Examination of the onset and decay of turbulence in pipe flow

The crisis (or critical) Reynolds number (${\text{Re}}_c$) is established at 1870, describing the threshold beyond which the lifetimes of turbulent puffs prior to the relaminarization extend from $O\left({10}^4\right)\text{to}O\left({10}^6\right)$ time units ($D/U_m$), where $D$ and $U_m$ denote the pipe diameter and mean velocity, respectively. To analyze the role of inplane motion for sustaining turbulence, fully resolved direct numerical simulations have been performed to generate a localized, equilibrium turbulent puff at $\text{Re}=1920$. Employing our approach based on proper orthogonal decomposition, the research confirms that azimuthal motion significantly contributes to the transition to turbulence. Notably, at supercritical Reynolds numbers ($\text{Re}>{\text{Re}}_c$) ranging from $\text{Re}=1920$ to $\text{Re}=2100$, reducing azimuthal motion energy by 80% substantially shortens the lifetime of turbulent puffs. It has been shown that the relaminarization of turbulent puffs at subcritical Reynolds numbers, $\text{Re}=1720\text{–}1840$, clearly implies an exponential time decay of turbulence energy. The expression for the decay rate was obtained as a best-fit curve of direct numerical simulations.