰。结果表明：得到了结构不稳定的对称流动。在Hopf分叉下观察到定常流动到定常流动的分叉现象。在Hopf分叉上方，观察到了几次双周期行为。发现吸引子的几何形状很简单，类似于罗斯勒带模型。在此基础上，对吸引子进行了初步研究，得到了直流道流动的模型方程，并对理论过程进行了初步研究，得到了直流道流动的模型方程，该理论方法可望适用于弯曲管道流动。

We studied flows in a curved duct of a rectangular cross-section having an aspect ratio of 2nd curvature ratio of 8. In particular, periodic and quasi-periodic flow occurring above the Hopf bifurcation were calculated numerically. The symbolic dynamics was applied to the multi periodic flows which include a series of flow patterns. Results were as follows ; no flow patterns have stable equilibrium state, but a flow pattern consisting of each the pattern was stable. There existed the Dean number where asymmetric flow occurred intermittently. The corresponding attractor showed that fracial demension of the attractor was nearly two dimensional. A stretching and folding of the attractors was found along orbits, from which shaos seems to be strange attractor. the results were published in the Transaction of the Japan society of mechanical engineers. Furthermore, we simulated numerically the same flows by the method of the simultaneous displacement instead of the successive displacement. It was expected that the latter method decreased asymmetric distubance as possible. The results were as follws ; symmetric flows were obtained which were structurally unstable. Bifurcation of steady into steady into steady flow was observed below the Hopf bifurcation. Above the Hopf bifurcation, a few times of the double periodic behavior were observed. The geometry of the attractor was found to be simple, which resembled the Rossler band model. Consequently the attractor preliminary study, the model equation for the flow in a straight channel was obtained, and the theoretical procedure was preliminary study, the model equation for the flow in a straight channel was obtained, and the theoretical procedure was expected to be applicable to a curved duct flow.