Threshold amplitudes for transition to turbulence in a pipe

Although flow in a circular pipe is stable to infinitesimal perturbations, it can be excited to turbulence by finite perturbations whose minimal amplitude shrinks as $R \rightarrow \infty ~(R =$ Reynolds number). Laboratory experiments have appeared to disagree with one another and with theoretical predictions about the dependence of this minimal amplitude on $R$, with published results ranging approximately from $R^{-1/4}$ to $R^{-3/2}$. Here it is shown that these discrepancies can be explained by the use of different definitions of amplitude by different authors. An attempt is made to convert the existing results to a uniform definition of amplitude, the nondimensionalized $L^2$ definition common in the theoretical literature. Although subtleties in the physics raise some questions, agreement appears to be reached on a minimal amplitude that scales as $R^{-3/2 \pm 0.3}$ The research of AM, TM and LNT was supported by the Engineering and Physical Sciences Research Council of the UK under Grant GR/M308890. The work of TM was also supported by the Leverhulme Trus