Pipe flow instability for the `critical' Reynolds number in transition to turbulence

Experimental results obtained over a century have shown that the flow in a circular pipe becomes `naturally' turbulent at a `critical' Reynolds number of $\sim 2000$. In this work a theoretical explanation, based on the minimum energy of an axisymmetric deviation, is suggested for this `critical' value. For this purpose a temporal stability analysis is preformed for the pipe Poiseuille flow, which has been modified by a primary axially-independent axisymmetric and finite amplitude deviation (following the method by Gavarini, Bottaro and Nieustadt, {\it J. Fluid Mech.} {\bf 517}, 131, 2004, who analyzed the spatial case). The optimal modification is defined as the primary base-flow deviation, with a given additional energy density, that yields the maximum growth rate to the secondary disturbances. Optimal modifications are computed by a variational technique. It is found, that above the Reynolds number of $\sim 2000$ the minimum energy of the deviation, which is concentrated at the central part of the pipe, becomes a global minimum for triggering secondary instabilities. Below this critical number the minimum energy deviation, which is concentrated next to the pipe wall, is the one having a global minimum for instability. Previous experimental observations support these results.