  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.
