圆管湍流的Hopf纤维化
Hopf fibrations for turbulent pipe flows
摘要：
我们提出了一个推广的霍普夫纤维化商的流向平移对称性的湍流管流视为动力系统。特别是，我们利用相关联的高维状态空间，这是一个主要的纤维丛的几何结构。本文研究了与束中轨道动力学相位有关的同动帧速度$U_{d}$与泰勒假设之间的关系。作为一个应用，激光诱导荧光技术被用来捕捉平面荧光染料浓度场跟踪湍流管流在散装雷诺数$\mathfrak{\mathsf{Re}}=3200$。对实验数据的对称约化分析表明，染料浓度猝发的速度$u$与纤维束中相应轨道的动力学和几何相位有关。特别是，在简化的框架中，我们揭示了被动标量结构的模式变化动力学，这解释了观测到的速度$u\approx U_{d}+U_{g}$的强烈爆发事件的几何相速度$U_{g}\approx0.43U_{d}$与轨道的捆绑。
英文摘要：
We propose a generalization of Hopf fibrations to quotient the streamwise translation symmetry of turbulent pipe flows viewed as dynamical systems. In particular, we exploit the geometric structure of the associate high dimensional state space, which is that of a principal fiber bundle. The relation between the comoving frame velocity $U_{d}$ associated with the dynamical phase of an orbit in the bundle and the Taylor's hypothesis is investigated. As an application, Laser-Induced-Fluorescence techniques are exploited to capture planar fluorescent dye concentration fields tracing a turbulent pipe flow at the bulk Reynolds number $\mathfrak{\mathsf{Re}}=3200$. The symmetry reduction analysis of the experimental data reveals that the speed $u$ of dye concentration bursts is associated with the dynamical and geometric phases of the corresponding orbits in the fiber bundle. In particular, in the symmetry-reduced frame we unveil a pattern-changing dynamics of the passive scalar structures, which explains the observed speed $u\approx U_{d}+U_{g}$ of intense bursting events in terms of the geometric phase velocity $U_{g}\approx0.43U_{d}$ associated with the orbits in the bundle.
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