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The development of accurate transition and turbulence models is of fundamental interest for predicting flows in turbomachinery and improving their design. Recently developed optimization techniques, based on Lagrangian multipliers approach and direct-adjoint it-erations, allow one to investigate the basic mechanisms by which perturbations grow and turbulence is self-sustained in wall-bounded flows. In particular, employing the linearized NavierStokes equations, the lift-up effect has been found to be responsible of the tran-sient growth of the perturbations and of the by-pass transition. In the present work, the extension of the optimization approach to the nonlinear equations is described, both for the cases of transitional and turbulent flows. Optimal coherent structures are identified, such as large scale streaks and hairpin vortices, which carry a great part of the Reynolds stresses. Such coherent structures are a suitable starting point for the development of reduced-order models of turbulence.

Optimal perturbations in transitional and turbulent flows at moderate Reynolds numbers