Discovering sparse optimal finite-amplitude perturbations in nonlinear flows

Managing flow instabilities is a central challenge in fluid dynamics. Active flow control has been proposed as a means by which unsteady phenomena such as flow separation can be mitigated. However, the design of effective flow control strategies requires an understanding of which physical phenomena should be targeted with actuation. To this end, we propose an optimization framework for finding sparse finite-amplitude perturbations that maximize transient energy growth in nonlinear systems. Using a variational approach, we derive the first-order necessary conditions for optimality, which form the basis of our direct-adjoint looping numerical algorithm. We demonstrate the approach on a reduced-order model of a sinusoidal shear flow. Our framework identifies that energy injection into a single mode yields comparable energy amplification as the non-sparse optimal solution. This analysis establishes the possibility of using such methods to determine actuation strategies for flow control in the future.