( Reference Jiang, Lee, Chen, Smith and Linden2020 a) systematically investigated the development of a WWF in the early stage of K-, N- and O-regime transition, demonstrating qualitative similarity between the stages. The warping process and kinked profiles were frequently observed experimentally or numerically in transitional boundary layers (Wortmann Reference Wortmann1981 Laurien & Kleiser Reference Laurien and Kleiser1989 Rist & Fasel Reference Rist and Fasel1995 Lee Reference Lee1998).
This localized and intensified warped wave front (WWF) was considered as the source of subsequent hairpin-shaped vortices. a ‘kink’), subsequently develops into a hairpin-shaped, discrete vortex. This layer of concentrated high-shear, which appears as an inflection in vertical hydrogen bubble timelines (i.e. In the classical work of Hama and his colleagues, which employed hydrogen bubble visualization to observe structures within a transitional boundary layer (Hama, Long & Hegarty Reference Hama, Long and Hegarty1957 Hama & Nutant Reference Hama and Nutant1963), they observed that a Tollmien–Schlichting wave (TS) wave warps three dimensionally during amplification, acquiring a longitudinal vorticity component along its swept-back front. In addition, a direct numerical simulation (DNS) study of a turbulent spot and an experimentally determined low-Reynolds-number turbulent boundary layer are also presented for comparison with the transitional flows. In the present study, a comparative analysis is made between a zero-gradient (Blasius) boundary layer (ZPG) and a Falkner–Skan (FS) flow with an adverse pressure gradient (APG). The origin of hairpin vortices that many studies take as the dominant flow structure of turbulent boundary layers is not well understood (Marusic Reference Marusic2009). However, most research investigating the generation or regeneration of turbulent structures usually does not take into account the early transition stage where three-dimensional (3-D) waves play important roles. There have been many reviews related to the coherent structures in transitional and well-developed turbulent boundary layers (Smith Reference Smith1984 Robinson Reference Robinson1991 Kachanov Reference Kachanov1994 Rempfer Reference Rempfer2003 Lee & Wu Reference Lee and Wu2008 Jiménez Reference Jiménez2018 Lee & Jiang Reference Lee and Jiang2019 Marusic & Monty Reference Marusic and Monty2019). This problem has attracted much attention because of the fundamental significance in understanding the process of turbulence production and the considerable difficulties encountered in identifying the building blocks of coherent structures in laminar–turbulent transition. The origin of wall-bounded turbulence has been one of the most classic research problems for the fluid mechanics community. 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. 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. It is hypothesized that the amplification and lift-up of a 3-D wave causes the development of high-shear layers and a WWF. It is observed that the APG case undergoes a more rapid evolution, precipitating a stronger viscous–inviscid interaction within the boundary layer. The study illustrates that a $\varLambda$-vortex develops from a 3-D warped wave front (WWF), which undergoes multiple folding processes. The underlying vortex dynamics was investigated using a proposed method of Lagrangian-averaged enstrophy.
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. 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 order to develop a deeper understanding of the spatiotemporal wave-warping process, we present numerical studies of both K-regime transition and bypass transition. However, the process of the evolution from a three-dimensional (3-D) wave to a $\varLambda$-vortex is not fully understood. Laminar–turbulent transition in boundary layers is characterized by the generation and metamorphosis of flow structures.
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