Scaling of the minimal energy for turbulence transition in pipe flow

Abstract

Predicting the transition of turbulence in pipe flow remains a fundamental problem in fluid dynamics. We use a variational approach to compute nonlinear optimal perturbations to the laminar flow at Reynolds number $Re\le 5000$. As $Re$ increases, optimal perturbations remain structurally similar, but increasingly localize while their thickness scales as ${\delta }_r\propto R{e}^{-1/3}$. They grow via the Orr mechanism, followed by a phase of strong nonlinear interaction of oblique waves and a lift-up phase. The energy gain during the Orr phase increases linearly with $Re$ and is independent of the initial perturbation energy, $E_0$. The energy gain during the oblique and lift-up phases is governed by nonlinearities and scales as $\propto R{e}^2$. We find that regardless of the Reynolds number, transition occurs if the energy of the perturbation exceeds a constant threshold. As a result, the minimum perturbation energy required to cause transition in pipe flow scales as $\mathcal{O}(R{e}^{-3})$.