Laminar–turbulent transition in boundary layers is characterized by the generation and metamorphosis of flow structures. However, the process of the evolution from a three-dimensional (3-D) wave to a $\varLambda$-vortex is not fully understood. In order to develop a deeper understanding of the spatiotemporal wave-warping process, we present numerical studies of both K-regime transition and bypass transition. A qualitative comparison of flow visualizations between a K-regime zero pressure gradient (ZPG) case and an adverse pressure gradient (APG) case is done, based on the method of Lagrangian tracking of marked particles. In bypass transition, the development of a 3-D wave packet before the breakdown into a turbulent spot was visualized for both the linear and nonlinear stages. The underlying vortex dynamics was investigated using a proposed method of Lagrangian-averaged enstrophy. The study illustrates that a $\varLambda$-vortex develops from a 3-D warped wave front (WWF), which undergoes multiple folding processes. It is observed that the APG case undergoes a more rapid evolution, precipitating a stronger viscous–inviscid interaction within the boundary layer. It is hypothesized that the amplification and lift-up of a 3-D wave causes the development of high-shear layers and a WWF. In order to seek a relationship between transitional and turbulent boundary layers, Lagrangian methods were also applied to an experimental data set from a turbulent boundary layer at low Reynolds number. Similarity of flow behaviours are observed, which further supports the hypothesis that the amplification of a 3-D wave precipitates low-speed streaks and rotational structures in wall-bounded flows.

A study of low-speed streaks (LSSs) embedded in the near-wall region of a turbulent boundary layer is performed using selective visualization and analysis of time-resolved tomographic particle image velocimetry (tomo-PIV). First, a three-dimensional velocity field database is acquired using time-resolved tomo-PIV for an early turbulent boundary layer. Second, detailed time-line flow patterns are obtained from the low-order reconstructed database using ‘tomographic visualizations’ by Lagrangian tracking. These time-line patterns compare remarkably well with previously observed patterns using hydrogen bubble flow visualization, and allow local identification of LSSs within the database. Third, the flow behaviour in proximity to selected LSSs is examined at varying wall distances ($10 < y^+ < 100$) and assessed using time-line and material surface evolution, to reveal the flow structure and evolution of a streak, and the flow structure evolving from streak development. It is observed that three-dimensional wave behaviour of the detected LSSs appears to develop into associated near-wall vortex flow structures, in a process somewhat similar to transitional boundary layer behaviour. Fourth, the presence of Lagrangian coherent structures is assessed in proximity to the LSSs using a Lagrangian-averaged vorticity deviation process. It is observed that quasi-streamwise vortices, adjacent to the sides of the streak-associated three-dimensional wave, precipitate an interaction with the streak. Finally, a hypothesis based on the behaviour of soliton-like coherent structures is made which explains the process of LSS formation, bursting behaviour and the generation of hairpin vortices. Comparison with other models is also discussed.

The beginning of laminar–turbulent transition is usually associated with a wave-like disturbance, but its evolution and role in precipitating the development of other flow structures are not well understood from a structure-based view. Nonlinear parabolized stability equations (NPSE) were solved numerically to simulate the transition of K-regime, N-regime and O-regime. However, only the K-regime transition was examined experimentally using both hydrogen bubble visualization and time-resolved tomographic particle image velocimetry (tomo-PIV). Based on the ‘NPSE visualization’ and ‘tomographic visualization’, at least four common characteristics of the generic transition process were identified: (i) inflectional regions representing high-shear layers (HSL) that develop in vertical velocity profiles, accompanied by ejection–sweep behaviours; (ii) low-speed streak (LSS) patterns, manifested in horizontal timelines, that seem to consist of several three-dimensional (3-D) waves; (iii) a warped wave front (WWF) pattern, displaying multiple folding processes, which develops adjacent to the LSS in the near-wall region, prior to the appearance of 𝛬-vortices; (iv) a coherent 3-D wave front, similar to a soliton, in the upper boundary layer, accompanied by regions of depression along the flanks of the wave. It was determined that the amplification and lift-up of a 3-D wave causes the development of the HSL, WWF and multiple folding behaviour of material surfaces, that all contribute to the development of a 𝛬-vortex. The amplified 3-D wave is hypothesized as a soliton-like coherent structure. Based on our results, a path to transition is proposed, which hypothesizes the function of the WWF in boundary-layer transition.