About the enstrophy change of the Reynolds-Orr solution in the channel flow

TL;DR

This paper investigates how using enstrophy and a tilted coordinate system in the Reynolds-Orr equation affects predictions of flow stability in channel flow, aiming to reconcile theory with experimental critical Reynolds numbers.

Contribution

It combines enstrophy-based norms and a tilted coordinate system in the Reynolds-Orr analysis to improve flow stability predictions in channel flow.

Findings

Enstrophy-based analysis predicts higher critical Reynolds numbers.

The combined method refines classical Reynolds-Orr solutions.

Results show better agreement with numerical simulations.

Abstract

The plane Poiseuille flow is one of the elementary flow configurations. Although its laminar-turbulent transition mechanism is investigated intensively in the last century, the significant difference in the critical Reynolds number between the experiments and theory lacks a clear explanation. In this paper, an attempt is made to reduce this gap by analysing the Reynolds-Orr equation solution. Recent literature results showed that the usage of enstrophy (the volume integral of the vorticity) instead of the kinetic energy as the norm of perturbations predicts higher Reynolds numbers in the two-dimensional case. Its usage in three dimensions is discussed in the paper. In addition, other research showed an improvement of the original Reynolds-Orr energy equation using the weighted norm in a tilted coordinate system. Here, these two methods are combined. The zero enstrophy growth constraint is applied to the classical Reynolds-Orr equation, and then the solution is further refined in the tilted coordinate system. The results are compared to direct numerical simulations from the literature.