Genesis of Streamwise-Localized Solutions from Globally Periodic Traveling Waves in Pipe Flow
Genesis of Streamwise-Localized Solutions from Globally Periodic Traveling Waves in Pipe Flow
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DOI：
10.1103/physrevlett.112.164501
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发表时间：
2014-04-22
影响因子：
8.6
通讯作者：
Kerswell, R. R.
中科院分区：
文献类型：
--
作者：
Chantry, M.;Willis, A. P.;Kerswell, R. R.
过渡湍流动力系统方法的目的是使用简单的不变集（精确解）及其稳定和不稳定流形在相空间中为动力学构建一个支架。在湍流与层流共存的大型（现实）区域中，这需要确定精确的局部解。在有壁剪切流中，最近在管流中发现了第一个解，但问题仍然是它们如何与许多已知的流向周期解相联系。在这里，我们证明了第一个本地化的解决方案的起源是在一个已知的全球行波，具有双重旋转对称的管轴的调制的hoppy-破缺的Hopf分岔。类似的行为被发现为一个全球性的三重旋转对称波，这一次导致两个本地化的相对周期轨道。这显然意味着，许多全球性解决方案应该通过这种分歧产生更现实的本地化对应方案，这为它们的产生提供了一条建设性的道路。
The aim in the dynamical systems approach to transitional turbulence is to construct a scaffold in phase space for the dynamics using simple invariant sets (exact solutions) and their stable and unstable manifolds. In large (realistic) domains where turbulence can coexist with laminar flow, this requires identifying exact localized solutions. In wall-bounded shear flows, the first of these has recently been found in pipe flow, but questions remain as to how they are connected to the many known streamwise-periodic solutions. Here we demonstrate that the origin of the first localized solution is in a modulational symmetry-breaking Hopf bifurcation from a known global traveling wave that has twofold rotational symmetry about the pipe axis. Similar behavior is found for a global wave of threefold rotational symmetry, this time leading to two localized relative periodic orbits. The clear implication is that many global solutions should be expected to lead to more realistic localized counterparts through such bifurcations, which provides a constructive route for their generation.