The minimal seed in a boundary layer flow: a frequency-domain approach

We identify the nonlinear optimal mechanisms at the edge of the turbulent regime through solution of the frequency-domain Harmonic-Balanced Navier-Stokes equations. Specifically, we seek the optimal harmonic or multi-harmonic forcing mechanisms that maximise the skin friction coefficient asymptotically in time, which can be viewed as the minimal forcing seed in the frequency domain. By retaining a small and finite number of harmonics that nonlinearly interact and distort the mean flow, but also transfer energy among them, a low-dimensional self-consistent dynamical description of the turbulent regime obtained. At the edge of turbulence, the skinfriction coefficient reaches the empirical turbulent values, and the velocity profiles depart from the law of the wall by developing a logarithmic region, as predicted by the model.