In the past two decades, our understanding of the transition to turbulence in shear flows with linearly stable laminar solutions has greatly improved. Regarding the susceptibility of the laminar flow, two concepts have been particularly useful: the edge states and the minimal seeds. In this nonlinear picture of the transition, the basin boundary of turbulence is set by the edge state’s stable manifold and this manifold comes closest in energy to the laminar equilibrium at the minimal seed. We begin this paper by presenting numerical experiments in which three-dimensional perturbations are too energetic to trigger turbulence in pipe flow but they do lead to turbulence when their amplitude is reduced. We show that this seemingly counter-intuitive observation is in fact consistent with the fully nonlinear description of the transition mediated by the edge state. In order to understand the physical mechanisms behind this process, we measure the turbulent kinetic energy production and dissipation rates as a function of the radial coordinate. Our main observation is that the transition to turbulence relies on the energy amplification away from the wall, as opposed to the turbulence itself, whose energy is predominantly produced near the wall. This observation is further supported by the similar analyses on the minimal seeds and the edge states. Furthermore, we show that the time-evolution of production-over-dissipation curves provide a clear distinction between the different initial amplification stages of the transition to turbulence from the minimal seed.

中文翻译：

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混沌的上缘和管道流动的过渡能量

在过去的二十年中，我们对具有线性稳定层流解决方案的剪切流向湍流过渡的理解有了很大的提高。关于层流的敏感性，两个概念特别有用：边缘状态和最小种子。在这种过渡的非线性图中，湍流的盆地边界由边缘状态的稳定歧管设定，并且该歧管在能量上最接近最小种子处的层状平衡。我们通过提出数值实验开始本文，其中三维扰动的能量太高，无法触发管道流动中的湍流，但是当振幅减小时，它们的确会导致湍流。我们表明，这种看似违反直觉的观察实际上与边缘状态介导的过渡的完全非线性描述相一致。为了了解此过程背后的物理机制，我们测量了湍动能的产生和耗散率，它是径向坐标的函数。我们的主要观察结果是，向湍流的过渡依赖于远离壁的能量放大，而不是湍流本身，其能量主要在壁附近产生。对最小种子和边缘状态的类似分析进一步支持了该观察结果。此外，