Can minimal absorbing zones predict turbulent mean profiles?

Understanding the onset of turbulence in shear flows remains challenging. A rarely used concept - the “absorbing zone,” defined as the subset of infinite‑dimensional state space that attracts all trajectories starting outside it and thus contains every attractor - is revisited. Using the Reynolds–Orr identity, a kinetic‑energy variational problem is derived to prove the existence of, and compute, the minimal absorbing zone around a so-called shift flow via gradient optimization. The method is applied to plane Couette and Poiseuille flows. It was hypothesized that, because a chaotic turbulent trajectory wanders quasi‑randomly and uniformly within its admissible region, the geometric center of the smallest absorbing zone (the optimal shift flow) would approximate the turbulent mean profile. Instead, excessively sharp near‑wall gradients are obtained, and quantitative discrepancies with direct numerical simulations and canonical laws of the wall are observed. Nevertheless, novel insight into shear‑flow dynamics is provided, and a route to improved global stability estimates is suggested. With further development, direct computation of turbulent mean states without empirical turbulence models or time‑dependent simulations may be enabled.