Critical Reynolds Number for a Natural Transition to Turbulence in Pipe Flows

Experimental results obtained over more than a century have shown that laminar flow in a circular pipe becomes naturally turbulent at a critical Reynolds number of $\mathrm{Re}\approx 2000$. In this Letter a theoretical explanation, based on the minimum energy of an axisymmetric deviation (from the developed pipe flow profile), is suggested for this critical value. It is shown that for $\mathrm{Re}>1840$ the minimum energy of the deviation, associated with the central part of the pipe, becomes a global minimum for triggering secondary instabilities. For $\mathrm{Re}<1840$ the global minimum energy deviation is located next to the pipe wall. Previous experimental observations support this explanation.

Figure 1

Energy density neutral curves vs axial wave-number for various Reynolds numbers.

Figure 2

Optimal deviation distributions associated with the minima of the $E_n$ curves in Fig. 1. The blue solid lines correspond to the lower ${\alpha }_{\min }$ branch and the red dashed lines correspond to the higher ${\alpha }_{\min }$ branch.

Figure 3

(a) Minimum deviation energy density vs Re. (b) Minimal amplitude vs $1/\mathrm{Re}$. (c) Two parts of the energy corresponding to the deviations in (a). (d) phase velocity and axial wave-number corresponding to the deviations in (a).