Identifying efficient routes to laminarization: an optimization approach

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 turbulent regimes. In this work, we introduce the concept of the minimal seed for relaminarization-the closest point to a reference state in the turbulent 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 multi-step penalty method to compute it. Applying this framework to a nine-mode model of a sinusoidal shear flow, we compute the minimal seeds for both transition to turbulence and relaminarization. While both of these minimal seeds lie infinitesimally close to the laminar-turbulent boundary-the 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 the optimal perturbation for triggering transition is primarily in the direction of the mode representing streamwise vortices (rolls), whereas the optimal perturbation for relaminarization is distributed across multiple modes without strong contributions in the roll or streak directions. By analyzing trajectories originating from these minimal seeds, we find that both transition and laminarization behavior are controlled by the stable and unstable manifolds of a periodic orbit on the edge of chaos. The laminarizing trajectory obtained from the minimal seed for relaminarization provides an efficient pathway out of turbulence and can inform the design and evaluation of flow control strategies aimed at inducing laminarization.