Patrick Keuchel | Searcharxiv

arXiv2026

Scaling of the minimal energy for turbulence transition in pipe flow

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 $Re\leq 5000$. As $Re$ increases, optimal perturbations remain structurally similar, but increasingly localize while their thickness scales as $\delta_r \propto Re^{-1/3}$. 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 $\propto Re^2$. 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 $\mathcal{O}(Re^{-3})$.

physics.flu-dyn↗

arXiv2025

Nonlinear optimal perturbation growth in pulsatile pipe flow

Pulsatile fluid flows through straight pipes undergo a sudden transition to turbulence that is extremely difficult to predict. The difficulty stems here from the linear Floquet stability of the laminar flow up to large Reynolds numbers, well above experimental observations of turbulent flow. This makes the instability problem fully nonlinear and thus dependent on the shape and amplitude of the flow perturbation, in addition to the Reynolds and Womersley numbers and the pulsation amplitude. This problem can be tackled by optimizing over the space of all admissible perturbations to the laminar flow. In this paper, we present an adjoint optimization code, based on a GPU implementation of the pseudo-spectral Navier-Stokes solver nsPipe, which incorporates an automatic, optimal check-pointing strategy. We leverage this code to show that the flow is susceptible to two distinct instability routes: One in the deceleration phase, where the flow is prone to oblique instabilities, and another during the acceleration phase with similar mechanisms as in steady pipe flow. Instability is energetically more likely in the deceleration phase. Specifically, localised oblique perturbations can optimally exploit nonlinear effects to gain over nine orders of magnitude in energy at a peak Reynolds number of $Re_{\max}\approx 4000$. These oblique perturbations saturate into regular flow patterns that decay in the acceleration phase or break down to turbulence depending on the flow parameters. In the acceleration phase, optimal perturbations are substantially less amplified, but generally trigger turbulence if their amplitude is sufficiently large.

physics.flu-dyn↗