Identifying efficient routes to laminarization: An optimization approach
Phys. Rev. Fluids 11, 023902 – Published 10 February, 2026
DOI: https://doi.org/10.1103/3dwk-nnn1
Abstract
The nonlinear and chaotic nature of turbulent flows poses a major challenge for designing effective control strategies to maintain or induce low-drag laminar states. Traditional linear methods often fail to capture the complex dynamics governing transitions between laminar and chaotic regimes. In this work, we introduce and investigate the concept of the minimal seed for relaminarization—the closest point to a reference state in the chaotic region of the state space that triggers a direct transition to laminar flow without a chaotic transient. We formulate the identification of this optimal perturbation as a fully nonlinear optimization problem and develop a numerical framework based on a multistep penalty method to compute it. Applying this framework to the nine-mode Moehlis-Faisst-Eckhardt model of a sinusoidal shear flow, which displays coexisting laminar and chaotic states, we compute the minimal seeds for both transition and relaminarization. While both of these minimal seeds lie infinitesimally close to the laminar-chaotic boundary—the so-called edge of chaos—they are generally unrelated and lie in distant and qualitatively distinct regions of state space, thereby providing different insights into the flow's underlying structure. We find that while the optimal perturbation for triggering transition is primarily in the direction of the mode representing streamwise vortices, or rolls, the optimal perturbation for relaminarization is distributed across multiple modes in this model, without strong contributions in the roll or streak directions. By analyzing the trajectories originating from the minimal seeds, we find that both the transition and laminarization behavior for this system are controlled by a common mechanism—the stable and unstable manifolds of a periodic orbit on the edge of chaos. The laminarizing trajectory obtained from the minimal seed for relaminarization also provides an efficient pathway out of the chaotic region of state space to the laminar state and can therefore inform the design and evaluation of flow control strategies aimed at inducing laminarization.
