Unraveling Transition and Turbulence Using Nonlinear Optimization

Subcritical transition in shear flows is due to both nonnormality and nonlinearity. Nonlinear optimization of perturbations of a given base flow is a powerful tool coupling nonnormality and nonlinearity, allowing to explore the phase space at a finite distance from the reference flow and study its nonlinear stability. This tool allows to compute: (i) minimal energy thresholds for transition, (ii) the most amplified coherent structures in transitional and turbulent flows, (iii) heteroclinic orbits connecting different invariant solutions of the Navier-Stokes equations, (iv) efficient ways for controlling transition to turbulence of finite-amplitude perturbations. The present work presents the nonlinear optimization framework and discusses its different applications to the study and control of transition and turbulence in shear flows.