School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom.
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, United Kingdom.
Phys Rev Lett. 2014 Apr 25;112(16):164501. doi: 10.1103/PhysRevLett.112.164501. Epub 2014 Apr 22.
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.
动力系统方法在转捩湍流中的目标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支架。在可以共存层流的大（现实）区域中，这需要识别精确的局部化解。在壁面剪切流中，最近在管流中发现了其中的第一个，但仍存在疑问，即它们如何与许多已知的流向周期性解相关联。在这里，我们证明了第一个局部化解的起源在于从具有关于管轴双重旋转对称性的已知全局行波的调制对称破缺 Hopf 分岔。对于具有三重旋转对称性的全局波，也发现了类似的行为，这导致了两个局部相对周期轨道。显然的是，许多全局解应该通过这种分岔产生更多现实的局部解，这为它们的生成提供了一种建设性的途径。
