This is a Reply to Comment arXiv:0707.2642 by Hof et al. on Letter arXiv:physics/0608292 which was subsequently published in Phys Rev Lett, 98, 014501 (2007).
In our letter it was reported that in pipe flow the median time $τ$ for relaminarisation of localised turbulent disturbances closely follows the scaling $τ\sim 1/(Re_c-Re)$. This conclusion was based on data from collections of 40 to 60 independent simulations at each of six different Reynolds numbers, Re. In the Comment, Hof et al. estimate $τ$ differently for the point at lowest Re. Although this point is the most uncertain, it forms the basis for their assertion that the data might then fit an exponential scaling $τ\sim \exp(A Re)$, for some constant A, supporting Hof et al. (2006) Nature, 443, 59. The most certain point (at largest Re) does not fit their conclusion and is rejected. We clarify why their argument for rejecting this point is flawed. The median $τ$ is estimated from the distribution of observations, and it is shown that the correct part of the distribution is used. The data is sufficiently well determined to show that the exponential scaling cannot be fit to the data over this range of Re, whereas the $τ\sim 1/(Re_c-Re)$ fit is excellent, indicating critical behaviour and supporting experiments by Peixinho & Mullin 2006.
A traveling wave bifurcation analysis of turbulent pipe flow
使用来自动力系统理论的各种技术，我们严格地研究了[Barkley et al.，Nature，526：550-553，2015]，其描述了经由复杂性降低的PDE系统的湍流管流的上升。湍流的快速发展是由反应扩散动力学耦合到中心线速度，这与伯格斯型平流和缓慢的再层化项。应用空间动力学分析和几何奇异摄动理论，我们证明了在湍流和层流定态之间存在异宿环，并建立了一个由各种行波组成的分叉级联。最复杂的行为可以发现在中间雷诺数制度，其中的行波表现出任意长的类似于混沌的动态指示的发病。我们的分析提供了一个系统的数学方法来确定过渡到时空湍流结构，也可能适用于其他模型中产生的流体动力学。
Drag reduction and subcritical turbulence in controlled pipe flows
Politecnico di Milano（米兰理工大学）；Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille（里尔大学、法国国家科学研究中心、法国航空发展局、巴黎国立高等工艺学院、里尔中央理工学院）；Massachusetts Institute of Technology（麻省理工学院）
