Identifying efficient routes to laminarization: an optimization approach

Turbulence control remains a major challenge in fluid dynamics, with significant implications for drag reduction and energy efficiency. In this work, we study the minimal seed for relaminarization—the smallest perturbation to a turbulent flow state that induces a return to the laminar flow without a chaotic transient. We identify this via a fully nonlinear optimization framework and apply the method to the nine-mode Moehlis-Faisst-Eckhardt model of sinusoidal shear flow. We also consider the minimal seed for relaminarization and the resulting trajectory in the context of the minimal seed for transition – the smallest perturbation of the laminar state that triggers a transition to turbulence. While both of these minimal seeds lie infinitesimally close to the laminar-turbulent edge, they are generally unrelated and lie in distant and qualitatively distinct regions of the state space, thereby providing different insights into the flow's underlying structure. Furthermore, the laminarizing trajectory obtained from the minimal seed for relaminarization provides an efficient pathway out of the turbulent region of state space to the laminar state and can therefore inform the design and evaluation of flow control strategies for inducing laminarization. These optimal perturbations also yield insights into which combinations of modes should be excited or suppressed to trigger transition or laminarization, respectively, offering a new perspective on flow control.