Reply to Comment on 'Critical behaviour in the relaminarization of localized turbulence in pipe flow'

TL;DR

This paper defends the original finding that the median relaminarization time in pipe flow scales as 1/(Re_c - Re), supporting the critical behaviour hypothesis over an exponential model, by clarifying the data analysis and rejecting flawed arguments.

Contribution

It clarifies the correct data analysis method for relaminarization times and refutes claims that support an exponential scaling, reaffirming the critical behaviour model.

Findings

Median relaminarization time follows a 1/(Re_c - Re) scaling.

Exponential scaling cannot fit the data over the examined Re range.

The original critical behaviour conclusion is supported by the data.

Abstract

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 $\tau$ for relaminarisation of localised turbulent disturbances closely follows the scaling $\tau\sim 1/(Re_c-Re)$. 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 $\tau$ 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 $\tau\sim \exp(A Re)$, 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 $\tau$ 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 $\tau\sim 1/(Re_c-Re)$ fit is excellent, indicating critical behaviour and supporting experiments by Peixinho & Mullin 2006.