Sensitivity analysis and optimization of initial conditions in two-dimensional unsteady vortical flows

Vortical flow interactions and instabilities can be harnessed for practical applications, for instance reducing the energy in complex trailing wakes. In many scenarios, the high dimensionality and non-linearity of the flow poses challenges for predicting the sensitivity of late-time flow metrics to key geometric parameters governing the initial flow state. Here we investigate these challenges and possible solutions in the context of 2D vortical flows. First, we implement both a tangent and adjoint methodology in a Julia code. The computation of the derivative terms in the vorticity-velocity Navier-Stokes equations is efficiently handled using automatic differentiation (AD) functions from Julia packages. Using the sensitivities obtained from these methods, we investigate the feasibility of performing gradient-based optimization of the initial vortical structures, in order to minimize key global cost functions evaluated within a later time window. Additionally, we explore the application of a reduced-order model (ROM) approach to address cases of unsteady vortical flows with chaotic behaviors where the traditional tangent or adjoint methods fail. The ROM models the dynamic behavior of sensitivities in the phase space, which helps determine the modal functions causing instability in the sensitivity functions. We will outline the design of this ROM and show preliminary results on its application to highly nonlinear flow problems, compared with traditional sensitivity techniques.