Streamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected to high levels of free-stream turbulence. Using the steady boundary-layer approximation, the upstream disturbances experiencing maximum spatial energy growth are numerically calculated. The calculations use techniques commonly employed when solving optimal-control problems for distributed parameter systems. The calculated optimal disturbances consist of streamwise vortices developing into streamwise streaks. The maximum spatial energy growth was found to scale linearly with the distance from the leading edge. Based on these results, a simple model for prediction of transition location is proposed. Available experiments have been used to correlate the single constant appearing in the model.

A scenario of transition to turbulence likely to occur during the development of natural disturbances in a flat-plate boundary layer is studied. The perturbations at the leading edge of the flat plate that show the highest potential for transient energy amplification consist of streamwise aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. Direct numerical simulations are used to follow the nonlinear evolution of these streaks and to verify secondary instability calculations. The theory is based on a linear Floquet expansion and focuses on the temporal, inviscid instability of these flow structures. The procedure requires integration in the complex plane, in the coordinate direction normal to the wall, to properly identify neutral modes belonging to the discrete spectrum. The streak critical amplitude, beyond which streamwise travelling waves are excited, is about 26% of the free-stream velocity. The sinuous instability mode (either the fundamental or the subharmonic, depending on the streak amplitude) represents the most dangerous disturbance. Varicose waves are more stable, and are characterized by a critical amplitude of about 37%. Stability calculations of streamwise streaks employing the shape assumption, carried out in a parallel investigation, are compared to the results obtained here using the nonlinearly modified mean fields; the need to consider a base flow which includes mean flow modification and harmonics of the fundamental streak is clearly demonstrated.

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Streamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected to freestream turbulence. Using the steady boundary-layer approximation, we numerically calculate the upstream disturbances experiencing maximum spatial energy growth. The calculations use techniques commonly employed when solving optimal-control problems for distributed parameter systems. The calculated optimal disturbances consist of streamwise vortices developing into streamwise streaks. The maximum possible energy growth was found to scale linearly with the distance from the leading edge. Based on this result, we propose a simple model for prediction of transition location. Available experiments have been used to correlate the single constant appearing in the model.