Calculation of global optimal initial and boundary perturbations for the linearised incompressible Navier-Stokes equations

This study considers numerical methods for computation of optimal boundary and initial perturbations to incompressible flows. Similar to previous work, constrained Lagrangian functionals are built and gradient optimisation methods are applied to optimise perturbations that maximise the energy of perturbations in the computational domain at a given time horizon. Unlike most of the previous work in this field we consider both optimal initial and boundary condition problems and demonstrate how each can be transformed into an eigenvalue problem. It is demonstrated analytically and numerically that both optimisation and eigenvalue approaches converge to the same outcome, even though the optimisation approach may converge more slowly owing to the large number of inflection points. In a case study, these tools are used to calculate optimal initial and boundary perturbations to the Batchelor vortex. It is observed that when the flow is asymptotically stable, the optimal inflow perturbation is similar to the most unstable local eigenmode, while when the flow is stable/weakly unstable, the spatial distribution of the optimal inflow perturbation is similar to the local optimal initial perturbation.