The critical point of the transition to turbulence in pipe flow

Abstract

Reynolds proposed that after sufficiently long times, the flow in a pipe should settle to a steady condition: below a critical Reynolds number, flows should (regardless of initial conditions) always return to laminar, while above, eddying motion should persist. As shown, even in pipes several thousand diameters long, the spatio-temporal intermittent flow patterns observed at the end of the pipe strongly depend on the initial conditions, with no indication of an approach to a (statistical) steady state. Exploiting the fact that turbulent puffs do not age, we continuously recreate the puff sequence exiting the pipe at the entrance, thus introducing periodic boundary conditions for the flow pattern. This procedure allows us to study the evolution of the flow patterns for arbitrary long times. We find that after times in excess of $10^7$ advective time units, a statistical steady state is reached. Though the resulting flows remain spatio-temporally intermittent, puff splitting and decay rates eventually reach a balance so that the turbulent fraction fluctuates around a well defined level which only depends on $Re$. We find that at lower $Re$ (here 2020), flows eventually always laminarize, while for higher $Re$ ($>=2060$) turbulence persists. The critical point for pipe flow hence lies in the interval $2020<Re_c<2060$, which is in good agreement with the recently proposed value of $Re_c=2040$. The latter estimate was based on single puff statistics and entirely neglected puff interactions. Unlike typical contact processes where such interactions strongly affect the critical point, in pipe flow it is only marginally influenced. Interactions on the other hand, are responsible for the approach to the statistical steady state. As shown, they strongly affect the resulting flow patterns, where they cause `puff clustering', with the clusters traveling across the puff pattern in a wave like fashion.