Using Automatic Differentiation to Search for Minimal Seeds in Channel Flow

A minimal seed is defined as the smallest amplitude perturbation that can cause a transition from a linearly-stable state to another, a simple concept but extremely difficult to find in high-dimensional dynamical systems such as sheared fluid flows. In this work we present a method for finding said minimal seeds efficiently in the setting of pressure-driven channel flow but our methodology is applicable to other settings. We use automatic differentiation to calculate partial derivatives of the energy growth at some target time with respect to initial perturbations to the laminar state permitting us to efficiently optimise over these to find that which maximises the energy growth. We then examine how the initial and final states and energy growth change with perturbation amplitude. In particular we are looking for a significant step in the energy growth accompanied by a qualitatively different final state indicating transition. We will start by discussing the simpler 2D problem and then move up to the 3D setting.